# Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional
Sobolev spaces via interpolation and extrapolation

Oscar Domínguez  O. Domínguez, Université Lyon 1, Institut Camille Jordan, 43
blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France; Departamento de
Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas,
Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Mario Milman  M.
Milman, Instituto Argentino de Matematica
Buenos Aires
Argentina<EMAIL_ADDRESS>https://sites.google.com/site/mariomilman/

###### Abstract.

The real interpolation spaces between $L^{p}({\mathbb{R}}^{n})$ and
$\dot{H}^{t,p}({\mathbb{R}}^{n})$ (resp. $H^{t,p}({\mathbb{R}}^{n})$), $t>0,$
are characterized in terms of fractional moduli of smoothness, and the
underlying seminorms are shown to be “the correct” fractional generalization
of the classical Gagliardo seminorms. This is confirmed by the fact that,
using the new spaces combined with interpolation and extrapolation methods, we
are able to extend the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit
formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to
fractional Sobolev spaces. On the other hand, we disprove a conjecture of [5]
suggesting fractional convergence results given in terms of classical
Gagliardo seminorms. We also solve a problem proposed in [5] concerning sharp
forms of the fractional Sobolev embedding.

###### Key words and phrases:

Triebel–Lizorkin spaces, fractional Sobolev spaces, limits, interpolation,
extrapolation

###### 1991 Mathematics Subject Classification:

Primary: 26A33, 46B20, 46B70; Secondary: 42B05, 42B10

##  1\. Preamble

###  1.1. The Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae

In their celebrated paper [3], Bourgain–Brezis–Mironescu studied limits of
Gagliardo semi-norms defined by111We assume that $p>1,$ since, as in [5], our
main focus will be the fractional Sobolev spaces defined via
$(-\Delta)^{\frac{t}{2}}.$ The case $p=1$ requires a separate treatment.

(1.1)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}=\left(\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\frac{\left|f(x)-f(y)\right|^{p}}{\left|x-y\right|^{n+sp}}dxdy\right)^{1/p},\qquad
s\in(0,1),\quad p\in(1,\infty),$

and showed that while

(1.2) $\lim_{s\rightarrow
1^{-}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}<\infty\iff f\quad\text{is
constant},$

the Gagliardo seminorms can be normalized so that the $L^{p}$ norm of the
gradient can be recovered: For $f\in W_{p}^{1}({\mathbb{R}}^{n})$, it holds

(1.3) $\lim_{s\rightarrow
1^{-}}(1-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}=C(p,n)\|\nabla
f\|_{L^{p}({\mathbb{R}}^{n})},$

where $C(p,n)$ is a constant that depends only on $p$ and $n,$ and can be
computed explicitly.

Later, Maz’ya–Shaposhnikova [24] used a different normalization to deal with
the case where the (smoothness) parameter $s$ tends to zero, and proved that

(1.4) $\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}=c(p,n)\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

These results have sparked intense research interest and have been used and
extended in many different directions (for recent related material,
applications and references cf. [5], [6], [7], [10], [12], [15], [34]). The
original proofs were somewhat complicated, and the process of finding the
correct normalizations involved ad-hoc considerations. For example, the
correct normalization for (1.4) was found using the sharp form of the Sobolev
embedding and, conversely, the limit result (1.3) suggested the sharp
Fractional Sobolev inequality obtained in [4].

In [25], [20], it was observed that these limiting results can be naturally
understood in the more general context of scales of interpolation spaces.
Given a pair of compatible spaces $(X_{0},X_{1}),$ interpolation methods (cf.
[2]) give rise to scales of spaces, in particular, the scale
$(X_{0},X_{1})_{s,q},$ $s\in(0,1),q\in[1,\infty],$ produced by the real
method222See Section 4 for background information on interpolation and
extrapolation., will play an important rôle in this paper. We colloquially
refer to $X_{0}$ and $X_{1}$ as “end point spaces”, and we think that, as the
parameter $s$ varies from $0$ to $1,$ we are moving from one end point space
in the scale, to the other. In this setting, the limit theorems take the form
of a continuity result that can be informally interpreted as saying that “with
appropriate normalizations we can recover the end point spaces” (cf. Theorem
4.7),

(1.5) $\lim_{s\rightarrow
0^{+}}c_{s,q}\left\|f\right\|_{(X_{0},X_{1})_{s,q}}=\sup_{t>0}K(t,f;X_{0},X_{1});\quad\text{
}\lim_{s\rightarrow
1^{-}}c_{s,q}\left\|f\right\|_{(X_{0},X_{1})_{s,q}}=\sup_{t>0}\frac{K(t,f;X_{0},X_{1})}{t}.$

For many pairs of spaces $(X_{0},X_{1}),$ we actually have estimates: For
$f\in X_{0}\cap X_{1},$333Throughout the paper, the symbol $f\lesssim g$
indicates the existence of a constant $c>0$, which is independent of all
essential parameters, such that $f\leq c\,g$. The symbol $f\approx g$ means
that $f\lesssim g$ and $g\lesssim f$.

(1.6) $\sup_{t>0}K(t,f;X_{0},X_{1})\approx\left\|f\right\|_{X_{0}},\text{ \ \
\ \ \ \
}\sup_{t>0}\frac{K(t,f;X_{0},X_{1})}{t}\approx\left\|f\right\|_{X_{1}}.$

Furthermore, if the pair is “exact” then one can replace the equivalences
signs in (1.6) by equalities444From this point of view, the constants that
appear in (1.3) and (1.4) are connected to the computation of the underlying
$K$-functionals. (cf. [25], [20]).

From the interpolation point of view, (1.3) and (1.4) are not separate
limiting results, but part of the recovery of end point norms of a particular
interpolation scale. Moreover, we note that the limiting formulae (1.5) are
somewhat stronger, in the sense that we are free to choose the index $q,$
which indeed, is not dependent on the original pair of spaces, as it is the
case in (1.3) or (1.4), where $q=p.$

In fact, the normalization constants produced by interpolation are
“universal”, in the sense that they are independent of the originating pair of
spaces and have an explicit formula555There is a method to define and compute
the normalizations in order insure the continuity of the interpolation scales.
In particular, the normalizations for other methods of interpolation are also
known. For example, for the complex method of Calderón, $c_{s}=1;$ for the
$J-$method, $c_{s,q}=\left((1-s)sq^{\prime}\right)^{-1/q^{\prime}},$ where
$\frac{1}{q}+\frac{1}{q^{\prime}}=1.$ In the case of quasi-Banach spaces some
adjustments are necessary (cf. [16, pag. 35] for more details).

(1.7)
$c_{s,q}=\left\\{\begin{array}[c]{cc}\left(s(1-s)q\right)^{\frac{1}{q}},&\quad
s\in(0,1),\quad q\in[1,\infty),\\\ 1,&q=\infty.\end{array}\right.$

The correct formulae for the normalizations originate from considerations
arising in the extrapolation theory of [16], where one needs to work with
continuous scales, and must pay close attention to the constants that appear
in the estimates. It turns out that normalized interpolation norms have
certain crucial monotonicity properties that will play a role in what follows.
In particular, it was shown in [16, (3.2), page 19] that, if $q\leq r,$ then

$c_{s,r}\left\|f\right\|_{(X_{0},X_{1})_{s,r}}\leq
c_{s,q}\left\|f\right\|_{(X_{0},X_{1})_{s,q}}.$

Furthermore, the usual reiteration formulae of interpolation theory, as
exemplified by Holmstedt’s reiteration formula (cf. [1, Theorem 2.1, page
307], [16, (3.15), page 33], [19, Lemmas 2.1–2.3, pages 61–63], [20], etc.)
can be sharpened, when the norms have been normalized according to (1.7).

Combining (1.5) and the fact that

(1.8)
$(L^{p}(\mathbb{R}^{n}),\dot{W}_{p}^{1}(\mathbb{R}^{n}))_{s,p}=\dot{W}^{s,p}(\mathbb{R}^{n}),$

with hidden equivalence constants independent of $s,$ we can derive versions
of (1.3) and (1.4). The proof of (1.8) hinges upon the known computation of
the underlying $K$-functional in terms of a $p-$modulus of continuity (cf. [1,
Theorem 4.12, page 339] and the references therein),

$\displaystyle K(u,f;L^{p}(\mathbb{R}^{n}),\dot{W}_{p}^{1}(\mathbb{R}^{n}))$
$\displaystyle:=\inf_{f=f_{0}+f_{1}}\\{\left\|f_{0}\right\|_{L^{p}(\mathbb{R}^{n})}+u\left\|\nabla
f_{1}\right\|_{L^{p}(\mathbb{R}^{n})}\\}$
$\displaystyle\approx\sup_{\left|h\right|\leq
u}\|\Delta_{h}^{1}f\|_{L^{p}(\mathbb{R}^{n})},$

where

$\Delta_{h}^{1}f(x)=\Delta_{h}f(x)=f(x+h)-f(x),\qquad x,h\in{\mathbb{R}}^{n}.$

The higher order differences $\Delta_{h}^{k},\,k\in\mathbb{N},$ are defined
inductively,

$\Delta_{h}^{k+1}=\Delta_{h}\Delta_{h}^{k},$

and we can write

(1.9) $\Delta_{h}^{k}f(x)=\sum_{j=0}^{k}(-1)^{j}{\binom{k}{j}}f(x+(k-j)h).$

The corresponding $K$-functionals for higher order Sobolev spaces
$\dot{W}^{k}_{p}({\mathbb{R}}^{n})$ are given by (cf. [1, Theorem 4.12, pag.
339], [20], and the references therein)

(1.10) $\displaystyle
K(u^{k},f;L^{p}(\mathbb{R}^{n}),\dot{W}^{k}_{p}(\mathbb{R}^{n}))$
$\displaystyle\approx\sup_{\left|h\right|\leq
u}\|\Delta_{h}^{k}f\|_{L^{p}(\mathbb{R}^{n})},$ (1.11) $\displaystyle
K(u^{k},f;L^{p}(\mathbb{R}^{n}),W^{k}_{p}(\mathbb{R}^{n}))$
$\displaystyle\approx\min\\{1,u^{k}\\}\left\|f\right\|_{L^{p}(\mathbb{R}^{n})}+\sup_{\left|h\right|\leq
u}\|\Delta_{h}^{k}f\|_{L^{p}(\mathbb{R}^{n})}.$

Combining (1.10) and (1.11) with (1.5), we can prove
Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova higher order limit results in a
unified fashion666The proof of property (1.6) related to
$(L^{p}({\mathbb{R}}^{n}),\dot{W}^{k}_{p}({\mathbb{R}}^{n}))$ can be seen,
e.g., in [27, Proposition 3, pag 139]. (cf. [20]).

Very recently, Brazke, Schikorra and Yung [5], using heavy Harmonic Analysis
machinery, obtained a new approach to (1.3) and suggested a possible extension
to fractional Sobolev spaces, through the use of Triebel–Lizorkin spaces. The
paper [5] contains a number of interesting new inequalities and poses a number
of intriguing open problems. It was their paper that provided the initial
impetus for the present work. However, in order to solve some of the main
questions asked in [5] we will rely on interpolation and extrapolation
methods. Interestingly, as we hope to show, interpolation theory will provide
us with the crucial spaces, and the appropriate scalings, to resolve the
outstanding issues.

The first problem proposed in [5, page 5] that we shall discuss here is the
extension of (1.3) to the fractional case. The interpolation method provides
the strategy, although to implement it requires the introduction of new spaces
as we now explain.

Let $t>0,$ and consider the fractional Sobolev spaces via the fractional
Laplacian777$\dot{H}^{t,p}(\mathbb{R}^{n})$ is also known as the space of
Riesz potentials, while $H^{t,p}(\mathbb{R}^{n})$ is the space of Bessel
potentials (cf. [27]).

$\dot{H}^{t,p}(\mathbb{R}^{n})=\\{f:\left\|f\right\|_{\dot{H}^{t,p}(\mathbb{R}^{n})}=\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}(\mathbb{R}^{n})}<\infty\\},$
$H^{t,p}(\mathbb{R}^{n})=\\{f:\left\|f\right\|_{H^{t,p}(\mathbb{R}^{n})}=\left\|f\right\|_{L^{p}(\mathbb{R}^{n})}+\left\|f\right\|_{\dot{H}^{t,p}(\mathbb{R}^{n})}<\infty\\},$

and the corresponding interpolation pairs
$(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n})),$
$(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n})).$ As a consequence
of $L^{p}$-boundedness of the Riesz transforms,

(1.12) $\|(-\Delta)^{\frac{1}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\approx\|\nabla
f\|_{L^{p}({\mathbb{R}}^{n})},\qquad p\in(1,\infty),$

and thus $\dot{H}^{1,p}({\mathbb{R}}^{n})=\dot{W}^{1}_{p}({\mathbb{R}}^{n})$
and $H^{1,p}({\mathbb{R}}^{n})=W^{1}_{p}({\mathbb{R}}^{n})$.

In view of the previous discussion the program we shall follow should be
clear. Indeed, a direct application of (1.5) yields the following extension of
the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limits: Let
$t>0,s\in(0,1),p>1,$ then for $f\in H^{t,p}({\mathbb{R}}^{n})$,

(1.13) $\lim_{s\rightarrow
1^{-}}(1-s)^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{s,p}}\approx\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})},\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{s,p}}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})},$

with a corresponding result for the inhomogeneous pair
$(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))$:

(1.14) $\lim_{s\rightarrow
1^{-}}(1-s)^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{s,p}}\approx\left\|f\right\|_{H^{t,p}(\mathbb{R}^{n})},\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{s,p}}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

As we noted before, the case $t=1$ in (1.13) (resp. (1.14)) corresponds to
(1.3) and (1.4), while integer values can be seen to correspond to the higher
order limit theorems of [20]. However, to obtain meaningful statements for
arbitrary $t>0,$ we are required to find explicit descriptions of the
indicated interpolation spaces, and control the corresponding interpolation
norms within constants independent of the parameter $s.$ The difficulty here
is due to the fact that the nonlocal operator $(-\Delta)^{\frac{t}{2}}$ is
involved. We will overcome these obstructions using a tool from Approximation
Theory: Fractional differences. Fractional differences give an appropriate
extension of (1.9) and lead to a natural extension of (1.10) and (1.11).

We now introduce fractional differences and a class of seminorms that extends
the classical Gagliardo seminorms (1.1) to the fractional case.

###  1.2. Interpolation and fractional differences: The Butzer seminorms

Butzer and his collaborators (cf. [8, 9] and the references therein)
apparently introduced the idea of modifying (1.9) in order to define
fractional differences of any order $t>0$ so that a related fractional
differentiation approach can be developed. For $t>0,$ we let $\Delta_{h}^{t}$
denote the Butzer operator888The concept of fractional differences was already
used by Butzer et al. [8] to introduce fractional order moduli of smoothness,
which has recently become a powerful tool in approximation theory, e.g., in
the study of sharp inequalities between moduli of smoothness
(Jackson–Marchaud–Ulyanov inequalities), cf. [29], [30], [21], [22] and the
references therein. given by

(1.15)
$\Delta_{h}^{t}f(x)=\sum_{j=0}^{\infty}(-1)^{j}{\binom{t}{j}}f(x+(t-j)h),$

where by definition

${\binom{t}{j}}=\frac{t(t-1)\ldots(t-j+1)}{j!},\qquad{\binom{t}{0}}=1.$

Obviously if $t=k\in{{\mathbb{N}}}$, then ${\binom{t}{j}=0}$ for all $j\geq
k+1,$ and (1.15) coincides with the classical differences (1.9). Remarkably,
the $K$-functionals, we are seeking to compute in relation with (1.13)-(1.14),
can be formally obtained999For the details of the computation we refer to
[21], with [8] and [33] as a forerunners. by replacing $\Delta_{h}^{k}$ by
$\Delta_{h}^{t}$ on the right hand side of (1.10) and (1.11), to obtain

$K(u^{t},f;L^{p}(\mathbb{R}^{n}),\dot{H}^{t,p}(\mathbb{R}^{n}))\approx\sup_{\left|h\right|\leq
u}\left\|\Delta_{h}^{t}f\right\|_{L^{p}(\mathbb{R}^{n})},$
$K(u^{t},f;L^{p}(\mathbb{R}^{n}),H^{t,p}(\mathbb{R}^{n}))\approx\min\\{1,u^{t}\\}\left\|f\right\|_{L^{p}(\mathbb{R}^{n})}+\sup_{\left|h\right|\leq
u}\left\|\Delta_{h}^{t}f\right\|_{L^{p}(\mathbb{R}^{n})}.$

Armed with the formulae for the $K$-functionals we can give a complete
description of the interpolation spaces
$(L^{p},\dot{H}^{t,p}({\mathbb{R}}^{n}))_{s,q}$ (resp.
$(L^{p},H^{t,p}({\mathbb{R}}^{n}))_{s,q}$) (cf. Lemma 4.8 below) in terms of
the following (semi)norms.

######  Definition 1.1.

Let101010In this paper we shall be interested in the case $1<p<\infty.$ $1\leq
p\leq\infty,0<s<t,$ then we define the Butzer seminorms, as follows111111Note
also that (1.16) enables us to introduce Sobolev spaces of any order $s>0$
(not necessarily $s<1$ like in (2.1)), via higher order differences. For
instance, one can introduce for $s\in(0,2)$,
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),2}=\bigg{(}\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\frac{|f(x+2h)-2f(x+h)+f(x)|^{p}}{|h|^{sp+N}}\,dx\,dh\bigg{)}^{\frac{1}{p}}.$

(1.16)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}:=\left\\{\begin{array}[c]{cc}\big{(}\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\frac{|\Delta_{h}^{t}f(x)|^{p}}{|h|^{sp+n}}\,dx\,dh\big{)}^{\frac{1}{p}},&\quad
p<\infty\\\ \sup_{(x,h)}\frac{|\Delta_{h}^{t}f(x)|}{|h|^{s}},&\quad
p=\infty.\end{array}\right.$

We let121212The appearence of the prefactor $(s(t-s)p)^{-\frac{1}{p}}$ in
front of $\|f\|_{L^{p}({\mathbb{R}}^{n})}$ is dictated by standard
normalizations in interpolation theory. This will become clear later, cf.
Lemma 4.8.

(1.17)
$\|f\|_{W^{s,p}({\mathbb{R}}^{n}),t}:=(s(t-s)p)^{-\frac{1}{p}}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

Observe that when $t=1$ we recover the classical Gagliardo seminorms (cf.
(1.1))

(1.18)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),1}=\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})},\qquad
s\in(0,1).$

Furthermore, if $s<t<1$ then $\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$
defines an equivalent semi-norm on $\dot{W}^{s,p}({\mathbb{R}}^{n})$, i.e.,

(1.19)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}.$

Note, however, that the constants of equivalence blow up, as $s$ approaches
$t$ (cf. Lemma 4.10 below):

(1.20)
$\frac{1}{(t-s)^{\frac{1}{\max\\{p,2\\}}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\lesssim\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\lesssim\frac{1}{(t-s)^{\frac{1}{\min\\{p,2\\}}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}.$

By direct computation we characterize the interpolation norms (cf. Lemma 4.8
below): Suppose that $1<p<\infty,0<s<1$ and $t>0$ then

$\left\|f\right\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{s,p}}\approx\left\|f\right\|_{\dot{W}^{st,p}({\mathbb{R}}^{n}),t},$

with hidden constants of equivalence independent of $s$ (but depending on
$n,p$ and $t$).

In particular, to derive the fractional
Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae, we simply
replace the interpolation seminorms
$\|\cdot\|_{(L^{p},\dot{H}^{t,p}({\mathbb{R}}^{n}))_{s,p}}$ by
$\|\cdot\|_{\dot{W}^{st,p}({\mathbb{R}}^{n}),t}$ in (1.13) (resp., replace
$\|\cdot\|_{(L^{p},H^{t,p}({\mathbb{R}}^{n}))_{s,p}}$ with
$\|\cdot\|_{\dot{W}^{st,p}({\mathbb{R}}^{n}),t}+(s(1-s))^{-\frac{1}{p}}\|\cdot\|_{L^{p}({\mathbb{R}}^{n})}$
in (1.14)$).$ For example, the general fractional homogeneous version reads
now

(1.21) $\lim_{s\rightarrow
t^{-}}(t-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}(\mathbb{R}^{n})},\qquad\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

This answers completely the question raised in [5, page 5] asking for a
possible fractional extension of (1.3) (see also (1.12)). Note that (1.21) and
(1.20) show that we cannot use the classical Gagliardo seminorms for this
purpose.

Further evidence of the important rôle of the Butzer seminorms in our project
is provided by the following result, which answers another question raised in
[5].

###  1.3. The fractional Bourgain–Brezis–Mironescu convergence theorem via
Butzer seminorms

We consider the extension to fractional Sobolev spaces of the
Bourgain–Brezis–Mironescu convergence theorem [3],

######  Theorem 1.2.

Let $p\in(1,\infty)$, assume that $f_{k}\in\mathcal{S}({\mathbb{R}}^{n})$,
$k\in\mathbb{N}$, and

$f_{k}\rightharpoonup f\quad\text{weakly in}\quad
L^{p}({\mathbb{R}}^{n})\quad\text{as}\quad k\rightarrow\infty.$

Let $\\{s_{k}\\}_{k\in{{\mathbb{N}}}}\subset(0,1)$ be such that $s_{k}\uparrow
1$ and, moreover, assume that

$\Lambda:=\sup_{k\in{{\mathbb{N}}}}\Big{(}\|f_{k}\|_{L^{p}({\mathbb{R}}^{n})}+(1-s_{k})^{\frac{1}{p}}\|f_{k}\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n})}\Big{)}<\infty.$

Then $f\in W_{p}^{1}({\mathbb{R}}^{n})$ and there exists $C=C(p,n)>0$ such
that

$\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|\nabla f\|_{L^{p}({\mathbb{R}}^{n})}\leq
C\Lambda.$

In fact, $f_{k}\rightarrow f$ strongly in $L_{loc}^{p}({\mathbb{R}}^{n}).$

In [5] the authors propose a plan of attack for a possible fractional
extension of Theorem 1.2. The idea is based on the use of the Triebel-Lizorkin
spaces $\dot{F}^{s}_{p,q}({\mathbb{R}}^{N})$ (we refer to Section 2 for the
definitions). Indeed, based on this idea, a new proof of Theorem 1.2 was
presented in [5], using the following sharp Sobolev-type inequality:

######  Theorem 1.3.

Let $s\in(0,1),p\in(1,\infty)$ and $\Lambda>1$. Assume
$1-s\leq\frac{1}{2\Lambda}$ and let $1-\bar{r}=\Lambda(1-s)$. Then there
exists $C$, which depends only on $n,p$ and $\Lambda$, such that

$\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+(1-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\Big{)},\qquad
r\in[0,\bar{r}].$

With this result at their disposal the proof of Theorem 1.2 given in [5] now
proceeds by combining Theorem 1.3 (taking limits as $r\rightarrow 1^{-})$,
with (1.12) and the Littewood–Paley estimate

$\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}\approx\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})},\qquad
1<p<\infty,$

and then applying the Rellich–Kondrachov theorem.

In view of this, the following conjecture was formulated in [5].

######  Open Problem 1.4 ([5, Question 1.11]).

Let $p\in(1,\infty)$, assume that

$f_{k}\rightharpoonup f\quad\text{weakly in}\quad
L^{p}({\mathbb{R}}^{n})\quad\text{as}\quad k\rightarrow\infty.$

Let $t\in(0,1)$ and $(s_{k})_{k\in{{\mathbb{N}}}}\subset(0,t)$ such that
$s_{k}\uparrow t$ and assume that

$\Lambda:=\sup_{k\in{{\mathbb{N}}}}\Big{(}\|f_{k}\|_{L^{p}({\mathbb{R}}^{n})}+(t-s_{k})^{\frac{1}{p}}\|f_{k}\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n})}\Big{)}<\infty.$

Then $f\in H^{t,p}({\mathbb{R}}^{n})$ and there exists $C=C(p,n,t)>0$ such
that

$\lim_{\bar{t}\uparrow
t}\limsup_{k\rightarrow\infty}\|(-\Delta)^{\frac{\bar{t}}{2}}f_{k}\|_{L^{p}({\mathbb{R}}^{n})}\leq
C\Lambda.$

Note that in the case $t=1,$ the conjectured result is exactly Theorem 1.2.
However, as we have indicated above, the standard Gagliardo seminorms are “too
far" from the “exact” interpolation seminorms
$\|\cdot\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n}),t}\,$ (cf. (1.16)) for
$0<t<1$. We use this insight to give a counterexample to Open Problem 1.4 (cf.
Section 3, Proposition 3.1 for the precise statement, and Section 5 for a
proof).

The problem thus remains: What spaces should we use to formulate and prove a
fractional extension of Theorem 1.2? Once again interpolation, via the Butzer
seminorms, comes to our rescue and we are essentially able to prove that the
conjectured result in Open Problem 1.4 is true if in its statement we replace
“Gagliardo seminorms” by “Butzer seminorms” (cf. Section 3, Theorem 3.2 for
the precise statement, and Section 5 for a complete proof).

###  1.4. Sharp Sobolev inequalities

Another problem proposed in [5] deals with Sobolev type inequalities. Let
$0<s<t<1$, an elementary version of Sobolev’s inequality asserts that

(1.22)
$\|f\|_{\dot{W}^{s,2}({\mathbb{R}}^{n})}\leq\gamma_{s,t}\Big{(}\|f\|_{L^{2}({\mathbb{R}}^{n})}+\|f\|_{\dot{W}^{t,2}({\mathbb{R}}^{n})}\Big{)}.$

Here $\gamma_{s,t}$ is a positive constant which depends, in particular, on
the smoothness parameters $s$ and $t$. Furthermore, at least formally, (1.22)
is still valid in the limiting values $s=0$ and $t=1$. According to (1.2),
(1.3) and (1.4), the validity of the previous assertion should be reflected in
the behavior of the equivalence constant $\gamma_{s,t}$ that appears in
(1.22), in terms of the blows up $\min\\{t,1-t\\}^{\frac{1}{2}}$ and
$\min\\{s,1-s\\}^{\frac{1}{2}}$. Indeed, the following result was obtained in
[5, Corollary 1.3].

######  Theorem 1.5.

Let $0<s<t<1$ and $f\in\mathcal{S}({\mathbb{R}}^{n})$. Then there exists
$C=C(n)>0$, such that

$\min\\{s,1-s\\}^{\frac{1}{2}}\|f\|_{\dot{W}^{s,2}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{2}({\mathbb{R}}^{n})}+\min\\{t,1-t\\}^{\frac{1}{2}}\|f\|_{\dot{W}^{t,2}({\mathbb{R}}^{n})}\Big{)}.$

It is well known that the $L^{p}$-counterpart of (1.22) is also true, i.e.,

$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C_{s,t}\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|f\|_{\dot{W}^{t,p}({\mathbb{R}}^{n})}\Big{)}$

provided that $0<s<t<1$ and $1<p<\infty$. Accordingly, a similar result, in
the spirit of Theorem 1.5, was expected. However, the set of harmonic analysis
techniques developed in [5] do not work outside the Hilbert setting given by
$p=2$. This leads to the following question, explicitly raised in [5]:

######  Open Problem 1.6.

Let $p\in(1,\infty),0<\theta<s<t<1$ and $f\in\mathcal{S}({\mathbb{R}}^{n})$.
Does there exist $C=C(n,p,\theta)>0$, such that

(1.23)
$\min\\{s,1-s\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\min\\{t,1-t\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{t,p}({\mathbb{R}}^{n})}\Big{)}?$

We provide a positive answer to this problem. Indeed, in Section 3, Theorem
3.9, we state a somewhat stronger result which is then proved in Section 5.

We now turn to explain the organization of the paper. The brief Section 2
contains the basic background and notation concerning the function spaces we
shall consider; Section 3 contains the statements of the main results, which
we then prove in Section 5. The interpolating Section 4 contains detailed
information with complete proofs concerning the interpolation and
extrapolation tools that constitute the basic methods we use in this paper.
Finally, in the Appendix, we collect further results and applications. In
particular, we prove an extension of the classical fractional Sobolev
inequalities of Bourgain–Brezis–Mironescu [4] and Maz’ya–Shaposhnikova [24],
that were one of the original motivations for the limiting theorems (1.3) and
(1.4).

Acknowledgments. We are grateful to Petru Mironescu for his constant interest
and encouragement during the preparation of this paper and to Sergey Tikhonov
for precious information concerning fractional differences and moduli of
smoothness.

The first named author has been partially supported by the LABEX MILYON
(ANR-10-LABX-0070) of Université de Lyon, within the program “Investissement
d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency
(ANR), and by MTM2017-84058-P (AEI/FEDER, UE) .

##  2\. Background on function spaces

Let $s\in(0,1)$ and $p\in(1,\infty)$, the _(fractional) Sobolev space_
$\dot{W}^{s,p}({\mathbb{R}}^{n})$ is equipped with the _Gagliardo seminorm_

(2.1)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}:=\bigg{(}\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{sp+n}}\,dx\,dy\bigg{)}^{1/p}.$

The _Riesz potential space_ $\dot{H}^{s,p}({\mathbb{R}}^{n})$ is endowed with

$\|f\|_{\dot{H}^{s,p}({\mathbb{R}}^{n})}:=\|(-\Delta)^{\frac{s}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}.$

The spaces $\dot{H}^{s,p}({\mathbb{R}}^{n})$ make sense for any
$s\in{\mathbb{R}}$. In particular,
$\dot{H}^{k,p}({\mathbb{R}}^{n}),\,k\in{{\mathbb{N}}},$ coincides with the
_classical Sobolev space_ $\dot{W}_{p}^{k}({\mathbb{R}}^{n})$ and

(2.2)
$\|f\|_{\dot{H}^{k,p}({\mathbb{R}}^{n})}\approx\|f\|_{\dot{W}_{p}^{k}({\mathbb{R}}^{n})}:=\|\nabla^{k}f\|_{L^{p}({\mathbb{R}}^{n})},\qquad
p\in(1,\infty).$

We let
$H^{s,p}({\mathbb{R}}^{n})=L^{p}({\mathbb{R}}^{n})\cap\dot{H}^{s,p}({\mathbb{R}}^{n})$
(the _space of Bessel potentials_) and
$\|f\|_{H^{s,p}({\mathbb{R}}^{n})}=\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|(-\Delta)^{\frac{s}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}$.

The _(homogeneous) Triebel–Lizorkin space_
$\dot{F}_{p,q}^{s}({\mathbb{R}}^{n}),\,s\in{\mathbb{R}},\,p\in(0,\infty),\,q\in(0,\infty]$,
is formed by all $f\in\dot{\mathcal{S}}^{\prime}({\mathbb{R}}^{n})$ such that

$\|f\|_{\dot{F}_{p,q}^{s}({\mathbb{R}}^{n})}:=\bigg{\|}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsq}|\Delta_{j}f|^{q}\bigg{)}^{1/q}\bigg{\|}_{L^{p}({\mathbb{R}}^{n})}<\infty,$

with the obvious modification if $q=\infty$. Here,
$\\{\Delta_{j}f:j\in{{\mathbb{Z}}}\\}$ is the standard dyadic Littlewood–Paley
decomposition of $f$. The inhomogeneous counterparts,
$F_{p,q}^{s}({\mathbb{R}}^{n}),$ can be introduced similarly.

Interchanging the roles of $L^{p}({\mathbb{R}}^{n})$ and
$\ell_{q}({{\mathbb{Z}}})$ in the definition of
$\dot{F}_{p,q}^{s}({\mathbb{R}}^{n})$ we arrive at the Besov spaces.
Specifically, the _(homogeneous) Besov space_
$\dot{B}_{p,q}^{s}({\mathbb{R}}^{n}),\,s\in{\mathbb{R}},p,q\in(0,\infty]$, is
formed by all $f\in\dot{\mathcal{S}}^{\prime}({\mathbb{R}}^{n})$ such that

$\|f\|_{\dot{B}_{p,q}^{s}({\mathbb{R}}^{n})}:=\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsq}\|\Delta_{j}f\|_{L^{p}({\mathbb{R}}^{n})}^{q}\bigg{)}^{1/q}<\infty,$

with the obvious modifications if $q=\infty$.

We shall use the following standard notations. Given a quasi-Banach space $X$
and $\lambda>0$, we will denote by $\lambda X$ the space endowed with

$\|f\|_{\lambda X}=\lambda\|f\|_{X},\qquad f\in X.$

Let $Y$ be a quasi-Banach space and let $\beta>0$. We will use the notation
$\lambda X\hookrightarrow\beta Y$ to indicate that there exists a positive
constant $C$, independent of $\lambda$ and $\beta$, such that

$\beta\|f\|_{Y}\leq C\lambda\|f\|_{X},\qquad f\in X.$

In particular, if $\lambda=\beta=1$ then we simply write $X\hookrightarrow Y$.
By $X=Y$ we mean that $X\hookrightarrow Y$ and $Y\hookrightarrow X$.

We collect some well-known relations between the function spaces described
above (cf. [2], [27] and [32]).

######  Lemma 2.1.

  1. (i)

Let $s\in{\mathbb{R}},\,p\in(0,\infty)$ and $q\in(0,\infty]$. Then

$\dot{B}_{p,\min\\{p,q\\}}^{s}({\mathbb{R}}^{n})\hookrightarrow\dot{F}_{p,q}^{s}({\mathbb{R}}^{n})\hookrightarrow\dot{B}_{p,\max\\{p,q\\}}^{s}({\mathbb{R}}^{n}).$

In particular,

$\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})=\dot{B}_{p,p}^{s}({\mathbb{R}}^{n}).$

  2. (ii)

Let $s\in(0,1)$ and $p\in(1,\infty)$. Then

$\dot{W}^{s,p}({\mathbb{R}}^{n})=\dot{B}_{p,p}^{s}({\mathbb{R}}^{n}).$

  3. (iii)

Let $s\in{\mathbb{R}}$ and $p\in(1,\infty)$. Then

$\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})=\dot{H}^{s,p}({\mathbb{R}}^{n}).$

##  3\. Main results

In this section we provide a list of statements of the main results we have
obtained. The proofs will be given in Section 5.

We start outlining some details of a counterexample that provides a negative
answer to Open Problem 1.4. To avoid technicalities, we switch temporarily
from ${\mathbb{R}}^{n}$ to the unit circle ${\mathbb{T}}$131313Spaces of
periodic functions are defined similarly as their analogs on
${\mathbb{R}}^{n}$, simply replacing $L^{p}({\mathbb{R}}^{n})$ by
$L^{p}({\mathbb{T}})$.. However, our construction can be easily modified to
deal with ${\mathbb{R}}^{n}$ and $p>\frac{2n}{n+1}$; in this connection see
Remark 5.1 below.

######  Proposition 3.1.

Let $p\in(1,\infty)$ and $t\in(0,1)$. Let $f$ be formally associated to a
Fourier series as follows,

(3.1) $f(x)\sim\sum_{\nu=1}^{\infty}\nu^{-t-1+\frac{1}{p}}\cos(\nu x),\qquad
x\in{\mathbb{T}}.$

Then, $f\in L^{p}({\mathbb{T}})$ but $f\not\in H^{t,p}({\mathbb{T}})$.
Furthermore, given any sequence $(s_{k})_{k\in{{\mathbb{N}}}}\subset(0,t)$
with $s_{k}\uparrow t$,

$\sup_{k\in{{\mathbb{N}}}}\,(t-s_{k})^{\frac{1}{p}}\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}})}<\infty.$

In fact, we are able to go far beyond, and we show that a correct formulation
of Open Problem 1.4 is obtained by means of replacing the classical seminorms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}$ by the Butzer seminorms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ (cf. (1.16) and (1.19)).

######  Theorem 3.2.

Let $p\in(1,\infty)$, assume that

$f_{k}\rightharpoonup f\quad\text{weakly in}\quad
L^{p}({\mathbb{R}}^{n})\quad\text{as}\quad k\rightarrow\infty.$

Let $t\in(0,1)$ and $(s_{k})_{k\in{{\mathbb{N}}}}\subset(0,t)$ be such that
$s_{k}\uparrow t.$ Assume that

$\Lambda:=\sup_{k\in{{\mathbb{N}}}}\Big{(}\|f_{k}\|_{L^{p}({\mathbb{R}}^{n})}+(t-s_{k})^{\frac{1}{p}}\|f_{k}\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n}),t}\Big{)}<\infty.$

Then $f\in H^{t,p}({\mathbb{R}}^{n})$ and there exists $C=C(n,p,t)>0$ such
that

$\lim_{\bar{t}\uparrow
t}\limsup_{k\rightarrow\infty}\|(-\Delta)^{\frac{\bar{t}}{2}}f_{k}\|_{L^{p}({\mathbb{R}}^{n})}\leq
C\Lambda.$

To prove Theorem 3.2 we need an extension of Theorem 1.3 formulated in terms
of the Butzer seminorms $\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$. It
turns out that our method is flexible enough to incorporate both, the case
$s\rightarrow 0+$ , as well as to deal with inhomogeneous Triebel–Lizorkin
spaces.

######  Theorem 3.3.

Let $0<s<t\leq 1,\,p\in(1,\infty)$ and $\Lambda>1$.

  1. (1)

Assume $t-s\leq\frac{t}{2\Lambda}$ and let $t-\bar{r}=\Lambda(t-s)$. Then,
there exists $C$, which depends only on $n,p,t$ and $\Lambda$, such that

(3.2) $\|f\|_{F_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+(t-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\Big{)}.$

There is a corresponding result for homogeneous Triebel–Lizorkin spaces,

(3.3) $\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+(t-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\Big{)},\qquad
r\in[0,\bar{r}].$

  2. (2)

Assume $s<\frac{1}{2\Lambda}$ and let $\bar{r}=\frac{1}{\Lambda}s$. Then there
exists $C$, which depends only on $n,p,t$ and $\Lambda$, such that

(3.4) $\|f\|_{F_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+s^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\Big{)}.$

Likewise, for homogeneous Triebel–Lizorkin spaces we have

(3.5) $\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+s^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\Big{)},\qquad
r\in[0,\bar{r}].$

We apply our method to provide an extension of (1.3)-(1.4) in terms of the
fractional Laplacian and Butzer seminorms.

######  Theorem 3.4.

Let $t\in(0,1]$ and $p\in(1,\infty)$. Assume $f\in H^{t,p}({\mathbb{R}}^{n})$.
Then

(3.6) $\lim_{s\rightarrow
t^{-}}(t-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}$

and

(3.7) $\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

The corresponding results for inhomogeneous spaces also hold true:

(3.8) $\lim_{s\rightarrow
t^{-}}(t-s)^{\frac{1}{p}}\|f\|_{W^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{H^{t,p}({\mathbb{R}}^{n})}$

and

(3.9) $\lim_{s\rightarrow
0^{+}}s^{\frac{1}{p}}\|f\|_{W^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

######  Remark 3.5.

Observe that (3.6) and (3.7) with $t=1$ give back, up to equivalence
constants, the classical formulae (1.3) and (1.4).

As indicated by Theorems 3.2 and 3.4, the classical seminorms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}$ are not the optimal choices for
dealing with Sobolev-type inequalities involving $H^{t,p}({\mathbb{R}}^{n})$.
This phenomenon is illustrated in the following result.

######  Theorem 3.6.

Let $1<p<\infty$ and $0<r<t<1$. Then there exists $C=C(n,p,r,t)>0,$ such that

(3.10) $\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C((s-r)(t-s))^{\frac{1}{\max\\{p,2\\}}}\bigg{(}\frac{1}{(s-r)^{\frac{1}{p}}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(t-s)^{\frac{1}{p}}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}\bigg{)},$

for every $s\in(r,t)$. In the limiting cases, $r=0$ and $t=1$, we have

(3.11) $\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C(t-s)^{\frac{1}{\max\\{p,2\\}}}\bigg{(}\frac{1}{s^{\frac{1}{p}}}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\frac{1}{(t-s)^{\frac{1}{p}}}\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\bigg{)}$

for $0<t<1$ and

(3.12) $\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C(s-r)^{\frac{1}{\max\\{p,2\\}}}\bigg{(}\frac{1}{(s-r)^{\frac{1}{p}}}\|(-\Delta)^{\frac{r}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}+\frac{1}{(1-s)^{\frac{1}{p}}}\|\nabla
f\|_{L^{p}({\mathbb{R}}^{n})}\bigg{)}$

for $0<r<1$, respectively.

As an immediate consequence of (3.11) and (3.12), we obtain

######  Corollary 3.7.

Let $0<r<t<1$. Then

(3.13)
$\sup_{s\in[r,t)}(t-s)^{\frac{1}{p}-\frac{1}{\max\\{p,2\\}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\Big{)}$

and

(3.14)
$\sup_{s\in(r,t]}(s-r)^{\frac{1}{p}-\frac{1}{\max\\{p,2\\}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|(-\Delta)^{\frac{r}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}+\|\nabla
f\|_{L^{p}({\mathbb{R}}^{n})}\Big{)}.$

######  Remark 3.8.

Theorem 3.6 is a considerable improvement of Theorem 1.6 in [5]: Let
$1<p<\infty$ and $0\leq r<t\leq 1$. Then there exists $C=C(n,p)>0$ such that

(3.15) $\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\,\bigg{(}\frac{1}{(s-r)^{\frac{1}{p}}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(t-s)^{\frac{1}{p}}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}\bigg{)},\qquad
s\in(r,t).$

The estimates provided in Theorem 3.6 sharpens (3.15) due to the appearance of
one of the additional prefactors
$((s-r)(t-s))^{\frac{1}{\max\\{p,2\\}}},(t-s)^{\frac{1}{\max\\{p,2\\}}}$ and
$(s-r)^{\frac{1}{\max\\{p,2\\}}}$. Note that the constant $C$ in (3.15) does
not depend on $r$ and $t$, but this is not the case in (3.10). The reason
behind this additional flexibility in (3.15) is explained by its non-
optimality.

On the other hand, (3.13) sharpens the following inequality obtained in
Corollary 1.7 in [5]: Let $1<p<\infty$ and $0<r<t\leq 1$. Then there exists
$C=C(n,p,r)>0$ such that

(3.16)
$\sup_{s\in[r,t)}(t-s)^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\Big{)}.$

In particular, this inequality with $t=1$ gives one of the estimates in the
classical Bourgain–Brezis–Mironescu formula (1.3). However, (3.16) is not
optimal if $t\in(0,1)$. In fact, as was already observed in [5, Remark 1.8],
the inequality (3.16) is only useful with $p<2$. Specifically, if $p\geq 2$
and $0<t<1$, it is well known that

(3.17) $\sup_{s\in[r,t)}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\Big{)}.$

This obstruction can be overcome via the inequality (3.13). Specifically, if
$p\in(1,2)$ then the prefactor $(t-s)^{\frac{1}{p}}$ on the left-hand side of
(3.16) is now improved by $(t-s)^{\frac{1}{p}-\frac{1}{2}}$ in (3.13). On the
other hand, if $p\in[2,\infty),$ then (3.13) coincides with the optimal
estimate (3.17). On the other hand, inequality (3.14), with $p<2,$ seems to be
new.

We give a positive answer to Open Problem 1.6. In fact, we obtain a stronger
version by means of removing the $\theta$-dependance from the constant $C,$
and working with inhomogeneous norms on both sides of (1.23). Furthermore, our
method works with the more general family of seminorms formed by
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),\alpha}$ (cf. (1.16)) for
arbitrary order of smoothness $\alpha>0$.

######  Theorem 3.9.

Let $p\in(1,\infty),\,\alpha>0,$ and $0<s<t<\alpha$. Then there exists a
positive constant $C$, independent of $s$ and $t$, such that

$\|f\|_{L^{p}({\mathbb{R}}^{n})}+\min\\{s,\alpha-s\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),\alpha}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\min\\{t,\alpha-t\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{t,p}({\mathbb{R}}^{n}),\alpha}\Big{)}.$

In particular, if $\alpha=1$ then (cf. (1.18))

$\|f\|_{L^{p}({\mathbb{R}}^{n})}+\min\\{s,1-s\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\min\\{t,1-t\\}^{\frac{1}{p}}\|f\|_{\dot{W}^{t,p}({\mathbb{R}}^{n})}\Big{)}$

for all $0<s<t<1$.

##  4\. Interpolation/extrapolation: A primer

The goal of this section is to present our interpolation/extrapolation based
methodology. These methods will be applied in Section 5 to prove all the
results that were stated in Section 3.

###  4.1. Revisiting some classical interpolation formulae

All the results contained in this subsection are well known. However, for the
purposes of this paper, it is important to establish these results with sharp
control on the dependance with respect to the parameters involved. For the
convenience of the reader, who may not be familiar with interpolation theory,
we give a self-contained exposition, since it is hard to find the material
organized in a manner that fits our needs in this paper.

Let $(A_{0},A_{1})$ be a quasi-semi-normed pair. The _real interpolation
space_ $(A_{0},A_{1})_{\theta,q},\,\theta\in(0,1),\,q\in(0,\infty]$, is formed
by all the elements $f\in A_{0}+A_{1},$ such that

(4.1)
$\|f\|_{(A_{0},A_{1})_{\theta,q}}:=\bigg{(}\int_{0}^{\infty}\left(t^{-\theta}K(t,f;A_{0},A_{1})\right)^{q}\frac{dt}{t}\bigg{)}^{1/q}<\infty,$

where $K(t,f;A_{0},A_{1})$ denotes the _$K$ -functional_ for the pair
$(A_{0},A_{1}),$ defined for $t>0,$ by

$K(t,f;A_{0},A_{1}):=\inf_{f=f_{0}+f_{1}}\Big{(}\|f_{0}\|_{A_{0}}+t\|f_{1}\|_{A_{1}}\Big{)}.$

We shall simply write $K(t,f)$ when the underlying pair of spaces
$(A_{0},A_{1})$ is understood from the context. We refer to the standard
references on interpolation theory [1, 2, 31]. Sometimes it is more convenient
to work with the modified $K$-functional given by

$K_{p}(t,f;A_{0},A_{1}):=K_{p}(t,f)=\inf_{f=f_{0}+f_{1}}\Big{(}\|f_{0}\|_{A_{0}}^{p}+t^{p}\|f_{1}\|_{A_{1}}^{p}\Big{)}^{\frac{1}{p}}$

for $p\in(0,\infty)$. Clearly $K(t,f)\approx K_{p}(t,f)$ with equivalence
constants depending only on $p$. Furthermore, it is plain that

(4.2) $(A_{0},A_{1})_{\theta,q}=(A_{1},A_{0})_{1-\theta,q}$

with equality of norms.

Let $A$ be a Banach space and $p\in(0,\infty)$. The vector-valued Lebesgue
spaces $L^{p}({\mathbb{R}}^{n};A)$ are formed by all strongly measurable
functions $f:{\mathbb{R}}^{n}\rightarrow A$ such that

$\|f\|_{L^{p}({\mathbb{R}}^{n};A)}:=\bigg{(}\int_{{\mathbb{R}}^{n}}\|f(x)\|_{A}^{p}\,dx\bigg{)}^{1/p}<\infty.$

The next result is well known.

######  Lemma 4.1.

Let $(A_{0},A_{1})$ be a pair of Banach spaces and let
$s\in(0,1),p\in(0,\infty)$. Then

$(L^{p}({\mathbb{R}}^{n};A_{0}),L^{p}({\mathbb{R}}^{n};A_{1}))_{s,p}=L^{p}({\mathbb{R}}^{n};(A_{0},A_{1})_{s,p})$

and

$\|f\|_{(L^{p}({\mathbb{R}}^{n};A_{0}),L^{p}({\mathbb{R}}^{n};A_{1}))_{s,p}}\approx\|f\|_{L^{p}({\mathbb{R}}^{n};(A_{0},A_{1})_{s,p})}$

where the hidden equivalence constants depend only on $p$.

###### Proof.

It is plain that

$K(t,f;L^{p}({\mathbb{R}}^{n};A_{0}),L^{p}({\mathbb{R}}^{n};A_{1}))^{p}\approx\int_{{\mathbb{R}}^{n}}K_{p}(t,f(x);A_{0},A_{1})^{p}\,dx.$

Therefore using Fubini’s theorem,

$\displaystyle\|f\|_{(L^{p}({\mathbb{R}}^{n};A_{0}),L^{p}({\mathbb{R}}^{n};A_{1}))_{s,p}}^{p}$
$\displaystyle\approx\int_{0}^{\infty}t^{-sp}\int_{{\mathbb{R}}^{n}}K(t,f(x);A_{0},A_{1})^{p}\,dx\,\frac{dt}{t}$
$\displaystyle\approx\int_{{\mathbb{R}}^{n}}\|f(x)\|_{(A_{0},A_{1})_{s,p}}^{p}\,dx.$

∎

For $s\in{\mathbb{R}}$ and $p\in(0,\infty)$, the space
$\ell_{p}^{s}({{\mathbb{Z}}})$ is formed by all the scalar-valued sequences
$\xi=(\xi_{j})_{j\in{{\mathbb{Z}}}},$ such that

$\|\xi\|_{\ell_{p}^{s}({{\mathbb{Z}}})}:=\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}<\infty.$

Similarly, we can define the spaces $\ell_{p}^{s}({{\mathbb{N}}}_{0})$ where
${{\mathbb{N}}}_{0}={{\mathbb{N}}}\cup\\{0\\}$.

######  Lemma 4.2.

Let $-\infty<s_{0}<s_{1}<\infty,\,\theta\in(0,1),s=(1-\theta)s_{0}+\theta
s_{1},$ and let $p,q\in(0,\infty)$. Then

$(\ell_{q}^{s_{0}}({{\mathbb{Z}}}),\ell_{q}^{s_{1}}({{\mathbb{Z}}}))_{\theta,p}=\ell_{p}^{s}({{\mathbb{Z}}}),$

with

$\bigg{(}\frac{1}{\theta^{\frac{1}{\max\\{p,q\\}}}}+\frac{1}{(1-\theta)^{\frac{1}{\max\\{p,q\\}}}}\bigg{)}\|\xi\|_{\ell_{p}^{s}({{\mathbb{Z}}})}\lesssim\|\xi\|_{(\ell_{q}^{s_{0}}({{\mathbb{Z}}}),\ell_{q}^{s_{1}}({{\mathbb{Z}}}))_{\theta,p}}\lesssim\bigg{(}\frac{1}{\theta^{\frac{1}{\min\\{p,q\\}}}}+\frac{1}{(1-\theta)^{\frac{1}{\min\\{p,q\\}}}}\bigg{)}\|\xi\|_{\ell_{p}^{s}({{\mathbb{Z}}})}$

uniformly w.r.t. $\theta$. The corresponding result also holds true for
${{\mathbb{N}}}_{0}$-indexed sequences.

###### Proof.

It is well known and easy to see that

$K_{q}(t,\xi;\ell_{q}^{s_{0}}({{\mathbb{Z}}}),\ell_{q}^{s_{1}}({{\mathbb{Z}}}))=\bigg{(}\sum_{j=-\infty}^{\infty}[\min\\{2^{js_{0}},2^{js_{1}}t\\}|\xi_{j}|]^{q}\bigg{)}^{1/q}.$

By the monotonicity properties of the $K$-functional, the following estimates
hold uniformly w.r.t. $\theta\in(0,1)$,

$\displaystyle\|\xi\|_{(\ell_{q}^{s_{0}}({{\mathbb{Z}}}),\ell_{q}^{s_{1}}({{\mathbb{Z}}}))_{\theta,p}}$
$\displaystyle\approx\bigg{(}\sum_{l=-\infty}^{\infty}2^{l\theta(s_{1}-s_{0})p}K(2^{-l(s_{1}-s_{0})},\xi;\ell_{q}^{s_{0}}({{\mathbb{Z}}}),\ell_{q}^{s_{1}}({{\mathbb{Z}}}))^{p}\bigg{)}^{1/p}$
$\displaystyle\approx\bigg{(}\sum_{l=-\infty}^{\infty}2^{l\theta(s_{1}-s_{0})p}\bigg{(}\sum_{j=-\infty}^{\infty}[2^{js_{0}}\min\\{1,2^{(j-l)(s_{1}-s_{0})}\\}|\xi_{j}|]^{q}\bigg{)}^{p/q}\bigg{)}^{1/p}$
(4.3) $\displaystyle\approx A+B,$

where

$A:=\bigg{(}\sum_{l=-\infty}^{\infty}2^{l(\theta-1)(s_{1}-s_{0})p}\bigg{(}\sum_{j=-\infty}^{l}2^{js_{1}q}|\xi_{j}|^{q}\bigg{)}^{p/q}\bigg{)}^{1/p}$

and

$B:=\bigg{(}\sum_{l=-\infty}^{\infty}2^{l\theta(s_{1}-s_{0})p}\bigg{(}\sum_{j=l}^{\infty}2^{js_{0}q}|\xi_{j}|^{q}\bigg{)}^{p/q}\bigg{)}^{1/p}.$

We consider now two possible cases. Assume first $p\geq q.$ Therefore

$\displaystyle A$
$\displaystyle\geq\bigg{(}\sum_{l=-\infty}^{\infty}2^{l(\theta-1)(s_{1}-s_{0})p}\sum_{j=-\infty}^{l}2^{js_{1}p}|\xi_{j}|^{p}\bigg{)}^{1/p}$
$\displaystyle=\bigg{(}\sum_{j=-\infty}^{\infty}2^{js_{1}p}|\xi_{j}|^{p}\sum_{l=j}^{\infty}2^{l(\theta-1)(s_{1}-s_{0})p}\bigg{)}^{1/p}$
$\displaystyle\approx\frac{1}{(1-\theta)^{1/p}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}.$

By the sharp version of Hardy’s inequality (see, e.g., [28, p. 196]),

$\displaystyle A$
$\displaystyle=\bigg{(}\sum_{l=-\infty}^{\infty}\bigg{(}2^{l(\theta-1)(s_{1}-s_{0})q}\sum_{j=-\infty}^{l}2^{js_{1}q}|\xi_{j}|^{q}\bigg{)}^{p/q}\bigg{)}^{1/p}$
$\displaystyle\lesssim\frac{1}{(1-\theta)^{1/q}}\bigg{(}\sum_{l=-\infty}^{\infty}2^{lsp}|\xi_{l}|^{p}\bigg{)}^{1/p}.$

Hence the term $A$ can be estimated by

(4.4)
$\frac{1}{(1-\theta)^{1/p}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}\lesssim
A\lesssim\frac{1}{(1-\theta)^{1/q}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}.$

Furthermore, similar estimates also hold for $B$, namely,

(4.5)
$\frac{1}{\theta^{1/p}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}\lesssim
B\lesssim\frac{1}{\theta^{1/q}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}.$

Combining (4.3), (4.4) and (4.5), we arrive at the desired result under $p\geq
q$.

Suppose now $p<q$. Then

$\displaystyle A$
$\displaystyle\leq\bigg{(}\sum_{l=-\infty}^{\infty}2^{l(\theta-1)(s_{1}-s_{0})p}\sum_{j=-\infty}^{l}2^{js_{1}p}|\xi_{j}|^{p}\bigg{)}^{1/p}$
$\displaystyle\approx\frac{1}{(1-\theta)^{1/p}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}.$

On the other hand, by a sharp version of Copson’s inequality (see, e.g., [23])
we find that

$A\gtrsim\frac{1}{(1-\theta)^{1/q}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}.$

Hence

$\frac{1}{(1-\theta)^{1/q}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p}\lesssim
A\lesssim\frac{1}{(1-\theta)^{1/p}}\bigg{(}\sum_{j=-\infty}^{\infty}2^{jsp}|\xi_{j}|^{p}\bigg{)}^{1/p},$

with hidden constants of equivalence uniform w.r.t. $\theta\in(0,1)$. The term
$B$ can be estimated similarly. Combining estimates yields the desired result
for $p<q$. ∎

We close this subsection by recalling the well-known characterization of
$\dot{B}_{p,q}^{s}({\mathbb{R}}^{n})$ as an interpolation space between
$L^{p}({\mathbb{R}}^{n})$ and $\dot{W}^{k}_{p}({\mathbb{R}}^{n})$; see, e.g.,
[1, Chapter 5, Corollary 4.13, page 341] and [2, Theorem 6.3.1, page 147].

######  Lemma 4.3.

Let $s\in(0,1),\,k\in{{\mathbb{N}}}$ and $p\in(1,\infty)$. Then141414The
limiting values $p=1,\infty$ are also adimissible.

$(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{k}({\mathbb{R}}^{n}))_{s,q}=\dot{B}_{p,q}^{sk}({\mathbb{R}}^{n}).$

###  4.2. Extrapolation theory

Generally speaking, the extrapolation theory of Jawerth and Milman [16]
develops methods to recover the endpoint spaces of a given parametrized scale
of quasi-Banach spaces $\\{A_{\theta}:\theta\in(0,1)\\}.$ In particular, it
contains a careful analysis of the asymptotic behavior of the related quasi-
norms $\|\cdot\|_{A_{\theta}}$ as $\theta\rightarrow 0^{+}$ or
$\theta\rightarrow 1^{-}$ . In [16] it is shown that
$(\theta(1-\theta)p)^{\frac{1}{p}}$ is the appropriate normalization factor
for the real interpolation method $(A_{0},A_{1})_{\theta,p}$ (cf. (4.1)). This
assertion is well-illustrated by the following results.

######  Lemma 4.4.

Let $p\in(0,\infty)$. Then there exists $C>0$, which depends only on $p$, such
that, for every $\theta\in(0,1)$ and $f\in A_{0}\cap A_{1}$,

$\|f\|_{(A_{0},A_{1})_{\theta,p}}\leq
C\Big{(}\frac{1}{\theta^{1/p}}\|f\|_{A_{0}}+\frac{1}{(1-\theta)^{1/p}}\|f\|_{A_{1}}\Big{)}.$

###### Proof.

Since $K(t,f)\leq\min\\{\|f\|_{A_{0}},t\|f\|_{A_{1}}\\}$, it follows that

$\displaystyle\|f\|_{(A_{0},A_{1})_{\theta,p}}^{p}$
$\displaystyle=\int_{0}^{1}t^{-\theta
p}K(t,f)^{p}\frac{dt}{t}+\int_{1}^{\infty}t^{-\theta p}K(t,f)^{p}\frac{dt}{t}$
$\displaystyle\leq\|f\|_{A_{1}}^{p}\int_{0}^{1}t^{(1-\theta)p}\frac{dt}{t}+\|f\|_{A_{0}}^{p}\int_{1}^{\infty}t^{-\theta
p}\frac{dt}{t}$
$\displaystyle=\|f\|_{A_{1}}^{p}\frac{1}{(1-\theta)p}+\|f\|_{A_{0}}^{p}\frac{1}{\theta
p}.$

∎

######  Lemma 4.5.

  1. (i)

Assume $\theta\in(0,1)$ and $0<p\leq q<\infty$. Then

$(\theta(1-\theta)p)^{1/p}(A_{0},A_{1})_{\theta,p}\hookrightarrow(\theta(1-\theta)q)^{1/q}(A_{0},A_{1})_{\theta,q}.$

  2. (ii)

Assume further that $(A_{0},A_{1})$ is ordered (i.e., $A_{1}\hookrightarrow
A_{0}$). Let $0<\theta<\eta<1$ and let $p,q\in(0,\infty)$. Then, the norm of
the embedding151515The assumption that the couple $(A_{0},A_{1})$ is ordered
is not restrictive, i.e., a corresponding embedding result still holds for
general couples $(A_{0},A_{1})$. However, this embedding is simplified when
dealing with ordered couples, which will be the only case of interest in this
paper.

$(A_{0},A_{1})_{\eta,p}\hookrightarrow(A_{0},A_{1})_{\theta,q}$

does not exceed

$\left\\{\begin{array}[c]{cl}\frac{(\eta(1-\eta))^{1/p}}{(\theta(1-\theta))^{1/q}}&\text{if}\quad
p\leq q,\\\ &\\\
\frac{1}{(\eta-\theta)^{1/q-1/p}}+\frac{\eta^{1/p}(1-\eta)^{1/p}}{\theta^{1/q}}&\text{if}\quad
p>q,\end{array}\right.$

uniformly w.r.t. $\eta$ and $\theta$.

###### Proof.

The formula that was stated in (i) can be found in [16, (3.2), p. 19].
Concerning (ii) with $p=q$, i.e.,

(4.6)
$(\eta(1-\eta))^{1/p}(A_{0},A_{1})_{\eta,p}\hookrightarrow(\theta(1-\theta))^{1/p}(A_{0},A_{1})_{\theta,p},$

we refer to [19, Corollary 2.1]. Furthermore the case $p\leq q$ in (ii) can be
obtained as a direct consequence of (i) and (4.6).

Next we concentrate on the case $p>q$ in (ii). Since $A_{1}\hookrightarrow
A_{0}$, it is plain to see that $K(t,f)\approx\|f\|_{A_{0}}$ for $t>1$. Then

(4.7)
$\|f\|_{(A_{0},A_{1})_{\theta,q}}\approx\bigg{(}\int_{0}^{1}(t^{-\theta}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q}+\frac{1}{\theta^{1/q}}\|f\|_{A_{0}}.$

On the one hand, applying Hölder’s inequality, we can estimate the integral
given on the right-hand side of (4.7) by

(4.8)
$\bigg{(}\int_{0}^{1}(t^{-\theta}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q}\lesssim\frac{1}{(\eta-\theta)^{1/q-1/p}}\bigg{(}\int_{0}^{1}(t^{-\eta}K(t,f))^{p}\frac{dt}{t}\bigg{)}^{1/p}.$

On the other hand, using monotonicity properties of the $K$-functional, the
second term in the right-hand side of (4.7) can be dominated as follows

(4.9)
$\|f\|_{A_{0}}\lesssim\min\\{\eta,1-\eta\\}^{1/p}\bigg{(}\int_{0}^{\infty}(t^{-\eta}K(t,f))^{p}\frac{dt}{t}\bigg{)}^{1/p}.$

Inserting (4.8) and (4.9) into (4.7), we get

$\|f\|_{(A_{0},A_{1})_{\theta,q}}\lesssim\bigg{(}\frac{1}{(\eta-\theta)^{1/q-1/p}}+\frac{\min\\{\eta,1-\eta\\}^{1/p}}{\theta^{1/q}}\bigg{)}\|f\|_{(A_{0},A_{1})_{\eta,p}}.$

∎

The previous result tells us that the standard interpolation norm
$\|\cdot\|_{(A_{0},A_{1})_{\theta,p}}$ given in (4.1) can be renormalized by
$(\theta(1-\theta)p)^{1/p}\|\cdot\|_{(A_{0},A_{1})_{\theta,p}}$ so that sharp
estimates w.r.t. the interpolation parameter $\theta$ can now be achieved.
Another important example of this phenomenon occurs in reiteration formulas.
Indeed, we recall the following sharp versions of reiteration properties for
the real method.

######  Lemma 4.6 ([19, Theorems 2.11 and 2.12] and [20, Theorem 3]).

Let $s_{0},s_{1},\theta\in(0,1)$ and $p,q\in(0,\infty)$.

  1. (i)

Let $s=(1-\theta)s_{0}+\theta s_{1}$. The following embeddings hold uniformly
w.r.t. $\theta$

$\displaystyle(\theta(1-\theta))^{-1/\min\\{p,q\\}}(A_{0},A_{1})_{s,p}$
$\displaystyle\hookrightarrow((A_{0},A_{1})_{s_{0},q},(A_{0},A_{1})_{s_{1},q})_{\theta,p}$
$\displaystyle\hookrightarrow(\theta(1-\theta))^{-1/\max\\{p,q\\}}(A_{0},A_{1})_{s,p}.$

  2. (ii)

The following embeddings hold uniformly w.r.t. $s_{1}$ and $\theta$

$\displaystyle
s_{1}^{1/p-1/q}[(1-s_{1})^{-1/q}+(1-\theta)^{-1/\min\\{p,q\\}}](A_{0},A_{1})_{\theta
s_{1},p}$
$\displaystyle\hookrightarrow(A_{0},(A_{0},A_{1})_{s_{1},q})_{\theta,p}\hookrightarrow
s_{1}^{1/p}(1-\theta)^{-1/\max\\{p,q\\}}(A_{0},A_{1})_{\theta s_{1},p}.$

A key role in our arguments is played by the continuity properties of general
interpolation scales obtained in [25] (see also [20]). In particular, we will
make use of the following result for the real interpolation method.

######  Theorem 4.7.

Let $p>0$ and $f\in A_{0}\cap A_{1}$. Then

(4.10) $\lim_{\theta\rightarrow
1^{-}}(\theta(1-\theta)p)^{\frac{1}{p}}\|f\|_{(A_{0},A_{1})_{\theta,p}}=\sup_{t>0}\frac{K(t,f;A_{0},A_{1})}{t}$

and

(4.11) $\lim_{\theta\rightarrow
0^{+}}(\theta(1-\theta)p)^{\frac{1}{p}}\|f\|_{(A_{0},A_{1})_{\theta,p}}=\sup_{t>0}K(t,f;A_{0},A_{1}).$

###  4.3. Butzer seminorms via interpolation

A key result in our approach is that the family of semi-norms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ introduced in (1.16) can be
generated (with sharp constants) via interpolation of the classical pair
$(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))$. This is the
assertion contained in the following

######  Lemma 4.8.

Let $\theta\in(0,1),t\in(0,\infty)$ and $p\in(1,\infty)$. Then

(4.12)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\approx\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n}),t}$

and

(4.13)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\approx\|f\|_{W^{\theta
t,p}({\mathbb{R}}^{n}),t}$

where hidden equivalence constants are independent of $\theta$ (but depending
on $n,p$ and $t$).

######  Remark 4.9.

Letting $t=1$ in the previous result (cf. (2.2) and (1.18)), we have

(4.14)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{\theta,p}}\approx\|f\|_{\dot{W}^{\theta,p}({\mathbb{R}}^{n})}$

and, cf. (1.17),

$\|f\|_{(L^{p}({\mathbb{R}}^{n}),W_{p}^{1}({\mathbb{R}}^{n}))_{\theta,p}}\approx\frac{1}{(\theta(1-\theta)p)^{\frac{1}{p}}}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|f\|_{\dot{W}^{\theta,p}({\mathbb{R}}^{n})}.$

In particular, (4.14) was already shown in [25, Lemma 1].

###### Proof of Lemma 4.8.

We shall employ the characterizations of the corresponding $K$-functional
given by

(4.15)
$K(u^{t},f;L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))\approx\frac{1}{u^{n}}\int_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}\,dh\approx\sup_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}$

for $u>0$; this characterization is well known in the classical case
$t=k\in{{\mathbb{N}}}$ (see, e.g., [1, Chapter 5, (4.42), p. 341] and [2,
Theorem 6.7.3]), for the general case $t\in(0,\infty)$ we refer to [21,
(1.10)] (with [8] and [33] as a forerunners).

Inserting the first equivalence in (4.15) into the definition of interpolation
space (cf. (4.1)), we have

$\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\approx\bigg{(}\int_{0}^{\infty}u^{-\theta
tp-
np}\bigg{(}\int_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}\,dh\bigg{)}^{p}\frac{du}{u}\bigg{)}^{1/p}.$

Consequently, the desired result (4.12) would be established once we are able
to prove (cf. (1.16))

(4.16) $\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n}),t}\approx\bigg{(}\int_{0}^{\infty}u^{-\theta tp-
np}\bigg{(}\int_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}\,dh\bigg{)}^{p}\frac{du}{u}\bigg{)}^{1/p}$

uniformly w.r.t. $\theta\in(0,1)$.

To deal with the estimate $\gtrsim$ in (4.16), we apply Hölder’s inequality
and change the order of integration so that

$\displaystyle\int_{0}^{\infty}u^{-\theta tp-
np}\bigg{(}\int_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}\,dh\bigg{)}^{p}\frac{du}{u}$
$\displaystyle\lesssim\int_{0}^{\infty}u^{-\theta
tp-n}\int_{|h|<u}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}^{p}\,dh\frac{du}{u}$
$\displaystyle\approx\int_{{\mathbb{R}}^{n}}|h|^{-\theta
tp-n}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}^{p}\,dh=\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n}),t}^{p}.$

Concerning the estimate $\lesssim$ in (4.16), we use basic monotonicity
properties and the second estimate given in (4.15) to get

$\displaystyle\|f\|_{\dot{W}^{\theta t,p}({\mathbb{R}}^{n}),t}^{p}$
$\displaystyle=\int_{{\mathbb{R}}^{n}}\frac{\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}^{p}}{|h|^{\theta
tp+n}}\,dh$ $\displaystyle=\sum_{j=-\infty}^{\infty}\int_{2^{j-1}<|h|\leq
2^{j}}\frac{\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}^{p}}{|h|^{\theta
tp+n}}\,dh$ $\displaystyle\lesssim\sum_{j=-\infty}^{\infty}2^{-j\theta
tp}\sup_{|h|\leq 2^{j}}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}^{p}$
$\displaystyle\approx\sum_{j=-\infty}^{\infty}2^{-j\theta
tp}\bigg{(}\frac{1}{2^{jn}}\int_{|h|<2^{j}}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}\,dh\bigg{)}^{p}.$

Next we focus on (4.13). The $K$-functional for the pair
$(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))$ can be estimated as (see
[33, (4.2)])

$K(u,f;L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))\approx\min\\{1,u\\}\|f\|_{L^{p}({\mathbb{R}}^{n})}+\sup_{|h|\leq
u^{\frac{1}{t}}}\|\Delta_{h}^{t}f\|_{L^{p}({\mathbb{R}}^{n})}$

for $u>0$ and $f\in L^{p}({\mathbb{R}}^{n})$. Therefore, by (4.15),

$K(u,f;L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))\approx\min\\{1,u\\}\|f\|_{L^{p}({\mathbb{R}}^{n})}+K(u,f;L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n})).$

By elementary computations we find

$\displaystyle\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}^{p}$
$\displaystyle\approx\bigg{(}\int_{0}^{\infty}u^{-\theta
p}\min\\{1,u\\}^{p}\frac{du}{u}\bigg{)}\|f\|_{L^{p}({\mathbb{R}}^{n})}^{p}+\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}^{p}$
$\displaystyle=\frac{1}{\theta(1-\theta)p}\|f\|_{L^{p}({\mathbb{R}}^{n})}^{p}+\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}^{p}.$

Therefore, by (4.12),

$\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}^{p}\approx\frac{1}{\theta(1-\theta)p}\|f\|_{L^{p}({\mathbb{R}}^{n})}^{p}+\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n}),t}^{p}.$

∎

The following result shows that the seminorm
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ is equivalent to the classical
Gagliardo seminorm $\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}$, but the
constants of equivalence blow up as $s\rightarrow t^{-}$.

######  Lemma 4.10.

Let $0<s<t<1$ and $1<p<\infty$. Then

(4.17)
$\frac{1}{(t-s)^{\frac{1}{\max\\{p,2\\}}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\lesssim\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\lesssim\frac{1}{(t-s)^{\frac{1}{\min\\{p,2\\}}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})},$

where the hidden equivalence constants are independent of $s$.

###### Proof.

Recall the relationships between $\dot{H}^{t,p}({\mathbb{R}}^{n})$ and
$\dot{B}_{p,q}^{t}({\mathbb{R}}^{n})$ (cf. Lemma 2.1),

$\dot{B}_{p,\min\\{p,2\\}}^{t}({\mathbb{R}}^{n})\hookrightarrow\dot{H}^{t,p}({\mathbb{R}}^{n})\hookrightarrow\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n}),$

therefore, by interpolation, the following embeddings hold uniformly w.r.t.
$\theta\in(0,1)$

(4.18)
$(L^{p}({\mathbb{R}}^{n}),\dot{B}_{p,\min\\{p,2\\}}^{t}({\mathbb{R}}^{n}))_{\theta,p}\hookrightarrow(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}\hookrightarrow(L^{p}({\mathbb{R}}^{n}),\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n}))_{\theta,p}.$

Furthermore, by Lemma 4.3, we can rewrite

$(L^{p}({\mathbb{R}}^{n}),\dot{B}_{p,q}^{t}({\mathbb{R}}^{n}))_{\theta,p}=(L^{p}({\mathbb{R}}^{n}),(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{t,q})_{\theta,p},\qquad
q\in(0,\infty),$

with related equivalence constants independent of $\theta$. Combining this and
Lemma 4.6(ii), we obtain the embeddings

$\displaystyle(1-\theta)^{-1/\min\\{p,q\\}}(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{\theta
t,p}$
$\displaystyle\hookrightarrow(L^{p}({\mathbb{R}}^{n}),\dot{B}_{p,q}^{t}({\mathbb{R}}^{n}))_{\theta,p}$
$\displaystyle\hookrightarrow(1-\theta)^{-1/\max\\{p,q\\}}(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{\theta
t,p}.$

In light of Lemma 4.8 (see also (4.14)), the previous embeddings turn out to
be equivalent to

(4.19) $(1-\theta)^{-1/\max\\{p,q\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}\lesssim\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{B}_{p,q}^{t}({\mathbb{R}}^{n}))_{\theta,p}}\lesssim(1-\theta)^{-1/\min\\{p,q\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}.$

Combining (4.18) and (4.19), we find

$(1-\theta)^{-1/\max\\{p,2\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}\lesssim\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\lesssim(1-\theta)^{-1/\min\\{p,2\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}.$

Equivalently (cf. (4.12))

$(1-\theta)^{-1/\max\\{p,2\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}\lesssim\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n}),t}\lesssim(1-\theta)^{-1/\min\\{p,2\\}}\|f\|_{\dot{W}^{\theta
t,p}({\mathbb{R}}^{n})}$

which, after the change of variables $\theta\leftrightarrow\frac{s}{t},$
yields the desired result (4.17). ∎

##  5\. Proofs of the main results

###  5.1. Proof of Proposition 3.1

We now recall some results that we need from the theory of Fourier series with
monotonic coefficients.

Suppose that the Fourier series of $f\in L^{1}({\mathbb{T}})$ is given by

(5.1) $f(x)\sim\sum_{\nu=1}^{\infty}c_{\nu}\cos(\nu x),\qquad
x\in{\mathbb{T}},$

with

(5.2) $c_{\nu}\geq c_{\nu+1}\geq\cdots\geq 0\qquad\text{and}\qquad
c_{\nu}\rightarrow 0.$

A well-known theorem of Hardy–Littlewood (see, e.g., [35, p. 129, Vol. II])
asserts that $f\in L^{p}({\mathbb{T}})$ if and only if

$\sum_{\nu=1}^{\infty}\nu^{p-2}c_{\nu}^{p}<\infty.$

Furthermore

(5.3)
$\|f\|_{L^{p}({\mathbb{T}})}^{p}\approx\sum_{\nu=1}^{\infty}\nu^{p-2}c_{\nu}^{p}.$

This result has been further extended in [11] in order to deal with other
spaces of smooth functions (namely, Besov spaces and Sobolev spaces), as well
as more general classes of Fourier series with monotone-type coefficients (the
so called, general monotone class). In particular, it was shown in [11,
Theorem 4.25] that if $f$ is given by (5.1) with coefficients satisfying (5.2)
then it belongs to
$H^{t,p}({\mathbb{T}}),\,t\in{\mathbb{R}},\,p\in(1,\infty),$ if and only if

$\sum_{\nu=1}^{\infty}\nu^{tp+p-2}c_{\nu}^{p}<\infty.$

Moreover,

(5.4)
$\|f\|_{H^{t,p}({\mathbb{T}})}^{p}\approx\sum_{\nu=1}^{\infty}\nu^{tp+p-2}c_{\nu}^{p}.$

Let $t>0$ and assume $c_{\nu}=\nu^{-t-1+\frac{1}{p}},\,\nu\in{{\mathbb{N}}}$.
By virtue of (5.3) and (5.4)

(5.5)
$\|f\|_{L^{p}({\mathbb{T}})}^{p}\approx\sum_{\nu=1}^{\infty}\nu^{p-2}\nu^{-tp-p+1}=\sum_{\nu=1}^{\infty}\nu^{-tp-1}<\infty$

and

$\|f\|_{H^{t,p}({\mathbb{T}})}^{p}\approx\sum_{\nu=1}^{\infty}\nu^{tp+p-2}\nu^{-tp-p+1}=\sum_{\nu=1}^{\infty}\nu^{-1}=\infty,$

respectively. Consequently, the Fourier series $f$ defined by (5.1) belongs to
$L^{p}({\mathbb{T}})$ but not to $H^{t,p}({\mathbb{T}})$.

To conclude, it remains to show that

(5.6) $\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}})}\leq C(t-s_{k})^{-\frac{1}{p}}$

uniformly w.r.t. $k\in{{\mathbb{N}}}$. To proceed, we argue as follows. In
view of (4.14),

(5.7)
$\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}})}\approx\bigg{(}\int_{0}^{\infty}u^{-s_{k}p}K(u,f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}},$

with hidden constants of equivalence independent of $k$. Next we estimate
$K(u,f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))$.

For $u>0$, the classical moduli of smoothness of $f\in L^{p}({\mathbb{T}})$ is
defined by

(5.8) $\omega(f,u)_{p}:=\sup_{|h|\leq
u}\|\Delta_{h}f\|_{L^{p}({\mathbb{T}})}.$

Assume further that $f$ satisfies (5.1) and (5.2). Then it is known that
$\omega(f,u)_{p}$ can be estimated in terms of the Fourier coefficients of
$f$. Namely,

(5.9) $\omega(f,u)_{p}\approx\bigg{(}\sum_{\nu=1}^{\infty}\min\\{1,\nu
u\\}^{p}\nu^{p-2}c_{\nu}\bigg{)}^{\frac{1}{p}},\qquad u\in(0,1);$

(cf. [29, Theorem 6.2] and [14, Theorem 6.1] where this is actually done even
in the more general context of general monotone Fourier coefficients.)

Specializing (5.9) to $f$ given by (5.1), yields

$\omega(f,l^{-1})_{p}\approx
l^{-1}\bigg{(}\sum_{\nu=1}^{l}\nu^{(1-t)p-1}\bigg{)}^{\frac{1}{p}}+\bigg{(}\sum_{\nu=l}^{\infty}\nu^{-tp-1}\bigg{)}^{\frac{1}{p}}\approx\frac{1}{(t(1-t))^{\frac{1}{p}}}l^{-t}$

for every $l\in{{\mathbb{N}}}$ and $t\in(0,1)$. According to (4.15) (with
$t=1$) and (5.8), the previous estimate is equivalent to

(5.10)
$K(l^{-1},f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))\approx\frac{1}{(t(1-t))^{\frac{1}{p}}}l^{-t}.$

Furthermore, we have the trivial estimate

(5.11)
$K(u,f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))\leq\|f\|_{L^{p}({\mathbb{T}})}.$

Putting together (5.7), (5.10), (5.11), applying monotonicity properties of
$K$-functionals, and taking into account that $s_{k}\uparrow t$, we obtain

$\displaystyle\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}})}^{p}$
$\displaystyle\lesssim\int_{0}^{1}u^{-s_{k}p}K(u,f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))^{p}\frac{du}{u}+\frac{1}{s_{k}p}\|f\|_{L^{p}({\mathbb{T}})}^{p}$
$\displaystyle\approx\sum_{l=1}^{\infty}l^{s_{k}p-1}K(l^{-1},f;L^{p}({\mathbb{T}}),\dot{W}_{p}^{1}({\mathbb{T}}))^{p}+\|f\|_{L^{p}({\mathbb{T}})}^{p}$
$\displaystyle\approx\sum_{l=1}^{\infty}l^{-(t-s_{k})p-1}+\|f\|_{L^{p}({\mathbb{T}})}^{p}$
$\displaystyle\approx\frac{1}{t-s_{k}}+\|f\|_{L^{p}({\mathbb{T}})}^{p}$
$\displaystyle\lesssim\frac{1}{t-s_{k}}.$

This concludes the proof of (5.6). ∎

######  Remark 5.1.

Similar ideas can be used to establish the counterpart of Proposition 3.1 in
${\mathbb{R}}^{n}$. We will only sketch the proof. Consider the monotone
function

$F_{0}(u)=\left\\{\begin{array}[c]{cl}u^{-t-n+\frac{n}{p}}&\text{if}\quad
u>1,\\\ &\\\ 1&\text{if}\quad u\in(0,1],\end{array}\right.$

and define the radial function $f(x)=f_{0}(|x|),\,x\in{\mathbb{R}}^{n},$ where
$f_{0}$ is the inverse Fourier–Hankel transform of $F_{0}$, i.e.,

$f_{0}(u)=\frac{2}{\Gamma\big{(}\frac{n}{2}\big{)}(2\sqrt{\pi})^{n}}\int_{0}^{\infty}F_{0}(\xi)j_{n/2-1}(u\xi)\xi^{n-1}\,d\xi,$

where $j_{\alpha}(u)=\Gamma(\alpha+1)(u/2)^{-\alpha}J_{\alpha}(u)$ is the
normalized Bessel function ($j_{\alpha}(0)=1$), $\alpha\geq-1/2$, and
$J_{\alpha}$ is the classical Bessel function of the first kind of order
$\alpha$. Applying the analogue of Hardy–Littlewood theorem for
$L^{p}({\mathbb{R}}^{n}),\,p>\frac{2n}{n+1},$ (cf. [13, Theorem 1] and [14,
(4.10)]), we have

$\|f\|_{L^{p}({\mathbb{R}}^{n})}^{p}\approx\int_{0}^{\infty}u^{np-n-1}F_{0}(u)^{p}\,du=\int_{0}^{1}u^{np-n-1}\,du+\int_{1}^{\infty}u^{-tp-1}\,du<\infty,$

which gives the Euclidean setting counterpart of (5.5). On the other hand,
using the counterpart of (5.4) for $H^{t,p}({\mathbb{R}}^{n})$, which may be
found in [11, Theorem 4.8], we have

$\displaystyle\|f\|_{H^{t,p}({\mathbb{R}}^{n})}^{p}$
$\displaystyle\approx\int_{0}^{1}u^{np-n-1}F_{0}^{p}(u)\,du+\int_{1}^{\infty}u^{tp+np-n-1}F_{0}(u)^{p}\,du$
$\displaystyle=\int_{0}^{1}u^{np-n-1}\,du+\int_{1}^{\infty}\frac{du}{u}=\infty.$

The analog of (5.9) for the moduli of smoothness
$\omega(f,u)_{p}=\sup_{|h|\leq u}\|\Delta_{h}f\|_{L^{p}({\mathbb{R}}^{n})}$
(cf. (5.8)) was obtained in [14, Corollary 4.1 and (7.6)], namely,

$\omega(f,u)_{p}^{p}\approx
u^{p}\int_{0}^{1/u}\xi^{p+np-n-1}F_{0}^{p}(\xi)\,d\xi+\int_{1/u}^{\infty}\xi^{np-n-1}F_{0}^{p}(\xi)\,d\xi.$

In particular (since $t\in(0,1)$),

$\omega(f,u)_{p}\approx u^{t}\qquad\text{for}\qquad u\in(0,1).$

Following now line by line the arguments given in the proof of Proposition
3.1, one can show that there exists $C>0$ such that

$\|f\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n})}\leq
C(t-s_{k})^{-\frac{1}{p}}\qquad\text{for every}\qquad k\in{{\mathbb{N}}}.$

###  5.2. Proof of Theorem 3.3

We give a unified proof in the inhomogeneous setting (i.e., (3.2) and (3.4)).
It follows from Lemma 4.8 that

$(s(t-s))^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}+(s(t-s))^{\frac{1}{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

Consequently, both (3.2) & (3.4) will be established if we are able to show
that

(5.12)
$\|f\|_{F_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}\lesssim(s(t-s))^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}.$

To prove this inequality, we invoke the method of retractions (cf. [31]) which
shows that the interpolation space
$(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}$ can be
isomorphically identified (with constants of equivalence independent of $s$)
with
$(L^{p}({\mathbb{R}}^{n};\ell_{2}({{\mathbb{N}_{0}}})),L^{p}({\mathbb{R}}^{n};\ell_{2}^{t}({{\mathbb{N}_{0}}})))_{\frac{s}{t},p}$.
In view of the interpolation formula given in Lemma 4.1,

$\|f\|_{(L^{p}({\mathbb{R}}^{n};\ell_{2}({{\mathbb{N}_{0}}})),L^{p}({\mathbb{R}}^{n};\ell_{2}^{t}({{\mathbb{N}_{0}}})))_{\frac{s}{t},p}}\approx\|f\|_{L^{p}({\mathbb{R}}^{n};(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{s}{t},p})},$

which reduces161616Here we must take into account that
$F_{p,2}^{\bar{r}}({\mathbb{R}}^{n})$ can be identified with the vector-valued
space $L^{p}({\mathbb{R}}^{n};\ell_{2}^{\bar{r}}({{\mathbb{N}_{0}}}))$.
(5.12) to the following sharp embedding at the level of sequence spaces

(5.13)
$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{s}{t},p}\hookrightarrow\ell_{2}^{\bar{r}}({{\mathbb{N}_{0}}}).$

Since (cf. Lemma 4.2)

$(\bar{r}(t-\bar{r}))^{\frac{1}{2}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{\bar{r}}{t},2}=\ell_{2}^{\bar{r}}({{\mathbb{N}_{0}}}),$

the embedding (5.13) turns out to be equivalent to

(5.14)
$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{s}{t},p}\hookrightarrow(\bar{r}(t-\bar{r}))^{\frac{1}{2}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{\bar{r}}{t},2}.$

It remains to show the validity of (5.14). To do this, we distinguish two
possible cases. Suppose first that $p\leq 2$, then by Lemma 4.5(ii) (note that
$\bar{r}<s$),

$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{s}{t},p}\hookrightarrow(\bar{r}(t-\bar{r}))^{\frac{1}{2}}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{\bar{r}}{t},2}.$

Suppose now that $p>2$. Applying again Lemma 4.5(ii), we have

(5.15)
$\bigg{(}\frac{1}{(s-\bar{r})^{1/2-1/p}}+\frac{s^{1/p}(t-s)^{1/p}}{\bar{r}^{1/2}}\bigg{)}(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{s}{t},p}\hookrightarrow(\ell_{2}({{\mathbb{N}_{0}}}),\ell_{2}^{t}({{\mathbb{N}_{0}}}))_{\frac{\bar{r}}{t},2}.$

Furthermore, it is easy to check that under the assumptions (1) & (2) in
Theorem 3.3, the constant in (5.15) behaves like

$\frac{1}{(s-\bar{r})^{1/2-1/p}}+\frac{s^{1/p}(t-s)^{1/p}}{\bar{r}^{1/2}}\approx(s(t-s))^{\frac{1}{p}}(\bar{r}(t-\bar{r}))^{-\frac{1}{2}},$

which yields the desired embedding (5.14).

For the homogeneous setting (i.e., (3.3) and (3.5)), observe that it is enough
to deal with $r=\bar{r}$, since for $r\in(0,\bar{r})$ we have

$\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\leq\|f\|_{\dot{F}_{p,2}^{0}({\mathbb{R}}^{n})}+\|f\|_{\dot{F}_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}\leq
C(\|f\|_{L^{p}({\mathbb{R}}^{n})}+\|f\|_{\dot{F}_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}).$

Henceforth, we concentrate on (3.3) and (3.5) with $r=\bar{r}$. The
methodology for the inhomogeneous setting given above (i.e., (3.2) and (3.4))
yields (cf. (5.12))

(5.16)
$\|f\|_{\dot{F}_{p,2}^{\bar{r}}({\mathbb{R}}^{n})}\lesssim(s(t-s))^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}.$

Indeed, in this case the analog of (5.13) reads

(5.17)
$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},p}\hookrightarrow\ell_{2}^{\bar{r}}({{\mathbb{Z}}})$

where

$\bar{r}=\left\\{\begin{array}[c]{cl}t-\Lambda(t-s),&\text{if}\quad
s\rightarrow t^{-},\\\ &\\\ \frac{1}{\Lambda}s,&\text{if}\quad s\rightarrow
0^{+}.\end{array}\right.$

Here the couple $(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))$ is
not ordered and we cannot invoke Lemma 4.5(ii) directly to deal with (5.17).
However, this difficulty can be overcome as follows. We write
${{\mathbb{Z}}}={{\mathbb{N}}}_{0}\cup{{\mathbb{N}}}_{-}$ where
${{\mathbb{N}}}_{-}:=\\{-j:j\in{{\mathbb{N}}}\\}$. Since
$\ell_{2}^{t}({{\mathbb{N}}}_{0})\hookrightarrow\ell_{2}({{\mathbb{N}}}_{0})$
and
$\ell_{2}({{\mathbb{N}}}_{-})\hookrightarrow\ell_{2}^{t}({{\mathbb{N}}}_{-})$,
it follows from (5.13) that

(5.18)
$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{N}}}_{0}),\ell_{2}^{t}({{\mathbb{N}}}_{0}))_{\frac{s}{t},p}\hookrightarrow\ell_{2}^{\bar{r}}({{\mathbb{N}}}_{0})$

and, by (4.2), and a simple change of variables,

(5.19)
$(s(t-s))^{\frac{1}{p}}(\ell_{2}({{\mathbb{N}}}_{-}),\ell_{2}^{t}({{\mathbb{N}}}_{-}))_{\frac{s}{t},p}\hookrightarrow\ell_{2}^{\bar{r}}({{\mathbb{N}}}_{-}).$

Let $\xi=(\xi)_{j\in{{\mathbb{Z}}}}$ and consider the related sequences
$\xi^{1}=(\xi_{j})_{j\in{{\mathbb{N}}}_{0}}$ and
$\xi^{2}=(\xi_{j})_{j\in{{\mathbb{N}}}_{-}}$. By triangle inequality and
(5.18), (5.19), we derive

$\displaystyle\|\xi\|_{\ell_{2}^{\bar{r}}({{\mathbb{Z}}})}$
$\displaystyle\leq\|\xi^{1}\|_{\ell_{2}^{\bar{r}}({{\mathbb{N}}}_{0})}+\|\xi^{2}\|_{\ell_{2}^{\bar{r}}({{\mathbb{N}}}_{-})}$
$\displaystyle\leq
C(s(t-s))^{\frac{1}{p}}(\|\xi^{1}\|_{(\ell_{2}({{\mathbb{N}}}_{0}),\ell_{2}^{t}({{\mathbb{N}}}_{0}))_{\frac{s}{t},p}}+\|\xi^{2}\|_{(\ell_{2}({{\mathbb{N}}}_{-}),\ell_{2}^{t}({{\mathbb{N}}}_{-}))_{\frac{s}{t},p}})$
$\displaystyle\leq
2C(s(t-s))^{\frac{1}{p}}\|\xi\|_{(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},p}},$

where the last step follows immediately from the interpolation property
applied to the canonical projections $\xi\mapsto\xi^{i},\,i=1,2$. The proof of
(5.17) is complete establishing that (5.16) holds. Now the rest of the proof
follows line by line the arguments provided for the inhomogeneous case. ∎

###  5.3. Proof of Theorem 3.2

Without loss of generality, we may assume that $t-s_{k}\leq\frac{t}{4}$.
According to (3.3) there is $C=C(n,p,t)>0$ such that, for every $r\leq
t-2(t-s_{k})$,

$\|f_{k}\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\leq
C\Big{(}\|f_{k}\|_{L^{p}({\mathbb{R}}^{n})}+(t-s_{k})^{\frac{1}{p}}\|f_{k}\|_{\dot{W}^{s_{k},p}({\mathbb{R}}^{n}),t}\Big{)}\leq
C\Lambda.$

Taking limits as $k\rightarrow\infty,$ and noting that
$\lim_{k\rightarrow\infty}t-2(t-s_{k})=t$, the previous estimate yields

$\limsup_{k\rightarrow\infty}\Big{(}\|f_{k}\|_{L^{p}({\mathbb{R}}^{n})}+\|f_{k}\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}\Big{)}\lesssim\Lambda\qquad\text{for
all}\qquad r\in(0,t).$

Therefore, one can apply [5, Lemma 2.6] to derive $f\in
H^{t,p}({\mathbb{R}}^{n})$ and

$\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}\lesssim\Lambda.$

∎

######  Remark 5.2.

It may be instructive to revisit the counterexample provided in Proposition
3.1 to show the important role played by the seminorms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ in Theorem 3.2. Consider the
Fourier series $f$ defined by (3.1). It was shown in Proposition 3.1 that the
condition

$\sup_{k\in{{\mathbb{N}}}}\,(t-s_{k})^{\frac{1}{p}}\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}})}<\infty$

is satisfied but $f\not\in H^{t,p}({\mathbb{T}})$. Next we check that this
example does not contradict Theorem 3.2 since

(5.20)
$\limsup_{k\rightarrow\infty}\,(t-s_{k})^{\frac{1}{p}}\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}}),t}=\infty.$

Indeed, applying [29, Theorem 6.2] one can estimate, for every
$l\in{{\mathbb{N}}}$,

$\displaystyle K(l^{-t},f;L^{p}({\mathbb{T}}),\dot{H}^{t,p}({\mathbb{T}}))$
$\displaystyle\approx
l^{-t}\bigg{(}\sum_{\nu=1}^{l}\nu^{tp+p-2}\nu^{-tp-p+1}\bigg{)}^{1/p}+\bigg{(}\sum_{\nu=l}^{\infty}\nu^{p-2}\nu^{-tp-p+1}\bigg{)}^{1/p}$
$\displaystyle\approx l^{-t}(1+\log l)^{1/p}$

and thus, by Lemma 4.8,

$\displaystyle\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}}),t}^{p}$
$\displaystyle\approx\|f\|_{(L^{p}({\mathbb{T}}),\dot{H}^{t,p}({\mathbb{T}}))_{\frac{s_{k}}{t},p}}^{p}$
$\displaystyle\gtrsim\sum_{l=0}^{\infty}2^{ls_{k}p}K(2^{-lt},f;L^{p}({\mathbb{T}}),\dot{H}^{t,p}({\mathbb{T}}))^{p}$
$\displaystyle\approx\sum_{l=0}^{\infty}2^{-l(t-s_{k})p}(1+l)$
$\displaystyle\gtrsim(t-s_{k})^{-1}\sum_{l=\left\lfloor{\frac{1}{t-s_{k}}}\right\rfloor}^{\infty}2^{-l(t-s_{k})p}$
$\displaystyle\approx\frac{(t-s_{k})^{-1}}{1-2^{-(t-s_{k})p}}\approx(t-s_{k})^{-2}.$

Hence

$(t-s_{k})^{1/p}\|f\|_{\dot{W}^{s_{k},p}({\mathbb{T}}),t}\gtrsim(t-s_{k})^{-1/p}$

which yields (5.20) (since $s_{k}\uparrow t$).

###  5.4. Proof of Theorem 3.4

Let $(A_{0},A_{1})=(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))$.
Recall that for this couple for $f\in$ $H^{t,p}({\mathbb{R}}^{n}),$ (cf. [33,
Corollary 10, page 75], [21, Section 1.3, Property 1])

$\sup_{t>0}\frac{K(t,f;L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))}{t}\approx\|(-\Delta)^{\frac{t}{2}}f\|_{L^{p}({\mathbb{R}}^{n})}$

and

$\sup_{t>0}K(t,f;L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}.$

Therefore (3.6) (respectively, (3.7)) is an immediate consequence of Lemma 4.8
and (4.10) (respectively, (4.11)).

Concerning the inhomogeneous setting (i.e., (3.8) and (3.9)), the proofs are
immediate consequences of their homogeneous counterparts (i.e., (3.6) and
(3.7)) applied to (1.17).

∎

###  5.5. Proof of Theorem 3.6

Assume $0<r<t<1$. Applying Lemma 4.4 to the Triebel–Lizorkin pair
$(A_{0},A_{1})=(\dot{F}_{p,2}^{r}({\mathbb{R}}^{n}),\dot{F}_{p,2}^{t}({\mathbb{R}}^{n}))$,
one has

$\|f\|_{(\dot{F}_{p,2}^{r}({\mathbb{R}}^{n}),\dot{F}_{p,2}^{t}({\mathbb{R}}^{n}))_{\theta,p}}\lesssim\frac{1}{\theta^{1/p}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(1-\theta)^{1/p}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}$

for every $\theta\in(0,1)$. Since (cf. Lemma 2.1(i))

$\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})\hookrightarrow\dot{B}_{p,\max\\{p,2\\}}^{r}({\mathbb{R}}^{n})\qquad\text{and}\qquad\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})\hookrightarrow\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n}),$

the previous inequality implies

(5.21)
$\|f\|_{(\dot{B}_{p,\max\\{p,2\\}}^{r}({\mathbb{R}}^{n}),\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n}))_{\theta,p}}\lesssim\frac{1}{\theta^{1/p}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(1-\theta)^{1/p}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}.$

Next we compute the interpolation norm given in the left-hand side of (5.21).
Indeed, since $r,t\in(0,1)$, we can invoke Lemma 4.3 to state

$\dot{B}_{p,\max\\{p,2\\}}^{r}({\mathbb{R}}^{n})=(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{r,\max\\{p,2\\}}$

and

$\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n})=(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{t,\max\\{p,2\\}}.$

Therefore, in light of Lemmas 4.6(i) and (4.14), we get, uniformly w.r.t.
$\theta$,

$\displaystyle\|f\|_{(\dot{B}_{p,\max\\{p,2\\}}^{r}({\mathbb{R}}^{n}),\dot{B}_{p,\max\\{p,2\\}}^{t}({\mathbb{R}}^{n}))_{\theta,p}}$
$\displaystyle\approx\|f\|_{((L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{r,\max\\{p,2\\}},(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{t,\max\\{p,2\\}})_{\theta,p}}$
$\displaystyle\gtrsim(\theta(1-\theta))^{-1/\max\\{p,2\\}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{W}_{p}^{1}({\mathbb{R}}^{n}))_{(1-\theta)r+\theta
t,p}}$
$\displaystyle\approx(\theta(1-\theta))^{-1/\max\\{p,2\\}}\|f\|_{\dot{W}^{(1-\theta)r+\theta
t,p}({\mathbb{R}}^{n})}.$

Combining this and (5.21), we arrive at

(5.22)
$\frac{1}{(\theta(1-\theta))^{1/\max\\{p,2\\}}}\|f\|_{\dot{W}^{(1-\theta)r+\theta
t,p}({\mathbb{R}}^{n})}\lesssim\frac{1}{\theta^{1/p}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(1-\theta)^{1/p}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}.$

Given $s\in(r,t)$, we choose $\theta\in(0,1)$ such that $s=(1-\theta)r+\theta
t$ and make the corresponding change of variables in (5.22), i.e.,

$\frac{1}{((s-r)(t-s))^{1/\max\\{p,2\\}}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}\lesssim\frac{1}{(s-r)^{1/p}}\|f\|_{\dot{F}_{p,2}^{r}({\mathbb{R}}^{n})}+\frac{1}{(t-s)^{1/p}}\|f\|_{\dot{F}_{p,2}^{t}({\mathbb{R}}^{n})}.$

The proofs in the limiting cases $r=0$ and $t=1$ (i.e., (3.11) and (3.12)
respectively) are easier and we omit further details. ∎

###  5.6. Proof of Theorem 3.9

Since $H^{\alpha,p}({\mathbb{R}}^{n})\hookrightarrow L^{p}({\mathbb{R}}^{n})$,
it follows from Lemma 4.5(ii) that

$(s(\alpha-s))^{1/p}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{\alpha,p}({\mathbb{R}}^{n}))_{\frac{s}{\alpha},p}}\lesssim(t(\alpha-t))^{1/p}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{\alpha,p}({\mathbb{R}}^{n}))_{\frac{t}{\alpha},p}}.$

Combining with (4.13) yields

$\displaystyle\|f\|_{L^{p}({\mathbb{R}}^{n})}+(s(\alpha-s))^{1/p}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),\alpha}$
$\displaystyle\approx(s(1-s))^{1/p}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{\alpha,p}({\mathbb{R}}^{n}))_{\frac{s}{\alpha},p}}$
$\displaystyle\lesssim(t(1-t))^{1/p}\|f\|_{(L^{p}({\mathbb{R}}^{n}),H^{\alpha,p}({\mathbb{R}}^{n}))_{\frac{t}{\alpha},p}}$
$\displaystyle\approx\|f\|_{L^{p}({\mathbb{R}}^{n})}+(t(\alpha-t))^{1/p}\|f\|_{\dot{W}^{t,p}({\mathbb{R}}^{n}),\alpha}.$

The desired result now follows since $s(\alpha-s)\approx\min\\{s,\alpha-s\\}$.
∎

## Appendix A: Sharp versions of fractional Sobolev inequalities

For $p\in(0,\infty)$ and $q\in(0,\infty]$, the _Lorentz space_
$L^{p,q}({\mathbb{R}}^{n})$ is formed by all measurable functions $f$ defined
on ${\mathbb{R}}^{n}$ such that

$\|f\|_{L^{p,q}({\mathbb{R}}^{n})}:=\bigg{(}\int_{0}^{\infty}\big{(}u^{\frac{1}{p}}f^{*}(u)\big{)}^{q}\frac{du}{u}\bigg{)}^{\frac{1}{q}}<\infty$

(with the usual modification if $q=\infty$). As usual, $f^{*}$ denotes the
non-increasing rearrangement of $f$. By $f^{**}$ we denote the maximal
function given by $f^{**}(u):=\frac{1}{u}\int_{0}^{u}f^{*}(v)\,dv$. We refer
to [1] and [2] for detailed accounts on Lorentz spaces.

Let $0<s<1,\,1\leq p<\frac{n}{s}$ and $p^{*}=\frac{np}{n-sp}$. The
Bourgain–Brezis–Mironescu–Maz’ya–Shasposhnikova formula (cf. [4] and [24])
claims that there exists $C=C(n,p)>0$ such that

(5.23) $\|f\|^{p}_{L^{p^{*},p}({\mathbb{R}}^{n})}\leq
C\frac{s(1-s)}{(n-sp)^{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}^{p};$

see also [20]. This inequality can be considered as a sharp version of the
classical Sobolev embedding

(5.24) $\dot{W}^{1}_{p}({\mathbb{R}}^{n})\hookrightarrow
L^{\frac{np}{n-p},p}({\mathbb{R}}^{n}),\qquad 1\leq p<n.$

Indeed, (5.24) follows from (5.23) by taking limits as $s\to 1^{-}$ (cf.
(1.3)).

Let $t\in(0,1)$ be fixed. Note that the formula (5.23) does not provide any
insight if $s\to t^{-}$. So the natural question here is: can we obtain an
analogue of (5.23) for fractional smoothness $t$? Or equivalently, what is the
fractional counterpart of (5.23) related to the classical embedding

(5.25) $\dot{H}^{t,p}({\mathbb{R}}^{n})\hookrightarrow
L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}),\qquad 1<p<\frac{n}{t}?$

The answer is given again in terms of the Butzer seminorms
$\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ (cf. (1.16)).

######  Theorem 5.3.

Let $0<s<t\leq 1,1<p<\frac{n}{s}$ and $p^{*}=\frac{np}{n-sp}$. Assume $f\in
W^{s,p}({\mathbb{R}}^{n})$. Then there exists $C=C(n,p,t)>0$ such that

(5.26) $\|f\|^{p}_{L^{p^{*},p}({\mathbb{R}}^{n})}\leq
C\frac{s(t-s)}{(n-sp)^{p}}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}^{p}.$

In particular, if $t=1$ then one recovers (5.23).

######  Remark 5.4.

Let $t>0$ and $1<p<\frac{n}{t}$. Taking limits as $s\to t^{-}$ in (5.26) and
applying Theorem 3.4, we recover (5.25).

###### Proof of Theorem 5.3.

Case 1: Assume $t<\frac{n}{p}$ and $s\in(0,t)$. Under these assumptions, the
Hardy–Littlewood–Sobolev theorem (cf. [26, p. 139]) asserts

$\dot{H}^{t,p}({\mathbb{R}}^{n})\hookrightarrow
L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}).$

If we interpolate this embedding with the trivial one
$L^{p}({\mathbb{R}}^{n})\hookrightarrow L^{p}({\mathbb{R}}^{n})$, we arrive at

(5.27)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}\lesssim\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}$

uniformly with respect to $s\in(0,t)$. In light of Lemma 4.8,

(5.28)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}\approx\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

Next we show that

(5.29)
$\|f\|_{L^{p^{*},p}({\mathbb{R}}^{n})}\approx(s(t-s))^{\frac{1}{p}}\|f\|_{(L^{p}({\mathbb{R}}^{n}),L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}.$

Indeed, since
$L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n})=(L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n}))_{\frac{tp}{n},p}$
(see, e.g., [2, Theorem 5.2.1, p. 109]), we can write

$(L^{p}({\mathbb{R}}^{n}),L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}=(L^{p}({\mathbb{R}}^{n}),(L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n}))_{\frac{tp}{n},p})_{\frac{s}{t},p}$

with hidden constants of equivalence independent of $s$. Invoking now the
sharp reiteration formula given in Lemma 4.6(ii), we get

(5.30)
$(L^{p}({\mathbb{R}}^{n}),L^{\frac{np}{n-tp},p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}=(t-s)^{-\frac{1}{p}}(L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}.$

Furthermore, using the well-known characterization of the $K$-functional (cf.
[2, Theorem 5.2.1, p. 109])

$K(u,f;L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n}))\approx\bigg{(}\int_{0}^{u^{p}}(f^{*}(v))^{p}\,dv\bigg{)}^{\frac{1}{p}}$

and applying Fubini’s theorem, we obtain

$\displaystyle\|f\|_{(L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}}^{p}$
$\displaystyle=\int_{0}^{\infty}(u^{-\frac{sp}{n}}K(u,f;L^{p}({\mathbb{R}}^{n}),L^{\infty}({\mathbb{R}}^{n})))^{p}\frac{du}{u}$
$\displaystyle\approx\int_{0}^{\infty}u^{-\frac{sp}{n}}\int_{0}^{u}(f^{*}(v))^{p}\,dv\frac{du}{u}$
$\displaystyle=\int_{0}^{\infty}(f^{*}(v))^{p}\int_{v}^{\infty}u^{-\frac{sp}{n}}\frac{du}{u}dv$
$\displaystyle=\frac{n}{sp}\|f\|_{L^{p^{*},p}({\mathbb{R}}^{n})}^{p}.$

Inserting this into (5.30), we arrive at the desired estimate (5.29).

Combining now (5.27)–(5.29) we achieve (5.26).

Case 2: Assume that either171717In particular, this argument with
$t=\frac{n}{p}\in{{\mathbb{N}}}$ completes the proof of [20, Theorem 8].

$t>\frac{n}{p}\quad\text{and}\quad s\in\Big{(}0,\frac{n}{p}\Big{)}$

or

$t=\frac{n}{p}\quad\text{and}\quad s\to 0^{+}.$

We will make use of the well-known embedding (see, e.g., [27, p. 164])

$\dot{H}^{\frac{n}{p},p}({\mathbb{R}}^{n})\hookrightarrow\text{BMO}({\mathbb{R}}^{n})$

where $\text{BMO}({\mathbb{R}}^{n})$ is the space of bounded mean oscillation
functions of John–Nirenberg [18] endowed with the seminorm

$\|f\|_{\text{BMO}({\mathbb{R}}^{n})}:=\|f^{\\#}\|_{L^{\infty}({\mathbb{R}}^{n})}$

and $f^{\\#}$ is the sharp maximal function of $f\in
L^{1}_{\text{loc}}({\mathbb{R}}^{n})$, i.e.,

$f^{\\#}(x):=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f(x)-f_{Q}|\,dx,\qquad
x\in{\mathbb{R}}^{n},$

where the supremum runs over all cubes $Q$ in ${\mathbb{R}}^{n}$ with sides
parallel to the axes of coordinates and $f_{Q}=\frac{1}{|Q|}\int_{Q}f$.

In this case, the analog of (5.27) reads as

(5.31)
$\|f\|_{(L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}}\lesssim\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{\frac{n}{p},p}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}}$

for all $s\in(0,\frac{n}{p})$.

We make the following claim, uniformly with respect to $s\in(0,\frac{n}{p})$,

(5.32)
$(L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}\hookrightarrow
s^{-\frac{1}{p}}(n-sp)L^{p^{*},p}({\mathbb{R}}^{n}).$

Assuming momentarily the validity of this claim, then (5.26) follows easily
from (5.31) and (5.28) (with $t=\frac{n}{p}$).

It remains to show (5.32). To proceed with, we will make use of the known
estimate (cf. [17, Corollary 3.3])

$K(u,f;L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n}))\approx\bigg{(}\int_{0}^{u^{p}}(f^{\\#*}(v))^{p}\,dv\bigg{)}^{\frac{1}{p}}.$

Accordingly, by Fubini’s theorem,

$\displaystyle\|f\|_{(L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}}^{p}$
$\displaystyle=\int_{0}^{\infty}(u^{-\frac{sp}{n}}K(u,f;L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n})))^{p}\frac{du}{u}$
(5.33)
$\displaystyle\approx\int_{0}^{\infty}u^{-\frac{sp}{n}}\int_{0}^{u}(f^{\\#*}(v))^{p}\,dv\frac{du}{u}$
$\displaystyle=\frac{n}{sp}\int_{0}^{\infty}(v^{\frac{1}{p^{*}}}f^{\\#*}(v))^{p}\frac{dv}{v}.$

We can compare $f^{\\#}$ and $f^{**}$ via the following weak-type estimate
(cf. [1, Proposition 8.10, p. 398])

$(f-f_{\infty})^{**}(u)\lesssim\int_{u}^{\infty}f^{\\#*}(v)\frac{dv}{v}$

where $f_{\infty}:=\lim_{|Q|\to\infty}\frac{1}{|Q|}\int_{Q}f$. Note that
$f_{\infty}=0$ since $f\in L^{p}({\mathbb{R}}^{n})$. Accordingly, by Hardy’s
inequality (see, e.g., [28, p. 196]),

$\displaystyle\|f\|_{L^{p^{*},p}({\mathbb{R}}^{n})}$
$\displaystyle\leq\bigg{(}\int_{0}^{\infty}[u^{\frac{1}{p^{*}}}f^{**}(u)]^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{\infty}\bigg{(}u^{\frac{1}{p^{*}}}\int_{u}^{\infty}f^{\\#*}(v)\frac{dv}{v}\bigg{)}^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim
p^{*}\bigg{(}\int_{0}^{\infty}u^{\frac{p}{p^{*}}}(f^{\\#*}(u))^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\approx
s^{\frac{1}{p}}p^{*}\|f\|_{(L^{p}({\mathbb{R}}^{n}),\text{BMO}({\mathbb{R}}^{n}))_{\frac{sp}{n},p}}$

where we have used (5.33) in the last step. The proof of (5.32) is complete.

Case 3: Assume $t=\frac{n}{p}$ and $s\to t^{-}$. Let $t_{0}\in(0,t)$ be fixed
and choose $\theta\in(0,1)$ such that $s=(1-\theta)t_{0}+\theta t$. Then there
exists a constant $C>0$, which is independent of $s$, such that

(5.34)
$\|f\|_{(\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\leq
C\,\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$

Indeed, since (see, e.g., [2, Section 6.4, pp. 149–153])

$\dot{B}^{t_{0}}_{p,\min\\{p,2\\}}({\mathbb{R}}^{n})=(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{t_{0}}{t},\min\\{p,2\\}}$

and

$\dot{B}^{t_{0}}_{p,\min\\{p,2\\}}({\mathbb{R}}^{n})\hookrightarrow\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),$

we have

(5.35)
$((L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p(}{\mathbb{R}}^{n}))_{\frac{t_{0}}{t},\min\\{p,2\\}},\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}\hookrightarrow(\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}$

with related embedding constant independent of $s$. Furthermore, in virtue of
the sharp version of the reiteration formula given in Lemma 4.6(ii) and (4.2)
(and taking into account that $\theta\to 1^{-}$), the following holds

$(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{1-(1-\theta)(1-\frac{t_{0}}{t}),p}\hookrightarrow((L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{t_{0}}{t},\min\\{p,2\\}},\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}$

uniformly with respect to $\theta$, or equivalently, by Lemma 4.8,

(5.36)
$\|f\|_{((L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{t_{0}}{t},\min\\{p,2\\}},\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\lesssim\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

Putting together (5.35) and (5.36), the estimate (5.34) is achieved.

By the well-known fact (see, e.g., [32, Theorem 1(i), Section 5.2.3, p. 242])
that $(-\Delta)^{\frac{t_{0}}{2}}$ acts as an isomorphism from
$\dot{H}^{t_{0},p}({\mathbb{R}}^{n})$ onto $L^{p}({\mathbb{R}}^{n})$ and from
$\dot{H}^{t,p}({\mathbb{R}}^{n})$ onto
$\dot{H}^{t-t_{0},p}({\mathbb{R}}^{n})$, we easily derive that

$K(u,f;\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))\approx
K(u,(-\Delta)^{\frac{t_{0}}{2}}f;L^{p}({\mathbb{R}}^{n}),\dot{H}^{t-t_{0},p}({\mathbb{R}}^{n}))$

and thus

$\|f\|_{(\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\approx\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t-t_{0}}_{p}({\mathbb{R}}^{n}))_{\theta,p}}$

where the hidden constants are independent of $\theta$. According to Lemma
4.8, the last estimate can be equivalently rewritten as

(5.37)
$\|f\|_{(\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}\approx\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{\dot{W}^{\theta(t-t_{0}),p}({\mathbb{R}}^{n}),t-t_{0}}.$

Since $0<\theta(t-t_{0})<t-t_{0}<t=\frac{n}{p}$, we can apply Case 1 so that
(take into account that $f\in W^{s,p}({\mathbb{R}}^{n})$ and, in particular,
$f\in H^{t_{0},p}({\mathbb{R}}^{n})$ since $t_{0}<s$)

(5.38)
$\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{L^{\frac{np}{n-(s-t_{0})p},p}({\mathbb{R}}^{n})}\lesssim(1-\theta)^{\frac{1}{p}}\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{\dot{W}^{\theta(t-t_{0}),p}({\mathbb{R}}^{n}),t-t_{0}}.$

As usual, we denote by $I_{\alpha},\,\alpha\in(0,n),$ the Riesz potential
operator

$(I_{\alpha}f)(x):=c_{n,\alpha}\int_{{\mathbb{R}}^{n}}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy,\qquad
x\in{\mathbb{R}}^{n}.$

By well-known mapping properties of $I_{\alpha}$ (cf. [27, Chapter V, Section
1.2])

(5.39) $I_{t_{0}}:L^{\frac{np}{n-(s-t_{0})p},p}({\mathbb{R}}^{n})\to
L^{p^{*},p}({\mathbb{R}}^{n}).$

However, our method requires to control the norm of this operator with respect
to $s$. Accordingly, to make the exposition self-contained, we provide below a
detailed proof of (5.39) using standard techniques. It follows from O’Neil’s
inequality [26, Lemma 1.5] that

$(I_{t_{0}}f)^{*}(u)\lesssim\int_{u}^{\infty}f^{**}(v)v^{\frac{t_{0}}{n}-1}\,dv$

and thus, by Hardy’s inequalities (cf. [28, p. 196]),

$\displaystyle\|I_{t_{0}}f\|_{L^{p^{*},p}({\mathbb{R}}^{n})}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{\infty}u^{\frac{p}{p^{*}}}\bigg{(}\int_{u}^{\infty}f^{**}(v)v^{\frac{t_{0}}{n}-1}\,dv\bigg{)}^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim
p^{*}\bigg{(}\int_{0}^{\infty}\big{(}u^{\frac{n-(s-t_{0})p}{np}}f^{**}(u)\big{)}^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim\frac{p^{*}}{1-\frac{1}{p}+\frac{s-t_{0}}{n}}\bigg{(}\int_{0}^{\infty}(u^{\frac{n-(s-t_{0})p}{np}}f^{*}(u))^{p}\frac{du}{u}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\approx\frac{1}{n-sp}\|f\|_{L^{\frac{np}{n-(s-t_{0})p},p}({\mathbb{R}}^{n})}$

where we have used $s\to t^{-}=\big{(}\frac{n}{p}\big{)}^{-}$ in the last
step. This proves (cf. (5.39))

(5.40) $\|I_{t_{0}}\|_{L^{\frac{np}{n-(s-t_{0})p},p}({\mathbb{R}}^{n})\to
L^{p^{*},p}({\mathbb{R}}^{n})}\lesssim(n-sp)^{-1}$

uniformly with respect to $s\to t^{-}$.

Since $I_{t_{0}}(-\Delta)^{\frac{t_{0}}{2}}f=f$, in light of (5.40), (5.38),
(5.37) and (5.34) we get

$\displaystyle\|f\|_{L^{p^{*},p}({\mathbb{R}}^{n})}$
$\displaystyle\lesssim(n-sp)^{-1}\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{L^{\frac{np}{n-(s-t_{0})p},p}({\mathbb{R}}^{n})}$
$\displaystyle\lesssim(n-sp)^{\frac{1}{p}-1}\|(-\Delta)^{\frac{t_{0}}{2}}f\|_{\dot{W}^{\theta(t-t_{0}),p}({\mathbb{R}}^{n}),t-t_{0}}$
$\displaystyle\approx(n-sp)^{\frac{1}{p}-1}\|f\|_{(\dot{H}^{t_{0},p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\theta,p}}$
$\displaystyle\lesssim(n-sp)^{\frac{1}{p}-1}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

The proof is finished. ∎

As an immediate consequence of Theorem 5.3 and the well-known inequality
between Lorentz spaces (cf. [28, p. 192])

$\Big{(}\frac{q}{p}\Big{)}^{\frac{1}{q}}\|f\|_{L^{p,q}({\mathbb{R}}^{n})}\leq\Big{(}\frac{r}{p}\Big{)}^{\frac{1}{r}}\|f\|_{L^{p,r}({\mathbb{R}}^{n})},\qquad
r<q,$

we obtain the following

######  Corollary 5.5.

Let $0<s<t\leq 1,1<p<\frac{n}{s}$ and $p^{*}=\frac{np}{n-sp}$. Assume $f\in
W^{s,p}({\mathbb{R}}^{n})$. Then there exists $C=C(n,p,t)>0$ such that

$\|f\|^{p}_{L^{p^{*}}({\mathbb{R}}^{n})}\leq
C\,s(t-s)(n-sp)^{1-p}\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}^{p}.$

## Appendix B: Sharp relationships between Sobolev–Triebel–Lizorkin spaces

The methodology proposed in [5] relies on sharp comparison estimates between
$\dot{W}^{s,p}({\mathbb{R}}^{n}),\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})$ and
$\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})$ using tools from harmonic analysis. We
now provide an approach to these results relying on the interpolation
techniques presented in Section 4.

Let $s\in(0,1)$ and $p\in(1,\infty)$. Recall that (cf. Lemma 2.1)

(5.41) $\dot{W}^{s,p}({\mathbb{R}}^{n})=\dot{F}_{p,p}^{s}({\mathbb{R}}^{n}),$
$\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})\hookrightarrow\dot{W}^{s,p}({\mathbb{R}}^{n}),\qquad
p\geq 2,$

and

$\dot{W}^{s,p}({\mathbb{R}}^{n})\hookrightarrow\dot{F}_{p,2}^{s}({\mathbb{R}}^{n}),\qquad
p\leq 2.$

The sharp behavior of the related equivalence constants in these embeddings
plays a key role in the Bourgain–Brezis–Mironescu method. In particular, note
that the family of semi-norms
$\\{\|\cdot\|_{\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})}:s\in(0,1)\\}$ converges to
$\|\cdot\|_{\dot{F}_{p,p}^{1}({\mathbb{R}}^{n})}$ as $s\rightarrow 1^{-}$,
however this is not the case for
$\\{\|\cdot\|_{\dot{W}^{s,p}({\mathbb{R}}^{n})}:s\in(0,1)\\}$ (cf. (1.2)). In
particular, the hidden equivalence constants in (5.41) must exhibit a certain
blow up as $s$ approaches the limiting values $0$ and $1$. The precise answer
is contained in the following

######  Theorem 5.6.

Let $0<s<t<\infty$ and $p\in(1,\infty)$. Then there exists a positive constant
$C$, which is independent of $s$, such that

  1. (a)

$\displaystyle
C^{-1}\bigg{(}\frac{1}{s^{\frac{1}{\max\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\max\\{p,2\\}}}}\bigg{)}\|f\|_{\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})}$
$\displaystyle\leq\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}$ (5.42)
$\displaystyle\leq
C\bigg{(}\frac{1}{s^{\frac{1}{\min\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\min\\{p,2\\}}}}\bigg{)}\|f\|_{\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})},$

  2. (b)

$C\bigg{(}\frac{1}{s^{\frac{1}{p}}}+\frac{1}{(t-s)^{\frac{1}{p}}}\bigg{)}\|f\|_{\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})}\leq\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t},\qquad
p\leq 2,$

  3. (c)

$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\leq
C\bigg{(}\frac{1}{s^{\frac{1}{p}}}+\frac{1}{(t-s)^{\frac{1}{p}}}\bigg{)}\|f\|_{\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})},\qquad
p\geq 2.$

######  Remark 5.7.

The previous result can be understood as a generalization to higher order
smoothness of [5, Theorems 1.2 and 1.5] (which corresponds to the classical
case $t=1$, cf. (1.18)).

###### Proof of Theorem 5.6.

(a): By Lemma 4.8, for every $s\in(0,t)$,

(5.43)
$\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}\approx\|f\|_{(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))_{\frac{s}{t},p}}.$

The interpolation space given in the right-hand side of the previous estimate
can be explicitly computed via the retraction method. Specifically,
$(L^{p}({\mathbb{R}}^{n}),\dot{H}^{t,p}({\mathbb{R}}^{n}))$ can be identified
with the vector-valued pair
$(L^{p}({\mathbb{R}}^{n};\ell_{2}({{\mathbb{Z}}})),\ell^{p}({\mathbb{R}}^{n};\ell_{2}^{t}({{\mathbb{Z}}})))$
and thus, by Lemma 4.1,

(5.44)
$\|(f_{j})\|_{(L^{p}({\mathbb{R}}^{n};\ell_{2}({{\mathbb{Z}}})),L^{p}({\mathbb{R}}^{n};\ell_{2}^{t}({{\mathbb{Z}}})))_{\frac{s}{t},p}}\approx\|(f_{j})\|_{L^{p}({\mathbb{R}}^{n};(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},p})}.$

Furthermore, according to Lemma 4.2,

$\displaystyle\bigg{(}\frac{1}{s^{\frac{1}{\max\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\max\\{p,2\\}}}}\bigg{)}\|\xi\|_{\ell_{p}^{s}({{\mathbb{Z}}})}$
$\displaystyle\lesssim\|\xi\|_{(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},p}}$
(5.45)
$\displaystyle\lesssim\bigg{(}\frac{1}{s^{\frac{1}{\min\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\min\\{p,2\\}}}}\bigg{)}\|\xi\|_{\ell_{p}^{s}({{\mathbb{Z}}})}.$

As a combination of (5.44) and (5.45),

$\displaystyle\bigg{(}\frac{1}{s^{\frac{1}{\max\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\max\\{p,2\\}}}}\bigg{)}\|(f_{j})\|_{L^{p}({\mathbb{R}}^{n};\ell_{p}^{s}({{\mathbb{Z}}}))}$
$\displaystyle\lesssim\|(f_{j})\|_{(L^{p}({\mathbb{R}}^{n};\ell_{2}({{\mathbb{Z}}})),L^{p}({\mathbb{R}}^{n};\ell_{2}^{t}({{\mathbb{Z}}})))_{\frac{s}{t},p}}$
(5.46)
$\displaystyle\lesssim\bigg{(}\frac{1}{s^{\frac{1}{\min\\{p,2\\}}}}+\frac{1}{(t-s)^{\frac{1}{\min\\{p,2\\}}}}\bigg{)}\|(f_{j})\|_{L^{p}({\mathbb{R}}^{n};\ell_{p}^{s}({{\mathbb{Z}}}))}.$

By the well-known fact that $\dot{F}_{p,p}^{s}({\mathbb{R}}^{n})$ is a retract
of $L^{p}({\mathbb{R}}^{n};\ell_{p}^{s}({{\mathbb{Z}}}))$, we can put together
(5.46) and (5.43) to arrive at (5.42).

(b): Let $p\leq 2$. Applying Lemmas 4.2 and 4.5(i),

(5.47)
$\|\xi\|_{\ell_{2}^{s}({{\mathbb{Z}}})}\approx(s(t-s))^{\frac{1}{2}}\|\xi\|_{(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},2}}\lesssim(s(t-s))^{\frac{1}{p}}\|\xi\|_{(\ell_{2}({{\mathbb{Z}}}),\ell_{2}^{t}({{\mathbb{Z}}}))_{\frac{s}{t},p}}.$

Taking into account that $\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})$ is a retract of
$L^{p}({\mathbb{R}}^{n};\ell_{2}^{s}({{\mathbb{Z}}}))$, it follows now from
(5.43), (5.44) and (5.47) that

$(s(t-s))^{-\frac{1}{p}}\|f\|_{\dot{F}_{p,2}^{s}({\mathbb{R}}^{n})}\lesssim\|f\|_{\dot{W}^{s,p}({\mathbb{R}}^{n}),t}.$

The proof of (c) follows similar ideas as those given in (b). ∎

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