# The invariant subspaces of periodic Fourier multipliers with application to
abstract evolution equations

Sebastian Król & Jarosław Sarnowski  Sebastian Król, Faculty of Mathematics
and Computer Science, Adam Mickiewicz University in Poznań, ul. Uniwersytetu
Poznańskiego 4, 61-614 Poznań, Poland<EMAIL_ADDRESS>Jarosław
Sarnowski, Faculty of Mathematics and Computer Science, Nicolaus Copernicus
University in Toruń, ul. Chopina 12/18, 87-100 Toruń, Poland
<EMAIL_ADDRESS>
###### Abstract.

By methods of harmonic analysis, we identify large classes of Banach spaces
invariant of periodic Fourier multipliers with symbols satisfying the
classical Marcinkiewicz type conditions. Such classes include general (vector-
valued) Banach function spaces $\Phi$ and/or the scales of Besov and Triebel-
Lizorkin spaces defined on the basis of $\Phi$.

We apply these results to the study of the well-posedness and maximal
regularity property of an abstract second-order integro-differential equation,
which models various types of elliptic and parabolic problems arising in
different areas of applied mathematics. In particular, under suitable
conditions imposed on a convolutor $c$ and the geometry of an underlying
Banach space $X$, we characterize the conditions on the operators $A$, $B$ and
$P$ on $X$ such that the following periodic problem

$\partial P\partial u+B\partial u+{A}u+c\ast u=f\qquad\textrm{in
}{\mathcal{D}}^{\prime}({\mathbb{T}};X)$

is well-posed with respect to large classes of function spaces. The obtained
results extend the known theory on the maximal regularity of such problem.

###### Key words and phrases:

integro-differential equations, maximal regularity, well-posedness, periodic
Fourier multipliers, Hardy-Littlewood maximal operator, Besov spaces, Triebel-
Lizorkin spaces, Rubio de Francia iteration algorithm

###### 1991 Mathematics Subject Classification:

42B37, 42A45, 45N05, 46N20, 43A15

##  1\. Introduction

Fourier multipliers with operator-valued symbols have found many applications
in the theory of abstract evolution equations, in particular, in connection
with solvability (well-posedness) and regularity of integro-differential
equations. A large class of such equations can be modelled by the following
abstract, degenerated second-order problem with a convolution term:

(AP) $(Pu^{\prime})^{\prime}+Bu^{\prime}+Au+c\ast u=f.\\\ $

Here, $A$, $B$, $P$ denote closed linear operators on a Banach space $X$ and
$c$ is an operator-valued function.

For particular forms of (AP), the studies of their well-posedness on diverse
vector-valued function spaces have been increased with occurring two seminal
papers by Amann [2] and Weis [69], where operator-valued counterparts of
classical multiplier theorems for Besov and Lebesgue-Bochner spaces on
$\mathbb{R}$ are provided. Those results indicated a right form of multiplier
conditions (see [31], [30]), which have been further adapted to different
situations; see, e.g. periodic multiplier results in [7, 8, 67], which are
relevant to this article.

In the literature one can extract two lines of research corresponding to such
studies: namely, when (AP) is considered in the euclidean setting, that is, on
$\mathbb{R}$ or $\mathbb{R}_{+}$, and in the periodic one, that is, on
${\mathbb{T}}:=\mathbb{R}/\mathbb{Z}$ (i.e., when the periodic conditions are
imposed). Each of these lines is represented by a long series of papers; to
mention a few representative results, see for the first one, e.g. [2, 69, 59,
6, 28, 11, 27, 5, 4, 51], and for the second one, e.g. [7, 8, 47, 53, 23, 54,
57, 24, 49, 40, 56, 38, 20, 21, 22] (as well as the references therein). Such
studies correspond to the well-known research program formulated by Amman in
[1, Section 3] and labelled as ’pairs of maximal regularity’. In both
settings, the basic idea for such studies is the same and relies on multiplier
theorems. Roughly, by the theory of vector-valued distributions, the well-
posedness and regularity questions for (AP), reduce to checking if
corresponding Fourier multipliers with operator-valued symbols are bounded in
a space under consideration. Such Fourier multiplier operators arise naturally
via the representation formula for corresponding solution operators associated
to a given form of (AP).

In the euclidean setting, the so-called phenomenon of the extrapolation of
$L^{p}$-maximal regularity, which can be simply considered as a special
variant of the well-posedness with respect to various Banach function spaces,
have been studied recently; see, e.g. [59, 11, 27, 37, 29, 51]. Beyond a
natural theoretical interest in this phenomenon, the maximal regularity with
respect to a more general function space is an important tool for the study of
associated non-linear problems; see, e.g. [52, 60, 11].

In the periodic case, the well-posedness and maximal regularity were addressed
mainly in the context of the classical Lebesgue, Besov, Triebel-Lizorkin
spaces; see, e.g. the corresponding series of the references mentioned above.
The aim of this article is to extend such results to a much wider context of
general Banach function spaces; see the main results of this paper, Theorems
6.5, 6.7, 6.9, and 6.12. In particular, we clarify the phenomenon of the
extrapolation of $L^{p}$-maximal regularity for several periodic evolution
equations modelled by (AP); see Theorems 6.7 and 6.12, as well as Section 7.
These results, in particular, provide counterpart of the euclidean line of
research mentioned above for the periodic situation. In addition, we provide a
convenient framework for such studies, which reveals an underlying structure,
allows to simplify and unify technicalities mainly resulting from the fact
that we deal with higher order Marcinkiewicz’s conditions, and allows to
handle different questions (distributional or strong solvability, maximal
regularity) in a unified manner. It is achieved with the help of two auxiliary
results Theorem 6.2 and Lemma 6.4. On the other-hand, it allows to extend many
results from the related literature in several ways, by showing that
assumptions usually made to get those results in the $L^{p}$-setting are
sufficient for a large class of Banach function spaces; see Section 7.

To establish such results we make a revision of underlying multiplier results
from [7, 8, 25] applied in the context of $L^{p}$ setting. Roughly, our main
periodic multiplier results, see Theorems 5.3 and 5.5, assert that the
standard multiplier conditions, which in the literature usually are imposed on
the symbol of a Fourier multiplier to get its boundedness on the classical
(vector-valued) Lebesgue, Besov or Triebel-Lizorkin spaces, are sufficient for
its boundedness on much larger classes of spaces. Such classes include general
Banach function spaces $\Phi$ and/or the scales of Besov and Triebel-Lizorkin
spaces defined on the basis of $\Phi$. In particular, these results extend [7,
Theorem 1.3], [8, Theorem 4.5] and [25, Theorem 3.2]. Our proofs differ from
the proofs presented in those papers. We rely on direct maximal function
estimates; see the proofs of Lemma 2.2 and Theorem 5.3.

We conclude with a remark on the strategy of the proof of our abstract
extrapolation result, Theorem 5.5. Since the theory of periodic distributions
presents a simplification in comparison to that on the real line, one could
expect the same in the context of multiplier theorems. In fact, we show that
such simplification is reflected mainly in the representation formulas for
periodic Fourier multipliers; see Section 3.1 for the further comments and
Lemma 5.1. In particular we do not address here problems which appear in the
euclidean setting; see, e.g. [44, Problems 3.2 and 3.3] and [51, Section 4].
However, at some points (for instance, when the interplay between the
regularity of the symbol and its Fourier transform is crucial), the euclidean
setting presents some benefits in comparison to the periodic one. For this
reason, instead of trying to prove some periodic results in a complete analogy
to corresponding ones known in the euclidean setting, we deduce them from
their euclidean counterparts via transference techniques; see the proof of
Theorem 5.5 (cf. also Lemma 2.1). The tools for such transference methods are
workout in Section 4, which may be of independent interest; see operator-
valued variants of Jodeit’s type theorem, Theorems 4.1 and 4.4, as well as
Lemmas 4.6 and 4.8.

The organization of the paper is well-reflected by the titles of the following
(sub)sections.

##  2\. Auxiliary results

###  2.1. Function spaces

We refer the reader to the monograph by Bennett and Sharpley [13] for the
background on Banach function spaces. Here, we mention only several facts we
use in the sequel.

Let $\Psi$ be a Banach function space over $(G,\mathrm{d}t)$, where $G$
denotes $\mathbb{R}$ or ${\mathbb{T}}$ equipped with the Lebesgue measure. It
means that $\Psi$ is a Banach space, which is an order ideal of
$L^{0}:=L^{0}(G,\mathrm{d}t)$, i.e. for every $f\in L^{0}$ and $g\in\Psi$ if
$|f|\leq|g|$, then $f\in\Psi$ and $\|f\|_{\Psi}\leq\|g\|_{\Psi}$. Here,
$L^{0}$ stands for the space of all complex measurable functions on $G$ (as
usual, any two functions equal almost everywhere are identified). Moreover,
$\Psi$ has Fatou’s property, and by the Lorentz-Luxemburg theorem [13, Theorem
2.7, p.10], $(\Psi^{\prime})^{\prime}=\Psi$ with equal norms. Here,
$\Psi^{\prime}$ denotes the (Köthe) dual (or associated space) of $\Psi$; see
[13].

We define the vector-valued variant of Banach function spaces $\Psi$ as
follows. Let $X$ be a Banach space with norm $|\cdot|_{X}$. Set

$\Psi(G;{X}):=\\{f:G\rightarrow X\textrm{ strongly
measurable}:\;\;|f|_{X}\in\Psi\\}$

and $\|f\|_{\Psi(G;X)}:=\||f|_{X}\|_{\Psi}$ for $f\in\Psi(X)$. Throughout, the
symbol $\Phi$ is reserved to denote a Banach function space over
$({\mathbb{T}},\mathrm{d}t)$. Note that if a function $e_{0}(\tau):=1$
$(\tau\in{\mathbb{T}})$ is in $\Phi$, then by the ideal property of $\Phi$ we
get that $L^{\infty}({\mathbb{T}};X)\subset\Phi({\mathbb{T}};X)=:\Phi(X)$. In
particular, if ${\mathcal{P}}({\mathbb{T}};X)$ denotes the set of all
$X$-valued polynomials on ${\mathbb{T}}$, i.e.

${\mathcal{P}}(X):={\mathcal{P}}({\mathbb{T}};X):=\left\\{\sum_{k=-N}^{N}e_{k}\otimes
x_{k}:N\in\mathbb{N},x_{k}\in X\right\\}$

then ${\mathcal{P}}(X)\subset\Phi(X)$. Here, $(e_{k}\otimes
x)(\tau):=e_{k}(\tau)x$, where $e_{k}(\tau):=\tau^{k}$ ($\tau\in{\mathbb{T}}$,
$k\in\mathbb{Z}$, $x\in X$).

Moreover, we introduce a variant of vector-valued Besov and Triebel-Lizorkin
spaces corresponding to a Banach function space $\Phi$.

Let ${\mathcal{D}}^{\prime}({\mathbb{T}};X):={\mathcal{L}}({\mathcal{D}},X)$,
where ${\mathcal{D}}:={\mathcal{D}}({\mathbb{T}})$ is a space of all complex-
valued infinitely differentiable functions on ${\mathbb{T}}$ equipped in the
usual locally convex topology. We refer to [63, Section 3] or [36] for the
backgrounds on the scalar distributions on ${\mathbb{T}}$, and to [8, Section
2] for their vector-valued counterpart. For instance, relying on [8,
Proposition 2.1], it is readily seen that for each
$\psi\in{\mathcal{C}}(\mathbb{R})$ with the compact support, the operator
$\psi(\Delta)$ given by

$\psi(\Delta)f:=\sum_{k\in\mathbb{Z}}e_{k}\otimes\psi(k)\hat{f}(k)\qquad(f\in{\mathcal{D}}^{\prime}({\mathbb{T}};X))$

is in ${\mathcal{L}}({\mathcal{D}}^{\prime}({\mathbb{T}};X))$.

Let $\\{\psi_{j}\\}_{j\in\mathbb{N}_{0}}$ be the resolution of the identity on
$\mathbb{R}$ generated by a function
$\psi\in{\mathcal{C}}^{\infty}(\mathbb{R})$ such that $\psi\equiv 1$ on
$[-1,1]$ and $\operatorname{supp}\psi\subset[-2,2]$, i.e.

$\psi_{0}:=\psi,\qquad\psi_{j}:=\psi(2^{-j}\cdot)-\psi(2^{-j+1}\cdot)\quad\textrm{
for }j\in\mathbb{N}.$

One can check that $\\{\psi_{j}(\Delta)\\}_{j\in\mathbb{N}_{0}}$ is the
resolution of the identity operator on
${\mathcal{D}}^{\prime}({\mathbb{T}};X)$, i.e. for every
$f\in{\mathcal{D}}^{\prime}({\mathbb{T}};X)$

$\sum_{j\leq N}\psi_{j}(\Delta)f=\psi(2^{-N}\Delta)f\rightarrow
f\qquad\textrm{ in }{\mathcal{D}}^{\prime}({\mathbb{T}};X)\textrm{ as
}N\rightarrow\infty.$

Let $\Phi$ be a Banach function space over $({\mathbb{T}},\mathrm{d}t)$. For
all $s\in\mathbb{R}$ and $q\in[1,\infty]$ we set (with usual modification when
$q=\infty$):

$B^{s,q}_{\Phi}({\mathbb{T}},X):=\left\\{f\in{\mathcal{D}}^{\prime}({\mathbb{T}};X):\,\,\|f\|_{B^{s,q}_{\Phi}({\mathbb{T}},X)}:=\left(\sum_{j=0}^{\infty}\|2^{sj}\psi_{j}(\Delta)f\|^{q}_{\Phi({\mathbb{T}};X)}\right)^{1/q}<\infty\right\\},$
$F^{s,q}_{\Phi}({\mathbb{T}},X):=\left\\{f\in{\mathcal{D}}^{\prime}({\mathbb{T}};X):\,\,\|f\|_{F^{s,q}_{\Phi}({\mathbb{T}},X)}:=\left\|\left(\sum_{j=0}^{\infty}|2^{sj}\psi_{j}(\Delta)f|^{q}_{X}\right)^{1/q}\right\|_{\Phi}<\infty\right\\}.$

For $G=\mathbb{R}$ and a general Banach function spaces $\Psi$ over
$(\mathbb{R},\mathrm{d}t)$, the corresponding generalized vector-valued Besov
$B^{s,q}_{\Psi}(\mathbb{R};X)$ and Triebel-Lizorkin spaces
$F^{s,q}_{\Psi}(\mathbb{R},X)$ were introduced in [51, Section 2]. In the case
when $G={\mathbb{T}}$, the vector-valued counterpart of the classical Besov
spaces $B^{s,q}_{p}({\mathbb{T}})$, i.e. for $\Phi=L^{p}$ over
$({\mathbb{T}},\mathrm{d}t)$, was introduced in [8]. In the both cases
($G={\mathbb{T}}$ or $G=\mathbb{R}$), under some additional assumption on
$\Psi$, one can show that the spaces $B^{s,q}_{\Psi}(G;X)$ and
$F^{s,q}_{\Psi}(G;X)$ share most of the properties of their well-known scalar
prototypes which correspond to $\Psi=L^{p}$ and $X=\mathbb{C}$. For our
further purposes, we only need a few basic properties of the spaces
$B^{s,q}_{\Phi}({\mathbb{T}};X)$ and $F^{s,q}_{\Phi}({\mathbb{T}};X)$
corresponding to a general Banach function space $\Phi$ over
$({\mathbb{T}},\mathrm{d}t)$; see Proposition 2.3 below. For their proofs we
need a preliminary result on the boundedness of Fourier multipliers
$\psi_{j}(\Delta)$, $j\in\mathbb{N}_{0}$, (and other ones) on the underlying
space $\Phi(X)$. Then, the proof of Proposition 2.3 can be carried out in an
analogy to the non-periodic case when $G=\mathbb{R}$ as it has been treated in
[51]; see [51, Lemma 3.6] and [51, Lemma 5.3]. However, it should be pointed
out that in a comparison to the proofs of some results in the case
$G=\mathbb{R}$, the proofs of their periodic counterparts admit an essential
simplification, which we indicate below; see also Subsection 3.1.

As it could be already noted above, we omit ’${\mathbb{T}}$’ in the symbols of
spaces over $({\mathbb{T}},\mathrm{d}t)$. Similarly, in the scalar case, i.e.
when $X=\mathbb{C}$, $\mathbb{C}$ is also omitted in the corresponding
symbols. For instance, $B^{s,q}_{\Phi}(X)$ stands for
$B^{s,q}_{\Phi}({\mathbb{T}};X)$ and ${\mathcal{C}}^{\infty}_{c}(\mathbb{R})$
denotes ${\mathcal{C}}^{\infty}_{c}(\mathbb{R};\mathbb{C})$, etc.

###  2.2. Preliminary results on boundedness of periodic multipliers

For a polynomially bounded sequence
$m:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Y)$ we write $\check{m}$ to denote the
corresponding periodic distribution in
${\mathcal{D}}^{\prime}({\mathcal{L}}(X,Y))$, i.e.

$\check{m}:=\sum_{k\in\mathbb{Z}}e_{k}\otimes m(k).$

######  Lemma 2.1.

Let $m:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Y)$ be such that $\check{m}$ is in
$L^{\infty}({\mathcal{L}}(X,Y))$. Then, for every Banach function space $\Phi$
over $({\mathbb{T}},\mathrm{d}t)$ such that $L^{\infty}\subset\Phi\subset
L^{1}$ the operator $m(\Delta)$ given by

$m(\Delta)f:=\sum_{k}e_{k}\otimes m(k)\hat{f}(k)\quad
f\in{\mathcal{D}}^{\prime}(X)$

is in ${\mathcal{L}}(\Phi(X),\Phi(Y))$ with
$\|m(\Delta)\|_{{\mathcal{L}}(\Phi(X),\Phi(Y))}\leq
c_{\Phi}\|\check{m}\|_{L^{\infty}}\|\chi_{\mathbb{T}}\|_{\Phi}$, where
$c_{\Phi}$ denotes the norm of embedding operator from $\Phi$ into $L^{1}$.

###### Proof.

A standard argument shows that $\Phi\hookrightarrow L^{1}$, i.e.
$\|g\|_{L^{1}}\leq c_{\Phi}\|g\|_{\Phi}$ ($g\in\Phi$). Since for every
$\eta\in L^{\infty}$, $g\in\Phi$ and $\tau\in{\mathbb{T}}$ we have

$|(\eta\ast g)(\tau)|\leq c_{\Phi}\|\eta\|_{L^{\infty}}\|g\|_{\Phi}$

we infer that

$\|\eta\ast g\|_{\Phi}\leq
c_{\Phi}\|\eta\|_{L^{\infty}}\|\chi_{\mathbb{T}}\|_{\Phi}\|g\|_{\Phi},$

where $\chi_{\mathbb{T}}$ is the characteristic function of ${\mathbb{T}}$.
Since for every $f\in L^{1}(X)$ we have

$|(m(\Delta)f)(\tau)|_{Y}=|(\check{m}\ast
f)(\tau)|_{Y}\leq\big{(}\|\check{m}\|_{{\mathcal{L}}(X,Y)}\ast|f|_{X}\big{)}(\tau)\qquad(\tau\in{\mathbb{T}}),$

the proof is complete. ∎

In particular, Lemma 2.1 shows that each operator $\psi_{j}(\Delta)$,
$j\in\mathbb{N}_{0}$, is bounded on $\Phi(X)$ if $L^{\infty}\subset\Phi\subset
L^{1}$. To show their uniform boundedness on $\Phi(X)$ we need an additional
assumption on the boundedness of the Hardy-Littlewood maximal operator on
$\Phi$.

Recall that the Hardy-Littlewood maximal operators $M_{\mathbb{T}}$ and
$M_{\mathbb{R}}$ are defined by

$M_{\mathbb{T}}f(\tau):=\sup_{\epsilon>0}\frac{1}{\epsilon}\int_{\Gamma(\tau,\epsilon)}|f(\zeta)|\,|\mathrm{d}\zeta|\qquad(\tau\in{\mathbb{T}})$

for $f\in L^{1}({\mathbb{T}})$, where
$\Gamma(\tau,\epsilon):={\mathbb{T}}\cap\\{z\in\mathbb{C}:|z-\tau|\leq\epsilon\\}$,
and

$M_{\mathbb{R}}f(t):=\sup_{\epsilon>0}\frac{1}{2\epsilon}\int_{[t-\epsilon,t+\epsilon]}|f(s)|\mathrm{d}s\qquad(t\in\mathbb{R})$

for $f\in L^{1}_{loc}(\mathbb{R})$. In the view of the standard identification
between the function $f$ on ${\mathbb{T}}$ with its $2\pi$-periodic extension
$\widetilde{f}$ on $\mathbb{R}$, $\widetilde{f}(t):=f(e^{it})$,
$t\in\mathbb{R}$, there exists a constant $c>0$ such that

$c^{-1}(M_{\mathbb{R}}\widetilde{f})(t)\leq(M_{\mathbb{T}}f)(e^{it})\leq
c(M_{\mathbb{R}}\widetilde{f})(t)\qquad(f\in
L^{1}({\mathbb{T}}),t\in\mathbb{R}).$

Note that the assumption that $M_{\mathbb{T}}$ is bounded on a Banach function
space $\Phi$ implies that $L^{\infty}\subset\Phi\subset L^{1}$. Indeed, if
$f\in\Phi\setminus\\{0\\}$, then there exists a constant $c>0$ and a
measurable subset $A$ of ${\mathbb{T}}$ such that $|f|\geq c\chi_{A}$, i.e.
$\chi_{A}\in\Phi$. Hence, by the boundedness of $M_{\mathbb{T}}$ on $\Phi$, we
get that $M_{\mathbb{T}}\chi_{A}\geq\frac{|A|}{2\pi}$. Consequently, by the
ideal property of $\Phi$, we obtain that $L^{\infty}\subset\Phi$.

The following lemma provides a periodic counterpart of [66, Chapter 2, (17)
p.57]. For its proof we need a vector-valued variant of Fejér’s theorem, which
asserts that for an arbitrary Banach space $X$ and $g\in
L^{1}({\mathbb{T}};X)$, if $S_{l}(g):=\sum_{|k|\leq l}e_{k}\otimes\hat{g}(k)$
for $l\in\mathbb{N}_{0}$, then

$\frac{1}{N+1}\sum_{l=0}^{N}S_{l}(g)\rightarrow g\quad\textrm{ as
}N\rightarrow\infty\textrm{ in }L^{1}({\mathbb{T}};X).$

Its proof follows the lines of the proof of its scalar prototype almost
verbatim.

######  Lemma 2.2.

_(i)_ Let $\eta\in{\mathcal{C}}_{c}(\mathbb{R};{\mathcal{L}}(X,Y))$ be such
that $\|{\mathcal{F}}^{-1}\eta(t)\|_{{\mathcal{L}}(X,Y)}\leq\phi(t)$,
$t\in\mathbb{R}$, for an even, radially decreasing, integrable function $\phi$
on $\mathbb{R}$. Then, there exists a constant $c>0$ such that for every $f\in
L^{1}({\mathbb{T}};X)$ we have

(1) $|\eta(\Delta)f(\tau)|_{Y}\leq
c\|\phi\|_{L^{1}(\mathbb{R})}(M_{\mathbb{T}}|f|_{X})(\tau)\qquad(\tau\in{\mathbb{T}}).$

_(ii)_ Let $\eta\in{\mathcal{C}}_{c}^{\infty}(\mathbb{R})$ and set
$\eta_{\epsilon}:=\eta(\epsilon\cdot)$, $\epsilon>0$. Then, for every Banach
function space $\Phi$ over $({\mathbb{T}},\mathrm{d}t)$ such that
$M_{\mathbb{T}}$ is bounded on $\Phi$, the operators
$\eta_{\epsilon}(\Delta)$, $\epsilon>0$, are uniformly in
${\mathcal{L}}(\Phi(X))$.

###### Proof.

(i) Let $f\in L^{1}({\mathbb{T}};X)$. It is readily seen that for every
$t\in\mathbb{R}$ the integral
$\int_{\mathbb{R}}({\mathcal{F}}^{-1}\eta)(s)\widetilde{f}(t-s)\mathrm{d}s=:({\mathcal{F}}^{-1}\eta\ast\widetilde{f})(t)$
is absolutely convergent. Suppose that $\operatorname{supp}\eta\subset[-K,K]$
for some $K\in\mathbb{Z}$. Note that for every $t\in\mathbb{R}$ and $l>K$ we
have

$\displaystyle\eta(\Delta)f(e^{it})$ $\displaystyle=\sum_{|k|\leq
l}e^{itk}\int_{\mathbb{R}}e^{-isk}{\mathcal{F}}^{-1}\eta(s)\mathrm{d}s\hat{f}(k)$
$\displaystyle=\int_{\mathbb{R}}{\mathcal{F}}^{-1}\eta(s)\sum_{|k|\leq
l}e^{i(t-s)k}\hat{f}(k)\mathrm{d}s.$

Therefore, following the notation introduced in Fejér’s theorem, for $N>K$ and
$t\in\mathbb{R}$ we get

(2)
$\frac{1}{N+1}\sum_{l=K}^{N}S_{l}(\eta(\Delta)f)(e^{it})=\int_{\mathbb{R}}{\mathcal{F}}^{-1}\eta(s)\frac{1}{N+1}\sum_{l=K}^{N}S_{l}(g_{t})(e^{is})\mathrm{d}s,$

where $g_{t}(\tau):=f(e^{it}\bar{\tau})$, $\tau\in{\mathbb{T}}$. By our
assumption on ${\mathcal{F}}^{-1}\eta$ it is straightforward to show that the
right-hand side of (2) converges to
$({\mathcal{F}}^{-1}\eta\ast\widetilde{f})(t)$ as $N\rightarrow\infty$. Since
the left-hand side is equal $\frac{N-K+1}{N}\eta(\Delta)f(e^{it})$, we infer
that for every $t\in\mathbb{R}$

$\left|\eta(\Delta)f(e^{it})\right|_{Y}=\left|({\mathcal{F}}^{-1}\eta\ast\widetilde{f})(t)\right|_{Y}\leq(\phi\ast|\widetilde{f}|_{X})(t).$

Further, note that for each $\epsilon>0$ there exists a function
$\phi_{\epsilon}:=\sum_{j=0}^{N}c_{j}\chi_{B_{j}}$, where $B_{j}$ denotes an
interval with the center in $0$ and $c_{j}>0$, such that
$0\leq\phi_{\epsilon}\leq\phi$,
$\|\phi-\phi_{\epsilon}\|_{L^{\infty}}<\epsilon$ and
$\|\phi-\phi_{\epsilon}\|_{L^{1}}<\epsilon$ . One can readily check that for
every $\delta>0$ there exists $\epsilon$ such that for every $t\in\mathbb{R}$
we have

$|(\phi\ast|\widetilde{f}|_{X})(t)|\leq((\phi-\phi_{\epsilon})\ast|\widetilde{f}|_{X})(t)+(\phi_{\epsilon}\ast|\widetilde{f}|_{X})(t)\leq\delta+\|\phi_{\epsilon}\|_{L^{1}}(M_{\mathbb{R}}|\widetilde{f}|_{X})(t).$

Since there exists $c>0$, independent of $f\in L^{1}({\mathbb{T}};X)$, such
that $(M_{\mathbb{R}}|\widetilde{f}|_{X})(t)\leq
c(M_{\mathbb{T}}|f|_{X})(e^{it})$ for every $t\in\mathbb{R}$, we get (1).

(ii) Note that
$({\mathcal{F}}^{-1}\eta_{\epsilon})(t)=\frac{1}{\epsilon}({\mathcal{F}}^{-1}\eta)(\frac{t}{\epsilon})$
for every $\epsilon>0$ and $t\in\mathbb{R}$. Moreover, there exists a constant
$C>0$ such that $|({\mathcal{F}}^{-1}\eta)(t)|\leq\frac{C}{1+t^{2}}$, which
yields
$|({\mathcal{F}}^{-1}\eta_{\epsilon})(t)|\leq\frac{\epsilon^{-1}C}{1+(\epsilon^{-1}t)^{2}}=:\phi_{\epsilon}(t)$
for every $t\in\mathbb{R}$ and $\epsilon>0$. Since
$\|\phi_{\epsilon}\|_{L^{1}}=C\pi$ for every $\epsilon$. Therefore, (1) gives
the desired claim. ∎

###  2.3. Fundamental properties of generalized Besov and Triebel-Lizorkin
spaces

Here, we collect the fundamental properties of such spaces, which play a role
in our further studies; see Proposition 2.3 below.

We start with some preliminaries. Note that for every $s\in\mathbb{R}$ the
space $B^{-s,1}_{\Phi}(X^{*})$ embeds into $(B^{s,\infty}_{\Phi}(X))^{*}$.
Indeed, this embedding is given by the following duality pairing: for each
$f\in B^{s,\infty}_{\Phi}(X)$ and $g\in B^{-s,1}_{\Phi^{\prime}}(X^{*})$ we
set

(3) $\langle
g,f\rangle:=\sum_{j,l\in\mathbb{N}_{0}}\int_{{\mathbb{T}}}\langle\psi_{l}(\Delta)g(t),\psi_{j}(\Delta)f(t)\rangle_{X^{*},X}\mathrm{d}t.$

Note that

$\langle g,f\rangle:=\sum_{r\in\\{\pm
1,0\\}}\sum_{l\in\mathbb{N}_{0}}\langle\breve{\psi}_{j}(\Delta)\psi_{l}(\Delta)g,\chi_{j}(\Delta)f\rangle_{\Phi^{\prime}(X^{*}),\Phi(X)},$

where $\breve{\psi}_{j}:=\psi_{j}(-\cdot)$ and
$\chi_{j}:=\psi_{j-1}+\psi_{j}+\psi_{j+1}$ $(j\in\mathbb{N}_{0})$ with
$\psi_{-1}\equiv 0$ if $j=0$.

######  Proposition 2.3.

Let $X$ be a Banach space and $\Phi$ be a Banach function space over
$({\mathbb{T}},\mathrm{d}t)$. Then, the following assertions hold.

  * (i)

If $L^{\infty}\subset\Phi\subset L^{1}$, then for every

$\mathbb{E}\in\left\\{B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}\,:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty)\right\\}$
(4)
${\mathcal{P}}(X)\subset\mathbb{E}(X)\hookrightarrow{\mathcal{D}}^{\prime}(X).$

In particular, $\mathbb{E}(X)$ is a Banach space.

  * (ii)

If $M_{\mathbb{T}}$ is bounded on $\Phi$, then for all $s\in\mathbb{R}$,
${\mathcal{P}}(X)$ is a dense subset of $B^{s,q}_{\Phi}(X)$ in the norm
topology if $q\in[1,\infty)$, and in the
$B^{-s,1}_{\Phi^{\prime}}(X^{*})$-topology of $B^{s,\infty}_{\Phi}(X)$ if
$q=\infty$.

More precisely, for every $f\in B^{s,q}_{\Phi}(X)$ we have that

$\sum_{0\leq j\leq N}\psi_{j}(\Delta)f=\psi(2^{-N}\Delta)f\rightarrow
f\quad\textrm{ as }N\rightarrow\infty.$

in $B^{s,q}_{\Phi}(X)$ for each $q<\infty$, and in the other case, i.e.
$q=\infty$, the convergence holds in the
$\sigma(B^{s,\infty}_{\Phi}(X),B^{-s,1}_{\Phi^{\prime}}(X^{*}))$-topology.

  * (ii’)

If $M_{\mathbb{T}}$ is bounded on $\Phi$ and its dual $\Phi^{\prime}$, then
${\mathcal{P}}(X)$ is a dense subset of $F^{s,q}_{\Phi}(X)$ and for ever $f\in
F^{s,q}_{\Phi}(X)$ we have that

$\sum_{0\leq j\leq N}\psi_{j}(\Delta)f=\psi(2^{-N}\Delta)f\rightarrow
f\quad\textrm{ as }N\rightarrow\infty.$

  * (iii)

If $M_{\mathbb{T}}$ is bounded on $\Phi$, then for every distribution
$f\in{\mathcal{D}}^{\prime}(X)$, $f$ belongs to $B^{s,q}_{\Phi}(X)$ if and
only if $\partial f$ belongs to $B^{s-1,q}_{\Phi}(X)$. Moreover, the function

(5) $B^{s,q}_{\Phi}(X)\ni f\mapsto\left\|\partial
f\right\|_{B^{s-1,q}_{\Phi}(X)}$

is an equivalent norm on $B^{s,q}_{\Phi}(X)$.

######  Remark 2.4.

(a) Compared to the case of the real line $\mathbb{R}$ (see [51, Lemma 3.6]),
note that in the periodic case there is a common dense subset, i.e.
${\mathcal{P}}(X)$, for all $\mathbb{E}(X)$ with

$\mathbb{E}\in\\{B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}\,:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty)\\}.$

(b) In contrast to [51, Lemma 3.6], in Proposition 2.3, in the case of Besov
spaces we do not assume that $M_{\mathbb{T}}$ is bounded on the dual of
$\Phi^{\prime}$.

By Lemma 2.2, the proof of Proposition 2.3, mimics the proof of the
corresponding statements on $\mathbb{R}$, [51, Lemmas 3.6 and 5.4]. For the
convenience of the reader we provide some auxiliary observations which should
be made.

###### The proof of Proposition 2.3.

(i) We start with the case when $\mathbb{E}=B^{s,q}_{\Phi}$. Since
${\mathcal{P}}\subset L^{\infty}\subset\Phi$, the left inclusion in (4) holds
readily. For the second one, note that for every $f\in B^{s,q}_{\Phi}(X)$ and
$\phi\in{\mathcal{D}}$ we have that

$\displaystyle\left|\left(\phi_{j}(\Delta)f\right)(\phi)\right|_{X}$
$\displaystyle=\left|\sum_{k\in\mathbb{Z}}\int_{\mathbb{T}}e_{k}\phi\mathrm{d}t\,\psi_{j}(k)\hat{f}(k)\right|_{X}$
$\displaystyle=2\pi\left|\sum_{k\in\mathbb{Z}}\chi_{j}(k)\hat{\phi}(-k)\psi_{j}(k)\hat{f}(k)\right|_{X}$
$\displaystyle\leq
2\pi\int_{\mathbb{T}}\left|\sum_{k\in\mathbb{Z}}e_{k}\otimes\psi_{j}(k)\hat{f}(k)\right|_{X}\left|\sum_{l\in\mathbb{Z}}\chi_{j}(l)\hat{\phi}(-l)e_{l}\right|\mathrm{d}t$
$\displaystyle\leq
2\pi\left\|\psi_{j}(\Delta)\hat{f}\right\|_{\Phi(X)}\left\|\chi_{j}(\Delta)\breve{\phi}\right\|_{\Phi^{\prime}}$

Here, $\breve{\phi}(e^{it})=\phi(e^{-it})$, $t\in[-\pi,\pi]$. Since
$\lim_{N\rightarrow\infty}\sum_{j=0}^{N}\phi_{j}(\Delta)f=f$ in
${\mathcal{D}}^{\prime}(X)$, we obtain that

$\displaystyle|f(\phi)|_{X}$
$\displaystyle\leq\sum_{j\in\mathbb{N}_{0}}\left\|2^{js}\psi_{j}(\Delta)f\right\|_{\Phi(X)}\|2^{-js}\chi_{j}(\Delta)\breve{\phi}\|_{\Phi^{\prime}}$
$\displaystyle\leq\left(\sum_{j\in\mathbb{N}_{0}}\left\|2^{jsq}\psi_{j}(\Delta)f\right\|^{q}_{\Phi(X)}\right)^{1/q}\left(\sum_{j\in\mathbb{N}_{0}}\|2^{-js}\chi_{j}(\Delta)\breve{\phi}\|_{\Phi^{\prime}}^{q^{\prime}}\right)^{1/q^{\prime}}$
$\displaystyle\leq\|f\|_{B^{s,q}_{\Phi}(X)}\left(\sum_{j\in\mathbb{N}_{0}}\|2^{-js}\chi_{j}(\Delta)\check{\phi}\|_{\Phi^{\prime}}^{q^{\prime}}\right)^{1/q^{\prime}}$

(with the usual modification when $q=\infty$). Take $\alpha>|s|+1$ and set
$\rho(t):=(1+t^{2})^{-\alpha}$, $t\in\mathbb{R}$. Then, for every
$j\in\mathbb{N}_{0}$ and $\tau\in{\mathbb{T}}$ we have

$\displaystyle
2^{(|s|+1)j}\left|{\mathcal{F}}^{-1}({\rho\chi_{j}}_{|\mathbb{Z}})(\tau)\right|$
$\displaystyle=\left|\sum_{k\in\mathbb{Z}}\tau^{k}\frac{2^{(|s|+1)j}}{(1+k^{2})^{\alpha}}\chi_{j}(k)\right|$
$\displaystyle\leq\sum_{2^{j-2}<k<2^{j+2}}4^{\alpha}2^{-\alpha j}\leq
4^{\alpha+1}$

Therefore, by Lemma 2.1(i), the operators
$(2^{(|s|+1)j}\chi_{j}\rho)(\Delta)$, $j\in\mathbb{N}_{0}$, restrict to
uniformly bounded operators in ${\mathcal{L}}(\Phi^{\prime})$.

Since
$(\rho^{-1}(\Delta)\breve{\phi})(\tau)=\sum_{k\in\mathbb{Z}}\tau^{k}(1+k^{2})^{\alpha}{\mathcal{F}}{\breve{\phi}}(k)$
$(\tau\in{\mathbb{T}})$ is in ${\mathcal{D}}\subset\Phi^{\prime}$, for every
$j\in\mathbb{N}_{0}$ we infer that

$\displaystyle\|2^{j|s|}\chi_{j}(\Delta)\breve{\phi}\|_{\Phi^{\prime}}$
$\displaystyle\leq
2^{-j}\|(2^{(|s|+1)j}\rho\chi_{j})(\Delta)(\rho^{-1})(\Delta)\breve{\phi}\|_{\Phi^{\prime}}$
$\displaystyle\leq
2^{-j}\|(2^{(|s|+1)j}\rho\chi_{j})(\Delta)\|_{{\mathcal{L}}(\Phi^{\prime})}\|\rho^{-1}(\Delta)\breve{\phi}]\|_{\Phi^{\prime}}$
$\displaystyle\leq
2^{-j}4^{\alpha+1}\|\rho^{-1}(\Delta)\breve{\phi}]\|_{\Phi^{\prime}}.$

It yields $B^{s,q}_{\Phi}(X)\hookrightarrow{\mathcal{D}}^{\prime}(X)$. The
proof in the case when $\mathbb{E}=F^{s,q}_{\Phi}$ follows closely the
arguments presented above. Therefore, we omit it.

(ii) Note that for every $j\in\mathbb{N}$, $\psi_{j}(t)=\eta(2^{-j}t)$, where
$\eta(t):=\psi(t)-\psi(2t)$, $t\in\mathbb{R}$. Thus, by Lemma 2.2(ii), the
operators $\psi_{j}(\Delta)$, $j\in\mathbb{N}_{0}$, restrict to uniformly
bounded operators in ${\mathcal{L}}(\Phi(X))$. Now the proof mimics the lines
of the proof of the corresponding statement in [51, Lemma 3.6]. Therefore, we
omit it.

(ii’) Since $F^{s,q}_{\Psi}(X)=B^{s,q}_{\Psi}(X)$ for every $\Psi=L^{q}_{w}$
with $q\in(1,\infty)$ and $w\in A_{q}({\mathbb{T}})$, by the point $(ii)$, the
statement $(ii^{\prime})$ holds for such spaces. The proof for a general
$\Phi$ extrapolates from these particular ones via an adaptation of the Rubio
de Francia iteration algorithm.

Let ${\mathcal{R}}$ and ${\mathcal{R}}^{\prime}$ denote the following
sublinear, positive operators defined on $\Phi$ and $\Phi^{\prime}$,
respectively:

(6)
${\mathcal{R}}g:=\sum_{k=0}^{\infty}\frac{M_{\mathbb{T}}^{k}|g|}{(2\|M_{\mathbb{T}}\|_{\Phi})^{k}}\quad(g\in\Phi)\quad\textrm{
and
}\quad{\mathcal{R}}^{\prime}h:=\sum_{k=0}^{\infty}\frac{M_{\mathbb{T}}^{k}|h|}{(2\|M_{\mathbb{T}}\|_{\Phi^{\prime}})^{k}}\quad(h\in\Phi^{\prime})$

Here, $M_{\mathbb{T}}^{k}$ stands for the $k$-th iteration of the Hardy-
Littlewood operator $M_{\mathbb{T}}$, $M_{\mathbb{T}}^{0}:=I$, and
$\|M_{\mathbb{T}}\|_{\Phi}:=\sup_{\|g\|_{\Phi}\leq
1}\|M_{\mathbb{T}}g\|_{\Phi}$. Since

$M({\mathcal{R}}g)\leq
2\|M\|_{\Phi}{\mathcal{R}}g\quad(g\in\Phi)\quad\textrm{and}\qquad
M({\mathcal{R}}^{\prime}h)\leq
2\|M\|_{\Phi^{\prime}}{\mathcal{R}}h\quad(h\in\Phi^{\prime}),$

for every $g\in\Phi$ and $h\in\Phi^{\prime}$ the functions ${\mathcal{R}}g$
and ${\mathcal{R}}^{\prime}h$ belong to Muckenhoupt’s class
$A_{1}({\mathbb{T}})$. Consequently, for every $q\in(1,\infty)$, $g\in\Phi$,
$g\neq 0$, and $h\in\Phi^{\prime}$, $h\neq 0$, the function

(7) $w_{g,h,q}:=({\mathcal{R}}g)^{1-q}{\mathcal{R}}^{\prime}h$

is in Muckenhoupt’s class $A_{q}({\mathbb{T}})$ with the $A_{q}$-constant

(8) $[w_{g,h,q}]_{A_{q}}\leq 2^{q}\|M\|_{\Phi}^{q-1}\|M\|_{\Phi^{\prime}}.$

Moreover, note that if $g\in\Phi(X)$, then $f\in L^{q}_{w}(X)$ for
$w:=({\mathcal{R}}|f|_{X})^{1-q}{\mathcal{R}}^{\prime}h$ with an arbitrary
$h\in\Phi^{\prime}$, $h\neq 0$. Indeed, since
$|g|_{X}\leq{\mathcal{R}}|g|_{X}$, we get

$\int_{\mathbb{R}}|g|_{X}^{q}w\mathrm{d}t\leq\int_{\mathbb{R}}{\mathcal{R}}|g|_{X}{\mathcal{R}}^{\prime}h\mathrm{d}t\leq\|{\mathcal{R}}|g|_{X}\|_{\Phi}\|{\mathcal{R}}^{\prime}h\|_{\Phi^{\prime}}<\infty.$

By Lemma 2.2 combined with Muckenhoupt’s theorem (see e.g. [16]) we get that
if

$\mu_{w,q}:=\sup_{j\in\mathbb{N}_{0}}\|\psi_{j}(\Delta)\|_{{\mathcal{L}}(L^{q}_{w}(X))},$

then $\sup_{w\in{\mathcal{W}}}\mu_{w,q}<\infty$ for each ${\mathcal{W}}\subset
A_{q}({\mathbb{T}})$ with $\sup_{w\in{\mathcal{W}}}[w]_{A_{q}}<\infty$.

Now we are in a position to apply Rubio de Francia’s extrapolation argument.
Fix $q\in(1,\infty)$. Let $f\in F^{s,q}_{\Phi}(X)$ and $f_{N}:=\sum_{j\leq
N}\psi_{j}(\Delta)f$ for all $N\in\mathbb{N}$. Note that
$f_{N}\in{\mathcal{P}}(X)$ and

$\|f-f_{N}\|_{F^{s,q}_{\Phi}(X)}\lesssim\sum_{r=-1}^{1}\Big{\|}\big{(}\sum_{j\geq
N}2^{jsq}|\psi_{j+r}(\Delta)\psi_{j}(\Delta)f|_{X}^{q}\big{)}^{1/q}\Big{\|}_{\Phi}.$

For each $r\in\\{\pm 1,0\\}$ set

$g_{r}:=g_{N,r,f}:=\big{(}\sum_{j\geq
N}2^{jsq}|\psi_{j+r}(\Delta)\psi_{j}(\Delta)f|_{X}^{q}\big{)}^{1/q},\quad\textrm{and}$
$g:=g_{N,f}:=\big{(}\sum_{j\geq
N}2^{jsq}|\psi_{j}(\Delta)f|_{X}^{q}\big{)}^{1/q}\quad(N\in\mathbb{N}).$

Let $h\in\Phi^{\prime}$ with $h\neq 0$ and
$w:=w_{g,h,q}={\mathcal{R}}(g)^{1-q}{\mathcal{R}}^{\prime}h$. Then,

$\displaystyle\mu_{w,q}\|{\mathcal{R}}\|_{{\mathcal{L}}(\Phi)}\|g\|_{\Phi}\|{\mathcal{R}}^{\prime}\|_{{\mathcal{L}}(\Phi^{\prime})}\|h\|_{\Phi^{\prime}}$
$\displaystyle\geq\mu_{w,q}\|{\mathcal{R}}g\|_{\Phi}\|{\mathcal{R}}^{\prime}h\|_{\Phi^{\prime}}$
$\displaystyle\geq\mu_{w,q}\left(\int_{\mathbb{T}}{\mathcal{R}}g{\mathcal{R}}^{\prime}h\,\mathrm{d}t\right)^{\frac{1}{q}}\left(\int_{\mathbb{T}}{\mathcal{R}}g{\mathcal{R}}^{\prime}h\,\mathrm{d}t\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\geq\left(\int_{\mathbb{T}}g_{r}^{q}w\,\mathrm{d}t\right)^{\frac{1}{q}}\left(\int_{\mathbb{T}}{\mathcal{R}}g{\mathcal{R}}^{\prime}h\mathrm{d}t\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\geq\int_{\mathbb{T}}g_{r}{\mathcal{R}}^{\prime}h\,\mathrm{d}t$
$\displaystyle\geq\int_{\mathbb{T}}g_{r}|h|\,\mathrm{d}t.$

Since

$\sup\left\\{[w_{g_{N,f},h,q}]_{A_{q}}:f\in
F^{s,q}_{\Phi}(X),N\in\mathbb{N},h\in\Phi^{\prime}\setminus\\{0\\}\right\\}<\infty,$

we get that

$\mu:=\sup\left\\{\mu_{w,q}:\,w=w_{g_{N,f},h,q}\textrm{ with }f\in
F^{s,q}_{\Phi}(X),N\in\mathbb{N},h\in\Phi^{\prime}\setminus\\{0\\}\right\\}<\infty.$

Therefore, by the Lorentz-Luxemburg theorem, we infer that $g_{r}\in\Phi(X)$
($r\in\\{\pm 1,0\\})$ and for all $N\in\mathbb{N}$ we have

$\|g_{N,r,f}\|_{\Phi}\leq\mu\|{\mathcal{R}}\|_{{\mathcal{L}}(\Phi)}\|{\mathcal{R}}^{\prime}\|_{{\mathcal{L}}(\Phi^{\prime})}\|f-f_{N}\|_{\Phi}.$

Since, by Fatou’s property of $\Phi$, $f_{N}-f\rightarrow 0$ in $\Phi(X)$ as
$N\rightarrow\infty$, we get the desired claim.

$(iii)$ First note that for every $f\in{\mathcal{D}}^{\prime}(X)$ and
$j\in\mathbb{N}_{0}$, $\partial f=\rho(\Delta)f$, where $\rho(t)=it$
$(t\in\mathbb{R})$, and

$\displaystyle\|2^{(s-1)j}\psi_{j}(\Delta)\partial f\|_{\Phi(X)}$
$\displaystyle=\|2^{(s-1)j}(\rho\chi_{j})(\Delta)\psi_{j}(\Delta)f\|_{\Phi(X)}$
$\displaystyle\leq\|2^{-j}(\rho\chi_{j})(\Delta)\|_{{\mathcal{L}}(\Phi(X))}\|2^{sj}\psi_{j}(\Delta)f\|_{\Phi(X)}.$

Moreover, if we set $\eta(t):=t\psi(\frac{1}{2}t)-t\psi(4t)$
$(t\in\mathbb{R})$, then for every $j\geq 2$ and $t\in\mathbb{R}$, we have

$\eta(2^{-j}t)=-i2^{-j}(\rho\chi_{j})(t).$

Thus, by Lemma 2.2, we infer that
$\sup_{j\in\mathbb{N}_{0}}\|2^{-j}(\rho\chi_{j})(\Delta)\|_{{\mathcal{L}}(\Phi(X))}<\infty$,
and consequently $\partial f\in B^{s-1,q}_{\Phi}(X)$ if $f\in
B^{s,q}_{\Phi}(X)$.

Now, suppose that $f\in{\mathcal{D}}^{\prime}(X)$ with $\partial f\in
B^{s-1,q}_{\Phi}(X)$. Let
$\eta(t):=\frac{1}{t}\psi(\frac{1}{2}t)-\frac{1}{t}\psi(4t)$. Then, for every
$j\geq 2$ we have $2^{j}\chi_{j}(t)\frac{1}{\rho(t)}=-i\eta(2^{-j}t)$
$(t\in\mathbb{R})$ and

$\displaystyle\|2^{sj}\psi_{j}(\Delta)f\|_{\Phi(X)}$
$\displaystyle\leq\|2^{j}(\chi_{j}\frac{1}{\rho})(\Delta)\|_{{\mathcal{L}}(\Phi(X))}\|2^{(s-1)j}\psi_{j}(\Delta)\rho(\Delta)f\|_{\Phi(X)}$
$\displaystyle=\|\eta(2^{-j}\Delta)\|_{{\mathcal{L}}(\Phi(X))}\|2^{(s-1)j}\psi_{j}(\Delta)\partial
f\|_{\Phi(X)}.$

Therefore, Lemma 2.2 shows that $f\in B^{s,q}_{\Phi}(X)$. It completes the
proof. ∎

######  Remark 2.5.

(a) The norms $\|\cdot\|_{B^{s,q}_{\Phi}(X)}$(respectively,
$\|\cdot\|_{F^{s,q}_{\Phi}(X)}$) of course depend on the chosen function
$\psi$ generating the resolution of the identity involved in the definition of
the corresponding spaces $B^{s,q}_{\Phi}(X)$(respectively,
${F^{s,q}_{\Phi}}(X)$). However, the arguments presented in the proof of
Proposition 2.3(ii)(respectively (ii’)), show that the norms corresponding to
different generating functions are equivalent when the Hardy-Littlewood
maximal operator $M_{\mathbb{T}}$ is bounded on $\Phi$ (respectively, on
$\Phi$ and its dual $\Phi^{\prime}$). The fact that, in the case of Triebel-
Lizorkin spaces, one cannot drop, in general, the boundedness of
$M_{\mathbb{T}}$ on $\Phi^{\prime}$ is well-known from the classical
situation, i.e. $\Phi=L^{\infty}$; cf. [68, Subsection 2.1.4].

(b) Straightforward arguments show that $B^{-s,1}_{\Phi}(X^{*})$ is separating
on $B^{s,\infty}_{\Phi}(X)$ for every $s\in\mathbb{R}$. We apply it in Section
5 below.

##  3\. Fourier multipliers

###  3.1. The representation formula for periodic multipliers

Let $X$ and $Y$ be Banach spaces. By the representation result for the
periodic distributions ${\mathcal{D}}^{\prime}({\mathbb{T}};X)$ (see, e.g. [8,
Proposition 2.1], as well as [35, Chapter 12] or [63, Section 3] for its
scalar-valued prototype), it is easily seen that for an arbitrary sequence
$m:\mathbb{Z}\mapsto{\mathcal{L}}(X,Y)$ with a polynomial growth, the
following operator

(9) $m(\Delta):{\mathcal{D}}^{\prime}({\mathbb{T}},X)\ni
f\mapsto\sum_{k\in\mathbb{Z}}e_{k}\otimes
m(k)\hat{f}(k)\in{\mathcal{D}}^{\prime}({\mathbb{T}},Y)$

is well-defined and continuous (the series is convergent in
${\mathcal{D}}^{\prime}({\mathbb{T}};Y)$).

Let $\Phi$ be a Banach function space over $({\mathbb{T}},\mathrm{d}t)$ and

$\mathbb{E}\in\\{\Phi,B^{s,q}_{\Phi},F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty)\\}.$

A sequence $m:\mathbb{Z}\to{\mathcal{L}}(X,Y)$ is called a
$\mathbb{E}$-Fourier multiplier if $m(\Delta)$ restricts to an operator in
${\mathcal{L}}(\mathbb{E}({\mathbb{T}};X),\mathbb{E}({\mathbb{T}};Y))$. We set
${\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,Y)$ to denote the space of all
${\mathcal{L}}(X,Y)$-valued $\mathbb{E}$-Fourier multipliers, and we simply
write ${\mathcal{M}}_{p}({\mathbb{T}};X,Y)$ for
${\mathcal{M}}_{L^{p}}({\mathbb{T}};X,Y)$.

Now, let $\Psi$ be a Banach function space over $(\mathbb{R},\mathrm{d}t)$ and

$\mathbb{E}\in\\{\Psi,B^{s,q}_{\Psi},F^{s,r}_{\Psi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty)\\}.$

Let $m:\mathbb{R}\to{\mathcal{L}}(X,Y)$ be a bounded measurable function and

$D_{m}:=\\{f\in{\mathcal{S}}^{\prime}(\mathbb{R};X):{\mathcal{F}}f\in
L^{1}_{loc}(\mathbb{R};X)\text{ and
}m(\cdot){\mathcal{F}}f\in{\mathcal{S}}^{\prime}(\mathbb{R};Y)\\}$

Then, $m$ is called a $\mathbb{E}$-Fourier multiplier, if the following
operator

$m(D):D_{m}\cap\mathbb{E}(\mathbb{R};X)\ni
f\mapsto{\mathcal{F}}^{-1}(m(\cdot){\mathcal{F}}f)\in{\mathcal{S}}^{\prime}(\mathbb{R};Y)$

extends to an operator in
${\mathcal{L}}(\mathbb{E}(\mathbb{R};X),\mathbb{E}(\mathbb{R};Y))$. Similarly
to the periodic case, we set ${\mathcal{M}}_{\mathbb{E}}(\mathbb{R};X,Y)$ to
denote the space of all ${\mathcal{L}}(X,Y)$-valued $\mathbb{E}$-Fourier
multipliers, and we simply write ${\mathcal{M}}_{p}(\mathbb{R};X,Y)$ for
${\mathcal{M}}_{L^{p}}(\mathbb{R};X,Y)$.

######  Remark 3.1.

Note that in a contrast to the representation formulas for Fourier multipliers
on the real line, each periodic multiplier has a (canonical) representation
formula (cf. [51, Section 4]). More precisely, since it is not obvious how to
define, in general, the extension of the pointwise multiplication of a
function $m\in L^{\infty}(\mathbb{R};{\mathcal{L}}(X,Y))$ with a distribution
$f\in{\mathcal{S}}^{\prime}(\mathbb{R};X)$, only in special cases one can show
that the operator

$m(D):{\mathcal{S}}(\mathbb{R};X)\ni
f\mapsto{\mathcal{F}}^{-1}(m(\cdot){\mathcal{F}}f)\in{\mathcal{S}}^{\prime}(\mathbb{R};Y)$

extends to a continuous operator on whole
${\mathcal{S}}^{\prime}(\mathbb{R};X)$. For instance, recall that by means of
Schwartz’ kernel theorem, such extension holds for each
$m\in{\mathcal{O}}_{M}(\mathbb{R};{\mathcal{L}}(X,Y))$, that is, slowly
increasing, smooth functions. In this case, we have a formal expression for
such extension given for all $f\in{\mathcal{S}}^{\prime}(\mathbb{R};X)$ by

$m(D)f={\mathcal{F}}^{-1}\Theta(m,{\mathcal{F}}f),$

where $\Theta$ is a hypocontinuous, bilinear map from
${\mathcal{O}}_{M}(\mathbb{R};{\mathcal{L}}(X,Y))\times{\mathcal{S}}^{\prime}(\mathbb{R};X)$
into ${\mathcal{S}}^{\prime}(\mathbb{R};Y)$; see, e.g. Amann [2, Theorem 2.1].
Consequently, compared to the case of the real line, the study of periodic
integro-differential equations is conceptually more straightforward at this
point (cf. e.g. the presentation of [51, Proposition 4.2] with Theorem 5.5
below, see also [43, Problems 3.2 and 3.3]).

###  3.2. The multiplier conditions

For the convenience of the reader we recall the notion of Marcinkiewicz’s type
conditions used in the next sections.

Let $X$ and $Y$ denote Banach spaces and $\gamma\in\mathbb{N}$. We say that a
function $m:\mathbb{R}\rightarrow{\mathcal{L}}(X,Y)$ satisfies the
Marcinkiewicz condition of order $\gamma$ (in short, the
$\mathfrak{M}^{\gamma}$-condition), if
$m\in{\mathcal{C}}^{\gamma}(\dot{\mathbb{R}};{\mathcal{L}}(X,Y))$ and

$[m]_{\mathfrak{M}^{\gamma}}:=\max_{l=0,...,\gamma}\sup_{t\neq
0}\|t^{l}m^{(l)}(t)\|_{{\mathcal{L}}(X,Y)}<\infty\quad\quad{(\mathfrak{M}^{\gamma})}.$

Then, we write $m\in\mathfrak{M}^{\gamma}(\mathbb{R};{\mathcal{L}}(X,Y))$,

Furthermore, let
$\widetilde{\mathfrak{M}}^{\gamma}(\mathbb{R};{\mathcal{L}}(X,Y))$ denote a
subclass of ${\mathfrak{M}}^{\gamma}(\mathbb{R};{\mathcal{L}}(X,Y))$, which
consists symbols $m$ such that

$[m]_{\widetilde{\mathfrak{M}}^{\gamma}}:=\max_{l=0,...,\gamma}\sup_{t\neq
0}\|(1+|t|)^{l}m^{(l)}(t)\|_{{\mathcal{L}}(X,Y)}<\infty\quad\quad{(\widetilde{\mathfrak{M}}^{\gamma})}$

and $m(0^{+})=m(0^{-})$ (equivalently, $m$ has continuous extension at $0$).

The discrete counterpart is defined analogously. Namely, a sequence
$m:\mathbb{Z}\to{\mathcal{L}}(X,Y)$ satisfies the
$\mathfrak{M}^{\gamma}$-condition, if

$[m]_{\mathfrak{M}^{\gamma}}:=\max_{l=0,...,\gamma}\sup_{k\in\mathbb{Z}}\|k^{l}(\Delta^{l}m)(k)\|_{{\mathcal{L}}(X,Y)}<\infty,$

where $(\Delta^{l}m)(k)=\sum_{j=0}^{l}\binom{l}{j}(-1)^{l-j}m(k+j)$ for
$l\in\mathbb{N}_{0}$. Similarly, we write
$m\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(X,Y))$.

Furthermore, we say that $m:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Y)$ satisfies
the variational Marcinkiewicz condition, if

$[m]_{Var}:=\sup_{k\in\mathbb{Z}}\|m(k)\|_{{\mathcal{L}}(X,Y)}+\sup_{j\geq
0}\sum_{2^{j}\leq|k|<2^{j+1}}\|m(k+1)-m(k)\|_{{\mathcal{L}}(X,Y)}<\infty.$

Set $Var(\mathbb{Z};{\mathcal{L}}(X,Y))$ to denote the space of all sequences
which satisfy the above condition. Note that
$\mathfrak{M}^{1}(\mathbb{Z};{\mathcal{L}}(X,Y))\subset
Var(\mathbb{Z};{\mathcal{L}}(X,Y))$.

Finally, we say that $m:\mathbb{Z}\to{\mathcal{L}}(X,Y)$ satisfies the
$\mathfrak{M}^{\gamma}_{{\mathcal{R}}}$-condition and write
$m\in\mathfrak{M}^{\gamma}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X,Y))$, if
the sets $\\{k^{l}\Delta^{l}m(k)\\}$ are ${\mathcal{R}}$-bounded for all
$l\in\\{0,1,...,\gamma\\}$. We refer the reader, e.g. to [52] or [34], for the
background on ${\mathcal{R}}$-boundedness.

##  4\. Transference results

In this section we provide some crucial ingredients of the results presented
in the following sections. These ingredients allow to transfer directly some
results known in the real-line setting to the periodic one, rather than
rephrasing the proofs from $\mathbb{R}$ to ${\mathbb{T}}$. Such transference
methods reveal some natural questions, which maybe of independent interest;
see e.g. Remark 4.7(a).

###  4.1. The operator-valued counterpart of Jodeit’s theorem

The following result is an operator-valued counterpart of an extension of
Jodeit’s theorem ([46, Theorem 3.5]) provided by Asmar et al; see[10, Theorem
1.1].

######  Theorem 4.1.

Let $X,Y$ and $Z$ be Banach spaces and $p\in[1,\infty)$. Let $\Lambda\in
L^{\infty}(\mathbb{R};Y,Z)$ have a compact support. Then the following
assertions are equivalent.

  * (i)

$\Lambda\in{\mathcal{M}}_{p}(\mathbb{R};Y,Z)$;

  * (ii)

For every $m\in{\mathcal{M}}_{p}(\mathbb{Z};X,Y)$ the function

(10) $e(\Lambda,m)(t):=\sum_{n\in\mathbb{Z}}\Lambda(t-n)m(n)$

is in ${\mathcal{M}}_{p}(\mathbb{R};X,Z)$.

######  Remark 4.2.

Note that (according to the well-known results of Burkholder [26] and Bourgain
[18]), if $Y$ is not $U\\!M\\!D$ space, then not for every compactly
supported, scalar function
$\lambda\in{\mathcal{M}}_{p}(\mathbb{R};\mathbb{C})$, the function
$\Lambda(t):=\lambda(t)I_{Y}$ is in
${\mathcal{M}}_{p}(\mathbb{R};{\mathcal{L}}(Y))$ (here we assume that $Y=Z$
and put $I_{Y}$ for the identity operator on $Y$). Moreover, since the proof
of [10, Theorem 1.1] is based on several non-trivial ingredients (mainly from
[33] and [46]), for the convenience of the reader we outline how the above-
stated operator-valued extension of [10, Theorem 1.1] can be derived from its
scalar counterpart.

###### Proof.

Following [10] we split the proof into two steps. In the first one, we need to
show that if $\Lambda$ is a function in ${\mathcal{M}}_{p}(\mathbb{R};Y,Z)$
with $\operatorname{supp}\Lambda\subset[-\frac{1}{4}2^{N},\frac{1}{4}2^{N}]$
for some $N\in\mathbb{N}$, then the periodic extension of
$\Lambda(2^{N}\cdot)$ on $\mathbb{R}$ from $[-1,1]$, i.e. the function
$\Lambda^{\sharp}(t):=\sum_{n\in\mathbb{Z}}\Lambda(2^{N}(t-n))$, is in
${\mathcal{M}}_{p}(\mathbb{R};Y,Z)$ as well.

Since, as one can directly check the function $\Lambda(2^{N}\cdot)$ is in
${\mathcal{M}}_{p}(\mathbb{R};Y,Z)$ with

$\|\Lambda(2^{N}\cdot)\|_{{\mathcal{M}}_{p}(\mathbb{R};Y,Z)}=\|\Lambda\|_{{\mathcal{M}}_{p}(\mathbb{R};Y,Z)},$

the proof of the above statement would follow from a direct operator-valued
variant of de Leeuw’s [33, Theorem 4.5]. Such variant can be obtained
following essentially the lines of the proof of [33, Theorem 4.5]. For the
proof of one of its main ingredients, i.e. [33, Proposition 4.2], it seems to
be more convenient for a given function
$\phi\in{\mathcal{C}}_{c}(\mathbb{R};X)$ and $N\in\mathbb{N}$, to define
$\lambda_{N}$ as an element of $L^{p}(\mathbf{D};X)$ by

$\lambda_{N}(t):=N^{-1/p}\phi(t/N)\chi_{\frac{1}{N}\mathbb{Z}}(t)\qquad(t\in\mathbf{D}).$

Here, $\mathbf{D}$ stands for the discrete real line $\mathbb{R}$. Then, if
$\mu=(\mu(n)))_{n\in\mathbb{Z}}$ is a sequence in
$l^{p}(\mathbb{Z};{\mathcal{L}}(X,Y))$ with finitely many non-zero elements,
which we consider as an element of $L^{p}(\mathbf{D};{\mathcal{L}}(X,Y))$,
then

$\displaystyle\|\mu\star\lambda_{N}\|^{p}_{L^{p}(\mathbf{D};{\mathcal{L}}(X,Y))}$
$\displaystyle=\sum_{t\in\mathbf{D}}\left|\mu\star\lambda_{N}(t)\right|^{p}_{Y}=\sum_{n\in\mathbb{Z}}N^{-1}\left|\sum_{k\in\mathbb{Z}}\mu(k)\phi(\frac{n}{N}-k)\right|^{p}_{Y}.$

Moreover, in the context of [33, Corollary 4.3], if we set
$r(t):=\sum_{k\in\mathbb{Z}}\mu(k)e^{ikt}$, $t\in\mathbb{R}$, then

${\mathcal{F}}^{-1}(r(\cdot){\mathcal{F}}\phi)(t)=\sum_{k\in\mathbb{Z}}\mu(k)\phi(t-k)\qquad(t\in\mathbb{R}).$

Therefore, it is readily seen that

$\|\mu\star\lambda_{N}\|_{L^{p}(\mathbf{D};Y)}\rightarrow\|{\mathcal{F}}^{-1}(r(\cdot){\mathcal{F}}\phi)\|_{L^{p}(\mathbb{R};Y)}\qquad\textrm{
as }N\rightarrow\infty$

for every ${\mathcal{C}}_{c}(\mathbb{R};X)$. It gives the desired claim about
$\Lambda^{\sharp}$.

For the second step, we need the following operator-valued counterpart of
Jodeit’s result, [46, Theorem 3.5]:
Let $\delta(t):=\max(0,1-|t|)$, $t\in\mathbb{R}$. Then, for arbitrary Banach
spaces $X$ and $Y$ and every $m\in{\mathcal{M}}_{p}(\mathbb{Z};X,Y)$ the
function $e(\delta,m)$ is in ${\mathcal{M}}_{p}(\mathbb{R};X,Y)$. The proof of
this statement follows almost verbatim the lines of the proof of its scalar
prototype [46, Theorem 3.5] (see also Remark 4.3 below).

Now the implication $(i)\Rightarrow(ii)$, follows from this special variant as
in the proof of [10, Theorem 1.1]. Namely, since Fourier’s transform of the de
la Vallée Poussin kernel ${V}$ is a linear combination of the dilations of
$\delta$, i.e. ${\mathcal{F}}V:=2\delta(2\cdot)-\delta(4\cdot)$, a direct
rescaling argument shows that $e({\mathcal{F}}{{V}},m)$ is in
${\mathcal{M}}_{p}(\mathbb{R};X,Y)$ for every
$m\in{\mathcal{M}}_{p}(\mathbb{Z};X,Y)$. Furthermore, since
$e(\Lambda(2^{N}\cdot),m)=\Lambda^{\sharp}e({\mathcal{F}}{V},m)$ is in
${\mathcal{M}}_{p}(\mathbb{R};X,Z)$, again, a rescaling argument yields
$(ii)$.

Finally, note that for $X=Y$ and the sequence
$m:\mathbb{Z}\rightarrow{\mathcal{L}}(Y)$ with $m(0):=I_{Y}$ and $m(k):=0$ for
$k\neq 0$, then $e(\Lambda,m)=\Lambda$. Here, $I_{Y}$ stands for the identity
operator on $Y$. It gives the converse implication, that is
$(ii)\Rightarrow(i)$, and completes the proof. ∎

######  Remark 4.3.

In the sequel, we apply Theorem 4.1 only in the case when $\Lambda=\lambda
I_{Y}$ and $Y=Z$ for scalar functions
$\lambda\in{\mathcal{C}}^{\infty}_{c}(\mathbb{R})$. It is natural to reduce
the above proof only to its second step, i.e. to a direct proof of such
special operator-valued variant of Jodeit’s theorem. Compared to the proof of
[46, Theorem 3.5], where $\lambda=\delta$, it is not clear how to show that
the sum of $l^{1}$-norms of Fourier’s coefficients of the functions
$h_{k}:{\mathbb{T}}\rightarrow\mathbb{C}$ given by
$h_{k}(e^{it}):=({\mathcal{F}}^{-1}\lambda)(t+\pi k)$ for
$t\in[-\frac{\pi}{2},\frac{\pi}{2}]$ and $0$ elsewhere, is finite, i.e.
$\sum_{n,k\in\mathbb{Z}}|\widehat{h}_{k}(n)|<\infty$.

For the proof of our extrapolation result, Theorem 5.5 below, we need a
refined version of the above operator-valued extension of Jodeit’s theorem.
Roughly speaking, for a given multiplier sequence
$m\in{\mathcal{M}}_{p}(\mathbb{Z};X,Y)$ satisfying the Marcinkiewicz condition
$\mathfrak{M}^{3}$ on ${\mathbb{T}}$, we are looking for a summability kernel
$\Lambda=\lambda(\cdot)I_{Y}$ such that the function $e(\Lambda,m)$ satisfies
the corresponding Marcinkiewicz condition $\mathfrak{M}^{3}$ on $\mathbb{R}$.
As can be readily seen, not every function
$\lambda\in{\mathcal{C}}^{\infty}_{c}(\mathbb{R})$ yields such an extension of
$m$, and a more suitable choice of $\lambda$ has to be made.

An extension procedure which preserves the first and second order
Marcinkiewicz conditions was provided in [41, Theorem 5.7.9] (piecewise affine
extension, that is, corresponding to Jodeit’s summability kernel $\delta$) and
in [6, Lemma 3.5], respectively. The following result provides an alternative
and unified argument for such extension procedures.

######  Theorem 4.4.

Let $X$ and $Y$ be arbitrary Banach spaces. Then, for every
$\gamma\in\\{1,2,3\\}$ there exists a function $\lambda\in
C_{c}^{\infty}(\mathbb{R};\mathbb{C})$ such that for each
$m\in\mathfrak{M}^{\gamma}(\mathbb{Z};X,Y)$ the function
$e(\lambda,m):=\sum_{k\in\mathbb{Z}}\lambda(\cdot-k)m(k)$ is an extension of
$m$, which belongs to $\mathfrak{M}^{\gamma}(\mathbb{R};{\mathcal{L}}(X,Y))$.

###### Proof.

We give the proof for $\gamma=3$. The proofs for $\gamma=1$ and $\gamma=2$ can
be readily extracted from that one (see also Remark 4.5).

First, note that for any function $\lambda\in{\mathcal{C}}_{c}^{\infty}$ with
$\operatorname{supp}{\lambda}\subset[-1,3]$ and any sequence
$m:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Y)$, the function $e:=e(\lambda,m)$
can be expressed as follows:

$e(s)=m(k_{s})\lambda(u_{s})+m(k_{s}-1)\lambda(u_{s}+1)+m(k_{s}-2)\lambda(u_{s}+2)+m(k_{s}-3)\lambda(u_{s}+3),$

where $k_{s}:=\lfloor{s}\rfloor+1$, $u_{s}:=s-\lfloor{s}\rfloor-1\in[-1,0]$,
and $s\in\mathbb{R}$. Moreover, straightforward arguments show that

$\displaystyle se^{\prime}(s)=$ $\displaystyle s(\Delta
m)(k_{s}-1)\lambda^{\prime}(u_{s})+s(\Delta
m)(k_{s}-2)\Big{(}\lambda^{\prime}(u_{s})+\lambda^{\prime}(u_{s}+1)\Big{)}$
$\displaystyle\quad+s(\Delta
m)(k_{s}-3)\Big{(}\lambda^{\prime}(u_{s})+\lambda^{\prime}(u_{s}+1)+\lambda^{\prime}(u_{s}+2)\Big{)}$
$\displaystyle\quad+s\,m(k_{s}-3)\Big{(}\lambda^{\prime}(u_{s})+\lambda^{\prime}(u_{s}+1)+\lambda^{\prime}(u_{s}+2)+\lambda^{\prime}(u_{s}+3)\Big{)},$
$\displaystyle s^{2}e^{\prime\prime}(s)=$ $\displaystyle
s^{2}(\Delta^{2}m)(k_{s}-2)\lambda^{\prime\prime}(u_{s})+s^{2}(\Delta
m)(k_{s}-3)\Big{(}2\lambda^{\prime\prime}(u_{s})+\lambda^{\prime\prime}(u_{s}+1)\Big{)}$
$\displaystyle\quad+s^{2}m(k_{s}-2)\Big{(}3\lambda^{\prime\prime}(u_{s})+2\lambda^{\prime\prime}(u_{s}+1)+\lambda^{\prime\prime}(u_{s}+2)\Big{)}$
$\displaystyle\quad+s^{2}m(k_{s}-3)\Big{(}-2\lambda^{\prime\prime}(u_{s})-\lambda^{\prime\prime}(u_{s}+1)+\lambda^{\prime\prime}(u_{s}+3)\Big{)},$

and

$\displaystyle s^{3}e^{\prime\prime\prime}(s)=$ $\displaystyle
s^{3}(\Delta^{3}m)(k-3)\lambda^{\prime\prime\prime}(u_{s})+s^{3}m(k_{s}-1)\Big{(}3\lambda^{\prime\prime\prime}(u_{s})+\lambda^{\prime\prime\prime}(u_{s}+1)\Big{)}$
$\displaystyle\quad+s^{3}m(k_{s}-2)\Big{(}-3\lambda^{\prime\prime\prime}(u_{s})+\lambda^{\prime\prime\prime}(u_{s}+2)\Big{)}$
$\displaystyle\quad+s^{3}m(k_{s}-3)\Big{(}\lambda^{\prime\prime\prime}(u_{s})+\lambda^{\prime\prime\prime}(u_{s}+3)\Big{)}.$

Notice that there exist a constant $c>0$ such that $|s|<c|(k_{s}-j)|$,
$s^{2}<c(k_{s}-j)^{2}$ and $|s|^{3}<c|(k_{s}-j)|^{3}$ for $j=0,1,2$ and $|s|$
sufficiently large.

Therefore, if for all $u\in[-1,0]$ the function $\lambda$ satisfied

(11)
$\lambda^{\prime}(u)+\lambda^{\prime}(u+1)+\lambda^{\prime}(u+2)+\lambda^{\prime}(u+3)=0,$

then
$m\in\mathfrak{M}^{3}(\mathbb{Z};{\mathcal{L}}(X,Y))\subset\mathfrak{M}^{1}(\mathbb{Z};{\mathcal{L}}(X,Y))$
would clearly yield the boundedness of $e$ and $(\cdot)e^{\prime}$.

Similarly, if for all $u\in[-1,0]$ the function $\lambda$ satisfied

(12)
$\begin{cases}3\lambda^{\prime\prime}(u)+2\lambda^{\prime\prime}(u+1)+\lambda^{\prime\prime}(u+2)=0,\\\
-2\lambda^{\prime\prime}(u)-\lambda^{\prime\prime}(u+1)+\lambda^{\prime\prime}(u+3)=0,\end{cases}$

then
$m\in\mathfrak{M}^{3}(\mathbb{Z};{\mathcal{L}}(X,Y))\subset\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X,Y))$
would imply the boundedness of $(\cdot)^{2}e^{\prime\prime}$.

Finally, if for all $u\in[-1,0]$ the function $\lambda$ satisfied

(13)
$\begin{cases}3\lambda^{\prime\prime\prime}(u)+\lambda^{\prime\prime\prime}(u+1)=0,\\\
-3\lambda^{\prime\prime\prime}(u)+\lambda^{\prime\prime\prime}(u+2)=0,\\\
\lambda^{\prime\prime\prime}(u)+\lambda^{\prime\prime\prime}(u+3)=0,\end{cases}$

then $m\in\mathfrak{M}^{3}(\mathbb{Z};{\mathcal{L}}(X,Y))$ would give that
$(\cdot)^{3}e^{\prime\prime\prime}$ is bounded.

Notice that the conditions stated in (11), (12) and (13) are not mutually
contradictory. Indeed, one can check that $\eqref{lambda
3}\Rightarrow\eqref{lambda 2}\Rightarrow\eqref{lambda 1}$. Because $e$ must be
an extension of $m$, we have to additionally ensure that $\lambda(0)=1$,
$\lambda(1)=0$, $\lambda(2)=0$.

To construct a function $\lambda$, which obeys all of the above conditions, on
the interval $[-1,0]$ we take $\lambda$ as any smooth function with

$\lambda(-1)=0,\quad\lambda(0)=1,\quad\lambda^{\prime}(-1)=0,\quad\lambda^{\prime}_{-}(0)=3/2,\quad\lambda^{\prime\prime}_{-}(0)=1,\quad\lambda^{(j)}(k)=0$

for $j\geq 3$ and $k\in\\{-1,0\\}$. Then, on the interval $[0,3]$, we define
$\lambda$ in the following way:

$\displaystyle\lambda(u+1)$
$\displaystyle:=-3\lambda(u)+\frac{(u+1)^{2}}{2}+\frac{3}{2}(u+1)+1,$
$\displaystyle\lambda(u+2)$
$\displaystyle:=3\lambda(u)-2\frac{(u+1)^{2}}{2}-2(u+1),$
$\displaystyle\lambda(u+3)$
$\displaystyle:=-\lambda(u)+\frac{(u+1)^{2}}{2}+\frac{1}{2}(u+1)\qquad(u\in[0,1]).$

Now it is straightforward to check that for every $u\in[-1,0]$ we have

$\lambda^{\prime}(u+1)=-3\lambda^{\prime}(u)+u+\frac{5}{2},\quad\lambda^{\prime}(u+2)=3\lambda^{\prime}(u)-2u-4,\quad\lambda^{\prime}(u+3)=-\lambda^{\prime}(u)+u+\frac{3}{2},$
$\lambda^{\prime\prime}(u+1)=-3\lambda^{\prime\prime}(u)+1,\quad\lambda^{\prime\prime}(u+2)=3\lambda^{\prime\prime}(u)-2,\quad\lambda^{\prime\prime}(u+3)=-\lambda^{\prime\prime}(u)+1,$
$\lambda^{\prime\prime\prime}(u+1)=-3\lambda^{\prime\prime\prime}(u),\quad\lambda^{\prime\prime\prime}(u+2)=3\lambda^{\prime\prime\prime}(u),\quad\lambda^{\prime\prime\prime}(u+3)=-\lambda^{\prime\prime\prime}(u).$

Using the above expressions for $\lambda^{\prime}$, $\lambda^{\prime\prime}$
and $\lambda^{\prime\prime\prime}$, it is easy to verify that such function
$\lambda$ satisfies (11), (12) and (13). The additional conditions imposed on
derivatives of $\lambda$ at $0$ ensure that $\lambda$ is smooth. It completes
the proof. ∎

######  Remark 4.5.

The function $\lambda$ which the existence is proven above for $\gamma=3$
works as well for $\gamma=1$ and $\gamma=2$. However, note that if we are
interested merely in the case when $\gamma=1$ or $\gamma=2$, then one can
simplify the corresponding parts of the above proof by considering $\lambda$
with support in $[-1,1]$ or $[-1,2]$, respectively.

###  4.2. De Leeuw’s couples

Let $\Phi=\Phi({\mathbb{T}})$ and $\Psi=\Psi(\mathbb{R})$ be two Banach
function spaces over $({\mathbb{T}},\mathrm{d}t)$ and
$(\mathbb{R},\mathrm{d}t)$, respectively. If for every
$m\in{\mathcal{M}}_{\Psi}(\mathbb{R};X,Y)$ such that each $k\in\mathbb{Z}$ is
a Lebesgue point of $m$ its restriction to $\mathbb{Z}$ is in
${\mathcal{M}}_{\Phi}(\mathbb{Z};X,Y)$, then we call $(\Phi,\Psi)$ de Leeuw’s
couple.

The classical de Leeuw theorem [33] (see also its operator-valued counterpart
[41, Theorem 5.93]) shows that the couples
$(L^{p}({\mathbb{T}};X),L^{p}(\mathbb{R};X))$, $p\in[1,\infty)$, have such
property. Moreover, some further extensions of de Leeuw’s result done in the
literature provide other examples of such couples; see e.g. [15] and the
references therein. In particular, for our further purposes we need the
following operator-valued variant of a weighted extension of de Leeuw’s
theorem due to Berkson and Gillespie [14, Theorem 1.2].

######  Lemma 4.6.

Let $p\in(1,\infty)$. Let $\widetilde{w}$ be a $2\pi$-periodic weight and
$w(e^{it}):=\widetilde{w}(t)$ for $t\in\mathbb{R}$. Then, the pair
$(L^{p}_{w}({\mathbb{T}}),L^{p}_{\widetilde{w}}(\mathbb{R}))$ is de Leeuw’s
couple.

More precisely, for every Banach spaces $X$ and $Y$, if
$m\in{\mathcal{M}}_{L^{p}_{\widetilde{w}}}(\mathbb{R};X,Y)$ and each point
$k\in\mathbb{Z}$ is a Lebesgue point of $m$, then

$\|m_{|\mathbb{Z}}(\Delta)\|_{{\mathcal{L}}(L^{p}_{w}({\mathbb{T}};X),L^{p}_{w}({\mathbb{T}};Y))}\leq\|m(D)\|_{{\mathcal{L}}(L^{p}_{\widetilde{w}}(\mathbb{R};X),L^{p}_{\widetilde{w}}(\mathbb{R};Y))}.$

The proof can be obtained by an adaptation of the proof of its scalar
counterpart; see e.g. the presentation given in [3]. Below, we propose a
slightly different approach.

###### Proof.

Let $\phi(t):=e^{-\frac{\pi}{p}t^{2}}$ and $\psi(t):=e^{-\frac{\pi}{q}t^{2}}$
for $t\in\mathbb{R}$ and $\frac{1}{p}+\frac{1}{q}=1$. As in [42, Lemma 5.94]
one can show that

$\int_{[-\pi,\pi]}\langle
m(\Delta)f(t),g(t)\rangle_{Y,Y^{*}}\mathrm{d}t=\lim_{\epsilon\rightarrow
0^{+}}\epsilon\int_{\mathbb{R}}\langle
m(D)(\phi(\epsilon\cdot)f)(t),(\phi(\epsilon\cdot)g)(t)\rangle_{Y,Y^{*}}\mathrm{d}t$

for all trygonometric polynomials $f:{\mathbb{T}}\to X,g:{\mathbb{T}}\to
Y^{*}$. Furthermore, note that

$\displaystyle\epsilon\Big{|}\int_{\mathbb{R}}\langle m(D)$
$\displaystyle(\phi(\epsilon\cdot)f)(t),(\psi(\epsilon\cdot)g)(t)\rangle_{Y,Y^{*}}\mathrm{d}t\Big{|}$
$\displaystyle\leq\epsilon\|m(D)(\phi(\epsilon\cdot)f)\|_{L^{p}_{\widetilde{w}}(\mathbb{R};Y)}\|(\psi(\epsilon\cdot)g)\|_{L^{q}_{\widetilde{w}^{1-q}}(\mathbb{R};Y^{*})}$
$\displaystyle\leq\|m(D)\|\epsilon^{\frac{1}{p}}\|\phi(\epsilon\cdot)f)\|_{L^{p}_{\widetilde{w}}(\mathbb{R};X)}\epsilon^{\frac{1}{q}}\|(\psi(\epsilon\cdot)g)\|_{L^{q}_{\widetilde{w}^{1-q}}(\mathbb{R};Y^{*})}$
$\displaystyle\leq\|m(D)\|\Big{(}\frac{1}{2\pi}\int_{-\pi}^{\pi}2\pi\epsilon\sum_{n\in\mathbb{Z}}|\phi(\epsilon(2\pi
n+t))|^{p}|f(t)|_{X}^{p}w(t)\mathrm{d}t\Big{)}^{\frac{1}{p}}$
$\displaystyle\quad\times\Big{(}\frac{1}{2\pi}\int_{-\pi}^{\pi}2\pi\epsilon\sum_{n\in\mathbb{Z}}|\psi(\epsilon(2\pi
n+t))|^{q}|g(t)|_{Y^{*}}^{q}w^{1-q}(t)\mathrm{d}t\Big{)}^{\frac{1}{q}}$
$\displaystyle\leq\|m(D)\|\|\phi\|_{L^{p}(\mathbb{R})}\|f\|_{L^{p}_{w}(X)}\|\psi\|_{L^{q}(\mathbb{R})}\|g\|_{L^{q}_{w^{1-q}}(Y^{*})}$
$\displaystyle\leq\|m(D)\|\|f\|_{L^{p}_{w}(X)}\|g\|_{L^{q}_{w^{1-q}}(Y^{*})}.$

Since $L^{q}_{w^{1-q}}(Y^{*})$ is norming for $L^{p}_{w}(Y)$ we have

$\displaystyle\|m(\Delta)f\|_{L^{p}_{w}(Y)}$
$\displaystyle=\sup_{\|g\|_{L^{q}_{w^{1-q}}(Y^{*})}=1}\quad\int_{[-\pi,\pi]}\langle
m(\Delta)f(t),g(t)\rangle_{Y,Y^{*}}\mathrm{d}t$
$\displaystyle\leq\|m(D)\|\|f\|_{L^{p}_{w}(X)}.$

It finishes the proof. ∎

######  Remark 4.7.

(a) For our purposes (see, e.g. the proof of Theorem 5.5), for a given Banach
function space $\Phi$ over $({\mathbb{T}},\mathrm{d}t)$ we are interested in
constructing a Banach function space $\Psi$ over $(\mathbb{R},\mathrm{d}t)$
such that $(\Phi,\Psi)$ is a de Leeuw’s couple and, in addition, $\Psi$
inherits some analytic properties of $\Phi$. For instance, we need to know
that the Hardy-Littlewood operator $M_{\mathbb{R}}$ is bounded on $\Psi$ if
$M_{\mathbb{T}}$ is bounded on $\Phi$. We do not know if such $\Psi$ can be
contracted for Banach function spaces $\Phi$, which are considered in the
results of Section 5. Here, we only mention that applying Wiener’s amalgam
type construction we can define such a space $\Psi$ under additional density
and rearrangement type assumptions on $\Phi$.

It determines our strategy of the proofs of some results presented below (e.g.
Theorems 5.3 and 5.5), and does not allow to obtain some periodic variants of
results already known in the $\mathbb{R}$-setting via a direct transference.

(b) However, by straightforward arguments (see also [16, Theorem 2.10]), one
can show that for every Muckenhoupt weight $w\in A_{p}({\mathbb{T}})$ its
periodic extension $\widetilde{w}$ on $\mathbb{R}$ is in $A_{p}(\mathbb{R})$.
Therefore, by Muckenhoupt’s theorem and Lemma 4.6, for every $w\in
A_{p}({\mathbb{T}})$ and $p\in(1,\infty)$ the couple
$(L^{p}_{w}({\mathbb{T}}),L^{p}_{\widetilde{w}}(\mathbb{R}))$ is de Leeuw’s
couple and $M_{\mathbb{R}}$ is bounded on $L^{p}_{\widetilde{w}}(\mathbb{R})$.
To get desired results for general $\Phi$’s we adopt the Rubio de Francia
iteration algorithm to the periodic setting.

###  4.3. The extrapolation theorem

The following extrapolation theorem is a crucial ingredient of results
presented in the sequel.

######  Lemma 4.8.

Let $\Phi$ be a Banach function space over $({\mathbb{T}},\mathrm{d}t)$ such
that the Hardy-Littlewood operator $M_{\mathbb{T}}$ is bounded on $\Phi$ and
its dual $\Phi^{\prime}$.

  * (i)

Let $X$ be a Banach space. Then, for every $p\in(1,\infty)$

$\Phi({\mathbb{T}};X)\subset\bigcup_{w\in
A_{p}({\mathbb{T}})}L^{p}_{w}({\mathbb{T}};X)$

  * (ii)

Let $X$ and $Y$ be Banach spaces and $p\in(1,\infty)$. Assume that
$\\{T_{j}\\}_{j\in J}$ is a family of linear operators
$T_{j}:{\mathcal{P}}({\mathbb{T}};X)\rightarrow{\mathcal{D}}^{\prime}({\mathbb{T}};Y)$
such that for every ${\mathcal{W}}\subset A_{p}({\mathbb{T}})$ with
$\sup_{w\in{\mathcal{W}}}[w]_{A_{p}}<\infty$

$\sup_{w\in{\mathcal{W}}}\sup_{j}\|T_{j}\|_{{\mathcal{L}}(L^{p}_{w}({\mathbb{T}};X),L^{p}_{w}({\mathbb{T}};Y))}<\infty.$

Then, each $T_{j}$ extends to a linear operator ${\mathcal{T}}_{j}$ on
$\bigcup_{w\in A_{p}({\mathbb{T}})}L^{p}_{w}({\mathbb{T}};X)\subset
L^{1}({\mathbb{T}};X)$ and has the restriction to an operator in
${\mathcal{L}}(\Phi({\mathbb{T}};X),\Phi({\mathbb{T}};Y))$. Moreover,

$\sup_{j\in
J}\|{\mathcal{T}}_{j}\|_{{\mathcal{L}}(\Phi({\mathbb{T}};X),\Phi({\mathbb{T}};Y))}<\infty.$

The proof follows the lines of the proof of [51, Theorem 3.1] almost verbatim.
Therefore, we leave it for the reader.

######  Remark 4.9.

(a) Recall that the Hardy-Littlewood maximal operator $M_{\mathbb{T}}$ is
bounded on $L^{p}_{w}({\mathbb{T}})$ for all $p\in(1,\infty)$ and each
Muckenhoupt weight $w\in A_{p}({\mathbb{T}})$; see, e.g. [16, Theorem 5.2,
Corollary 5.3].

(b) By the reverse Hölder inequality for weights in $A_{p}({\mathbb{T}})$ one
can show that for every $p\in(1,\infty)$ and $w\in A_{p}({\mathbb{T}})$ there
exists $q>p$ such that $L^{p}_{w}({\mathbb{T}})\hookrightarrow
L^{q}({\mathbb{T}})$; see the proof of [50, Theorem 4.1]. In the context of a
comment stated below [50, Remark 4.4], note that for each $p\in(1,\infty)$ one
can construct a weight $w$ on $[0,2\pi)$, which is in the class
$A_{p}([0,2\pi])$(as it is defined in [50]), but the function
${\mathbb{T}}\ni\tau\mapsto w(\arg\tau)$ is not in $A_{p}({\mathbb{T}})$.

By Lemma 4.8(i) we get that

(14) $\bigcup\Phi({\mathbb{T}};X)=\bigcup_{p>1}L^{p}({\mathbb{T}};X),$

where the first union is taken over all Banach function spaces $\Phi$ over
$({\mathbb{T}},\mathrm{d}t)$ such that $M_{\mathbb{T}}$ is bounded on $\Phi$
and $\Phi^{\prime}$.

We conclude this section with the following consequence of Lemmas 4.6 and 4.8.

######  Proposition 4.10.

Let $X$ be a Banach space with the UMD property. Then, for every Banach
function space $\Phi$ over $({\mathbb{T}},\mathrm{d}t)$ such that the Hardy-
Littlewood operator $M_{\mathbb{T}}$ is bounded on $\Phi$ and $\Phi^{\prime}$,
the family $\\{\chi_{I}(\Delta):I\subset\mathbb{R}\textrm{ an interval}\\}$ is
uniformly bounded in ${\mathcal{L}}(\Phi(X))$.

###### Proof.

Recall that the Hilbert transform $H_{\mathbb{R}}=-i\operatorname{sgn}(D)$ is
bounded on $L^{2}(\mathbb{R};X)$ and its kernel satisfies the Calderón-Zygmund
conditions; see, e.g. [42, Theorem 5.1, p. 374]. In particular, for each
${\mathcal{W}}\subset A_{2}(\mathbb{R})$ with
$\sup_{w\in{\mathcal{W}}}[w]_{A_{2}}<\infty$,
$\sup_{w\in{\mathcal{W}}}\|H_{\mathbb{R}}\|_{{\mathcal{L}}(L^{2}(\mathbb{R};X))}<\infty$;
see, e.g. [62] or [45]. By Lemma 4.6 we infer that the periodic Hilbert
transform $H_{\mathbb{T}}:=-i\operatorname{sgn}(\Delta)$ has the analogues
boundedness property as $H_{\mathbb{R}}$ does. Therefore, by Lemma 4.8 we get
that $H_{\mathbb{T}}$ is in ${\mathcal{L}}(\Phi(X))$. Since for every
$a,b\in\mathbb{Z}$ with $a<b$ we have

$\chi_{\\{a,...,b\\}}(k)=\frac{1}{2}\operatorname{sgn}(k-a)-\frac{1}{2}\operatorname{sgn}(k-b)+\chi_{\\{a\\}}(k)+\chi_{\\{b\\}}(k)\qquad(k\in\mathbb{Z})$

and
$\|\operatorname{sgn}(\cdot-n)(\Delta)\|_{{\mathcal{L}}(\Phi(X))}=\|H_{\mathbb{T}}\|_{{\mathcal{L}}(\Phi(X))}$
for $n\in\mathbb{Z}$, we get the desired claim. ∎

##  5\. The boundedness results for periodic Fourier multipliers

This section provides the boundedness results for periodic Fourier multipliers
on spaces introduced in Section 2. We treat the case of Besov and Triebel-
Lizorkin spaces separately to the case of Banach function spaces $\Phi$.

In the both subsections, our purpose is to show that the typical multiplier
conditions, sufficient for the boundedness on
$B^{s,q}_{L^{p}}$-spaces/$F^{s,q}_{L^{p}}$-spaces/$L^{p}$-spaces, ensure the
boundedness on $B^{s,q}_{\Phi}$-spaces/$F^{s,q}_{\Phi}$-spaces/$\Phi$-spaces
corresponding to general Banach functions spaces $\Phi$ (see Theorems 5.3 and
5.5 below).

###  5.1. Multipliers on generalized Besov and Triebel-Lizorkin spaces

In the case of Besov spaces, it is readily seen from the definition of their
norms $\|\cdot\|_{B^{s,q}_{\Phi}}$, that the problem whether a sequence $m$ is
a multiplier on $B^{s,q}_{\Phi}(X)$ reduces to showing that its dyadic parts
$\psi_{j}m(\Delta)$, $j\in\mathbb{N}_{0}$, are uniformly bounded on the
underlying Banach function space $\Phi(X)$. Note that, in the contrast to the
real line case, no extension procedures are needed in the periodic setting;
see Subsection 3.1. Moreover, the boundedness of each dyadic part
$m\psi_{j}(\Delta)$ of $m(\Delta)$ is automatic (see Lemma 2.1), and the
problem reduces to the uniform boundedness of their norms. For this reason we
postpone the study of the boundedness of general multipliers on Banach
function spaces $\Phi$ to the next section.

The following lemma makes the above observation rigorous.

######  Lemma 5.1.

Let $X$ and $Y$ be Banach spaces. Let $\Phi$ be a Banach function space over
$({\mathbb{T}},\mathrm{d}t)$. Then the following statements hold.

Let $m:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Y)$ be such that

(15)
$\mu:=\sup_{j\in\mathbb{N}_{0}}\|(\psi_{j}m)(\Delta)\|_{{\mathcal{L}}(\Phi(X),\Phi(Y))}<\infty.$

Then, for every $s\in\mathbb{R}$ and $q\in[1,\infty]$,
$m\in{\mathcal{M}}_{B^{s,q}_{\Phi}}({\mathbb{T}};X,Y)$.

More precisely, the multiplier $m(\Delta)$ restricts to an operator in
${\mathcal{L}}(B^{s,q}_{\Phi}(X),{\mathcal{L}}(B^{s,q}_{\Phi}(Y))$ and for
every $f\in{B}^{s,q}_{\Phi}(X)$

(16)
$m(\Delta)f=\sum_{j=0}^{\infty}(\psi_{j}m)(\Delta)f=\lim_{N\rightarrow\infty}\sum_{k\in\mathbb{Z}}e_{k}\otimes\psi(2^{-N}k)m(k)\hat{f}(k)$

with the convergence in ${B}^{s,q}_{\Phi}(Y)$ if $q<\infty$, and in the
$B^{-s,1}_{\Phi^{\prime}}(Y^{*})$-topology if $q=\infty$.

Moreover, if $q=\infty$, then the restriction of $m(\Delta)$ to
${B}^{s,\infty}_{\Phi}(X)$ is
$\sigma({B}^{s,\infty}_{\Phi}(X),B^{-s,1}_{\Phi^{\prime}}(X^{*}))$-to-$\sigma({B}^{s,\infty}_{\Phi}(Y),B^{-s,1}_{\Phi^{\prime}}(Y^{*}))$-continuous.

###### Proof.

Since $\psi_{j}(\Delta)m(\Delta)f=(\chi_{j}m)(\Delta)\psi_{j}(\Delta)f$ for
every $f\in{\mathcal{D}}^{\prime}(X)$ and $j\in\mathbb{N}_{0}$, where
$\chi_{j}:=\psi_{j-1}+\psi_{j}+\psi_{j+1}$ with $\psi_{-1}\equiv 0$, the
condition (15) readily gives that
$m\in{\mathcal{M}}_{B^{s,q}_{\Phi}}({\mathbb{T}};X,Y)$ for all
$s\in\mathbb{R}$ and $q\in[1,\infty]$. The point is to show the claimed
representation formula for $m(\Delta)$ and its continuity when $q=\infty$.

For $q<\infty$, since $\\{\psi_{j}(\Delta)\\}_{j\in\mathbb{N}_{0}}$ is the
resolution of the identity operator on ${\mathcal{D}}^{\prime}(X)$ and, by
Lemma 2.3, ${\mathcal{P}}(X)$ is a dense subset of $B^{s,q}_{\Phi}(X)$, it is
sufficient to show that the operators $\sum_{0\leq l\leq
N}(\psi_{l}m)(\Delta)$, $N\in\mathbb{N}$, are uniformly in
${\mathcal{L}}(B^{s,q}_{\Phi}(X),B^{s,q}_{\Phi}(Y))$. Let $f\in
B^{s,q}_{\Phi}(X)$. Then,

$\displaystyle\left\|\sum_{0\leq l\leq
N}(\psi_{l}m)(\Delta)f\right\|_{B^{s,q}_{\Phi}(Y)}$
$\displaystyle=\left(\sum_{j\in\mathbb{N}_{0}}\left\|2^{js}\sum_{0\leq|l-j|\leq
1\atop 0\leq l\leq
N}(\psi_{l}m)(\Delta)\psi_{j}(\Delta)f\right\|^{q}_{\Phi(Y)}\right)^{\frac{1}{q}}$
$\displaystyle\leq 3\mu\,\|f\|_{B^{s,q}_{\Phi}(X)}.$

Let $q=\infty$. We start with the continuity statement. First note that, for
all $j\in\mathbb{N}_{0}$, the adjoint operator to the operator
$(\psi_{j}m)(\Delta)\in{\mathcal{L}}(\Phi(X),\Phi(Y))$ on functions
$g\in\Phi^{\prime}(Y^{*})\subset[\Phi(Y)]^{*}$ (see (3)) acts as the
multiplier $m(-\cdot)^{*}(\Delta)$. Here, $m(s)^{*}$ for $s\in\mathbb{R}$
stands for the adjoint of $m(s)\in{\mathcal{L}}(X,Y)$. Let $f\in
B^{s,\infty}_{\Phi}(X)$ and $g\in B^{-s,1}_{\Phi^{\prime}}(Y^{*})$. Since for
$j,l\in\mathbb{N}_{0}$

$\langle\psi_{j}(\Delta)g,\psi_{l}(\Delta)[m(\Delta)f]\rangle_{\Phi^{\prime}(Y^{*}),\Phi(Y)}=\langle\psi_{j}(\Delta)[m(-\cdot)^{*}(\Delta)g],\psi_{l}(\Delta)f\rangle_{\Phi^{\prime}(X^{*}),\Phi(X)},$

it suffices to show that $m(-\cdot)^{*}(\Delta)g$ is in
$B^{-s,1}_{\Phi^{\prime}}(X^{*})$. For, since
$\Phi(X)\subset[\Phi^{\prime}(X^{*})]^{*}$ is a norming subspace of
$\Phi^{\prime}(X^{*})$, for each $j\in\mathbb{N}_{0}$ there exists
$h_{j}\in\Phi(X)$ such that

$\|\psi_{j}(\Delta)[m(-\cdot)^{*}(\Delta)g]\|_{\Phi^{\prime}(X^{*})}\leq\left|\langle\psi_{j}(\Delta)[m(-\cdot)^{*}(\Delta)g],h_{j}\rangle_{\Phi^{\prime}(X^{*}),\Phi(X)}\right|+2^{sj-j}.$

However,

$\displaystyle|\langle\psi_{j}(\Delta)[m(-\cdot)^{*}(\Delta)g],h_{j}\rangle_{\Phi^{\prime}(X^{*}),\Phi(X)}|$
$\displaystyle=\left|\langle\psi_{j}(\Delta)g,(\chi_{j}m)(\Delta)h_{j}\rangle_{\Phi^{\prime}(X^{*}),\Phi(X)}\right|$
$\displaystyle\leq 3\mu\|\psi_{j}(\Delta)g\|_{\Phi^{\prime}(Y^{*})},$

which gives the desired claim. The convergence of the series in (16) in the
$B^{-s,1}_{\Phi^{\prime}}(Y^{*})$-topology follows from similar arguments to
those presented above. We omit it. ∎

In the context of Subsection 3.1, it is interesting to compare the proof of
the above boundedness principle for periodic Fourier multipliers with the
proof of the corresponding result for $\mathbb{R}$, see [51, Theorem 3.8]; cf.
also [44, Problems 3.2 and 3.3].

We complete Lemma 5.1 with the following supplementary observation on the
convergence of the series in (16) under additional assumptions on the geometry
of the underlying spaces $X$ and $Y$.

######  Corollary 5.2.

Let $X$ and $Y$ be Banach spaces and suppose that $Y$ has the $U\\!M\\!D$
property. Let $\Phi$ be a Banach function space on
$({\mathbb{T}};\mathrm{d}t)$ such that $M_{\mathbb{T}}$ is bounded on $\Phi$
and $\Phi^{\prime}$. Then, for every $q\in[1,\infty)$, $s\in\mathbb{R}$,
$m\in{\mathcal{M}}_{B^{s,q}_{\Phi}}({\mathbb{T}};X,Y)$ and $f\in
B^{s,q}_{\Phi}(X)$

(17) $m(\Delta)f=\lim_{N\rightarrow\infty}\sum_{|k|\leq N}e_{k}\otimes
m(k)\hat{f}(k)$

with the convergence in $B^{s,q}_{\Phi}(Y)$. If, in addition, $Y$ is a Banach
function space and $m(\Delta)f\in\Phi(Y)$ (for instance, when $s>0$), then the
above convergence holds pointwise almost everywhere on ${\mathbb{T}}$.

###### Proof.

By Lemma 5.1 and Proposition 4.10, we infer that $\chi_{[-N,N]}(\Delta)$,
$N\in\mathbb{N}$, are uniformly in ${\mathcal{L}}(B^{s,q}_{\Phi}(Y))$. Since
(17) holds for every $f\in{\mathcal{P}}(X)$ and ${\mathcal{P}}(X)$ is dense in
$B^{s,q}_{\Phi}(X)$ (see Lemma 2.3), by Lemma 5.1, we get (17) for every $f\in
B^{s,q}_{\Phi}(X)$

For the additional statement, note that by (14), $m(\Delta)f\in L^{p}(Y)$ for
some $p>1$. By Rubio de Francia’s vector-valued counterpart of Carleson’s
theorem (see [61]), we get the pointwise convergence in (17). It completes the
proof. ∎

######  Theorem 5.3.

Let $X$ and $Y$ be Banach spaces. Let $\Phi$ denote a Banach function space
over $({\mathbb{T}},\mathrm{d}t)$ such that the Hardy-Littlewood operator
$M_{\mathbb{T}}$ is bounded on $\Phi$.

Then the following assertions hold.

  * (i)

For every $\mathbb{E}:={B}^{s,q}_{\Phi}$ with $s\in\mathbb{R}$ and
$q\in[1,\infty]$, we have that

$\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X,Y))\subset{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,Y).$

  * (ii)

Let $X$ and $Y$ have the $U\\!M\\!D$ property and, in addition,
$M_{\mathbb{T}}$ is bounded on $\Phi^{\prime}$. Then, for every
$\mathbb{E}:={B}^{s,q}_{\Phi}$ with $s\in\mathbb{R}$ and $q\in[1,\infty]$, we
have that

$Var(\mathbb{Z};{\mathcal{L}}(X,Y))\subset{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,Y).$

  * (iii)

If, in addition, $M_{\mathbb{T}}$ is bounded on $\Phi^{\prime}$, then the
above statements $(i)$ and $(ii)$ hold for every $\mathbb{E}=F^{s,q}_{\Phi}$
with $s\in\mathbb{R}$ and $q\in(1,\infty)$.

###### Proof.

$(i)$ Let $\\{\psi_{j}\\}_{j\in\mathbb{N}_{0}}$ be the resolution of the
identity on $\mathbb{R}$. By Lemma 4.4 there exists an extension
$\widetilde{m}$ of $m$ on $\mathbb{R}$ such that
$\widetilde{m}\in\mathfrak{M}^{2}(\mathbb{R};{\mathcal{L}}(X,Y))$. Therefore,
to get the uniform boundedness of the operators
$\psi_{j}m(\Delta)=(\psi_{j}\widetilde{m})(\Delta)$, $j\in\mathbb{N}$, in
${\mathcal{L}}(\Phi(X),\Phi(Y))$, by Lemma 2.2, it is sufficient to show that
there exists an even, radially decreasing functions $\phi_{j}$,
$j\in\mathbb{N}$, on $\mathbb{R}$ such that
$\sup_{j\in\mathbb{N}}\|\phi_{j}\|_{L^{1}}<\infty$ and
$\|{\mathcal{F}}^{-1}(\psi_{j}\widetilde{m})(t)\|_{{\mathcal{L}}(X,Y)}\leq\phi_{j}(t)$
for all $t\in\mathbb{R}$. The fact that
$(\psi_{0}m)(\Delta)=\psi_{0}\widetilde{m}(\Delta)$ is in
${\mathcal{L}}(\Phi({\mathbb{T}};X),\Phi({\mathbb{T}};Y))$ follows directly
from, e.g. Lemma 2.1. But, the existence of such majorants it is exactly what
the proof of [51, Proposition 4.2(i)] shows
($\phi_{j}(t)=\frac{C[\widetilde{m}]_{\mathfrak{M}^{2}}2^{j}}{1+2^{2j}t^{2}}$
$(t\in\mathbb{R})$). Therefore, by Lemma 2.2 and Lemma 5.1 we get the desired
claim.

(ii) First note that for every $j\geq 1$ and $f\in{\mathcal{D}}^{\prime}(X)$
we can write

$\displaystyle\chi_{[2^{j-1},2^{j+1}]}(\Delta)m(\Delta)$
$\displaystyle\psi_{j}(\Delta)f$
$\displaystyle=\sum_{l=2^{j-1}}^{2^{j+1}}e_{l}\otimes
m(2^{j+1})\psi_{j}(l)\hat{f}(l)$
$\displaystyle\quad+\sum_{k=2^{j-1}}^{2^{j+1}-1}\sum_{l=2^{j-1}}^{k}e_{l}\otimes[m(k)-m(k+1)]\psi_{j}(l)\hat{f}(l)$
$\displaystyle=m(2^{j+1})\chi_{[2^{j-1},2^{j+1}]}(\Delta)\psi_{j}(\Delta)f$
$\displaystyle\quad+\sum_{k=2^{j-1}}^{2^{j+1}-1}[m(k)-m(k+1)]\chi_{[2^{j-1},k]}(\Delta)\psi_{j}(\Delta)f$

Therefore,

$\displaystyle\big{\|}\chi_{[2^{j-1},2^{j+1}]}(\Delta)\psi_{j}(\Delta)m(\Delta)f\big{\|}_{\Phi(Y)}$
$\displaystyle\quad\leq\left\|m(2^{j+1})\right\|_{{\mathcal{L}}(X,Y)}\left\|\chi_{[2^{j-1},2^{j+1}]}(\Delta)\psi_{j}(\Delta)f\right\|_{\Phi(X)}$
$\displaystyle\quad\quad+\sup_{2^{j-1}\leq k\leq
2^{j+1}-1}\|\chi_{[2^{j-1},k]}(\Delta)\psi_{j}(\Delta)f\|_{\Phi(X)}\Big{\|}\sum_{k=2^{j-1}}^{2^{j+1}-1}[m(k)-m(k+1)]\Big{\|}_{{\mathcal{L}}(X,Y)}.$

Similarly, the analogous estimate holds for
$\|\chi_{[-2^{j+1},-2^{j-1}]}(\Delta)m(\Delta)\psi_{j}(\Delta)f\|_{\Phi(Y)}$.
Since $\operatorname{supp}\psi_{j}\subset\\{2^{j-1}\leq|t|\leq 2^{j+1}\\}$
($j\geq 1$), and $m\in Var(\mathbb{Z};{\mathcal{L}}(X,Y))$, by Proposition
4.10, there exists a constant $\mu>0$ such that for all $f\in\Phi(X)$ and
$j\geq 1$ we have

$\|\psi_{j}(\Delta)m(\Delta)f\|_{\Phi(Y)}\leq\mu[m]_{Var}\|\psi_{j}(\Delta)f\|_{\Phi(Y)}.$

Therefore, Lemma 5.1 completes the proof.

(iii) The proof of this part follows arguments presented in the proof of
Proposition 2.3. We sketch some details. First, since
$F^{s,q}_{\Psi}(X)=B^{s,q}_{\Psi}(X)$ for every $\Psi=L^{q}_{w}$
($q\in(1,\infty),w\in A_{p}({\mathbb{T}})$), $m(\Delta)$ restricts to an
operator in ${\mathcal{L}}(F^{s,q}_{\Psi}(X),F^{s,q}_{\Psi}(Y))$ for such
$\Psi^{\prime}s$. Its norm is bounded by

$\mu_{w,q}:=\sup_{j\in\mathbb{N}_{0}}\|(\psi_{j}m)(\Delta)\|_{{\mathcal{L}}(\Psi(X),\Psi(Y))},$

$\sup_{w\in{\mathcal{W}}}\mu_{w,q}<\infty$ for each ${\mathcal{W}}\subset
A_{q}({\mathbb{T}})$ with $\sup_{w\in{\mathcal{W}}}[w]_{A_{q}}<\infty$.
Furthermore, for every $f\in F^{s,q}_{\Psi}(X)$ and $j\in\mathbb{N}_{0}$

(18) $\|\psi_{j}(\Delta)m(\Delta)f\|_{\Psi(Y)}\leq
3\mu_{w,q}\|\psi_{j}(\Delta)f\|_{\Psi(X)}.$

Fix $\Phi$ and $f\in{\mathcal{P}}(X)$. Let

$Gf:=\left(\sum_{j\in\mathbb{N}_{0}}|2^{sj}\psi_{j}(\Delta)f(\cdot)|^{q}_{X}\right)^{1/q}.$

and similarly for $G(m(\Delta)f)$. Of course, $Gf$ and $G(m(\Delta)f)$ are in
$\Phi$; see, e.g. (4). Let $h\in\Phi^{\prime}$, $h\neq 0$, and set

$w:=w_{Gf,h,q}:={\mathcal{R}}(Gf)^{1-q}{\mathcal{R}}^{\prime}h.$

Then $Gf\in L^{q}_{w}$, i.e. $f\in F^{s,q}_{L^{q}_{w}}({\mathbb{T}};X)$ and
$m(\Delta)f\in F^{s,q}_{L^{q}_{w}}({\mathbb{T}};X)$. By (18) and a similar
argument as in the proof of Proposition 2.3(ii) give

$\displaystyle
3\mu_{w,q}\|{\mathcal{R}}\|_{{\mathcal{L}}(\Phi)}\|Gf\|_{\Phi}\|{\mathcal{R}}^{\prime}\|_{{\mathcal{L}}(\Phi^{\prime})}\|h\|_{\Phi^{\prime}}$
$\displaystyle\geq
3\mu_{w,q}\|{\mathcal{R}}Gf\|_{\Phi}\|{\mathcal{R}}^{\prime}h\|_{\Phi^{\prime}}$
$\displaystyle\geq
3\mu_{w,q}\left(\int_{\mathbb{R}}{\mathcal{R}}(Gf){\mathcal{R}}^{\prime}h\,\mathrm{d}t\right)^{\frac{1}{q}}\left(\int_{\mathbb{T}}{\mathcal{R}}(Gf){\mathcal{R}}^{\prime}h\,\mathrm{d}t\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\geq\left(\int_{\mathbb{T}}G(m(\Delta)f)^{q}w\,\mathrm{d}t\right)^{\frac{1}{q}}\left(\int_{\mathbb{T}}{\mathcal{R}}(Gf){\mathcal{R}}^{\prime}h\mathrm{d}t\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\geq\int_{\mathbb{T}}G(m(\Delta)f)h\,\mathrm{d}t$

Since

$\sup\left\\{[w_{Gf,h,q}]_{A_{q}}:{f\in{\mathcal{P}}(X),h\in\Phi^{\prime}}\right\\}<\infty,$
$\mu:=\sup\left\\{3\mu_{w,q}:w=w_{Gf,h,q}\textrm{ with
}f\in{\mathcal{P}}(X),h\in\Phi^{\prime}\right\\}<\infty.$

Therefore, for every $f\in{\mathcal{P}}(X)$

$\|m(\Delta)f\|_{F^{s,q}_{\Phi}({\mathbb{T}};Y)}\leq\mu\|{\mathcal{R}}\|_{{\mathcal{L}}(\Phi)}\|{\mathcal{R}}^{\prime}\|_{{\mathcal{L}}(\Phi^{\prime})}\|f\|_{F^{s,q}_{\Phi}(X)}.$

Since ${\mathcal{P}}(X)$ is dense in $F^{s,q}_{\Phi}({\mathbb{T}};X)$ (see
Lemma 2.3), $m(\Delta)$ restricts to an operator in
${\mathcal{L}}(F^{s,q}_{\Phi}(X),F^{s,q}_{\Phi}(Y))$. ∎

######  Remark 5.4.

Note that [8, Theorem 4.2] shows that the boundedness of $M_{\mathbb{T}}$ on
$\Phi^{\prime}$ cannot be drop in the part $(ii)$ in general. Indeed,
$Var(\mathbb{Z};{\mathcal{L}}(X))\nsubseteq{\mathcal{M}}_{B^{s,\infty}_{\Phi}}({\mathbb{T}};X,X)$
for $\Phi=L^{\infty}$ and $s\in(0,1)$, when $X$ is not isomorphic to a Hilbert
space.

###  5.2. Multipliers on general Banach function spaces

The main aim of this section is to provide a result on the extrapolation of
the boundedness of periodic Fourier multipliers; see Theorem 5.5. More
precisely, we identify the classes of multipliers $m$ for which a priori
knowledge that $m$ is in ${\mathcal{M}}_{L^{p}}({\mathbb{T}};X,Y)$ for some
$p\in(1,\infty)$ implies that $m\in{\mathcal{M}}_{\Phi}({\mathbb{T}};X,Y)$ for
a large class of Banach function spaces $\Phi$. In the next section, we apply
this result to prove the phenomenon of the extrapolation of the
$L^{p}$-maximal regularity for a large class of abstract evolution equations;
see Theorems 6.7, 6.9 and 6.12.

######  Theorem 5.5.

Let $X$ and $Y$ be Banach spaces. Let $\Phi$ be a Banach function space over
$({\mathbb{T}},\mathrm{d}t)$ such that $M_{\mathbb{T}}$ is bounded on $\Phi$
and its dual $\Phi^{\prime}$. Then the following assertions hold.

  * (i)

Assume that $m\in{\mathfrak{M}}^{3}(\mathbb{Z};{\mathcal{L}}(X,Y))$ and
$m\in{\mathcal{M}}_{L^{p}}({\mathbb{T}};X,Y)$ for some $p\in(1,\infty)$. Then,
$m\in{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,Y)$ for every

$\mathbb{E}\in\left\\{\Phi,\,B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}:\,s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty)\right\\}.$

  * (ii)

Assume that
$m\in{\mathfrak{M}}_{\mathcal{R}}^{1}(\mathbb{Z};{\mathcal{L}}(X,Y))$ and, in
addition, $X$ and $Y$ have the $U\\!M\\!D$ property. Then, the conclusion of
$(i)$ holds.

###### Proof.

$(i)$ The case $\mathbb{E}=B^{s,q}_{\Phi}$ and $\mathbb{E}=F^{s,q}_{\Phi}$
follows from Theorem 5.3. Therefore, let $\mathbb{E}=\Phi$. By Theorem 4.4
there exists an extension $\widetilde{m}:=e(\Lambda,m)$ of $m$ on $\mathbb{R}$
such that
$\widetilde{m}\in\widetilde{\mathfrak{M}}^{3}(\mathbb{R};{\mathcal{L}}(X,Y))$.
Moreover, Theorem 4.1 yields
$\widetilde{m}\in{\mathcal{M}}_{L^{p}}(\mathbb{R};X,Y)$. Consequently,
$\widetilde{m}(D)$ is a Calderón-Zygmund operator; see [66, Proposition 4.4.2,
p.254], or [51, Lemma 4.1]. In particular, by [62, Theorem 1.6], for every
$q\in(1,\infty)$ and for every $\widetilde{{\mathcal{W}}}\subset
A_{q}(\mathbb{R})$ with
$\sup_{\widetilde{w}\in\widetilde{{\mathcal{W}}}}[\widetilde{w}]_{A_{q}(\mathbb{R})}<\infty$
we have that

$\sup_{\widetilde{w}\in\widetilde{{\mathcal{W}}}}\|\widetilde{m}(D)\|_{{\mathcal{L}}(L^{q}_{\widetilde{w}}(X),L^{q}_{\widetilde{w}}(Y))}<\infty.$

Let ${\mathcal{W}}\subset A_{q}({\mathbb{T}})$ be such that
$\sup_{w\in{\mathcal{W}}}[w]_{A_{q}({\mathbb{T}})}<\infty$. By $\widetilde{w}$
we denote the periodic extension of $w$ on $\mathbb{R}$. Set
$\widetilde{{\mathcal{W}}}:=\\{\widetilde{w}:\,w\in{\mathcal{W}}\\}$. Then
simple argumentation shows that $\widetilde{{\mathcal{W}}}\subset
A_{q}(\mathbb{R})$ and there exists a constant $C>0$ such that

$\sup_{\widetilde{w}\in\widetilde{{\mathcal{W}}}}[\widetilde{w}]_{A_{q}(\mathbb{R})}\leq
C\sup_{w\in{\mathcal{W}}}[w]_{A_{q}({\mathbb{T}})}<\infty.$

By Lemma 4.6 we get that

$\sup_{w\in{\mathcal{W}}}\|m(\Delta)\|_{{\mathcal{L}}(L^{q}_{w}(X),L^{q}_{w}(Y))}\leq\sup_{\widetilde{w}\in\widetilde{{\mathcal{W}}}}\|\widetilde{m}(D)\|_{{\mathcal{L}}(L^{q}_{\widetilde{w}}(X),L^{q}_{\widetilde{w}}(Y))}<\infty.$

Therefore, by Lemma 4.8 we conclude that $\widetilde{m}_{|\mathbb{Z}}=m$ is in
${\mathcal{M}}_{\Phi}({\mathbb{T}};X,Y)$.

$(ii)$ By Theorem 4.4 we find
$\lambda\in{\mathcal{C}}^{\infty}_{c}(\mathbb{R})$ such that
$\widetilde{m}:=e(\Lambda,m)$ (for $\Lambda:=\lambda(\cdot)I_{Y}$) extends $m$
and is in $\mathfrak{M}^{1}(\mathbb{R};{\mathcal{L}}(X,Y))$. By Kahane’s
contraction principle (see [52, Proposition 2.5]), we easily get that
$\widetilde{m}$ satisfies, in fact, the
$\mathfrak{M}_{\mathcal{R}}^{1}$-condition. Now [37, Theorem 3.5.(a)] shows
that for every ${\mathcal{W}}\subset A_{q}(\mathbb{R})$ with
$\sup_{w\in{\mathcal{W}}}[w]_{A_{q}(\mathbb{R})}<\infty$

$\sup_{w\in{\mathcal{W}}}\|\widetilde{m}(D)\|_{{\mathcal{L}}(L^{p}_{w}(\mathbb{R};X),L^{p}_{w}(\mathbb{R};Y))}<\infty.$

Therefore, Lemma 4.6, Lemma 4.8 and similar reasoning as in (i) give that $m$
is in ${\mathcal{M}}_{\Phi}({\mathbb{T}};X,Y)$. The case of the Besov spaces
follows now directly from Lemma 5.1. For the Triebel-Lizorkin case, note that
the functions $\psi_{j}\widetilde{m}$ satisfy the
$\mathfrak{M}^{1}_{\mathcal{R}}$-condition uniformly in $j\in\mathbb{N}_{0}$.
Therefore, in this case, the proof mimics the arguments presented already in
the proof of Theorem 5.3(iii). ∎

######  Remark 5.6.

(a) In the context of applications, see Theorem 6.5(c), Theorem 5.5(i) can be
read as an extrapolation of the $L^{p}$-maximal regularity property. Indeed,
as Theorem 6.2 shows, for Fourier multipliers $m$ related to some evolution
equations (see Subsection 7.2), the fact that
$m\in{\mathcal{M}}_{L^{p}}(\mathbb{R};X,Y)$ implies that $m$ satisfies
Marcinkiewicz’s condition $(\mathfrak{M}^{\gamma})$ of an arbitrary order
$\gamma\in\mathbb{N}$.

(b) The assumptions of Theorem 5.5 should be confronted with [9, Theorem 1],
which says that for every Banach space $X$, which is not isomorphic to a
Hilbert space, there is a sequence $m:\mathbb{Z}\rightarrow{\mathcal{L}}(X)$
satisfying the $\mathfrak{M}^{\gamma}$-condition for every
$\gamma\in\mathbb{N}$, but $m$ is not in
${\mathcal{M}}_{L^{p}}(\mathbb{Z};X,X)$ ($p\in(1,\infty)$).

##  6\. Applications

In this section we apply the results developed above to study the solvability
and maximal regularity of an abstract second-order integro-differential
equation, see (AEE) below. We start with some preliminaries.

Let $A,B,P$ be closed operators on a Banach space $X$. By $D_{A},D_{B},D_{P}$
we denote their domains equipped with the corresponding graph norms. Let
${\mathcal{A}}$, ${\mathcal{B}}$ and ${\mathcal{P}}$ denote the evaluations of
operators $A$, $B$ and $P$ on ${\mathcal{D}}^{\prime}(D_{A})$,
${\mathcal{D}}^{\prime}(D_{B})$ and ${\mathcal{D}}^{\prime}(D_{P})$,
respectively. More precisely,
$\mathcal{A}\in\mathcal{L}({\mathcal{D}}^{\prime}(D_{A}),{\mathcal{D}}^{\prime}(X))$
is given by $(\mathcal{A}u)(\phi)=A(u(\phi))$ for every
$u\in{\mathcal{D}}^{\prime}(D_{A})$ and $\phi\in{\mathcal{D}}$. The operators
${\mathcal{B}}$ and ${\mathcal{P}}$ are defined in the similar manner.
Moreover, let $c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(Z,X))$, where $Z$ is a
Banach space continuously embedded in $X$.

Let us consider the following abstract degenerated, second-order problem with
the convolution term:

(AEE) $\partial\mathcal{P}\partial u+\mathcal{B}\partial u+\mathcal{A}u+c\ast
u=f\quad(\text{in }{\mathcal{D}}^{\prime}(X))$

where $f\in{\mathcal{D}}^{\prime}(X)$ is a given $X$-valued distribution. The
convolution term $c\ast u$ we interpret as the Fourier multiplier, i.e. $c\ast
u:=\hat{c}(\Delta)u$. For a given $f\in{\mathcal{D}}^{\prime}(X)$, a
distribution $u\in{\mathcal{D}}^{\prime}(X)$ is called the distributional
solution of (AEE) if
$u\in{\mathcal{D}}^{\prime}(D_{A})\cap{\mathcal{D}}^{\prime}(Z)$, $\partial
u\in{\mathcal{D}}^{\prime}(D_{B})\cap{\mathcal{D}}^{\prime}(D_{P})$ and (AEE)
holds in ${\mathcal{D}}^{\prime}(X)$.

Set

(19) $Y:=D_{A}\cap D_{B}\cap D_{P}\cap Z$

with the norm

$|y|_{Y}:=\max(|y|_{A},|y|_{B},|y|_{P},|y|_{Z})\quad(y\in Y).$

Our first result shows how the structure of (AEE) affects the relation between
the regularities of the symbols of Fourier multipliers, which are involved in
the study of solvability of (AEE), which we address below. To make this result
applicable to the different situations (see Section 7), we need an abstract
joint multiplier condition on two symbols. For $\gamma\in\mathbb{N}$ we say
that a sequence $d:\mathbb{Z}\rightarrow{\mathcal{L}}(Z,X)$ satisfies the
$\mathfrak{M}^{\gamma}$-condition with respect to a sequence
$a:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Z)$, if

$[d]_{\mathfrak{M}^{\gamma}(a)}:=\max_{l=0,...,\gamma}\sup_{k\in\mathbb{Z}}\bigl{\|}k^{l}(\Delta^{l}d)(k)a(k+l)\bigr{\|}_{{\mathcal{L}}(X)}<\infty.\quad$

We write $d\in\mathfrak{M}^{\gamma}(a)$, when the above holds.

######  Remark 6.1.

Of course, if $d\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(Z,X))$ and
$a\in l^{\infty}(\mathbb{Z};{\mathcal{L}}(X,Z))$, then
$d\in\mathfrak{M}^{\gamma}(a)$. However, under further information on the
boundedness of $a$ one can provide more suitable conditions on $d$ to show
that $d\in\mathfrak{M}^{\gamma}(a)$. For example, if, in addition, we have
that $(\cdot)a\in l^{\infty}(\mathcal{L}(X,Z))$, then
$(\cdot)^{-1}d\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(Z,X))$ yields
$d\in\mathfrak{M}^{\gamma}(a)$. To see it, notice that

$\sup_{l=0,...,\gamma}\sup_{k\in\mathbb{Z}}\|k^{l-1}(\Delta^{l}d)(k)\|_{{\mathcal{L}}(Z,X)}<\infty\quad\textrm{
if and only if
}\quad[(\cdot)^{-1}d]_{\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(Z,X))}<\infty.$

Furthermore, note that
$d\in\mathfrak{M}^{\gamma-1}(\mathbb{Z};{\mathcal{L}}(Z,X))$ implies
$(\cdot)^{-1}d\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(Z,X))$, but
the converse does not hold in general. For instance, for $\gamma=2$ and the
sequence $d(k):=k$ ($k\in\mathbb{Z}$) we have that
$(\cdot)^{-1}d\in\mathfrak{M}^{2}$ but $d$ does not satisfy the
$\mathfrak{M}^{1}$-condition. We apply these observations below.

We say that a sequence $d:\mathbb{Z}\to{\mathcal{L}}(Z,X)$ satisfies the
variational Marcinkiewicz condition with respect to a sequence
$a:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Z)$, and write $d\in Var(a)$, if

$[d]_{Var(a)}:=[d]_{\mathfrak{M}^{0}(a)}+\sup_{j\geq
0}\sum_{2^{j}\leq|k|<2^{j+1}}\|\Delta d(k)a(k+1)\|_{{\mathcal{L}}(X)}<\infty$

We say that a sequence $d:\mathbb{Z}\to{\mathcal{L}}(Z,X)$ satisfies the
$\mathfrak{M}^{\gamma}_{{\mathcal{R}}}$-condition with respect to a sequence
$a:\mathbb{Z}\rightarrow{\mathcal{L}}(X,Z)$ (and write
$d\in\mathfrak{M}^{\gamma}_{{\mathcal{R}}}(a)$), if for each
$l=0,1,...,\gamma$ the set
$\\{k^{l}(\Delta^{l}d)(k)a(k+l)\\}_{k\in\mathbb{Z}}$ is
${\mathcal{R}}$-bounded in ${\mathcal{L}}(X)$.

######  Theorem 6.2.

Let $A$, $B$, $P$, $c$ and $Y$ be as stated above. Assume that for every
$k\in\mathbb{Z}\setminus\\{0\\}$ the operator

$b(k):=-k^{2}P+ikB+A+\hat{c}(k)\in{\mathcal{L}}(Y,X)$

is invertible. Let

$a(k):=b(k)^{-1},\ a_{0}(k):=kBa(k),\ a_{1}(k):=k^{2}Pa(k),\
a_{2}(k):=ka(k)\quad(k\in\mathbb{Z}\setminus\\{0\\})$

and $a(0)=a_{0}(0)=a_{1}(0)=a_{2}(0)=0\in{\mathcal{L}}(X)$. Then, the
following assertions hold.

  * (i)

Assume that $a\in l^{\infty}({\mathcal{L}}(X,Z))$ and $a_{0},a_{1}\in
l^{\infty}({\mathcal{L}}(X))$. Then, for every $\gamma\in\\{1,2,3\\}$, if
$\hat{c}\in\mathfrak{M}^{\gamma}(a)$ (respectively, $\hat{c}\in Var(a)$), then
$a\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(X,Y))$,
$a_{0},a_{1}\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(X))$
(respectively, $a\in Var(\mathbb{Z};{\mathcal{L}}(X,Y)$, $a_{0},a_{1}\in
Var(\mathbb{Z};{\mathcal{L}}(X))$).

In addition, if the sequence $a_{2}\in l^{\infty}({\mathcal{L}}(X))$, then
$a_{2}\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(X))$ (respectively,
$a_{2}\in Var(\mathbb{Z};{\mathcal{L}}(X)$).

  * (ii)

The statement (i) holds in its ${\mathcal{R}}$-bound reformulation, that is,
if, in addition, the sequences $a$, $a_{0}$ and $a_{1}$ are
${\mathcal{R}}$-bounded, then for every $\gamma\in\\{1,2,3\\}$ the fact that
$\hat{c}$ satisfies the $\mathfrak{M}_{\mathcal{R}}^{\gamma}$-condition with
respect to a sequence $a$, implies that the sequences $a$, $a_{0}$ and $a_{1}$
satisfy the $\mathfrak{M}_{\mathcal{R}}^{\gamma}$-condition.

If, in addition, $a_{2}$ is ${\mathcal{R}}$-bounded then it satisfies the
$\mathfrak{M}_{\mathcal{R}}^{\gamma}$-condition.

###### Proof.

First we show that $a\in l^{\infty}({\mathcal{L}}(X,Y))$. Indeed, note that
each condition imposed on $\hat{c}$ implies that $\hat{c}(\cdot)a(\cdot)\in
l^{\infty}({\mathcal{L}}(X))$. Since

$I_{X}:=b(k)a(k)=-k^{2}Pa(k)+ikBa(k)+Aa(k)+\hat{c}(k)a(k),$

where $I_{X}$ denotes the identity operator on $X$, we get $Aa(\cdot)\in
l^{\infty}({\mathcal{L}}(X))$. Moreover, $Z\hookrightarrow X$ implies that
$a\in l^{\infty}({\mathcal{L}}(X))$, which gives our claim that $a\in
l^{\infty}({\mathcal{L}}(X,Y))$.

(i) First we prove the statement for $a$ and $\gamma\in\\{1,2,3\\}$. For
$\gamma=1$, note that by Leibniz’ rule for difference operators we have

(19) $(\Delta a)(k)=-a(k)(\Delta b)(k)a(k+1)\quad(k\in\mathbb{Z}),$

and one easily gets that

(20) $(\Delta b)(k)=-(2k+1)P+iB+(\Delta\hat{c})(k)\quad(k\in\mathbb{Z}).$

Thus, by our assumptions on $a_{0}$, $a_{1}$ and $\hat{c}$,

(21) $(k(\Delta
b)(k)a(k+1))_{k\in\mathbb{Z}}\in\l^{\infty}({\mathcal{L}}(X)).$

Consequently, since $a\in l^{\infty}({\mathcal{L}}(X,Y))$,

$(k(\Delta a)(k))_{k\in\mathbb{Z}}\in\l^{\infty}({\mathcal{L}}(X,Y)).$

For $\gamma=2$, since the $\mathfrak{M}^{2}$-condition implies the
$\mathfrak{M}^{1}$-condition, it is sufficient to show that

$(k^{2}(\Delta^{2}a)(k))_{k\in\mathbb{Z}}\in\l^{\infty}({\mathcal{L}}(X,Y)).$

For, note that for all $k\in\mathbb{Z}$ we have

(22) $(\Delta^{2}a)(k)=-2(\Delta a)(k)(\Delta
b)(k+1)a(k+2)-a(k)(\Delta^{2}b)(k)a(k+2).$

Therefore, by the step for $\gamma=1$ (see (21)), it is enough to check that

(23) $(k^{2}(\Delta^{2}b)(k)a(k+2))_{k\in\mathbb{Z}}\in
l^{\infty}({\mathcal{L}}(X)),$

But this again follows directly from our assumptions, since

$(\Delta^{2}b)(k)=-2P+(\Delta^{2}\hat{c})(k)\quad(k\in\mathbb{Z}).$

For $\gamma=3$, similarly it suffices to show that

$\left(k^{3}(\Delta^{3}a)(k)\right)_{k\in\mathbb{Z}}\in\l^{\infty}({\mathcal{L}}(X,Y)).$

For, note that for all $k\in\mathbb{Z}$ we have

(24) $\displaystyle(\Delta^{3}a)(k)$ $\displaystyle=-3(\Delta^{2}a)(k)(\Delta
b)(k+2)a(k+3)-a(k)(\Delta^{3}b)(k)a(k+3)$ $\displaystyle\quad-3(\Delta
a)(k)(\Delta^{2}b)(k+1)a(k+3).$

By the steps for $\gamma=1,2$ (see (21) and (23)), it is sufficient to show
the boundedness of

$\left(k^{3}(\Delta^{3}b)(k)a(k+3)\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X).$

However, since $\Delta^{3}b=\Delta^{3}\hat{c}$, it follows directly from the
assumption on $\hat{c}$. This finishes the proof of the statement about $a$.

Now we turn to the sequence $a_{0}$. For $\gamma=1$, note that

(25) $(\Delta a_{0})(k)=iBa(k+1)+ikB(\Delta a)(k).$

Thus, combining (19) and (21) with the boundedness of $a_{0}$, we get

$\left(k(\Delta a_{0})(k)\right)_{k\in\mathbb{Z}}\in
l^{\infty}({\mathcal{L}}(X)).$

For $\gamma=2$, note that

$(\Delta^{2}a_{0})(k)=2iB(\Delta
a)(k+1)+ikB(\Delta^{2}a)(k)\qquad(k\in\mathbb{Z}).$

Therefore, the boundedness of

$\left(k^{2}(\Delta^{2}a_{0})(k)\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$

follows from (22), (23), and the assumption on $a_{0}$.
Finally, for $\gamma=3$, note that

$(\Delta^{3}a_{0})(k)=3iB(\Delta^{2}a)+ikB(\Delta^{3}a)(k)\qquad(k\in\mathbb{Z}).$

Hence, the boundedness of

$\left(k^{3}(\Delta^{3}a_{0})(k)\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$

is implied by the formula (24) and our assumptions on $a_{0}$ and $\hat{c}$.
It completes the proof of the statement about $a_{0}$. The fact that
$a_{1}\in\mathfrak{M}^{\gamma}(\mathbb{Z};{\mathcal{L}}(X))$ follows from very
similar arguments. Therefore, we omit it.
Now we assume that $\hat{c}\in Var(\mathbb{Z};a)$. Then, by (19) and (20), for
every $j\in\mathbb{N}$ we have that

$\displaystyle\sum_{2^{j}\leq|k|<2^{j+1}}\|\Delta a(k)\|_{{\mathcal{L}}(X,Y)}$
$\displaystyle\leq\|a\|_{l^{\infty}({\mathcal{L}}(X,Y))}\Big{(}\sum_{2^{j}\leq|k|<2^{j+1}}\frac{1}{|k|}\|k(2k+1)Pa(k+1)\|_{{\mathcal{L}}(X)}$
$\displaystyle\quad+\sum_{2^{j}\leq|k|<2^{j+1}}\frac{1}{|k|}\|kBa(k+1)\|_{{\mathcal{L}}(X)}$
$\displaystyle\quad+\sum_{2^{j}\leq|k|<2^{j+1}}\|(\Delta\hat{c})(k)a(k+1)\|_{{\mathcal{L}}(X)}\Big{)}.$

Hence, by the boundedness of $a_{0},a_{1}$ and the condition imposed on
$\hat{c}$, we get that $a\in Var({\mathcal{L}}(X,Y))$. To show that $a_{0}\in
Var(\mathbb{Z};{\mathcal{L}}(X))$, using the formulas (19),(20) and (25), for
every $j\in\mathbb{N}$ we get that

$\displaystyle\sum_{2^{j}\leq|k|<2^{j+1}}$ $\displaystyle\|\Delta
a_{0}(k)\|_{{\mathcal{L}}(X,Y)}$
$\displaystyle=\sum_{2^{j}\leq|k|<2^{j+1}}\|Ba(k+1)+kB(\Delta
a)(k)\|_{{\mathcal{L}}(X,Y)}$
$\displaystyle\leq\sum_{2^{j}\leq|k|<2^{j+1}}\frac{1}{|k|}\|kBa(k+1)\|_{{\mathcal{L}}(X)}$
$\displaystyle\quad+\|a_{0}\|_{\l^{\infty}({\mathcal{L}}(X,Y))}\Big{(}\sum_{2^{j}\leq|k|<2^{j+1}}\frac{1}{|k|}\|k(2k+1)Pa(k+1)\|_{{\mathcal{L}}(X)}$
$\displaystyle\quad+\sum_{2^{j}\leq|k|<2^{j+1}}\frac{1}{|k|}\|kBa(k+1)\|_{{\mathcal{L}}(X)}$
$\displaystyle\quad+\sum_{2^{j}\leq|k|<2^{j+1}}\|(\Delta\hat{c})(k)a(k+1)\|_{{\mathcal{L}}(X)}\Big{)}.$

Therefore, the claim about $a_{0}$ follows from assumptions imposed on
$a_{0},a_{1}$ and $\hat{c}$. In a similar manner we prove that $a_{1}\in
Var(\mathbb{Z};{\mathcal{L}}(X))$. The proof of the additional statement about
$a_{2}$ mimics that for $a_{0}$ (formally, note that $a_{2}=a_{0}$, when
$B=I$).

(ii) The proof of the ${\mathcal{R}}$-bounded version follows from the same
arguments as presented in (i) and the fact that for any Banach spaces $X,Y,Z$,
if $\tau,\sigma\subset{\mathcal{L}}(X,Y)$ and $\rho\subset{\mathcal{L}}(Y,Z)$
are ${\mathcal{R}}$-bounded, then the families $\tau+\sigma$ and
$\tau\circ\rho$ are ${\mathcal{R}}$-bounded. ∎

######  Remark 6.3.

By the open mapping theorem, the assumption on the operators $b(k)$ made in
Theorem 6.2, i.e. the invertibility of $b(k)$ considered as an operator in
${\mathcal{L}}(Y,X)$, is equivalent to say that $b(k)$ considered as an
operator on $X$ is bijective and its inverse is in ${\mathcal{L}}(X)$. Indeed,
the norm $|\cdot|_{Y}$ is stronger then the graph norm of $b(k)$.

We call a distributional solution of (AEE), the strong solution, if $u\in
W^{1,1}(X)$ with $u(t)\in D_{A}$, $u^{\prime}(t)\in D_{B}\cap D_{P}$ for a.e.
$t\in\mathbb{R}$, and

$Au,\,Bu^{\prime},\,c\ast u\in L^{1}(X),\textrm{ and }Pu^{\prime}\in
W^{1,1}(X).$

Recall that $W^{1,1}(X)\subset{\mathcal{C}}(X)$. Moreover, note that the
existence of a strong solution of (AEE) requires that $f\in L^{1}(X)$, and
then (AEE) reads as

$(Pu^{\prime}(t))^{\prime}+Bu^{\prime}(t)+Au(t)+(c\ast u)(t)=f(t)\quad\textrm{
for a.e. }t\in\mathbb{R}$

with $u(0)=u(2\pi)$ and $(Pu^{\prime})(0)=(Pu^{\prime})(2\pi)$. The ’prime’ in
the symbols $(Pu^{\prime}(t))^{\prime}$ and $Bu^{\prime}(t)$ refers to the
classical derivative, which exists almost everywhere on $\mathbb{R}$ in the
topology of $X$. In the context of Theorem 6.2, note that the multiplier
sequence $a$ corresponds to the solution operator of (AEE), $f\mapsto
a(\Delta)f$, the sequences $a_{0}$ and $a_{1}$ correspond to the summands
${\mathcal{B}}\partial u$ and $\partial{\mathcal{P}}\partial u$ in (AEE), and
$a_{2}$ relates to the strong differentiability of $a(\Delta)f$.

In the following lemma we abstract some facts which allow to adapt multiplier
results from the previous section to the study of the solvability of (AEE).

######  Lemma 6.4.

(i) Assume that $A$, $B$ and $P$ are closed, linear operators on a Banach
spaces $X$ and $c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(Z,X))$, where $Z$ is
a Banach space with $Z\hookrightarrow X$. Then, the distributional solution
$u$ of (AEE) is a strong one if and only if

$u,\,\partial u,\,\mathcal{P}\partial u,\,\partial\mathcal{P}\partial
u,\,\mathcal{B}\partial u,\,\mathcal{A}u,\,c\ast u\in L^{1}(X).$

(ii) Assume that for every function $f=e_{k}\otimes x$, where $k\in\mathbb{Z}$
and $x\in X$, the problem (AEE) has a unique distributional solution. Then,
for every $k\in\mathbb{Z}$ the operator $b(k)=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective with the bounded inverse.

###### Proof.

$(i)$ The necessity is readily seen. Combining Lebesgue’s differentiation
theorem and closedness of operators $A$, $B$ and $P$ it is straightforward to
show that this condition is sufficient. For instance, if $u$ is a
distributional solution of (AEE) such that $u\in W^{1,1}(X)$ and the
distribution ${\mathcal{P}}\partial u$ is represented by $v\in L^{1}(X)$, then
for every $\phi\in{\mathcal{D}}$

$\int_{{\mathbb{T}}}v\phi\mathrm{d}t=({\mathcal{P}}\partial
u)(\phi)=P\left(\int_{\mathbb{T}}u^{\prime}\phi\mathrm{d}t\right),$

and both integrals are convergent in $X$. The Lebesgue differentiation theorem
and closedness of $P$ give that for a.e. $t\in[0,2\pi]$, $u^{\prime}(t)\in
D_{P}$ and $Pu^{\prime}(t)=v(t)$. Since $v\in L^{1}(X)$, we get that
$Pu^{\prime}\in L^{1}(X)$. $(ii)$ For the surjectivity, first note that if
$u\in{\mathcal{D}}^{\prime}(X)$ is a distributional solution of (AEE) then
$\hat{u}(k)\in D_{A}\cap Z$ for all $k\in\mathbb{Z}$ and $\hat{u}(k)\in
D_{B}\cap D_{P}$ for all $k\in\mathbb{Z}\setminus\\{0\\}$. Indeed, note that
$\partial u\in{\mathcal{D}}^{\prime}(D_{B})$ if and only if
$u-\hat{u}(0)\in{\mathcal{D}}^{\prime}(D_{B})$, and similarly for $P$.
Therefore, if $u$ is the corresponding solution of (AEE) for $f=e_{k}\otimes
x$, where $x\in X$ and $k\neq 0$, then $\hat{u}(k)\in Y$, and in the case when
$k=0$, $\hat{u}(0)\in D_{A}\cap Z$. Testing the both sides of (AEE) on
$e_{-k}\in{\mathcal{D}}$ ($k\in\mathbb{Z}$) we get that $b(k)\hat{u}(k)=x$,
which yields the suriectivity of $b(k)$.

Suppose now that for some $k\in\mathbb{Z}$, $b(k)y=0$ for some $y$ in the
domain of $b(k)$. Then, the function $u:=e_{k}\otimes y$ satisfies

$\partial{\mathcal{P}}\partial u+{\mathcal{B}}\partial u+{\mathcal{A}}u+c\ast
u=0.$

Therefore, the postulated uniqueness yields $u\equiv 0$, that is, $y=0$.
Consequently, the operators $b(k)$, $k\in\mathbb{Z}$, are bijective.

Finally, since $b(k)\in{\mathcal{L}}(Y,X)$ for $k\neq 0$, and
$b(0)\in{\mathcal{L}}(D_{A}\cap Z,X)$, and $Y,D_{A}\cap Z\hookrightarrow X$,
the boundedness of the inverse of $b(k)$ on $X$ follows from the open mapping
theorem. It finishes the proof of $(ii)$. ∎

For $\mathbb{E}\in\\{\Phi,B^{s,q}_{\Phi},F^{s,q}_{\Phi}:\Phi\textrm{ a Banach
function space over }({\mathbb{T}},\mathrm{d}t)\\}$, we say that the problem
(AEE) has $\mathbb{E}$-maximal regularity, if for every $f\in\mathbb{E}(X)$
there exists a unique distributional solution $u$ of (AEE) such that

$u-\hat{u}(0)\in\mathbb{E}(Y)\quad\textrm{and}\quad\partial\mathcal{P}\partial
u,\,\mathcal{B}\partial u,\,\mathcal{A}u,\,c\ast u\in\mathbb{E}(X).$

Moreover, for $\mathbb{E}\subset L^{1}$, we say that (AEE) is
$\mathbb{E}$-well-posed, if for every $f\in\mathbb{E}(X)$ the problem (AEE)
has a unique strong solution such that

(26)
$u,\,u^{\prime},\,Pu^{\prime},\,(Pu^{\prime})^{\prime},\,Bu^{\prime},\,Au,\,c\ast
u\in\mathbb{E}(X).$

Note that by Lemma 6.4(i), if $\mathbb{E}\subset L^{1}$ and (AEE) has
$\mathbb{E}$-maximal regularity, then (AEE) is $\mathbb{E}$-well-posed if and
only if

$u,\,\partial u,\,{\mathcal{P}}\partial u\in\mathbb{E}(X).$

The following result is the main result of this section. In the points (a) and
(b) we address the questions of the maximal regularity and well-posedness of
the problem (AEE) under different assumptions on the geometry of the
underlying Banach space $X$, as well as multiplier conditions imposed on
corresponding multiplier symbols. The point (c) clarifies the phenomenon of
extrapolation of $L^{p}$-maximal regularity and $L^{p}$-well-posedness for
such problem.

For the simplicity of its formulation, let ${\mathcal{L}}_{M}$ denote the
family of all Banach function spaces over $({\mathbb{T}},\mathrm{d}t)$ on
which the Hardy-Littlewood operator $M_{\mathbb{T}}$ is bounded, i.e.

${\mathcal{L}}_{M}:=\\{\Phi\textrm{ a Banach space over
}({\mathbb{T}},\mathrm{d}t):M_{\mathbb{T}}\textrm{ is bounded on }\Phi\\}.$

######  Theorem 6.5.

Let $X$ and $Z$ be Banach spaces such that $Z\hookrightarrow X$. Let $A$, $B$
and $P$ be closed, linear operators on a Banach space $X$,
$c\in{\mathcal{D}}^{\prime}({\mathbb{T}};{\mathcal{L}}(Z,X))$ and $Y$ has the
meaning specified in (19). For every $k\in\mathbb{Z}$ let

$b(k):=-k^{2}P+ikB+A+\hat{c}(k).$

_(a)_ Assume that for every $k\in\mathbb{Z}$ the operator $b(k)$ is bijective
and $a\in l^{\infty}({\mathcal{L}}(X,Z))$ and $a_{0},a_{1}\in
l^{\infty}({\mathcal{L}}(X))$, where

$a(k):=b(k)^{-1},\quad a_{0}(k):=ikBa(k),\quad
a_{1}(k):=-k^{2}Pa(k)\quad(k\in\mathbb{Z}\setminus\\{0\\}).$

and $a(0)=a_{0}(0)=a_{1}(0)=0$. Then, the following assertions hold.

  * (a1)

If $\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};a)$, then for every

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Psi},F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

the problem (AEE) has the $\mathbb{E}$-maximal regularity. In addition, if the
sequence $(ka(k))_{k\in\mathbb{Z}}$ is bounded in ${\mathcal{L}}(X)$ and
$\mathbb{E}\subset L^{1}$, then (AEE) is $\mathbb{E}$-well-posed.

  * (a2)

If $\hat{c}\in Var(\mathbb{Z};a)$ and, in addition, $X$ has
$U\\!M\\!D$-property, then the conclusion of _(a1)_ holds for each

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime}\in{\mathcal{L}}_{M}\bigr{\\}}.$

_(b)_ Assume that for every $k\in\mathbb{Z}$ the operator $b(k)$ is bijective,
the sequences $a$, and $a_{0}$, $a_{1}$ are ${\mathcal{R}}$-bounded in
${\mathcal{L}}(X,Z)$ and ${\mathcal{L}}(X)$, respectively, and $X$ has the
$U\\!M\\!D$ property.

If $\hat{c}\in\mathfrak{M}_{\mathcal{R}}^{1}(\mathbb{Z};a)$, then for every

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime}\in{\mathcal{L}}_{M}\bigr{\\}}$

the problem (AEE) has $\mathbb{E}$-maximal regularity.

In addition, if $(ka(k))_{k\in\mathbb{Z}}$ is ${\mathcal{R}}$-bounded in
${\mathcal{L}}(X)$, then for every $\mathbb{E}\subset L^{1}$, the problem
(AEE) is $\mathbb{E}$-well-posed.

In particular, the last statement holds for each

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Phi},\,F^{s,r}_{\Phi}:s>0,q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime}\in{\mathcal{L}}_{M}\bigr{\\}}.$

_(c)_ Assume that the problem (AEE) has $L^{p}$-maximal regularity for some
$p\in(1,\infty)$ (respectively, (AEE) is $L^{p}$-well-posed).

If $\hat{c}\in\mathfrak{M}^{3}(\mathbb{Z};a)$, then for every

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Psi},\,F^{s,q}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}$

it has $\mathbb{E}$-maximal regularity (respectively, in addition, if
$\mathbb{E}\subset L^{1}$, it is $\mathbb{E}$-well-posed).

###### Proof.

(a1) Combining Theorem 6.2 with Theorem 5.3(i) and (iii) we infer that
$a\in{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,Y)$ and
$a_{0},a_{1},Aa(\cdot)\in{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,X)$. In
particular, for every $f\in\mathbb{E}(X)$, if we put
$u:=a(\Delta)f+b(0)^{-1}\hat{f}(0)$, then
$u\in{\mathcal{D}}^{\prime}(D_{A})\cap{\mathcal{D}}^{\prime}(Z)$ and $\partial
u\in{\mathcal{D}}^{\prime}(Y)\subset{\mathcal{D}}^{\prime}(D_{B})\cap{\mathcal{D}}^{\prime}(D_{P})$.
It is easy to check that $u$ is a distributional solution of (AEE). Since the
Fourier coefficients of distributions are uniquely determined, the injectivity
of the operators $b(k)$, $k\in\mathbb{Z}$, gives that $u$ is the unique
solution of (AEE).

Moreover, note that

$\partial{\mathcal{P}}\partial u=a_{1}(\Delta)f,\,{\mathcal{B}}\partial
u=a_{0}(\Delta)f,\,{\mathcal{A}}u=[Aa(\cdot)](\Delta)f\in\mathbb{E}(X)$

and, since $f\in\mathbb{E}(X)$,

$c\ast u=\hat{c}(\Delta)a(\Delta)f=\partial{\mathcal{P}}\partial
u+{\mathcal{B}}\partial u+{\mathcal{A}}u-f\in\mathbb{E}(X).$

It proves the $\mathbb{E}$-maximal regularity of (AEE).

For the additional statement, again by Theorem 6.2 and Theorem 5.3(i), we get
that $a_{2}(k):=ika(k)$ and $a_{3}(k):=ikPa(k)$, $k\in\mathbb{Z}$, are in
${\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,X)$. Since $Y\hookrightarrow X$ and
$D_{A}\cap Z\hookrightarrow X$, we have that
$a\in{\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,X)$. Hence, if
$f\in\mathbb{E}(X)$, then for the corresponding solution
$u=a(\Delta)f+b(0)^{-1}\hat{f}(0)$ of (AEE) we get that

$u,\,\partial u=a_{2}(\Delta)f,\,{\mathcal{P}}\partial
u=a_{3}(\Delta)f\in\mathbb{E}(X).$

If, in addition, $\mathbb{E}\subset L^{1}$, Lemma 6.4(i) shows that $u$ is a
strong solution and (26), that is, the $\mathbb{E}$-well-posedness of (AEE).
It completes the proof of (a1).

Relying on Theorem 6.2, the parts (ii) and (iii) of Theorem 5.3, and Lemma
6.4(i), the proof of the statement (a2) follows the similar arguments
presented for the proof of (a1). Therefore, we omit them.

For the statement (b), note that Theorem 5.5(ii) reduces the proof of (b) to
the arguments provided in the proof of (a1).

Finally, for the proof of (c), by Lemma 6.4(ii), the $L^{p}$-maximal
regularity of (AEE) implies that for each $k\in\mathbb{Z}$, $b(k)$ is
bijective with a bounded inverse. That is,
$a(k)=b(k)^{-1}\in{\mathcal{L}}(X,Y)$ for all $k\in\mathbb{Z}\setminus\\{0\\}$
and $a(0)=b(0)^{-1}\in{\mathcal{L}}(X,D_{A}\cap Z)$ (see Remark 6.3(i)). It is
straighforward to check that the corresponding solution operator for (AEE) is
given by the Fourier multiplier $a(\Delta)$.

We show that $a\in l^{\infty}({\mathcal{L}}(X,Y))$ and $a_{0}$, $a_{1}\in
l^{\infty}({\mathcal{L}}(X))$. The $L^{p}$-maximal regularity implies that the
maps

$L^{p}(X)\ni f\mapsto a(\Delta)f\in L^{p}(Y)\textrm{ and }L^{p}(X)\ni
f\mapsto{\mathcal{B}}\partial a(\Delta)f,\,\,\partial{\mathcal{P}}\partial
a(\Delta)f\in L^{p}(X),$

are well-defined and (by the uniqueness) linear. By the closed graph theorem,
it is straightforward to see that these maps are bounded. Moreover, since
$a(\Delta)(e_{k}\otimes x)=e_{k}\otimes b(k)^{-1}x$ for every $x\in X$ and
$k\in\mathbb{Z}$, we infer that for $k\neq 0$

$|b^{-1}(k)x|_{Y}=\|a(\Delta)(e_{k}\otimes
x)\|_{L^{p}(Y)}\leq\|a(\Delta)\|_{{\mathcal{L}}(L^{p}(X),L^{p}(Y))}|x|_{X}$

and similarly

$|kBb(k)^{-1}x|_{X}\leq\|{\mathcal{B}}\partial
a(\Delta)\|_{{\mathcal{L}}(L^{p}(X))}|x|_{X},\,\,|k^{2}Pb(k)^{-1}x|_{X}\leq\|\partial{\mathcal{P}}\partial
a(\Delta)\|_{{\mathcal{L}}(L^{p}(X))}|x|_{X}.$

It gives our claim. Combining it with the assumption on $\hat{c}$, by Theorem
6.2 and Theorem 5.5(i), we are in the position to use the same arguments as in
(a1) to get the desired assertion on the $\mathbb{E}$-maximal regularity of
(AEE). For the additional statement, similarly as above, the $L^{p}$-well-
posedness of (AEE) implies that the sequences $a_{2}(k)=ika(k)$ and
$a_{3}(k)=ikPa(k)$ ($k\in\mathbb{Z}$) are in
${\mathcal{M}}_{\mathbb{E}}({\mathbb{T}};X,X)$. Hence, the proof follows the
corresponding lines of the proof of (a1). This completes the proof of (c). ∎

######  Remark 6.6.

Since for each
$\mathfrak{M}\in\\{\mathfrak{M}^{\gamma},\mathfrak{M}^{\gamma}_{\mathcal{R}},Var\\}$,
if $(kBa(k))_{k\in\mathbb{Z}},(k^{2}Pa(k))_{k\in\mathbb{Z}}$ are in
$\mathfrak{M}(\mathbb{Z},{\mathcal{L}}(X))$ then
$(Ba(k))_{k\in\mathbb{Z}},(kPa(k))_{k\in\mathbb{Z}},(Pa(k))_{k\in\mathbb{Z}}\in\mathfrak{M}(\mathbb{Z};{\mathcal{L}}(X))$,
the proof of Theorem 6.5 shows that the $\mathbb{E}$-maximal regularity of
(AEE) concluded in each of its statements (a), (b), (c) (no additional
assumptions on $(ka(k))_{k\in\mathbb{Z}}$ is required) gives that the problem

$\partial^{2}{\mathcal{P}}u+\partial{\mathcal{B}}u+{\mathcal{A}}u+c\ast u=f$

is $\mathbb{E}$-well-posed whenever $\mathbb{E}\subset L^{1}$. That is, for
each $f\in\mathbb{E}(X)$ there exists a unique distributional solution $u$
such that $u-\hat{u}(0)\in\mathbb{E}(Y)$, $Pu\in
W^{2,1}(X)\subset{\mathcal{C}}^{1}(X)$, $Bu\in
W^{1,1}(X)\subset{\mathcal{C}}(X)$ and

$(Pu)^{\prime\prime}(t)+(Bu)^{\prime}+Au(t)+c\ast u(t)=f(t)\qquad\textrm{ a.e.
}t\in[0,2\pi].$

In this context note that if
$\partial^{2}{\mathcal{P}}u,\partial{\mathcal{B}}u\in
B^{s,q}_{\Phi}(X)\subset\Phi(X)$ for some $s>0$, then by Proposition 2.3(iii),
we immediately get $Pu\in W^{2,1}(X)$ and $Bu\in W^{1,1}(X)$.

As was already mentioned in Remark 6.1 one can impose separate conditions on
$a$ and $\hat{c}$ which imply their joint condition
$\hat{c}\in\mathfrak{M}^{\gamma}(a)$ (or others). It leads to the following
characterisation of the maximal regularity and well-posedness of the problem
(AEE). Here, we assume that $Z=X$; see Remark 6.11 below. We start with the
characterization and extrapolation of the maximal regularity of (AEE).

######  Theorem 6.7.

Let $A$, $B$ and $P$ be closed, linear operators on a Banach space $X$. Let
$c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(X))$.

_(a)_ Assume that $\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$.
Then, the following assertions are equivalent.

  * (i)

The problem (AEE) has $\mathbb{E}$-maximal regularity for every $\mathbb{E}$
such that

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Psi},F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

  * (ii)

The problem (AEE) has $B^{s,q}_{L^{p}}$-maximal regularity for some
$p\in(1,\infty)$ and $q\in[1,\infty]$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$\left(b(k)^{-1}\right)_{k\in\mathbb{Z}},\quad(kBb(k)^{-1})_{k\in\mathbb{Z}},\quad(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}$

are bounded in ${\mathcal{L}}(X)$.

_(b)_ Assume that $X$ has the $U\\!M\\!D$ property and
$\hat{c}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$. Then,
the following assertions are equivalent.

  * (i)

The problem (AEE) has $\mathbb{E}$-maximal regularity for every $\mathbb{E}$
such that

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Psi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

  * (ii)

The problem (AEE) has $L^{p}$-maximal regularity for some $p\in(1,\infty)$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$(b(k)^{-1})_{k\in\mathbb{Z}},\quad(kBb(k)^{-1})_{k\in\mathbb{Z}},\quad(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}$

are ${\mathcal{R}}$-bounded in ${\mathcal{L}}(X)$ .

######  Remark 6.8.

(a) It is easily seen that for (AEE) with $B=I\in{\mathcal{L}}(X)$ one can
relax the assumption on $\hat{c}$ in the statement (a) and (b) of Theorem 6.7
to $(\cdot)^{-1}\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$ in
(a), and to
$(\cdot)^{-1}\hat{c}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$
in (b), respectively.

Furthermore, in the case when $P=I\in{\mathcal{L}}(X)$, one can further weaken
the assumption on $\hat{c}$, namely, to
$(\cdot)^{-2}\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$ in (a),
and to
$(\cdot)^{-2}\hat{c}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$
in (b), respectively; cf. also Remark 6.1.

(b) The assertion (ii) in Theorem 6.7(a) can be replaced with the following
one

  * (ii’)

The problem (AEE) has $\mathbb{E}$-maximal regularity for some $\mathbb{E}$
such that

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Psi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

Indeed, one can easily check that the maximal regularity of (AEE) with respect
to every such $\mathbb{E}$ implies the boundedness of the sequences stated
(iii). It is not clear if $\mathbb{E}$-maximal regularity of (AEE) for general
$\mathbb{E}$ yields the ${\mathcal{R}}$-boundedness of these sequences.

(c) In a similar manner to that in Theorem 6.7 one can formulate the statement
corresponding to $(a2)$ of Theorem 6.5. Its proof follows the same arguments.
We leave it to the interested reader.

###### Proof of Theorem 6.7.

(a) Of course (i)$\Rightarrow$(ii). For (ii)$\Rightarrow$(iii), since
$e_{k}\otimes x\in B^{s,q}_{L^{p}}(X)$ for every $k\in\mathbb{Z}$ and $x\in X$
(see, e.g. Proposition 2.3(i)), by Lemma 6.4(ii) we get that for all
$k\in\mathbb{Z}$ the operator $b(k)$ is bijective with inverse in
${\mathcal{L}}(X,Y)$ when $k\neq 0$, and in ${\mathcal{L}}(X,Z)$ when $k=0$.
Analogously as in the proof of Theorem 6.5(c), by the closed graph theorem, we
infer that the maps

$B^{s,q}_{L^{p}}(X)\ni f\mapsto a(\Delta)f\in B^{s,q}_{L^{p}}(Z)\textrm{ and
}$ $B^{s,q}_{L^{p}}(X)\ni f\mapsto{\mathcal{B}}\partial
a(\Delta)f,\,\,\partial{\mathcal{P}}\partial a(\Delta)f\in B^{s,q}_{L^{p}}(X)$

are bounded, where $a(k):=b^{-1}(k)$, $k\in\mathbb{Z}$. Since for every
$k\in\mathbb{Z}$ and $x\in X$

$\|e_{k}\otimes x\|_{B^{s,q}_{L^{p}}(X)}=\Big{(}\sum_{j\geq
0}2^{sjq}\psi_{j}(k)^{q}\Big{)}^{\frac{1}{q}}|x|_{X},$

we get the boundedness of desired sequences in (iii). The proof of
(iii)$\Rightarrow$(i) follows directly from Theorem 6.5(a1); see Remark 6.1.

The proof of (b) mimics the same arguments, we only recall here that each
sequence in ${\mathcal{M}}_{p}(\mathbb{Z},X,Y)$ (for arbitrary Banach spaces
$X$ and $Y$) is necessarily ${\mathcal{R}}$-bounded; see [31] or [7]. ∎

Now we formulate analogous result for the well-posedness of (AEE). Its proof
follows the lines of the proof of Theorem 6.7 with straightforward
modifications. Therefore, we omit it.

######  Theorem 6.9.

Let $A$, $B$ and $P$ be closed, linear operators on a Banach space $X$. Let
$c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(X))$.

_(a)_ Assume that
$(k^{-1}\hat{c}(k))_{k\in\mathbb{Z}}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$.
Then, the following assertions are equivalent.

  * (i)

The problem (AEE) is $\mathbb{E}$-well-posed for every $\mathbb{E}\subset
L^{1}$ such that

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Psi},F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}$

  * (ii)

The problem (AEE) is $B^{s,q}_{L^{p}}$-well-posed for some $p\in(1,\infty)$
and $q\in[1,\infty]$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$\left(kb(k)^{-1}\right)_{k\in\mathbb{Z}},\,\,(kBb(k)^{-1})_{k\in\mathbb{Z}},\,\,(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}$

are bounded in ${\mathcal{L}}(X)$.

_(b)_ Assume that $X$ has the $U\\!M\\!D$ property and
$(k^{-1}\hat{c}(k))_{k\in\mathbb{Z}}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$.
Then, the following assertions are equivalent.

  * (i)

The problem (AEE) is $\mathbb{E}$-well-posed for every $\mathbb{E}$ such that

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Psi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

  * (ii)

The problem (AEE) is $L^{p}$-well-posed for some $p\in(1,\infty)$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$(kb(k)^{-1})_{k\in\mathbb{Z}},\quad(kBb(k)^{-1})_{k\in\mathbb{Z}},\quad(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}$

are ${\mathcal{R}}$-bounded in ${\mathcal{L}}(X)$.

######  Remark 6.10.

The similar statements to those stated in Remark 6.8 hold in the context of
Theorem 6.9. In particular, for (AEE) with $P=I\in{\mathcal{L}}(X)$ in (AEE),
one can relax the assumption on $\hat{c}$ in the statements (a) and (b) of
Theorem 6.9 to
$(\cdot)^{-2}\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$ in (a)
and to
$(\cdot)^{-2}\hat{c}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$
in (b), respectively.

It is worth noticing that according to Remark 6.1, if
$\hat{c}\in\mathfrak{M}^{1}(\mathbb{Z};{\mathcal{L}}(X))$, then
$(\cdot)^{-1}\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$. Thus,
for $c$ such that $\hat{c}\in\mathfrak{M}^{1}(\mathbb{Z};{\mathcal{L}}(X))$
the equivalence in Theorem 6.9(a) also holds. This is a strictly stronger
condition, but in some situations it might be easier to verify.

######  Remark 6.11.

If we modify the notion of the well-posedness of (AEE) (which sometimes can be
motivated by a special form of the convolution therm; see, e.g. (27)) by
replacing the condition $u\in W^{1,1}(X)$ by a stronger one, e.g. $u\in
W^{1,1}(Z)$, where $Z$ is a Banach space such that $Z\hookrightarrow X$, then
the above characterizations, Theorems 6.7 and 6.9 (where $Z=X$) can be easily
adjusted to such modified setting. The proofs of such modifications have the
same pattern as those of Theorems 6.7 and 6.9. Here, we do not provide such
reformulation of the above characterization results, which corresponds to such
stronger notion of well-posedness of (AEE).

For our further purposes we present a counterpart of Theorems 6.7 and 6.9 for
a convolutor $c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(Z,X))$ with and
arbitrary $Z\hookrightarrow X$, which we apply in the next section.

######  Theorem 6.12.

Let $A$, $B$ and $P$ be closed, linear operators on a Banach space $X$. Let
$c\in{\mathcal{D}}^{\prime}({\mathcal{L}}(Z,X))$, where $Z$ is a Banach space
continuously embedded in $X$.

_(a)_ Assume that $\hat{c}\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(Z,X))$.
Then, the following assertions are equivalent.

  * (i)

The problem (AEE) has $\mathbb{E}$-maximal regularity (respectively, is
$\mathbb{E}$-well-posed) for every $\mathbb{E}$ such that

$\mathbb{E}\in\bigl{\\{}B^{s,q}_{\Psi},F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}$

(respectively, in addition, $\mathbb{E}\subset L^{1}$).

  * (ii)

The problem (AEE) has $B^{s,q}_{L^{p}}$-maximal regularity (respectively, is
$B^{s,q}_{L^{p}}$-well-posed) for some $p\in(1,\infty)$ and $q\in[1,\infty]$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$\left(b(k)^{-1}\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X,Z),\,\,(kBb(k)^{-1})_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X),\,\,(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$

are bounded (respectively, in addition,
$\left(kb(k)^{-1}\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$ is bounded).

_(b)_ Assume that $X$ has the $U\\!M\\!D$ property and
$\hat{c}\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(Z,X))$.
Then, the following assertions are equivalent.

  * (i)

The problem (AEE) has $\mathbb{E}$-maximal regularity (respectively, is
$\mathbb{E}$-well-posed) for every $\mathbb{E}$ such that

$\mathbb{E}\in\bigl{\\{}\Phi,\,B^{s,q}_{\Psi},\,F^{s,r}_{\Phi}:s\in\mathbb{R},q\in[1,\infty],r\in(1,\infty),\,\,\Phi,\Phi^{\prime},\Psi\in{\mathcal{L}}_{M}\bigr{\\}}.$

(respectively, in addition, $\mathbb{E}\subset L^{1}$).

  * (ii)

The problem (AEE) has $L^{p}$-maximal regularity (respectively, is
$\mathbb{E}$-well-posed) for some $p\in(1,\infty)$.

  * (iii)

For every $k\in\mathbb{Z}$ the operator $b(k):=-k^{2}P+ikB+A+\hat{c}(k)$ is
bijective and the sequences

$(b(k)^{-1})_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X,Z),\quad(kBb(k)^{-1})_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X),\quad(k^{2}Pb(k)^{-1})_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$

are ${\mathcal{R}}$-bounded (respectively, in addition,
$\left(kb(k)^{-1}\right)_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$ is
${\mathcal{R}}$-bounded).

##  7\. Particular forms of (AEE)

In this section we specialize our general results from the previous section to
particular forms of the abstract problem (AEE), which have been studied in the
literature. In particular, Theorems 6.7 and 6.9 extend many results from a
long series of articles, where such characterisations have been studied
progressively; [32, 58, 7, 8, 47, 53, 23, 54, 57, 24, 49, 40, 56, 38, 20, 21,
22] and the references therein.

###  7.1. Integro-differential equations

Here, we consider (AEE) for special classes of convolutors $c$, which arise in
the (abstract) reformulation of the integro-differential equations describing
physical processes in materials with fading memory. We start with equations,
where a convolutor $c$ is given by the so-called finite delay operators, that
is, $c$ is of the form

(27) $c\ast u:=Hu_{\cdot}+Gu^{\prime}_{\cdot},$

where $H,G\in{\mathcal{L}}(L^{p}({\mathbb{T}};X),X)$ for some
$p\in(1,\infty)$, $u_{t}(s):=u(t+s)$ and, if, for instance, $u\in
W^{1,p}({\mathbb{T}};X)$, $u^{\prime}_{t}(s)=u^{\prime}(t+s)$
($s\in[0,2\pi]$). Note that if for all $k\in\mathbb{Z}$ and $x\in X$ we set

$H_{k}x:=H(e_{k}\otimes x)\quad\textrm{ and }\quad G_{k}x:=G(e_{k}\otimes x),$

then $\widehat{Hu_{\cdot}}(k)=H_{k}\hat{u}(k)$ and
$\widehat{Gu_{\cdot}}(k)=G_{k}\hat{u}(k)$ for all $u\in L^{p}(X)$.
Consequently, the sequences $h:=(H_{k})_{k\in\mathbb{Z}}$ and
$g:=(G_{k})_{k\in\mathbb{Z}}$ are in ${\mathcal{M}}_{p}(\mathbb{Z};X,X)$. It
shows that one can extend the meaning of (27) to all
$u\in{\mathcal{D}}^{\prime}(X)$ putting

$c\ast u=h(\Delta)u+\partial g(\Delta)u=h(\Delta)u+g(\Delta)(\partial u),$

that is, $\hat{c}(k)=h(k)+ikg(k)$, $k\in\mathbb{Z}$.

This abstract reformulation of (27) leads to the following characterisation of
the maximal regularity and well-posedness for this class of equations.

######  Corollary 7.1.

Let $A$, $B$ and $P$ be closed, linear operators on a Banach space $X$. Let
$(h(k))_{k\in\mathbb{Z}}$, $(g(k))_{k\in\mathbb{Z}}\subset{\mathcal{L}}(X)$.

_(1)_ Assume that
$(\cdot)^{-\alpha}h\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$ and
$(\cdot)^{1-\alpha}g\in\mathfrak{M}^{2}(\mathbb{Z};{\mathcal{L}}(X))$ for some
$\alpha\in\\{0,1,2\\}$. Let $\hat{c}:=h+i(\cdot)g$. Then, the following
statements are true.

  * (i)

If $\alpha=0$, then the equivalence
$(i)\Leftrightarrow(ii)\Leftrightarrow(iii)$ of Theorem 6.7(a) holds for (AEE)
with $\hat{c}$.

  * (ii)

If $\alpha=1$, then the conclusion of Theorem 6.9(a) holds, and if, in
addition, $B=I$, then the conclusion of Theorem 6.7(a) holds too.

  * (iii)

If $\alpha=2$ and, in addition, $P=I$, then the conclusion of Theorem 6.7(a)
and Theorem 6.9(a) hold.

_(2)_ Let $X$ have the $U\\!M\\!D$ property. Assume that
$(\cdot)^{-\alpha}h\in\mathfrak{M}_{\mathcal{R}}^{1}(\mathbb{Z};{\mathcal{L}}(X))$
and
$(\cdot)^{1-\alpha}g\in\mathfrak{M}^{1}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(X))$
for some $\alpha\in\\{0,1,2\\}$. Then the analogous statements to those of the
point (1) hold, that is, the statements which are obtained by replacing the
assertion (a) of Theorems 6.7 and 6.9 with the corresponding (b) therein.

######  Remark 7.2.

(a) One can readily seen, that Corollary 7.1 extends several related results
from the literature, where the convolutors $c$ are given by finite delay
operators; see, e.g. [22], [21] and also references therein.

(b) Moreover, Theorem 7.1(2) covers [38, Theorem 3.4]. Indeed, recall that Fu
and Li in [38] studied the problem (AEE) with $P=I$ and the convolutor $c$
given by the infinite delay operators, that is,

$c\ast u=Hu_{\cdot}+Gu^{\prime}_{\cdot},$

where $H,G:\mathscr{B}\rightarrow X$ are bounded, linear operators,
$\mathscr{B}$ is a space of $X$-valued functions on
$\mathbb{R}_{-}:=(-\infty,0]$, which is axiomatically defined in [38], and
$u_{\cdot}$ is given as before by $u_{t}(s):=u(t+s)$ $(s\in\mathbb{R}_{-})$.
Since $e_{k}$, $k\in\mathbb{Z}$, considered as functions on $\mathbb{R}_{-}$,
belong to ${\mathcal{C}}_{b}(\mathbb{R}_{-})\subset\mathscr{B}$, one can
define sequences $h$ and $g$ similarly as above, i.e.
$h(k)x:=H(e_{k}(\cdot)x)$ and $g(k)x:=(e_{k}(\cdot)x)$, $k\in\mathbb{Z}$.
Then, $c\in{\mathcal{D}}^{\prime}(X)$ with $\hat{c}(k)=h(k)+ikg(k)$,
$k\in\mathbb{Z}$. The axioms of $\mathscr{B}$ easily give that $h$ and $g$ are