###### Abstract

We prove optimal regularity results in $L_{p}$-based function spaces in space
and time for a large class of linear parabolic equations with a nonlocal
elliptic operator in bounded domains with limited smoothness. Here the
nonlocal operator is given by a strongly elliptic and even pseudodifferential
operator of order $2a$ ($0<a<1$) with nonsmooth $x$-dependent coefficients.
The proofs are based on general results on maximal $L_{p}$-regularity and its
relation to $\mathcal{R}$-boundedness of the resolvent of the associated
(elliptic) operator. Finally, we apply these results to show existence of
strong solutions locally in time for a class of nonlinear nonlocal parabolic
equations, which include a fractional nonlinear diffusion equation and a
fractional porous medium equation after a transformation. The nonlinear
results are new for operators on domains with boundary; the linear results are
so when $P$ is $x$-dependent nonsymmetric.

Key words: Fractional Laplacian; even pseudodifferential operator; Dirichlet
problem; nonsmooth coefficients; maximal regularity; nonlinear nonlocal
parabolic equations; fractional porous medium equation.
MSC (2020): Primary: 35S15, 35R11, Secondary: 35K61, 35S16, 47G30, 60G52

##  1 Introduction

Optimal regularity results for solutions of linear parabolic equations are
essential for the construction of regular solutions of corresponding nonlinear
parabolic evolution equations with the aid of the contraction mapping
principle. Of particular importance are results for $L_{q}$-based Sobolev type
function space for general $q\in(1,\infty)$ (not necessarily $q=2$) since in
applications to nonlinear equations one uses Sobolev type embeddings for $q$
sufficiently large. This topic is intensively studied for parabolic
differential equations. But in the case of nonlocal operators in domains with
boundary there are only few results. This is of a particular challenge since
results on elliptic regularity in the standard spaces often fail.

In this contribution we study the heat equation for a nonlocal operator $P$ of
order $2a\in(0,2)$ (strongly elliptic and even),

$\begin{split}\partial_{t}u+Pu&=f\quad\text{ on }\Omega\times I,\
I=\,(0,T)\,,\\\ u&=0\quad\text{ on }(\mathbb{R}^{n}\setminus\Omega)\times
I,\\\ u|_{t=0}&=0.\end{split}$ (1.1)

Estimates of the solution in $L_{q}$-based function spaces were shown by the
second author [23, 24, 26] for $1<q<\infty$ in the case when $P$ is symmetric
and translation-invariant. The results are restricted to this case since the
proofs rely on a Markovian property obtainable in that case. However, interior
estimates (and global estimates on $\mathbb{R}^{n+1}$) could be shown by
another method in $x$-dependent cases [23]. We note that the works [23, 24]
assumed $\Omega$ to be $C^{\infty}$.

After the extension in Abels-Grubb [2] of the general treatment of boundary
problems for $P$ to cases with nonsmooth domains $\Omega$, the heat equation
results have been followed up in [26], the case $q\neq 2$ still limited to
symmetric operators with the Markovian property.

In the present work we address the question of solvability of (1.1) for
$x$-dependent operators $P=\operatorname{Op}(p(x,\xi))$ in an $L_{q}$-setting
($1<q<\infty$), when both $p$ and $\Omega$ are nonsmooth. The symbols are
assumed to be classical, strongly elliptic and even (this is short for an
alternating symmetry property of the homogeneous terms (3.1)), and the
resolvent estimates are obtained for a large class of nonsymmetric operators
not necessarily having the Markovian property. This includes the important
example $P=L^{a}$, where $L=-\nabla\cdot A(x)\nabla+b(x)$, $A(x)$ being a
smooth $(n\times n)$-matrix with positive lower bound, and
$\operatorname{Re}b(x)\geq 0$; $A(x)$ is assumed real for
$x\in\partial\Omega$. Related operators are treated in a general framework on
compact boundaryless manifolds by Roidos and Shao [36].

We draw on several tools: The interior regularity is obtained by the general
strategy introduced in [23] where a symbolic calculus is set up for symbols
with an extra parameter in the style of [27, 18, 19], allowing the
construction of a symbolic inverse (here nonsmooth results may be included by
a simple approximation). Another tool is that the resolvent estimates at the
boundary can be obtained from the $x$-independent case by the technique
presented in [2, Section 6]: Here the forward operator $P-\lambda$ is compared
to its principal part “frozen” at a boundary point $x_{0}$, and an estimate
can be obtained in a small neighborhood of $x_{0}$ by a scaling that flattens
the symbol of $P$ and the boundary.

Still another tool to obtain sharp solvability properties in $L_{q}$-spaces,
is to aim for $\mathcal{R}$-bounds on the resolvent of the Dirichlet
realization of $P$ on $\Omega$. This has to the best of our knowledge not been
attempted before for these fractional-order problems. Here we use the theory
laid out in e.g. Denk-Hieber-Prüss [11] and Prüss-Simonett [35].

The main linear results are:

###### Theorem 1.1

Let $0<a<1$, $\tau>2a$, $1<q<\infty$, and let $\Omega$ be bounded with
$C^{1+\tau}$-boundary. Let $P=\operatorname{Op}(p)$ with symbol $p(x,\xi)\in
C^{\tau}S^{2a}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, strongly elliptic and
even, and assume that the principal symbol $p_{0}(x,\xi)$ is real positive at
each boundary point $x\in\partial\Omega$. Denote the $L_{q}$-Dirichlet
realization by $P_{D}$.

Then the resolvent $(P_{D}-\lambda)^{-1}$ exists for $\lambda$ in a set
$V_{\delta,K}$ with $0<\delta<\frac{\pi}{2}$, $K\geq 0$,

$V_{\delta,K}=\\{\lambda\in\mathbb{C}\setminus\\{0\\}\mid\operatorname{arg}\lambda\in[\tfrac{\pi}{2}-\delta,\tfrac{3\pi}{2}+\delta],|\lambda|\geq
K\\},$ (1.2)

and the operator family $\\{\lambda(P_{D}-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\}\subset\mathcal{L}(L_{q}(\Omega))$ is $\mathcal{R}$-bounded.

The domain of $P_{D}$ is a so-called $a$-transmission space
$H_{q}^{a(2a)}(\overline{\Omega})$ [22],[2], denoted
$D_{q}(\overline{\Omega})$ for short.

###### Theorem 1.2

Assumptions as in Theorem 1.1. Let $1<p<\infty$.

For any $f\in L_{p}(I;L_{q}(\Omega))$, any $T>0$, the heat equation (1.1) has
a unique solution $u\in C^{0}(\overline{I};L_{q}(\Omega))$ satisfying

$u\in L_{p}(I;D_{q}(\overline{\Omega}))\cap H^{1}_{p}(I;L_{q}(\Omega)).$ (1.3)

This is maximal $L_{p}$-regularity, shown here for the first time for
nonsymmetric fractional-order Dirichlet problems with $x$-dependent symbols.

There is also a solvability result with a nonzero local Dirichlet condition,
when $q<(1-a)^{-1}$:

###### Theorem 1.3

In addition to the assumptions in Theorem 1.1, let $\tau>2a+1$, $q<(1-a)^{-1}$
and $1<p<\infty$. The nonhomogeneous heat problem

$\begin{split}\partial_{t}u+Pu&=f\text{ on }\Omega\times I,\\\
\gamma_{0}(u/d_{0}^{a-1})&=\psi\text{ on }\partial\Omega\times I,\\\
u&=0\text{ on }(\mathbb{R}^{n}\setminus\Omega)\times I,\\\
u|_{t=0}&=0,\end{split}$ (1.4)

has for $f\in L_{p}(I;L_{q}(\Omega))$, $\psi\in
L_{p}(I;B_{q,q}^{a+1-1/q}(\partial\Omega))\cap
H_{p}^{1}(I;B_{q,q}^{\varepsilon}(\partial\Omega))$ with $\psi(x,0)=0$
($\varepsilon>0$), and any $T>0$ a unique solution $u$ satisfying

$u\in L_{p}(I;H_{q}^{(a-1)(2a)}(\overline{\Omega}))\cap
H_{p}^{1}(I;L_{q}(\Omega)).$

From Theorem 1.2 we moreover deduce nonlinear results. Consider the parabolic
problem

$\displaystyle\partial_{t}u+a_{0}(x,u)Pu$ $\displaystyle=f(x,u)$
$\displaystyle\text{in }\Omega\times(0,T),$ (1.5) $\displaystyle u$
$\displaystyle=0$ $\displaystyle\text{on
}(\mathbb{R}^{n}\setminus\Omega)\times(0,T),$ $\displaystyle u|_{t=0}$
$\displaystyle=u_{1}$ $\displaystyle\text{in }\Omega,$

for some $T>0$.

###### Theorem 1.4

Let $\Omega$ be a bounded domain with $C^{1+\tau}$-boundary for some
$\tau>2a$, and let $1<p,q<\infty$ be such that
$(a+\tfrac{1}{q})(1-\tfrac{1}{p})-\tfrac{n}{q}>0$. If $n=1$, assume moreover
$\frac{1}{q}<a$. Let $P$ be as in Theorem 1.1. Moreover, for an open set
$U\subset\mathbb{R}$ with $0\in U$, let $a_{0}\in
C^{\max(1,\tau)}(\mathbb{R}^{n}\times U,\mathbb{R})$ with $a_{0}(x,s)>0$ for
all $s\in U$ and $x\in\mathbb{R}^{n}$, let $f\colon\mathbb{R}^{n}\times
U\to\mathbb{R}$ be continuous in $(x,u)$ and locally Lipschitz in $u$, and let
$u_{0}\in(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}\cap
C^{\tau}(\overline{\Omega})$ with $\overline{u_{0}(\Omega)}\subset U$.

Then there are $\varepsilon_{0},T>0$ such that for every
$u_{1}\in(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}$ with
$\|u_{0}-u_{1}\|_{X_{\gamma},1}\leq\varepsilon_{0}$, the system (1.5)
possesses a unique solution

$u\in L_{p}((0,T);D_{q}(\overline{\Omega}))\cap
H^{1}_{p}((0,T);L_{q}(\Omega)).$

This leads in particular to solvability results for nonlinear diffusion
equations, including problems of the type of the porous medium equation, see
Corollary 7.3ff.

Earlier works on (1.1) have mostly been concerned with $P=(-\Delta)^{a}$ and
$x$-independent singular integral operator generalizations. To mention a few:
There are results on Schauder estimates and Hölder properties, by e.g.
Felsinger and Kassmann [14], Chang-Lara and Davila [6], Jin and Xiong [30];
and quite precise results on regularity in anisotropic Hölder spaces by
Fernandez-Real and Ros-Oton [15], and Ros-Oton and Vivas [39]. For
$P=(-\Delta)^{a}$, Leonori, Peral, Primo and Soria [33] showed
$L_{q}(I;L_{r}(\Omega))$ estimates; Biccari, Warma and Zuazua [4] showed
$L_{q}(I;B^{2a}_{q,r,loc}(\Omega))$-estimates for certain $r$, and Choi, Kim
and Ryu have in [7] shown weighted $L_{q}$-estimates. Results on
${\mathbb{R}}^{n}$ with $x$-dependence have been obtained by by Dong, Jung and
Kim [13]. As recalled further above, we have earlier shown maximal
$L_{q}$-regularity results on ${\mathbb{R}}^{n}$, and for domains $\Omega$ in
the translation-invariant symmetric case, [23], [26]. Roidos and Shao [36]
show maximal $L_{p}$-regularity for operators like
$(-\nabla\cdot\mathfrak{a}(x)\nabla)^{a}$ on compact boundaryless manifolds.
The latter includes nonlinear applications such as the fractional porous
medium equation; this is also treated in Vázques, de Pablo, Quirós and
Rodríguez [44] on ${\mathbb{R}}^{n}$ in Hölder spaces.

Plan of the paper: Section 2 recalls definitions of function spaces and
pseudodifferential operators. Section 3 presents our hypotheses on $P$ and
sets up the Dirichlet realization. Section 4 collects the needed features of
$\mathcal{R}$-boundedness and their connection with maximal
$L_{p}$-regularity. In Section 5, we deduce the main results on resolvent
estimates for the Dirichlet realization. In Section 6, this is applied to
obtain the linear results for time-dependent problems. Finally, in Section 7
we show some consequences for nonlinear evolution problems, including
fractional nonlinear diffusion equations and fractional porous medium
equations.

##  2 Prerequisites

###  2.1 Function spaces

The space $C^{k}({\mathbb{R}}^{n})\equiv C^{k}_{b}({\mathbb{R}}^{n})$ consists
of $k$-times differentiable functions with bounded norms
$\|u\|_{C^{k}}=\sup_{|\alpha|\leq k,x\in{\mathbb{R}}^{n}}|D^{\alpha}u(x)|$
($k\in{\mathbb{N}}_{0}$), and the Hölder spaces $C^{\tau}({\mathbb{R}}^{n})$,
$\tau=k+\sigma$ with $k\in{\mathbb{N}}_{0}$, $0<\sigma<1$, also denoted
$C^{k,\sigma}({\mathbb{R}}^{n})$, consists of functions $u\in
C^{k}({\mathbb{R}}^{n})$ with bounded norms
$\|u\|_{C^{\tau}}=\|u\|_{C^{k}}+\sup_{|\alpha|=k,x\neq
y}|D^{\alpha}u(x)-D^{\alpha}u(y)|/|x-y|^{\sigma}$. The latter definition
extends to Lipschitz spaces $C^{k,1}({\mathbb{R}}^{n})$. There are similar
spaces over subsets of ${\mathbb{R}}^{n}$. There are also the Hölder-Zygmund
spaces $C^{s}_{*}({\mathbb{R}}^{n})\equiv
B^{s}_{\infty,\infty}({\mathbb{R}}^{n})$ defined for $s\in\mathbb{R}$ with
good interpolation properties, coinciding with $C^{s}({\mathbb{R}}^{n})$ when
$s\in\mathbb{R}_{+}\setminus\mathbb{N}$.

The halfspaces $\mathbb{R}^{n}_{\pm}$ are defined by
$\mathbb{R}^{n}_{\pm}=\\{x\in{\mathbb{R}}^{n}\mid x_{n}\gtrless 0\\}$, with
points denoted $x=(x^{\prime},x_{n})$, $x^{\prime}=(x_{1},\dots,x_{n-1})$;
$\mathbb{R}^{1}_{\pm}$ is denoted $\mathbb{R}_{\pm}$. For a given real
function $\zeta\in C^{1+\tau}(\mathbb{R}^{n-1})$ (some $\tau>0$), we define
the curved halfspace ${\mathbb{R}}^{n}_{\zeta}$ by

${\mathbb{R}}^{n}_{\zeta}=\\{x\in\mathbb{R}^{n}\mid
x_{n}>\zeta(x^{\prime})\\};$ (2.1)

it is a $C^{1+\tau}$-domain.

By a bounded $C^{1+\tau}$-domain $\Omega$ we mean the following:
$\Omega\subset{\mathbb{R}}^{n}$ is open, bounded and nonempty, and every
boundary point $x_{0}$ has an open neighborhood $U$ such that, after a
translation of $x_{0}$ to $0$ and a suitable rotation, $U\cap\Omega$ equals
$U\cap{\mathbb{R}}^{n}_{\zeta}$ for a function $\zeta\in
C^{1+\tau}(\mathbb{R}^{n-1})$ with $\zeta(0)=0$.

Restriction from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}_{\pm}$ (or from
${\mathbb{R}}^{n}$ to $\Omega$ resp.
$\complement\overline{\Omega}={\mathbb{R}}^{n}\setminus\overline{\Omega}$) is
denoted $r^{\pm}$, extension by zero from $\mathbb{R}^{n}_{\pm}$ to
$\mathbb{R}^{n}$ (or from $\Omega$ resp. $\complement\overline{\Omega}$ to
${\mathbb{R}}^{n}$) is denoted $e^{\pm}$. (The notation is also used for
$\Omega={\mathbb{R}}^{n}_{\zeta})$.) Restriction from
$\overline{\mathbb{R}}^{n}_{+}$ or $\overline{\Omega}$ to
$\partial{\mathbb{R}}^{n}_{+}$ resp. $\partial\Omega$ is denoted $\gamma_{0}$.

When $\Omega$ is a $C^{1+\tau}$-domain, we denote by $d(x)$ a function that is
$C^{1+\tau}$ on $\overline{\Omega}$, positive on $\Omega$ and vanishes only to
the first order on $\partial\Omega$ (i.e., $d(x)=0$ and $\nabla d(x)\neq 0$
for $x\in\partial\Omega$). It is bounded above and below by the distance
$d_{0}(x)=\operatorname{dist}(x,\partial\Omega)$; see further details in [2].

Throughout the paper, $q$ satisfies $1<q<\infty$. The Bessel-potential spaces
$H^{s}_{q}({\mathbb{R}}^{n})$ are defined for $s\in{\mathbb{R}}$ by

$H_{q}^{s}(\mathbb{R}^{n})=\\{u\in\mathcal{S}^{\prime}({\mathbb{R}}^{n})\mid\mathcal{F}^{-1}(\langle{\xi}\rangle^{s}\hat{u})\in
L_{q}(\mathbb{R}^{n})\\},$ (2.2)

where $\mathcal{F}$ is the Fourier transform
$\hat{u}(\xi)=\mathcal{F}u(\xi)=\int_{{\mathbb{R}}^{n}}e^{-ix\cdot\xi}u(x)\,dx$,
and the function $\langle{\xi}\rangle$ equals $(|\xi|^{2}+1)^{\frac{1}{2}}$.
For $q=2$, this is the scale of $L_{2}$-Sobolev spaces, where the index $2$ is
usually omitted. $\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ is the Schwartz
space of temperate distributions, the dual space of
$\mathcal{S}({\mathbb{R}}^{n})$; the space of rapidly decreasing
$C^{\infty}$-functions. (The spaces can be defined for other values of $q$,
but some properties we need are linked to $q\in(1,\infty)$.)

For $s\in{\mathbb{N}}_{0}=\\{0,1,2,\dots\\}$, the spaces
$H_{q}^{s}({\mathbb{R}}^{n})$ are also denoted $W_{q}^{s}({\mathbb{R}}^{n})$
or $W^{s,q}({\mathbb{R}}^{n})$ in the literature. We moreover need to refer to
the Besov spaces $B^{s}_{q,q}({\mathbb{R}}^{n})$, also denoted
$B^{s}_{q}({\mathbb{R}}^{n})$, that coincide with the $W^{s}_{q}$-spaces when
$s\in\mathbb{R}_{+}\setminus\mathbb{N}$. They necessarily enter in connection
with boundary value problems in an $H^{s}_{q}$-context, because they are the
correct range spaces for trace maps
$\gamma_{j}u=(\partial_{n}^{j}u)|_{x_{n}=0}$:

$\gamma_{j}\colon\overline{H}^{s}_{q}({\mathbb{R}}^{n}_{+}),\overline{B}^{s}_{q,q}({\mathbb{R}}^{n}_{+})\to
B_{q,q}^{s-j-\frac{1}{q}}({\mathbb{R}}^{n-1}),\text{ for }s-j-\tfrac{1}{q}>0,$
(2.3)

(cf. (2.4)), surjectively and with a continuous right inverse; see e.g. the
overview in the introduction to [17]. For $q=2$, the two scales $H_{q}^{s}$
and $B_{q,q}^{s}$ are identical, but for $q\neq 2$ they are related by strict
inclusions: $H^{s}_{q}\subset B^{s}_{q,q}\text{ when }q>2$, $H^{s}_{q}\supset
B^{s}_{q,q}\text{ when }q<2$.

In relation to $\Omega$, (2.3) gives rise to two scales of spaces for
$s\in{\mathbb{R}}$:

$\begin{split}\overline{H}_{q}^{s}(\Omega)&=\\{u\in\mathcal{D}^{\prime}(\Omega)\mid
u=r^{+}U\text{ for some }U\in H_{q}^{s}(\mathbb{R}^{n})\\},\text{ the {\it
restricted} space},\\\ \dot{H}_{q}^{s}(\overline{\Omega})&=\\{u\in
H_{q}^{s}({\mathbb{R}}^{n})\mid\operatorname{supp}u\subset\overline{\Omega}\\},\text{
the {\it supported} space;}\end{split}$ (2.4)

here $\operatorname{supp}u$ denotes the support of $u$ (the complement of the
largest open set where $u=0$). $\overline{H}_{q}^{s}(\Omega)$ is in other
texts often denoted $H_{q}^{s}(\Omega)$ or $H_{q}^{s}(\overline{\Omega})$, and
$\dot{H}_{q}^{s}(\overline{\Omega})$ may be indicated with a ring, zero or
twiddle; the current notation stems from Hörmander [29, App. B.2]. For
$1<q<\infty$, there is an identification of $\overline{H}_{q}^{s}(\Omega)$
with the dual space of $\dot{H}_{q^{\prime}}^{-s}(\overline{\Omega})$,
$\frac{1}{q^{\prime}}=1-\frac{1}{q}\,$, in terms of a duality extending the
sesquilinear scalar product
$\langle{f,g}\rangle=\int_{\Omega}f\,\overline{g}\,dx$.

In discussions of heat operator problems it will sometimes be convenient to
refer to anisotropic Bessel-potential spaces over
$\mathbb{R}^{n+1}=\\{(x,t)\mid x\in{\mathbb{R}}^{n},t\in\mathbb{R}\\}$. With
$d\in\mathbb{R}_{+}$, we define

$\\{\xi,\tau\\}\equiv(\langle{\xi}\rangle^{2d}+\tau^{2})^{1/(2d)},$ (2.5)

leading to the “order-reducing” operators (defined for all $s\in{\mathbb{R}}$)

$\Theta^{s}u=\operatorname{Op}(\\{\xi,\tau\\}^{s})u\equiv\mathcal{F}^{-1}_{(\xi,\tau)\to(x,t)}(\\{\xi,\tau\\}^{s}\mathcal{F}_{(x,t)\to(\xi,\tau)}u),$

Then we define:

$H^{(s,s/d)}_{q}({\mathbb{R}}^{n}\\!\times\\!\mathbb{R})=\Theta^{-s}L_{q}(\mathbb{R}^{n+1});$
(2.6)

for $1<q<\infty$, $s\in\mathbb{R}$. Note that the case $s=0$ gives
$L_{q}(\mathbb{R}^{n+1})$, and the case $s=d$ gives

$H^{(d,1)}_{q}({\mathbb{R}}^{n}\\!\times\\!\mathbb{R})=L_{q}(\mathbb{R};H_{q}^{d}({\mathbb{R}}^{n}))\cap
H^{1}_{q}(\mathbb{R};L_{q}({\mathbb{R}}^{n})).$ (2.7)

More on these spaces in [23].

###  2.2 Pseudodifferential operators and transmission spaces

Recall that the pseudodifferential operator (briefly expressed: $\psi$do) $P$
on ${\mathbb{R}}^{n}$ with symbol
$p\colon{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\to\mathbb{C}$ is defined as

$(Pu)(x)=\mathcal{F}^{-1}_{\xi\to
x}\bigl{(}p(x,\xi)(\mathcal{F}u)(\xi)\bigr{)}=\operatorname{Op}(p)u(x),$ (2.8)

where
$(\mathcal{F}u)(\xi)=\hat{u}(\xi)=\int_{{\mathbb{R}}^{n}}e^{-ix\cdot\xi}u(x)\,dx$
denotes the Fourier transform of $u$ for suitable
$u\colon{\mathbb{R}}^{n}\to\mathbb{C}$, and $\mathcal{F}^{-1}$ is the inverse
Fourier transform. Under suitable conditions on the symbol $p$, $P$ is well-
defined for $u\in\mathcal{S}({\mathbb{R}}^{n})$, and the definition extends to
much more general spaces. (Further details and references are given in [2],
[26].)

For $\tau>0$, $m\in\mathbb{R}$, the space
$C^{\tau}S^{m}_{1,0}(\mathbb{R}^{n}\times{\mathbb{R}}^{n})$ of
$C^{\tau}$-symbols of order $m$ consists of functions
$p\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{C}$ that are continuous
with respect to $(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ and
$C^{\infty}$ with respect to $\xi\in\mathbb{R}^{n}$, such that for every
$\alpha\in\mathbb{N}_{0}^{n}$ we have: $\partial_{\xi}^{\alpha}p(x,\xi)$ is in
$C^{\tau}(\mathbb{R}^{n})$ with respect to $x$ and satisfies for all
$\xi\in\mathbb{R}^{n}$, $\alpha\in{\mathbb{N}}_{0}^{n}$,

$\|\partial_{\xi}^{\alpha}p(\cdot,\xi)\|_{C^{\tau}(\mathbb{R}^{n})}\leq
C_{\alpha}\langle{\xi}\rangle^{m-|\alpha|},$ (2.9)

for some $C_{\alpha}>0$. The symbol space is a Fréchet space with the semi-
norms

$|p|_{k,C^{\tau}S^{m}_{1,0}(\mathbb{R}^{n}\times\mathbb{R}^{n})}:=\max_{|\alpha|\leq
k}\sup_{\xi\in\mathbb{R}^{n}}\langle{\xi}\rangle^{-m+|\alpha|}\|\partial_{\xi}^{\alpha}p(\cdot,\xi)\|_{C^{\tau}(\mathbb{R}^{{n}})}\quad\text{for
}k\in\mathbb{N}_{0}.$ (2.10)

For such symbols there holds:

$\operatorname{Op}(p)\colon H^{s+m}_{q}(\mathbb{R}^{n})\to
H^{s}_{q}(\mathbb{R}^{n})\quad\text{for all }|s|<\tau,$ (2.11)

where the operator norm for each $s$ is estimated by a semi-norm for some
$k\in\mathbb{N}_{0}$ (depending on $s$).

The space of $C^{\infty}$-symbols
$S^{m}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ of order
$m\in{\mathbb{R}}$ equals
$\bigcap_{\tau>0}C^{\tau}S^{m}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$.

The subspaces of _classical_ symbols
$C^{\tau}S^{m}(\mathbb{R}^{n}\times{\mathbb{R}}^{n})$ resp.
$S^{m}(\mathbb{R}^{n}\times{\mathbb{R}}^{n})$ consist of those functions in
$C^{\tau}S^{m}_{1,0}(\mathbb{R}^{n}\times{\mathbb{R}}^{n})$ resp.
$S^{m}_{1,0}(\mathbb{R}^{n}\times{\mathbb{R}}^{n})$ that moreover have
expansions into terms $p_{j}$ homogeneous in $\xi$ of degree $m-j$ for
$|\xi|\geq 1$, all $j\in\mathbb{N}_{0}$, such that for all
$\alpha\in{\mathbb{N}}_{0}^{n}$, $J\in{\mathbb{N}}_{0}$ there is some
$C_{\alpha,J}>0$ satisfying

$\|\partial_{\xi}^{\alpha}\bigl{(}p(\cdot,\xi)-{\sum}_{j<J}p_{j}(\cdot,\xi)\bigr{)}\|_{C^{\tau}(\mathbb{R}^{n})}\leq
C_{\alpha,J}\langle{\xi}\rangle^{m-J-|\alpha|}\quad\text{for all
}\xi\in\mathbb{R}^{n}.$ (2.12)

The operator $P=\operatorname{Op}(p)$ and the symbol $p$ are said to be
_elliptic_ , when, for a sufficiently large $R>0$ there is a $c>0$ such that

$|p(x,\xi)|\geq c|\xi|^{m}\quad\text{for all }|\xi|\geq
R,x\in{\mathbb{R}}^{n};$

this holds in the classical case if and only if (with some $c^{\prime}>0$)

$|p_{0}(x,\xi)|\geq c^{\prime}|\xi|^{m}\quad\text{for all }|\xi|\geq
1,x\in{\mathbb{R}}^{n}.$

A special role in the theory is played by the _order-reducing operators_.
There is a simple definition of operators $\Xi_{\pm}^{t}$ on
${\mathbb{R}}^{n}$, $t\in{\mathbb{R}}$,

$\Xi_{\pm}^{t}=\operatorname{Op}(\chi_{\pm}^{t}),\quad\chi_{\pm}^{t}=(\langle{\xi^{\prime}}\rangle\pm
i\xi_{n})^{t};$ (2.13)

they preserve support in $\overline{\mathbb{R}}^{n}_{\pm}$, respectively. The
functions $(\langle{\xi^{\prime}}\rangle\pm i\xi_{n})^{t}$ do not satisfy all
the estimates for $S^{t}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, but
definition (2.8) applies anyway. There is a more refined choice
$\Lambda_{\pm}^{t}$ [17, 22], with symbols $\lambda_{\pm}^{t}(\xi)$ that do
satisfy all the estimates for
$S^{t}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$; here
$\overline{\lambda_{+}^{t}}=\lambda_{-}^{t}$. The symbols have holomorphic
extensions in $\xi_{n}$ to the complex halfspaces
${\mathbb{C}}_{\mp}=\\{z\in{\mathbb{C}}\mid\operatorname{Im}z\lessgtr 0\\}$;
it is for this reason that the operators preserve support in
$\overline{\mathbb{R}}^{n}_{\pm}$, respectively. Operators with that property
are called “plus” resp. “minus” operators. There is also a pseudodifferential
definition $\Lambda_{\pm}^{(t)}$ adapted to the situation of a smooth domain
$\Omega$, cf. [22]. For nonsmooth domains, one applies the operators
$\Xi^{t}_{\pm}$ in localizations where a piece of $\Omega$ is carried over to
a piece of ${\mathbb{R}}^{n}_{+}$.

It is elementary to see by the definition of the spaces
$H_{q}^{s}(\mathbb{R}^{n})$ in terms of Fourier transformation, that the
operators define homeomorphisms for all $s$: $\Xi^{t}_{\pm}\colon
H_{q}^{s}(\mathbb{R}^{n})\overset{\sim}{\rightarrow}H_{q}^{s-t}(\mathbb{R}^{n})$,
$\Lambda^{t}_{\pm}\colon
H_{q}^{s}(\mathbb{R}^{n})\overset{\sim}{\rightarrow}H_{q}^{s-t}(\mathbb{R}^{n})$.
The special interest is that the “plus”/”minus” operators also define
homeomorphisms related to $\overline{\mathbb{R}}^{n}_{+}$ and
$\overline{\Omega}$, for all $s\in{\mathbb{R}}$:
$\Xi^{t}_{+}\colon\dot{H}_{q}^{s}(\overline{\mathbb{R}}^{n}_{+})\overset{\sim}{\rightarrow}\dot{H}_{q}^{s-t}(\overline{\mathbb{R}}^{n}_{+})$,
$r^{+}\Xi^{t}_{-}e^{+}\colon\overline{H}_{q}^{s}({\mathbb{R}}^{n}_{+})\overset{\sim}{\rightarrow}\overline{H}_{q}^{s-t}({\mathbb{R}}^{n}_{+})$,
with similar statements for $\Lambda^{t}_{\pm}$, and for $\Lambda^{(t)}_{\pm}$
relative to $\Omega$. Moreover, the operators $\Xi^{t}_{+}$ and
$r^{+}\Xi^{t}_{-}e^{+}$ identify with each other’s adjoints over
$\overline{\mathbb{R}}^{n}_{+}$, because of the support preserving properties.
There is a similar statement for $\Lambda^{t}_{+}$ and
$r^{+}\Lambda^{t}_{-}e^{+}$, and for $\Lambda^{(t)}_{+}$ and
$r^{+}\Lambda^{(t)}_{-}e^{+}$ relative to the set $\Omega$.

The special $\mu$-transmission spaces were introduced by Hörmander [28] for
$q=2$, and developed in detail for $1<q<\infty$ by Grubb [22]:

$\begin{split}H_{q}^{\mu(s)}(\overline{\mathbb{R}}^{n}_{+})&=\Xi_{+}^{-\mu}e^{+}\overline{H}_{q}^{s-\mu}({\mathbb{R}}^{n}_{+})=\Lambda_{+}^{-\mu}e^{+}\overline{H}_{q}^{s-\mu}({\mathbb{R}}^{n}_{+}),\quad\text{if
}s>\mu-\tfrac{1}{q^{\prime}},\\\
H_{q}^{\mu(s)}(\overline{\Omega})&=\Lambda_{+}^{(-\mu)}e^{+}\overline{H}_{q}^{s-\mu}(\Omega),\quad\text{if
}s>\mu-\tfrac{1}{q^{\prime}};\end{split}$ (2.14)

here $\mu>-1$. With $\mu=a$, they are the appropriate solution spaces for
homogeneous Dirichlet problems for the operators of order $2a$ that we shall
study. For problems with a nonhomogeneous local Dirichlet condition they enter
with $\mu=a-1$. There holds $H_{q}^{\mu(s)}(\overline{\Omega})\subset
H_{q}^{\mu(s^{\prime})}(\overline{\Omega})$ for $s>s^{\prime}$. In the first
line of (2.14), we have

$H_{q}^{\mu(s)}(\overline{\mathbb{R}}^{n}_{+})=\Xi_{+}^{-\mu}e^{+}\overline{H}_{q}^{s-\mu}({\mathbb{R}}^{n}_{+})=\Xi_{+}^{-\mu}\dot{H}_{q}^{s-\mu}(\overline{\mathbb{R}}^{n}_{+})=\dot{H}_{q}^{s}(\overline{\mathbb{R}}^{n}_{+}),\quad\text{if
}\mu+\tfrac{1}{q}>s>\mu-\tfrac{1}{q^{\prime}}.$ (2.15)

On the other hand, when $s>\mu+\tfrac{1}{q}$, $\Xi_{+}^{-\mu}$ is applied to
functions having a jump at $x_{n}=0$; this results in a singularity
$x_{n}^{\mu}$ at $x_{n}=0$.

The second line in (2.14) is valid in the case of a $C^{\infty}$-domain
$\Omega$. In the case where $\Omega$ is $C^{1+\tau}$, $\tau>0$, we have
instead a definition using local coordinates, based on the definition for the
case of a curved halfspace $\mathbb{R}^{n}_{\zeta}$ (2.1). Here we use the
diffeomorphism $F_{\zeta}$ mapping ${\mathbb{R}}^{n}_{\zeta}$ to
${\mathbb{R}}^{n}_{+}$ and its inverse $F_{\zeta}^{-1}$,

$F_{\zeta}(x)=(x^{\prime},x_{n}-\zeta(x^{\prime})),\quad
F^{-1}_{\zeta}(x)=(x^{\prime},x_{n}+\zeta(x^{\prime})),\quad$

defining, for $\mu-\frac{1}{q^{\prime}}<t<1+\tau$,

$u\in H_{q}^{\mu(t)}(\overline{\mathbb{R}}_{\zeta})\iff u\circ
F_{\zeta}^{-1}\in H_{q}^{\mu(t)}(\overline{\mathbb{R}}^{n}_{+}),$

with the inherited norm ($u\circ F_{\zeta}^{-1}$ is also denoted
$F_{\zeta}^{-1,*}u$). For a bounded $C^{1+\tau}$-domain $\Omega$, every point
$x_{0}\in\partial\Omega$ has a bounded open neighborhood
$U\subset{\mathbb{R}}^{n}$ and a $\zeta\in C^{1+\tau}(\mathbb{R}^{n-1})$, such
that after a suitable rotation, $\Omega\cap U={\mathbb{R}}^{n}_{\zeta}\cap U$.
$H_{q}^{\mu(t)}(\overline{\Omega})$ is now defined (cf. [2, Def. 4.3]) as the
set of functions $u\in H^{t}_{loc}(\Omega)$ such that for each $x_{0}$, with a
$\varphi\in C_{0}^{\infty}(U)$ with $\varphi\equiv 1$ in a neighborhood of
$x_{0}$, $(\varphi u)\circ F_{\zeta}^{-1}\in
H_{q}^{\mu(t)}(\overline{\mathbb{R}}^{n}_{+})$ (in the rotated situation).

A norm on $H_{q}^{\mu(t)}(\overline{\Omega})$ can be defined as follows: There
is a cover of $\overline{\Omega}$ by bounded open sets
$\\{U_{0},U_{1},\dots,U_{J}\\}$, and a partition of unity
$\\{\varrho_{j}\\}_{0\leq j\leq J}$ (with $\varrho_{j}\in
C_{0}^{\infty}(U_{j},[0,1])$, satisfying $\sum_{0\leq j\leq J}\varrho_{j}=1$
on $\overline{\Omega}$), where the $U_{j}$ for $j\geq 1$ are neighborhoods of
points $x_{j}\in\partial\Omega$ with $\Omega\cap
U={\mathbb{R}}^{n}_{\zeta_{i}}\cap U$ (after a rotation), $\zeta_{i}\in
C^{1+\tau}({\mathbb{R}}^{n})$, as described above. Moreover,
$\partial\Omega\subset\bigcup_{1\leq j\leq J}U_{j}$ and
$\overline{U}_{0}\subset\Omega$. Then

$\|u\|_{H_{q}^{\mu(t)}(\overline{\Omega})}=\bigl{(}\sum_{1\leq j\leq
J}\|(\varrho_{j}u)\circ
F_{\zeta_{j}}^{-1}\|^{q}_{H_{q}^{\mu(t)}(\overline{\mathbb{R}}^{n}_{+})}+\|\varrho_{0}u\|^{q}_{H^{t}_{q}({\mathbb{R}}^{n})}\bigr{)}^{\frac{1}{q}}$
(2.16)

is a norm on ${H_{q}^{\mu(t)}(\overline{\Omega})}$. (This way to define norms
over curved spaces is recalled e.g. in [20, Sect. 8.2].)

Further properties of the $\mu$-transmission spaces are described in detail in
[22], [25], [2] and [26].

##  3 The Dirichlet realization

Our main hypothesis on $P$ is:

###### Hypothesis 3.1

Let $0<a<1$, $\tau>2a$, and $P=\operatorname{Op}(p)$, where $p\in
C^{\tau}S^{2a}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ (is a classical
$C^{\tau}$-symbol of order $2a$). Moreover, $P$ is strongly elliptic, i.e.,
$\operatorname{Re}p_{0}(x,\xi)\geq c|\xi|^{2a}$ with $c>0$ for $|\xi|\geq 1$,
and has the evenness property:

$p_{j}(x,-\xi)=(-1)^{j}p_{j}(x,\xi)\text{ for all
}j\in\mathbb{N}_{0},\;|\xi|\geq 1,\;x\in{\mathbb{R}}^{n}.$ (3.1)

###### Remark 3.2

One of the convenient properties of the pseudodifferential calculus is that
for elliptic problems, the interior regularity of solutions is dealt with,
once and for all: When $P$ is classical elliptic (i.e., $p_{0}(x,\xi)\neq 0$
for $|\xi|\geq 1$) of order $m$, then for any open set
$\Omega\subset{\mathbb{R}}^{n}$, $Pu|_{\Omega}\in H^{s}_{q,loc}(\Omega)$
implies $u|_{\Omega}\in H^{s+m}_{q,loc}(\Omega)$. In the case $\tau=\infty$,
this holds for $s\in\mathbb{R}$ and was shown already by Seeley in [40] (see
also [41]) in $H^{s}_{q}$-spaces, and it extends to all scales of function
spaces, where pseudodifferential operators are continuous, as indicated in
[21]. For finite $\tau$ and, say, $u\in
H^{m-\tau+\varepsilon}_{q}({\mathbb{R}}^{n})$ for some $\varepsilon>0$, it
follows for $-\tau<s\leq\tau$ e.g. from Theorem 9 in Marschall [34] after $P$
is reduced to order zero and a standard localization procedure is applied.

Now some words on the special case where the symbol is real, $x$-independent
and has no lower-order terms, $p(x,\xi)=p_{0}(\xi)$. Denote by
$p^{h}\colon{\mathbb{R}}^{n}\to\mathbb{C},\xi\mapsto p^{h}(\xi)$ the
homogeneous function on ${\mathbb{R}}^{n}$ coinciding with $p_{0}$ for
$|\xi|\geq 1$. The operator $P^{h}=\operatorname{Op}(p^{h})$ is then a
complexified version of the real singular integral operator $\mathcal{L}$
studied in many works on generalizations of the fractional Laplacian (cf. e.g.
Ros-Oton et al. [37], [38]):

$\displaystyle\mathcal{L}u(x)$
$\displaystyle=PV\int_{{\mathbb{R}}^{n}}(u(x)-u(x+y))K(y)\,dy$
$\displaystyle=\int_{{\mathbb{R}}^{n}}(u(x)-\tfrac{1}{2}(u(x+y)+u(x-y)))K(y)\,dy.$
(3.2)

Here $K\colon{\mathbb{R}}^{n}\setminus\\{0\\}\to\mathbb{C}$ is homogeneous of
degree $-n-2a$ and smooth in ${\mathbb{R}}^{n}\setminus\\{0\\}$ when $p^{h}$
is so, and even: $K(-y)=K(y)$ for all $y\neq 0$. In the rotation-invariant
case, $\mathcal{L}=(-\Delta)^{a}$ when $K(y)=c_{n,a}|y|^{-n-2a}$ for a
suitable $c_{n,a}>0$. And more generally, this singular integral definition
coincides with our pseudodifferential definition of $P^{h}$, when
$K=(\mathcal{F}^{-1}p^{h})|_{{\mathbb{R}}^{n}\setminus\\{0\\}}$. Note here
that $P^{h}$ differs from $P_{0}=\operatorname{Op}(p_{0}(\xi))$ by the
operator $\mathcal{R}=\operatorname{Op}(r)$, where $r=p^{h}-p_{0}$ is bounded
and supported for $|\xi|\leq 1$, hence $\mathcal{R}$ maps e.g.
$H_{q}^{s}({\mathbb{R}}^{n})\to H_{q}^{t}({\mathbb{R}}^{n})$ for all
$s,t\in\mathbb{R}$; it is smoothing. So mapping properties and regularity
results for $\mathcal{L}$ follow from those for $P_{0}$ (or $P^{h}$).

Let $\Omega\subset{\mathbb{R}}^{n}$ be open and bounded with a
$C^{1+\tau}$-boundary. The homogeneous Dirichlet problem for $P$ is, for a
given function $f$ on $\Omega$ to find $u$ such that

$Pu=f\text{ on }\Omega,\quad u=0\text{ on }{\mathbb{R}}^{n}\setminus\Omega.$
(3.3)

(More precisely, one can write $r^{+}Pu$ instead of “$Pu$ on $\Omega$”.)

From the sesquilinear form $s(u,v)$ obtained by closure on
$\dot{H}^{a}(\overline{\Omega})$ of

$s(u,v)=\int_{\Omega}Pu\,\bar{v}\,dx,\quad u,v\in C_{0}^{\infty}(\Omega),$
(3.4)

one defines the Dirichlet realization $P_{D,2}$ in $L_{2}(\Omega)$ by the Lax-
Milgram lemma. For a general $1<q<\infty$, one likewise defines a Dirichlet
realization $P_{D,q}$ of $P$ in $L_{q}(\Omega)$, namely as the operator acting
like $r^{+}P$ with domain
$D(P_{D,q})=\\{u\in\dot{H}_{q}^{a}(\overline{\Omega})\mid r^{+}Pu\in
L_{q}(\Omega)\\}$. It is shown in [22] for $\tau=\infty$, [2] for general
$\tau>2a$, that these operators have nice solvability properties, and their
domains are found to equal $a$-transmission spaces

$D(P_{D,q})=\\{u\in\dot{H}_{q}^{a}(\overline{\Omega})\mid r^{+}Pu\in
L_{q}(\Omega)\\}=H_{q}^{a(2a)}(\overline{\Omega}).$ (3.5)

By the observations around (2.15),

$H_{q}^{a(2a)}(\overline{\Omega})=\dot{H}_{q}^{2a}(\overline{\Omega})\text{
when }a<\tfrac{1}{q};\quad
H_{q}^{a(2a)}(\overline{\Omega})\subset\dot{H}_{q}^{a+\frac{1}{q}-\varepsilon}(\overline{\Omega})\text{
when }a\geq\tfrac{1}{q},$ (3.6)

any $\varepsilon>0$. Moreover, $H_{q}^{a(2a)}(\overline{\Omega})$ is when
$a>\tfrac{1}{q}$ contained in
$\dot{H}_{q}^{2a}(\overline{\Omega})+d^{a}e^{+}\overline{H}^{a}_{q}(\Omega)$;
recall that $d(x)\sim\operatorname{dist}(x,\partial\Omega)$. There is also an
exact description when $1+\frac{1}{q}>a>\frac{1}{q}$, namely: When
$\tau=\infty$,
$H_{q}^{a(2a)}(\overline{\Omega})=\dot{H}_{q}^{2a}(\overline{\Omega})+d^{a}K_{0}B_{q,q}^{a-\frac{1}{q}}(\partial\Omega)$
by [25], where $K_{0}$ is a Poisson operator and
$B_{q,q}^{a-\frac{1}{q}}(\partial\Omega)$ is a Besov space; and this holds in
local coordinates when $\tau$ is finite. For brevity, we shall use the
notation

$D_{q}(\overline{\Omega})=H_{q}^{a(2a)}(\overline{\Omega}).$ (3.7)

In the following, we mostly consider a fixed $q$, and denote $P_{D,q}=P_{D}$.
$P_{D}$ has a resolvent that is compact in $L_{q}(\Omega)$, and as accounted
for in [26], the spectrum is contained in a convex sectorial region opening to
the right. Hence the resolvent set $\varrho(P_{D})$ contains an obtuse
sectorial region $V_{\delta,K}$ (the complement of a “keyhole region”). Here
we define $V_{\delta,K}$ for $0<\delta<\delta_{0}$ where
$0<\delta_{0}\leq\frac{\pi}{2}$, and $K\geq 0$, by

$V_{\delta,K}=\\{\lambda\in\mathbb{C}\setminus\\{0\\}\mid\operatorname{arg}\lambda\in[\tfrac{\pi}{2}-\delta,\tfrac{3\pi}{2}+\delta],|\lambda|\geq
K\\}.$ (3.8)

For the actual $P$, $\tfrac{\pi}{2}-\delta_{0}=\sup\\{|\arg p_{0}(x,\xi)|\mid
x\in{\mathbb{R}}^{n},|\xi|\geq 1\\}$.

In the $x$-independent real homogeneous case considered around (3.2) where
$P^{h}=\operatorname{Op}(p^{h})$ identifies with $\mathcal{L}$, the quadratic
form $s^{h}(u,u)$ (as in (3.4)) identifies with the form

$Q(u)=\tfrac{1}{2}\int_{\mathbb{R}^{2n}}|u(x)-u(y)|^{2}K(x-y)\,dxdy\quad\text{for
}u\in\dot{H}^{a}(\overline{\Omega}),$ (3.9)

acting on real $u$, cf. e.g. [37].

##  4 Auxiliary results on resolvent estimates

Consider a closed, densely defined linear operator
$A\colon\mathcal{D}(A)\subset X\to X$ on a UMD-space $X$. (Cf. e.g. Burkholder
[5] for the definition and characterizations of UMD-spaces; the
$H_{q}^{s}$-spaces are of this kind.) Numerous studies through the times show
that estimates of the resolvent $(A-\lambda)^{-1}$ lead to solvability
properties, in various function spaces, of the heat equation

$\partial_{t}u(t)+Au(t)=f(t)\text{ for }t\in I,\quad u(0)=0,$ (4.1)

where $I=(0,T)$ for $T\in(0,\infty)$ or $I=(0,\infty)$. A basic problem is to
show _maximal $L_{p}$-regularity_, namely that (4.1) for any $f\in L_{p}(I;X)$
has a unique solution $u\colon\overline{I}\to X$ satisfying

$\partial_{t}u\text{ and }Au\in L_{p}(I;X).$ (4.2)

We note that, if $I=(0,T)$ for some $T<\infty$, then this is equivalent to
$u\in H^{1}_{p}(I;X)\cap L_{p}(I;\mathcal{D}(A))$. This is usually relatively
easy to obtain for $p=2$ and a Hilbert space $X$; the difficulty when $p\neq
2$ and general $X$ for differential and pseudodifferential realizations is
linked to the fact that multiplier theorems valued in Hilbert spaces do not in
general extend to Banach spaces. The difficulty was overcome by a deeper
analysis in [27], [18] for operators in the Boutet de Monvel calculus
(including differential boundary value problems and nontrivial initial- and
boundary conditions), in a smooth setting. A nonsmooth case stemming from the
Stokes problem was treated in Abels [1].

To include nonsmooth settings in general, other tools have been introduced. We
shall in the present paper take advantage of the concept of
$\mathcal{R}$-boundedness, as developed through works of Da Prato and
Grisvard, Lamberton, Dore and Venni, Clément, Prüss, Hieber, Denk, Weiss,
Bourgain and others, and explained very nicely in Denk-Hieber-Prüss [11],
which applies it to vector-valued nonsmooth differential operator problems.
The theory is also included in the book Prüss-Simonett [35].

$\mathcal{R}$-boundedness of a family $\mathcal{T}$ of bounded linear
operators $T\colon X\to Y$ is defined as follows:

###### Definition 4.1

Let $X$ and $Y$ be Banach spaces, and let $\mathcal{T}$ be a family of
operators $T$ in $\mathcal{L}(X,Y)$. $\mathcal{T}$ is said to be
$\mathcal{R}$-bounded if there is a constant $C\geq 0$ and a $p\in[1,\infty)$
such that there holds: For each $N\in\mathbb{N}$,
$\\{T_{j}\\}_{j=1}^{N}\subset\mathcal{T}$, $\\{x_{j}\\}_{j=1}^{N}\subset X$,
and $\\{\varepsilon_{j}\\}_{j=1}^{N}$ belonging to a system of independent and
identically distributed symmetric $\\{-1,+1\\}$-valued random variables
$\varepsilon$ on some probability space $(\Omega,\mathcal{M},\mu)$,

$\|\sum_{j=1}^{N}\varepsilon_{j}T_{j}x_{j}\|_{L_{p}(\Omega,Y)}\leq
C\|\sum_{j=1}^{N}\varepsilon_{j}x_{j}\|_{L_{p}(\Omega,X)}.$ (4.3)

###### Remark 4.2

As the probability space and random variables, one can for example take
$(\Omega,\mathcal{M},\mu)=([0,1],\mathcal{B}([0,1]),\lambda)$, where
$\mathcal{B}([0,1])$ stands for the Borel $\sigma$-algebra, $\lambda$ for the
Lebesgue measure, and the random variables are given by the Rademacher
functions, as explained in detail e.g. in Denk [10].

An alternative formulation is given in Denk and Seiler [12]:

###### Definition 4.3

Let $p\in[1,\infty)$. Denote $Z_{N}=\\{(z_{1},\dots,z_{N})\mid
z_{j}\in\\{-1,+1\\}\text{ for all }j\\}$, a subset of $\mathbb{R}^{N}$. Let
$X$ and $Y$ be Banach spaces.

A subset $\mathcal{T}$ of the bounded linear operators $\mathcal{L}(X,Y)$ is
$\mathcal{R}$-bounded if there is a constant $C\geq 0$ such that for every
choice of $N\in\mathbb{N}$ and every choice of $x_{1},\dots,x_{N}$ in $X$ and
$T_{1},\dots,T_{N}$ in $\mathcal{T}$,

$\bigl{(}\sum_{z\in
Z_{N}}\|\sum_{j=1}^{N}z_{j}T_{j}x_{j}\|_{Y}^{p}\bigr{)}^{1/p}\leq
C\bigl{(}\sum_{z\in Z_{N}}\|\sum_{j=1}^{N}z_{j}x_{j}\|_{X}^{p}\bigr{)}^{1/p}.$
(4.4)

The finiteness for one $p\in[1,\infty)$ implies the finiteness for all other
$p\in[1,\infty)$. The best constant $C$, denoted
$\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})$ or just
$\mathcal{R}(\mathcal{T})$, is called the $\mathcal{R}$-bound of $\mathcal{T}$
(for some fixed $p$). An $\mathcal{R}$-bounded set is norm-bounded. Finite
families $\mathcal{T}$ are $\mathcal{R}$-bounded. Norm bounds and
$\mathcal{R}$-bounds are equivalent if $X$ and $Y$ are Hilbert spaces.

The $\mathcal{R}$-boundedness is preserved under addition and composition
([11, Prop. 3.4]):

###### Proposition 4.4

$1^{\circ}$ Let $X$ and $Y$ be Banach spaces, and let $\mathcal{T}$ and
$\mathcal{S}$ $\subset\mathcal{L}(X,Y)$ be $\mathcal{R}$-bounded. Then

$\mathcal{T}+\mathcal{S}=\\{T+S\mid T\in\mathcal{T},S\in\mathcal{S}\\}$

is $\mathcal{R}$-bounded, and
$\mathcal{R}\\{\mathcal{T}+\mathcal{S}\\}\leq\mathcal{R}\\{\mathcal{T}\\}+\mathcal{R}\\{\mathcal{S}\\}$.

$2^{\circ}$ Let $X$, $Y$ and $Z$ be Banach spaces, and let
$\mathcal{T}\subset\mathcal{L}(X,Y)$ and $\mathcal{S}\subset\mathcal{L}(Y,Z)$
be $\mathcal{R}$-bounded. Then

$\mathcal{S}\mathcal{T}=\\{ST\mid T\in\mathcal{T},S\in\mathcal{S}\\}$

is $\mathcal{R}$-bounded, and
$\mathcal{R}\\{\mathcal{S}\mathcal{T}\\}\leq\mathcal{R}\\{\mathcal{S}\\}\mathcal{R}\\{\mathcal{T}\\}$.

The fundamental interest of this concept is that it leads to a criterion for
maximal $L_{p}$-regularity, shown in [11, Theorem 4.4]:

###### Theorem 4.5

Let $1<p<\infty$ and $X$ be a UMD-space. Problem (4.1) has maximal
$L_{p}$-regularity on $I=\mathbb{R}_{+}$ if and only if
$V_{\delta,0}\subset\rho(A)$ and the family
$\\{\lambda(A-\lambda)^{-1}\mid\lambda\in V_{\delta,0}\\}$ in $\mathcal{L}(X)$
is $\mathcal{R}$-bounded for some $\delta\in(0,\frac{\pi}{2})$.

Note that $\mathcal{R}$-boundedness of
$\\{\lambda(A-\lambda)^{-1}\mid\lambda\in V_{\delta,K}\\}$ implies that for
some $k>K$, $\mathcal{R}$-boundedness holds for
$\\{\lambda(A+k-\lambda)^{-1}\mid\lambda\in V_{\delta,0}\\}$. Then the shifted
operator $A+k$ has maximal $L_{q}$-regularity on $\mathbb{R}_{+}$, and $A$
itself has it on finite intervals $I=(0,T)$.

We shall say that $A$ is $\mathcal{R}$-sectorial on $V_{\delta,K}$ when
$V_{\delta,K}\subset\rho(A)$ and

$\mathcal{R}_{\mathcal{L}(X)}\\{\lambda(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\}<\infty.$ (4.5)

One of the reasons that Theorem 4.5 is particularly useful, is that
$\mathcal{R}$-sectoriality is preserved under suitable perturbations of $A$.

###### Proposition 4.6

$1^{\circ}$ Let $A$ satisfy $V_{\delta,K}\subset\rho(A)$ and

$\|\lambda(A-\lambda)^{-1}\|_{\mathcal{L}(X)}\leq C\quad\text{for all
}\lambda\in V_{\delta,K}.$ (4.6)

Let $S\colon D(A)\to X$ be linear and satisfy

$\|Su\|_{X}\leq\alpha\|Au\|_{X}+\beta\|u\|_{X}\quad\text{for all }u\in D(A).$
(4.7)

Then when $\alpha$ is sufficiently small, there exists $K_{1}\geq K$ and
$C^{\prime}$ such that $V_{\delta,K_{1}}\subset\rho(A+S)$ and

$\|\lambda(A+S-\lambda)^{-1}\|_{\mathcal{L}(X)}\leq C^{\prime}\quad\text{for
all }\lambda\in V_{\delta,K_{1}}.$

$2^{\circ}$ Assume in addition that $\\{\lambda(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\}$ is $\mathcal{R}$-bounded. Then, for sufficiently small
$\alpha>0$ there is a $K_{2}\geq K$ such that
$V_{\delta,K_{2}}\subset\rho(A+S)$ and
$\\{\lambda(A+S-\lambda)^{-1}\mid\lambda\in V_{\delta,K_{2}}\\}$ is
$\mathcal{R}$-bounded.

###### Proof.

$1^{\circ}$ has been known for many years. A version is proved in [11, Theorem
1.5], which adapts straightforwardly to our sectorial sets. $2^{\circ}$ is an
adaptation of [11, Prop. 4.3] in a similar way. ∎

We shall supply these results with some further properties that are essential
for our studies here:

###### Lemma 4.7

Let $X,Y,Y_{0},Y_{1}$ be Banach spaces satisfying $Y_{1}\subset Y\subset
Y_{0}$, with continuous, dense injections. Assume that for some
$\theta\in(0,1)$, $C>0$,

$\|x\|_{Y}\leq C\|x\|_{Y_{0}}^{\theta}\|x\|_{Y_{1}}^{1-\theta}\quad\text{for
all }x\in Y_{1}.$

Then for any operator family $\mathcal{T}$ in
$\mathcal{L}(X,Y_{0})\cap\mathcal{L}(X,Y_{1})$, the $\mathcal{R}$-bound of the
operators considered as elements of $\mathcal{L}(X,Y)$ satisfies

$\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})\leq
C\mathcal{R}_{\mathcal{L}(X,Y_{0})}(\mathcal{T})^{\theta}\mathcal{R}_{\mathcal{L}(X,Y_{1})}(\mathcal{T})^{1-\theta}.$

###### Proof.

Follows from the definition of $\mathcal{R}$-boundedness:

When $\Omega$, $\varepsilon_{j}$, $T_{j}\in\mathcal{T}$ and $x_{j}$ are as in
Definition 4.1,

$\displaystyle\|\sum_{j=1}^{N}\varepsilon_{j}T_{j}x_{j}\|_{L_{p}(\Omega,Y)}$
$\displaystyle\leq
C\|\sum_{j=1}^{N}\varepsilon_{j}T_{j}x_{j}\|_{L_{p}(\Omega,Y_{0})}^{\theta}\|\sum_{j=1}^{N}\varepsilon_{j}T_{j}x_{j}\|_{L_{p}(\Omega,Y_{1})}^{1-\theta}$
$\displaystyle\leq
C\mathcal{R}_{\mathcal{L}(X,Y_{0})}(\mathcal{T})^{\theta}\mathcal{R}_{\mathcal{L}(X,Y_{1})}(\mathcal{T})^{1-\theta}\|\sum_{j=1}^{N}\varepsilon_{j}x_{j}\|_{L_{p}(\Omega,X)}.$

∎

###### Theorem 4.8

Let $A$ be a closed, densely defined linear operator in a Banach space $X$,
such that $A$ is $\mathcal{R}$-sectorial over $V_{\delta,K}$. Let $Y$ be a
Banach space satisfying $D(A)\subset Y\subset X$ with dense, continuous
injections, and assume that for some $\theta\in[0,1]$ and $C_{0}>0$,

$\|u\|_{Y}\leq C_{0}\|u\|_{X}^{\theta}\|u\|_{D(A)}^{1-\theta}\quad\text{for
all }u\in D(A).$ (4.8)

  1. $1^{\circ}$

With

$C_{1}=\mathcal{R}_{\mathcal{L}(X)}\\{\lambda(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\},$

one has for any $K_{1}\geq K$ with $K_{1}>0$ that the $\mathcal{R}$-bound of
$(A-\lambda)^{-1}$ over $V_{\delta,K_{1}}$ satisfies

$\mathcal{R}_{\mathcal{L}(X)}\\{(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}\leq 2C_{1}/K_{1}.$ (4.9)

  2. $2^{\circ}$

We have

$\mathcal{R}_{\mathcal{L}(X,D(A))}\\{(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\}=C_{2}<\infty,$ (4.10)

and when $S\in\mathcal{L}(D(A),X)$, then $\\{S(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K}\\}$ is $\mathcal{R}_{\mathcal{L}(X)}$-bounded.

  3. $3^{\circ}$

Let $S\in\mathcal{L}(Y,X)$. With $\theta$ as in (4.8), there is a constant $C$
such that for all $K_{1}\geq K$ with $K_{1}>0$,

$\mathcal{R}_{\mathcal{L}(X)}\\{S(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}\leq CK_{1}^{-\theta}.$

###### Proof.

$1^{\circ}$. Let $N\in\mathbb{N}$, $\\{x_{j}\\}_{j=1}^{N}\subset X$, and
$\\{\varepsilon_{j}\\}_{j=1}^{N}$ be as in Definition 4.1,
$\\{\lambda_{j}\\}_{j=1}^{N}\subset V_{\delta,K_{1}}$ and $p\in[1,\infty)$.
Then

$\displaystyle
K_{1}\|\sum_{j=1}^{N}\varepsilon_{j}(A-\lambda_{j})^{-1}x_{j}\|_{L_{p}(\Omega;X)}$
$\displaystyle\leq
2\|\sum_{j=1}^{N}\varepsilon_{j}\lambda_{j}(A-\lambda_{j})^{-1}x_{j}\|_{L_{p}(\Omega;X)}$
$\displaystyle\leq
2C_{1}\|\sum_{j=1}^{N}\varepsilon_{j}x_{j}\|_{L_{p}(\Omega;X)}$

by the contraction principle of Kahane (cf. e.g. Lemma 3.5 in [11]) since
$|\lambda_{j}|\geq K_{1}$. This yields the first statement.

$2^{\circ}$. Since $A(A-\lambda)^{-1}=I-\lambda(A-\lambda)^{-1}$, the
$\mathcal{R}_{\mathcal{L}(X)}$-boundedness of the family
$\\{A(A-\lambda)^{-1}\mid\lambda\in V_{\delta,K}\\}$ follows from that of
$\\{\lambda(A-\lambda)^{-1}\mid\lambda\in V_{\delta,K}\\}$; it moreover holds
for $(A-\lambda_{0})(A-\lambda)^{-1}$ for any $\lambda_{0}\in\mathbb{C}$. We
here use the sum rule Proposition 4.4 $1^{\circ}$.

Take $\lambda_{0}$ in the resolvent set of $A$; then $A-\lambda_{0}$ is a
homeomorphism of $D(A)$ onto $X$, so for $(A-\lambda)^{-1}$ viewed as the
composition of $(A-\lambda_{0})^{-1}\colon X\to D(A)$ and
$(A-\lambda_{0})(A-\lambda)^{-1}\colon X\to X$, we get (4.10) by the product
rule Proposition 4.4 $2^{\circ}$.

Moreover, we can write

$S(A-\lambda)^{-1}=S(A-\lambda_{0})^{-1}(A-\lambda_{0})(A-\lambda)^{-1}.$

Since $S(A-\lambda_{0})^{-1}\in\mathcal{L}(X)$, the last statement in
$2^{\circ}$ follows from the product rule.

$3^{\circ}$. Because of (4.8), we have by Lemma 4.7,

$\displaystyle\mathcal{R}_{\mathcal{L}(X,Y)}\\{(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}$ $\displaystyle\leq
C_{0}\mathcal{R}_{\mathcal{L}(X,X)}\\{(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}^{\theta}\mathcal{R}_{\mathcal{L}(X,D(A))}\\{(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}^{1-\theta}$ $\displaystyle\leq
C_{0}(C_{1}/K_{1})^{\theta}C_{2}^{1-\theta}=C_{3}K_{1}^{-\theta},$

where we used (4.9) and (4.10). If $S\in\mathcal{L}(Y,X)$ with norm $C_{4}$,
we then find by the product rule

$\mathcal{R}_{\mathcal{L}(X)}\\{S(A-\lambda)^{-1}\mid\lambda\in
V_{\delta,K_{1}}\\}\leq C_{4}C_{3}K_{1}^{-\theta}.$

∎

###### Remark 4.9

These general results will in the following be applied to the situation where
$A$ is the realization in $X=L_{q}(\Omega)$ of a pseudodifferential operator
$P$ satisfying Hypothesis 3.1, with domain $D(A)=D_{q}(\overline{\Omega})$
(3.7), $\Omega$ being open, bounded and $C^{1+\tau}$. The perturbation $S$
will often be taken as an operator of order $s<a+1/q$, $s\geq 0$, satisfying
$\|Su\|_{L_{q}(\Omega)}\leq c\|u\|_{\dot{H}^{s}_{q}(\overline{\Omega})}$.
Recall that
$H_{q}^{a(t)}(\overline{\Omega})=\dot{H}_{q}^{t}(\overline{\Omega})$ when
$t<a+1/q$. Since $s<a+1/q$, there is a $t$ with $s<t<a+1/q$, and there is an
interpolation inequality

$\|u\|_{\dot{H}_{q}^{s}(\overline{\Omega})}\leq
c\|u\|^{\theta}_{L_{q}(\Omega)}\|u\|^{1-\theta}_{\dot{H}_{q}^{t}(\overline{\Omega})}\quad\text{for
all }u\in\dot{H}_{q}^{t}(\overline{\Omega})$ (4.11)

with a $\theta\in(0,1)$ (more precisely, $\theta=1-s/t$, cf. Triebel [43,
1.3.3/5, 2.4.2]). Here
$\|u\|_{\dot{H}_{q}^{t}(\overline{\Omega})}=\|u\|_{H_{q}^{a(t)}(\overline{\Omega})}\leq
c\|u\|_{H_{q}^{a(2a)}(\overline{\Omega})}=c\|u\|_{D_{q}(\overline{\Omega})}$.
Thus

$\|u\|_{\dot{H}_{q}^{s}(\overline{\Omega})}\leq
c^{\prime}\|u\|^{\theta}_{L_{q}(\Omega)}\|u\|^{1-\theta}_{D_{q}(\overline{\Omega})}\quad\text{for
all }u\in D_{q}(\overline{\Omega}).$ (4.12)

This also implies that for any $\varepsilon>0$ there is a constant
$C_{\varepsilon}$ such that

$\|u\|_{\dot{H}_{q}^{s}(\overline{\Omega})}\leq\varepsilon\|u\|_{D_{q}(\overline{\Omega})}+C_{\varepsilon}\|u\|_{L_{q}(\Omega)}\quad\text{for
all }u\in D_{q}(\overline{\Omega}),$ (4.13)

showing that $S$ satisfies (4.7) with arbitrarily small $\alpha>0$.

If $s<a$, we can take $t=a$ in the interpolation.

##  5 Resolvent $\mathcal{R}$-bounds for the Dirichlet problem

In the following we shall show how resolvent $\mathcal{R}$-bounds can be
obtained for a general class of $x$-dependent operators $P$ from the knowledge
in some special cases.

First consider pseudodifferential operators on ${\mathbb{R}}^{n}$ without
boundary conditions. They can be handled in a way based directly on symbolic
calculus, as in [18] and [23] (when $\tau=\infty$).

###### Proposition 5.1

Let $P=\operatorname{Op}(p)$ with $p\in
S^{d}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, homogeneous of order $d>0$ in
$\xi$ satisfying $\operatorname{Re}p(x,\xi)\geq c|\xi|^{d}$ for all $|\xi|\geq
1$, $x\in{\mathbb{R}}^{n}$. Then for every $1<q<\infty$ and a suitable
constant $b$, the heat problem (4.1) for $A=P+b$ with
$D(P+b)=H^{d}_{q}({\mathbb{R}}^{n})$ and $X=L_{q}({\mathbb{R}}^{n})$ has
maximal $L_{q}$-regularity on $\mathbb{R}_{+}$, and hence $P+b$ is
$\mathcal{R}$-sectorial on $V_{\delta,0}$ for some $\delta>0$.

###### Proof.

For integer $d$, this follows from Theorem 3.1 (1) in [18], where mapping
properties in anisotropic Bessel-potential spaces
$H_{q}^{(s,s/d)}({\mathbb{R}}^{n}\times\mathbb{R})$ were established; they
hold for $P+\partial_{t}$ as well as its parametrix, as formulated for the
case with boundary in Theorem 3.4 there. The property of being supported for
$t\geq 0$ is preserved by these mappings, since the symbols are holomorphic in
$\tau$ for $\operatorname{Im}\tau<0$. Here $b$ can be chosen so that $P+b$ has
positive lower bound in $L_{2}$ (by the Gårding inequality); then there is a
solution operator, which also works in the $L_{q}$-setting, and

$f\in L_{q}({\mathbb{R}}^{n}\times\mathbb{R}_{+})\iff
u\in\dot{H}_{q}^{(d,1)}({\mathbb{R}}^{n}\times\overline{\mathbb{R}}_{+})=L_{q}(\mathbb{R}_{+};H^{d}_{q}({\mathbb{R}}^{n}))\cap\dot{H}^{1}_{q}(\overline{\mathbb{R}}_{+};L_{q}({\mathbb{R}}^{n})),$
(5.1)

where
$\dot{H}^{1}_{q}(\overline{\mathbb{R}}_{+};L_{q}({\mathbb{R}}^{n}))=\\{f\in
H^{1}_{q}(\mathbb{R}_{+};L_{q}({\mathbb{R}}^{n}))\mid f|_{t=0}=0\\}$.
Noninteger $d$ are included in the detailed presentation of symbol classes in
[23]. In that paper, the emphasis is on the regularity conclusion $\implies$
in (5.1); the existence is shown as in [18]. ∎

Next, there is a special result for operators on a bounded domain.

###### Proposition 5.2

Let $p^{h}\colon{\mathbb{R}}^{n}\to\mathbb{C}$ be smooth in
${\mathbb{R}}^{n}\setminus\\{0\\}$, strictly homogeneous of degree $2a>0$,
even, strongly elliptic and real (so $p^{h}(\xi)\geq c|\xi|^{2a}$ for all
$\xi\in\mathbb{R}^{n}$ with $c>0$). Let
$\overline{p}\colon{\mathbb{R}}^{n}\to\mathbb{C}$ be smooth, coinciding with
$p^{h}(\xi)$ for $|\xi|\geq 1$ and positive for all $\xi\in{\mathbb{R}}^{n}$.
Let $P^{h}=\operatorname{Op}(p^{h})$,
$\overline{P}=\operatorname{Op}(\overline{p})$. Let $\Omega$ be a bounded
$C^{1+\tau}$-domain, $\tau>2a$.

Then there are $\delta>0$ and $K\geq 0$ such that the $L_{q}$-Dirichlet
realization $P^{h}_{D}$ of $P^{h}$ on $\Omega$ is $\mathcal{R}$-sectorial on
$V_{\delta,0}$, and the $L_{q}$-Dirichlet realization $\overline{P}_{D}$ of
$\overline{P}$ on $\Omega$ is $\mathcal{R}$-sectorial on $V_{\delta,K}$.

###### Proof.

The operator $P^{h}$ is of the kind $\mathcal{L}$ considered around (3.2), its
$L_{2}(\Omega)$-Dirichlet realization being associated with the quadratic form
$Q$ recalled in (3.9). It is accounted for in [23] around (5.10) how the form
$Q(u)$ is a so-called Dirichlet form in the sense of Fukushima, Oshima and
Takeda [16] (also considered in Davies [9]). It has a Markovian property,
which assures that $-P^{h}_{D}$ generates a strongly continuous contraction
semigroup $T_{q}(t)$ not only in $L_{2}(\Omega)$ but also in $L_{q}(\Omega)$
for $1<q<\infty$, and $T_{q}(t)$ is bounded holomorphic (and the operators for
varying $q$ are consistent). By Lamberton [32], these properties imply that
the heat problem (4.10) with $A=P^{h}_{D}$ has maximal $L_{q}$-regularity, for
$1<q<\infty$ and all finite intervals $I$.

It is also shown in [32] that the constant $C$ in the estimates over
$\Omega\times I$, $I=(0,T)$,

$\|Au\|_{L_{q}(\Omega\times I)}+\|\partial_{t}u\|_{L_{q}(\Omega\times I)}\leq
C\|f\|_{L_{q}(\Omega\times I)},$ (5.2)

is independent of $T$. This allows us to conclude that (5.2) also holds with
$I=\mathbb{R}_{+}$. [32] applies to very general, also unbounded, sets
$\Omega$, and what we have said so far, only shows that $A=P^{h}_{D}$ has the
weak maximal-regularity property defined in Prüss-Simonett [35] p. 142 (is in
${}_{0}\mathcal{M}\mathcal{R}_{q}(\mathbb{R}_{+};L_{q}(\Omega))$ in their
notation).

Now since $\Omega$ is bounded, the quadratic form $Q(u)$ on
$\dot{H}^{a}(\overline{\Omega})$ moreover satisfies a Poincaré inequality (as
accounted for in Ros-Oton [37]) so that 0 is in the resolvent set of
$P^{h}_{D}$. Then by Cor. 3.5.3 in [35], $P^{h}_{D}$ has the full maximal-
regularity property (is in
$\mathcal{M}\mathcal{R}_{q}(\mathbb{R}_{+};L_{q}(\Omega))$ in their notation).
It means that $\|u\|_{L_{q}(I,D(A))}$ can be added to the left-hand side in
(5.2) with $I=\mathbb{R}_{+}$.

We then conclude from Theorem 4.5 that $P^{h}_{D}$ is $\mathcal{R}$-sectorial
on $V_{\delta,0}$ for some $\delta>0$.

Since $\overline{P}-P^{h}$ has bounded symbol supported in $|\xi|\leq 1$, it
defines a smoothing operator over $\Omega$. Then by Proposition 4.6
$2^{\circ}$, $\overline{P}_{D}$ is $\mathcal{R}$-sectorial on $V_{\delta,K}$
for some $K\geq 0$. ∎

###### Remark 5.3

It will also be used that $\mathcal{R}$-sectoriality is preserved under
suitable coordinate transformations (such as those used in [2]). This holds,
since composition with a single operator preserves $\mathcal{R}$-boundedness
(by Proposition 4.4 $2^{\circ}$).

Denote the ball $\\{|x-x_{0}|<r\\}$ in ${\mathbb{R}}^{n}$ by $B_{r}(x_{0})$;
if $x_{0}=0$, we just write $B_{r}$. The closure is denoted
$\overline{B}_{r}(x_{0})$. The balls in $\mathbb{R}^{n-1}$ will be denoted
$B^{\prime}_{r}(x^{\prime}_{0})$, or just $B^{\prime}_{r}$ if
$x^{\prime}_{0}=0$. By $\chi_{r,s}$ ($r>s>0$) we denote a function in
$C_{0}^{\infty}({\mathbb{R}}^{n},[0,1])$ such that
$\operatorname{supp}\chi_{r,s}\subset B_{r}$ and $\chi_{r,s}(x)=1$ for $x\in
B_{s}$. Denote in particular

$\chi_{2,1}=\eta,\quad\chi_{1,\frac{1}{2}}=\psi.$ (5.3)

We shall show:

###### Proposition 5.4

Let $\Omega$ be bounded with $C^{1+\tau}$-boundary, $\tau>2a$, and let
$1<q<\infty$. Let $P=\operatorname{Op}(p)$ satisfy Hypothesis 3.1. Assume
moreover that for $x\in\partial\Omega$, $p_{0}(x,\xi)$ is real $>0$. Let
$P_{0}=\operatorname{Op}(p_{0})$.

Consider a point $x_{0}\in\partial\Omega$, and denote
$p_{0}(x_{0},\xi)=\overline{p}(\xi)$ for all $\xi\in{\mathbb{R}}^{n}$,
$\operatorname{Op}(p_{0}(x_{0},\cdot))=\overline{P}$. Translate $x_{0}$ to
$0$, and let $U$ be a neighborhood of $0$ where, after a rotation,
$U\cap\Omega$ has the form $U\cap{\mathbb{R}}^{n}_{\zeta_{1}}$ for a function
$\zeta_{1}\in C^{1+\tau}(\mathbb{R}^{n-1},\mathbb{R})$ with $\zeta_{1}(0)=0$,
$\nabla\zeta_{1}(0)=0$. By a dilation we can assume that $U$ contains
$B^{\prime}_{2}\times[-M,M]$, where $M=\max_{|x^{\prime}|\leq
2}\\{|\zeta_{1}(x^{\prime})|,2\\}$.

Then there exists a $z\in(0,1]$ such that the following holds: There is a
bounded $C^{1+\tau}$-domain $\Sigma_{1}$ with
$B_{z}\cap\Omega=B_{z}\cap\Sigma_{1}$, and an operator $P_{1}$ satisfying
Hypothesis 3.1 such that for $u\in D_{q}(\overline{\Omega})$ supported in
$B_{z/4}$,

$\chi_{1/z,1/2z}(P_{0}-\lambda)u=\chi_{(1+\varepsilon)/z,1/z}(P_{1}-\lambda)u\text{
on }{\mathbb{R}}^{n}$ (5.4)

(some $\varepsilon>0$), where $P_{1,D}\colon D_{q}(\overline{\Sigma}_{1})\to
L_{q}(\overline{\Sigma}_{1})$ is $\mathcal{R}$-sectorial on $V_{\delta,K}$ for
some $K\geq 0$. Consequently, for any $\varphi\in C_{0}^{\infty}(B_{z/2})$

$\varphi(P_{0}-\lambda)u=\varphi(P_{1}-\lambda)u\text{ on }{\mathbb{R}}^{n}.$
(5.5)

###### Proof.

Departing from Proposition 5.2, we will show the formula by use of a scaling
argument, making it possible to find a small set where $P_{0}-\overline{P}$
and $\zeta_{1}(x^{\prime})$ have so small values that the resolvent estimates
for $\overline{P}$ can be carried over to $P_{0}$.

Introduce an auxiliary domain: Along with $\zeta_{1}\in
C^{1+\tau}(\mathbb{R}^{n-1},\mathbb{R})$, consider
$\zeta(x^{\prime})=\chi_{2,1}(x^{\prime})\zeta_{1}(x^{\prime})$ for all
$x^{\prime}\in\mathbb{R}^{n-1}$, coinciding with $\zeta_{1}$ when
$|x^{\prime}|\leq 1$ but vanishing for $|x^{\prime}|\geq 2$. We now choose a
$C^{1+\tau}$ set $\Sigma$ such that for $|x^{\prime}|\geq 2$, it is a subset
of the slab $\\{x\in{\mathbb{R}}^{n}\mid 0<x_{n}<2M\\}$ containing the
cylindrical set $\\{x\mid 2\leq|x^{\prime}|\leq 5,\;0<x_{n}<2M\\}$, and for
$|x^{\prime}|\leq 2$ it is the set
$V=\\{x\in{\mathbb{R}}^{n}\mid|x^{\prime}|\leq
2,\;\zeta(x^{\prime})<x_{n}<2M\\}$.

The diffeomorphism
$F_{\zeta}\colon(x^{\prime},x_{n})\mapsto(x^{\prime},x_{n}-\zeta(x^{\prime}))$
sends ${\mathbb{R}}^{n}_{\zeta}$ bijectively to ${\mathbb{R}}^{n}_{+}$; it
acts as the identity on points outside the cylinder
$B^{\prime}_{2}\times\mathbb{R}$, and maps $V$ to a set $V^{\prime}\subset
B^{\prime}_{2}\times[0,2M]$, which has the boundary piece
$B^{\prime}_{2}\times\\{0\\}$ in common with ${\mathbb{R}}^{n}_{+}$. Denote
$F_{\zeta}(\Sigma)=\Sigma^{\prime}$. Recall from [2, Section 6] that under the
diffeomorphism $F_{\zeta}$ on ${\mathbb{R}}^{n}$, a suitable operator $P$,
acting on functions defined on $\Sigma$ or $\mathbb{R}^{n}$, is carried over
to the operator $P_{\zeta}=F^{-1,\ast}_{\zeta}PF_{\zeta}^{\ast}$.

Now a slight variant of [2, Proposition 6.5] is needed:

###### Lemma 5.5

Let $\overline{p}\in S^{2a}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, $p\in
C^{\tau}S^{2a}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, and $1<q<\infty$. For
any $\varepsilon>0$ there exist $k\in\mathbb{N}$ and
$\varepsilon^{\prime}=\varepsilon^{\prime}(\overline{p},q)>0$ such that if

$|\overline{p}-p|_{k,C^{\tau}S^{2a}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})}\leq\varepsilon^{\prime}\quad\text{and}\quad\|\zeta\|_{C^{1+\tau}(\mathbb{R}^{n-1})}\leq\varepsilon^{\prime}.$
(5.6)

then
$\|\overline{P}-P_{\zeta}\|_{\mathcal{L}(D_{q}(\overline{\Sigma}^{\prime}),L_{q}(\Sigma^{\prime}))}\leq\varepsilon$.

Proof of Lemma 5.5: Proposition 6.5 in [2] shows this with $\Sigma^{\prime}$
replaced by ${\mathbb{R}}^{n}_{+}$ (the difficult part is the change of
variables, prepared there in Theorem 5.13). We note that in the latter
proposition it is assumed that $p$ is strongly elliptic and even. But for the
estimate in (5.6) this is not needed. To obtain the statement in the lemma, we
decompose a function $u\in D_{q}(\overline{\Sigma}^{\prime})$, by use of fixed
smooth cut-off functions, into three terms $u=u_{1}+u_{2}+u_{3}$, with
$\operatorname{supp}u_{1}\subset B^{\prime}_{4}\times[0,\frac{3}{2}M]$,
$\operatorname{supp}u_{2}\subset B^{\prime}_{4}\times[M,3M]$, and
$\operatorname{supp}u_{3}\subset\overline{\Sigma}^{\prime}\setminus(B^{\prime}_{3}\times[0,2M])$;
all three belonging to $D_{q}(\overline{\Sigma}^{\prime})$. The term $u_{1}$
can also be viewed as an element of $D_{q}(\overline{\mathbb{R}}^{n}_{+})$,
and the rule in [2, Proposition 6.5] pertaining to ${\mathbb{R}}^{n}_{+}$ and
${\mathbb{R}}^{n}_{\zeta}$ applies. This yields
$\|(\overline{P}-P_{\zeta})u_{1}\|_{L_{q}({\mathbb{R}}^{n})}\leq\varepsilon\|u_{1}\|_{D_{q}(\overline{\Sigma}^{\prime})}$
if $\varepsilon^{\prime}$ is sufficiently small. For the term $u_{2}$ there is
a similar rule pertaining to the halfspace $\\{x\mid x_{n}<2M\\}$ and the
curved halfspace $\\{x\mid x_{n}+\zeta(x^{\prime})<2M\\}$. For $u_{3}$ there
is a simpler rule since the variable $x$ is not shifted. The norm
$\|(\overline{P}-P_{\zeta})u_{3}\|_{L_{q}({\mathbb{R}}^{n})}$ will then be
dominated by the norms in (5.6) (times
$\|u_{3}\|_{D_{q}(\overline{\Sigma}^{\prime})}$), and so will, a fortiori, the
norm
$\|\overline{P}-P_{\zeta}\|_{\mathcal{L}(D_{q}(\overline{\Sigma}^{\prime}),L_{q}(\Sigma^{\prime}))}$.
This shows the lemma.

We continue the proof of Proposition 5.4, with
$P,\overline{P},P_{0},p,\overline{p}$ defined there. Here we have that
Proposition 5.2 applies to $\overline{P}$ considered over $\Sigma^{\prime}$.
Then Proposition 4.6 can be applied to $P_{0,\zeta}$ as a perturbation of
$\overline{P}$, when the norm difference is small enough, and by Lemma 5.5,
this can be obtained when the symbol estimates of $\overline{p}-p_{0}$ and
$C^{1+\tau}$-estimate of $\zeta$ in (5.6) are small enough. Thus we get for
such $p_{0}$ and $\zeta$ close to $\overline{p}$ and $0$, resp.:

$\|(P_{0,\zeta}-\lambda)v\|_{L_{q}(\Sigma^{\prime})}\geq
c_{0}|\lambda|\|v\|_{L_{q}(\Sigma^{\prime})}\quad\text{for all }\lambda\in
V_{\delta,K_{0}},\;v\in D_{q}(\overline{\Sigma}^{\prime}),$ (5.7)

and moreover, the family $\lambda(P_{0,\zeta,D}-\lambda)^{-1}$ is
$\mathcal{R}$-bounded for $\lambda\in V_{\delta,K_{0}}$ (here $P_{0,\zeta,D}$
stands for the Dirichlet realization of $P_{0,\zeta}$ on $\Sigma^{\prime}$).

For such $p_{0}$ and $\zeta$, a similar estimate can be concluded for $P_{0}$
over $\Sigma$, by changing variables back to $\Sigma$:

$\|(P_{0}-\lambda)v\|_{L_{q}(\Sigma)}\geq
c_{0}|\lambda|\|v\|_{L_{q}(\Sigma)}\quad\text{for all }\lambda\in
V_{\delta,K_{0}},\;v\in D_{q}(\overline{\Sigma});$ (5.8)

also $\mathcal{R}$-boundedness is preserved here, cf. Remark 5.3.

It will now be shown how this leads to estimates for $P_{0}$ with larger
symbol differences from $\overline{P}$, when applied to functions supported in
a sufficiently small ball around $0$.

Recalling that $\eta=\chi_{2,1}$, we define for $z>0$:

$\begin{split}\zeta_{z}(x^{\prime})&=z^{-1}\eta((x^{\prime},0))\zeta(zx^{\prime}),\\\
p_{z}(x,\xi)&=\eta(x)p_{0}(zx,\xi)+(1-\eta(x))p_{0}(0,\xi),\end{split}$ (5.9)

for all $x,\xi\in\mathbb{R}^{n}$, $x^{\prime}=(x_{1},\ldots,x_{n-1})$. Define
moreover

$q_{z}(x,\xi)=p_{0}(zx,z^{-1}\xi)-z^{-2a}p_{0}(zx,\xi).$ (5.10)

Because of the homogeneity of $p_{0}$,
$p_{0}(zx,z^{-1}\xi)=z^{-2a}p_{0}(zx,\xi)$ for all $|\xi|\geq 1$ and
$z\in(0,1]$ and therefore $q_{z}(x,\xi)=0$ for all $|\xi|\geq 1$, $z\in(0,1]$.
Hence $q_{z}\in
C^{\tau}S^{-\infty}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$.

For $v\colon{\mathbb{R}}^{n}\to\mathbb{C}$ and $z>0$ we shall write
$\sigma_{z}v\colon{\mathbb{R}}^{n}\to\mathbb{C}$ for the function
$(\sigma_{z}v)(x)=v(zx)$. We have with $P_{0}=\operatorname{Op}(p_{0})$ for
all $x\in\mathbb{R}^{n}$ and all suitable
$v\colon{\mathbb{R}}^{n}\to\mathbb{C}$:

$\sigma_{z}(P_{0}v)(x)=\int_{\mathbb{R}^{n}}e^{izx\cdot\xi}p_{0}(zx,\xi)\hat{v}(\xi)\text{\it\dj\hskip
1.0pt}\xi=\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}p_{0}(zx,z^{-1}\xi)\widehat{\sigma_{z}(v)}(\xi)\,\text{\it\dj\hskip
1.0pt}\xi,$ (5.11)

Then, by use of $q_{z}$

$\sigma_{z}(P_{0}v)(x)=z^{-2a}(\operatorname{Op}(p_{z})\sigma_{z}(v))(x)+(\operatorname{Op}(q_{z})\sigma_{z}(v))(x)\quad\text{for
all }|x|\leq 3.$ (5.12)

Denote $\operatorname{Op}(p_{z})=P_{z}$, $\operatorname{Op}(q_{z})=Q_{z}$, so
that (5.12) reads

$\sigma_{z}(P_{0}v)(x)=z^{-2a}(P_{z}\sigma_{z}(v))(x)+(Q_{z}\sigma_{z}(v))(x)\;\quad\text{for
all }|x|\leq 3.$ (5.13)

Recall from [2, Lemma 6.7] the technical lemma that serves to control
remainder terms:

###### Lemma 5.6

For any $k\in\mathbb{N}$ there is some $C>0$ such that for all $z\in(0,1]$

$\|\zeta_{z}\|_{C^{1+\tau}(\mathbb{R}^{n-1})}\leq
Cz^{\min(1,\tau)},\qquad|p_{z}-\overline{p}|_{k,C^{\tau}S^{2a}_{1,0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})}\leq
Cz^{\min(1,\tau)}.$ (5.14)

Define $\Sigma_{z}$ to be like $\Sigma$ for $|x^{\prime}|\geq 2$, and for
$|x^{\prime}|\leq 2$ to be of the form $\\{x\mid|x^{\prime}|\leq
2,\zeta_{z}(x^{\prime})<x_{n}<2M\\}$. Using Lemma 5.6, we can apply the same
argumentation as around (5.7) to the difference $\overline{P}-P_{z,\zeta_{z}}$
to show that for a sufficiently small $z\in(0,1]$, $P_{z,\zeta_{z}}$ has an
estimate

$\|(P_{z,\zeta_{z}}-\lambda)v\|_{L_{q}(\Sigma^{\prime}_{z})}\geq
c^{\prime}_{1}|\lambda|\|v\|_{L_{q}(\Sigma^{\prime}_{z})}\quad\text{for all
}\lambda\in V_{\delta,K_{1}},\;v\in D_{q}(\overline{\Sigma}_{z}).$

This is carried back by diffeomorphism to show that $P_{z}$ has an estimate

$\|(P_{z}-\lambda)v\|_{L_{q}(\Sigma_{z})}\geq
c_{1}|\lambda|\|v\|_{L_{q}(\Sigma_{z})}\quad\text{for all }\lambda\in
V_{\delta,K_{1}},\;v\in D_{q}(\overline{\Sigma}_{z}).$ (5.15)

Here the family $\lambda(P_{z,\zeta_{z},D}-\lambda)^{-1}$ is
$\mathcal{R}$-bounded for $\lambda\in V_{\delta,K_{1}}$, and then so is the
family $\lambda(P_{z,D}-\lambda)^{-1}$.

We _fix_ such a $z$ in the following!

Note that when $v$ is supported in $B_{2}$, then
$\|v\|_{D_{q}(\overline{\Sigma}_{z})}$ identifies with
$\|v\|_{D_{q}(\overline{\mathbb{R}}^{n}_{\zeta_{z}})}$. For functions $u$
supported in $B_{1}$, $\|u\|_{D_{q}(\overline{\Omega})}$ can be replaced by
$\|u\|_{D_{q}(\overline{\Sigma})}$, since $D_{q}(\overline{\Omega})$ is
defined here by the localization using $\zeta_{1}(x^{\prime})$, which equals
$\zeta(x^{\prime})$ for $|x^{\prime}|\leq 1$ (cf. the definition of the
transmission space by local coordinates).

Now we consider a function $u\in D_{q}(\overline{\Sigma})$ with support in
$B_{z/4}$. By the definition of $\zeta_{z}$, the function $\sigma_{z}u$ is in
$D_{q}(\overline{\Sigma}_{z})$, supported in $B_{1/4}$. We insert $u$ in
(5.13), replace $P_{0}$ by $P_{0}-\lambda$ by subtracting $\lambda\sigma_{z}u$
from both sides, and multiply the resulting equation by $\psi=\chi_{1,1/2}$,
so that the validity extends to $x\in{\mathbb{R}}^{n}$. We shall moreover
multiply the equation by $\psi^{\prime}=\chi_{(1+\varepsilon),1}$ for a small
$\varepsilon>0$; it satisfies $\psi^{\prime}\psi=\psi$. This gives, since
$\psi\sigma_{z}u=\sigma_{z}u$,

$\displaystyle\psi(x)$
$\displaystyle\sigma_{z}((P_{0}-\lambda)u)(x)=z^{-2a}\psi(x)(P_{z}\sigma_{z}(u))(x)-\lambda\psi(x)\sigma_{z}u(x)+\psi(x)(Q_{z}\sigma_{z}(u))(x)$
$\displaystyle=\psi^{\prime}(x)[z^{-2a}\psi(x)(P_{z}\sigma_{z}(u))(x)-\lambda\sigma_{z}u(x)+\psi(x)(Q_{z}\sigma_{z}(u))(x)]\quad\text{for
all }x\in{\mathbb{R}}^{n}.$

Here we can moreover use that

$\psi
P_{z}\sigma_{z}u=P_{z}(\psi\sigma_{z}u)+[\psi,P_{z}]\sigma_{z}u=P_{z}(\sigma_{z}u)+[\psi,P_{z}]\sigma_{z}u,$

so that we get

$\psi\sigma_{z}((P_{0}-\lambda)u)=\psi^{\prime}[z^{-2a}P_{z}\sigma_{z}u-\lambda\sigma_{z}u+\psi
Q_{z}\sigma_{z}u+z^{-2a}[\psi,P_{z}]\sigma_{z}u].$ (5.16)

Denote $S_{z}=\psi Q_{z}+z^{-2a}[\psi,P_{z}]$ — it is bounded from
$H_{q}^{\max\\{0,2a-1\\}}({\mathbb{R}}^{n})$ to $L_{q}({\mathbb{R}}^{n})$ —
then (5.16) takes the form

$\psi\sigma_{z}((P_{0}-\lambda)u)=\psi^{\prime}(z^{-2a}P_{z}+S_{z}-\lambda)\sigma_{z}u\quad\text{on
}{\mathbb{R}}^{n}.$ (5.17)

When $a\leq\frac{1}{2}$, $S_{z}$ is bounded in $L_{q}({\mathbb{R}}^{n})$, and
when $a>\frac{1}{2}$, it satisfies an inequality (4.13) since $0<2a-1<a$. Then
we can apply Proposition 4.6 with $A=z^{-2a}P_{z}$ and $S=S_{z}$ over
$\Sigma_{z}$, finding that the Dirichlet problem for $z^{-2a}P_{z}+S_{z}$ over
$\Sigma_{z}$ has the desired type of estimate for some $K_{2}$ sufficiently
large:

$\|(z^{-2a}P_{z}+S_{z}-\lambda)v\|_{L_{q}(\Sigma_{z})}\geq
c_{2}|\lambda|\|v\|_{L_{q}(\Sigma_{z})}\quad\text{for }\lambda\in
V_{\delta,K_{2}},$ (5.18)

for all $v\in D_{q}(\overline{\Sigma}_{z})$, with $\mathcal{R}$-boundedness of
the family
$\lambda((z^{-2a}P_{z}+S_{z})_{D}-\lambda)^{-1}\in\mathcal{L}(L_{q}(\Sigma_{z}))$
for $\lambda\in V_{\delta,K_{2}}$.

Finally, this will be scaled back to a replacement of $\sigma_{z}u$ (recall
that it is short for $x\mapsto u(zx)$) by $u$. The set $\Sigma_{z}$ will then
be replaced by a set $\Sigma_{1}=\\{y\in{\mathbb{R}}^{n}\mid
y/z\in\Sigma_{z}\\}$, where the important observation is that the piece where
$|x^{\prime}|<z$, $\zeta_{z}(x^{\prime})<x_{n}<2M$, is carried over to the
piece where $|y^{\prime}|<1$, $\zeta(y^{\prime})<y_{n}<2zM$, which coincides
with a piece of $\Omega$. The operator $z^{-2z}P_{z}+S_{z}$ is (by a formula
as in (5.11)) carried over to an operator we shall call $P_{1}$ (by a slight
abuse of notation);

$(z^{-2z}P_{z}+S_{z})\sigma_{z}v=P_{1}v,$

and $P_{1,D}$ now has the appropriate $\mathcal{R}$-sectoriality over
$\Sigma_{1}$. Note also that $\psi=\chi_{1,1/2}$ carries over to
$\chi_{1/z,1/2z}$, where $1/2z\geq 1/2$. Formula (5.17) then takes the form

$\chi_{1/z,1/2z}(P_{0}-\lambda)u=\chi_{(1+\varepsilon)/z,1/z}(P_{1}-\lambda)u,\text{
when }u\in D_{q}(\overline{\Omega})\text{ with }\operatorname{supp}u\subset
B_{z/4},$

showing (5.4). Multiplication by $\varphi$ on both sides gives (5.5), ending
the proof of Proposition 5.4. ∎

There is a related, slightly easier statement for interior points:

###### Proposition 5.7

Let $P$, $P_{0}$ and $\Omega$ be as in Proposition 5.4. Consider an interior
point $x_{0}\in\Omega$.

Then there exists a $z\in(0,1]$ and a $P_{1}$ satisfying Hypothesis 3.1 such
that the following holds: For $u\in D_{q}(\overline{\Omega})$ supported in
$B_{z/4}(x_{0})$, and $\varphi\in C_{0}^{\infty}(B_{z/2}(x_{0}))$, we have

$\varphi(P_{0}-\lambda)u=\varphi(P_{1}-\lambda)u\text{ on }{\mathbb{R}}^{n},$
(5.19)

where $P_{1}\colon H^{2a}_{q}({\mathbb{R}}^{n})\to L_{q}({\mathbb{R}}^{n})$ is
$\mathcal{R}$-sectorial on $V_{\delta,K}$ for some $K\geq 0$.

###### Proof.

Here we depart from Proposition 5.1 in a similar way. Consider an interior
point $x_{0}\in\Omega$. We have from Proposition 5.1 that there are $\delta>0$
and $K\geq 0$ such that $\overline{P}=\operatorname{Op}(p(x_{0},\cdot))$
satisfies an estimate

$\|(\overline{P}-\lambda)u\|_{L_{q}({\mathbb{R}}^{n})}\geq
c_{0}|\lambda|\|u\|_{L_{q}({\mathbb{R}}^{n})}\quad\text{for all }\lambda\in
V_{\delta,K},$

with $\mathcal{R}$-boundedness of $\lambda(\overline{P}-\lambda)^{-1}$ on
$V_{\delta,K}$. By a dilation, we can assume that
$\overline{B}_{4}(x_{0})\subset\Omega$. There is a version of Lemma 5.5
stating that the first inequality in (5.6) assures that
$\|\overline{P}-P_{0}\|_{\mathcal{L}(H^{2a}_{q}({\mathbb{R}}^{n}),L_{q}({\mathbb{R}}^{n}))}\leq\varepsilon$,
for a small enough $\varepsilon^{\prime}$. Then we get when $p_{0}$ is close
enough to $\overline{p}$ that for some $K^{\prime}\geq K$,

$\|(P_{0}-\lambda)u\|_{L_{q}({\mathbb{R}}^{n})}\geq
c_{0}|\lambda|\|u\|_{L_{q}({\mathbb{R}}^{n})}\quad\text{for all }\lambda\in
V_{\delta,K^{\prime}},$

with $\mathcal{R}$-boundedness of $\lambda(P_{0}-\lambda)^{-1}$ on
$V_{\delta,K^{\prime}}$.

Define $p_{z}$ and $q_{z}$ as in (5.9)ff. Solutions supported in balls
$B_{r}(x_{0})$ with $r<4$ are then simply in
$\dot{H}_{q}^{2a}(\overline{B}_{r}(x_{0}))$ (and no modification of a boundary
is needed). The result is now obtained by repeating the arguments from the
proof of Proposition 5.4, with ${\mathbb{R}}^{n}$ as the auxiliary domain
instead of $\Sigma_{z}$. ∎

Our aim is now to use these very local statements to control operators over
$\Omega$.

It was shown in [26] that the spectrum of the Dirichlet realization of $P$,
known in the $L_{2}$-setting to be a discrete set $\Sigma$ contained in a
sector opening to the right, is the same in the $L_{q}$-setting for all
$1<q<\infty$. So we know already that the resolvent equation

$(P-\lambda)u=f\text{ in }\Omega,\quad u=0\text{ in
}{\mathbb{R}}^{n}\setminus\Omega,$ (5.20)

has a unique solution for $\lambda$ in a suitable sector $V_{\delta,K}$; it is
the estimate of the solution operator for large $\lambda$ that we need to
show.

Resolvent estimates are easy to deduce in the $L_{2}$-setting from the
variational theory. We want to obtain them for general $q$, including
$\mathcal{R}$-boundedness, when $P$ has real positive principal symbol at the
boundary points.

###### Theorem 5.8

Let $\Omega$ be bounded with $C^{1+\tau}$-boundary, $\tau>2a$, and let
$1<q<\infty$. Let $P=\operatorname{Op}(p)$ satisfy Hypothesis 3.1, and assume
that the principal symbol $p_{0}(x,\xi)$ is real positive at each boundary
point $x\in\partial\Omega$. Then there are constants $\delta>0$, $c_{0}>0$ and
$K_{0}\geq 0$ such that $P-\lambda$ satisfies an estimate for all $u\in
D_{q}(\overline{\Omega})=H^{a(2a)}(\overline{\Omega})$:

$\|(P-\lambda)u\|_{L_{q}(\Omega)}\geq
c_{0}|\lambda|\|u\|_{L_{q}(\Omega)}\text{ when }\lambda\in V_{\delta,K_{0}},$
(5.21)

with $\mathcal{R}$-boundedness of the family
$\\{\lambda(P_{D}-\lambda)^{-1}\mid\lambda\in V_{\delta,K_{0}}\\}$ in
$\mathcal{L}(L_{q}(\Omega))$.

###### Proof.

We can assume $P=P_{0}$, since $P-P_{0}$ is a $\psi$do of order $2a-1$ to
which Proposition 4.6 can be applied as soon as the estimates are established
for $P_{0}$. (One here uses Remark 4.9, observing that
$s=(2a-1)_{+}=\max\\{2a-1,0\\}$ is $<a$.)

By Proposition 5.4, there is for every $x\in\partial\Omega$ a ball $B_{r}(x)$
and an auxiliary $C^{1+\tau}$-domain $\Sigma_{1}$ and $\mathcal{R}$-sectorial
operator $P_{1}$ on $D_{q}(\overline{\Sigma}_{1})$ such that
$\varphi(P_{0}-\lambda)u=\varphi(P_{1}-\lambda)u$ when $u\in
D_{q}(\overline{\Omega})$ with support in $B_{r/4}(x)$ and $\varphi\in
C_{0}^{\infty}(B_{r/2}(x))$; here the $B_{s}(x)$, $0<s\leq r$, are
neighborhoods of the kind $U_{j}$ ($j\geq 1$) described before (2.15). A
related statement holds for interior points $x$, by Proposition 5.7; here the
auxiliary domain $\Sigma_{1}$ is simply ${\mathbb{R}}^{n}$. Since
$\overline{\Omega}$ is compact, there is a finite cover $B_{r_{i}/4}(x_{i})$,
$i=1,\dots,N$, of $\overline{\Omega}$ by such balls. Introduce a partition of
unity $\\{\varrho_{i}\\}_{i=1,\dots,N}$ (with $\varrho_{i}\in
C_{0}^{\infty}(B_{r_{i}/4}(x_{i}),[0,1])$, satisfying $\sum_{1\leq i\leq
N}\varrho_{i}=1$ on $\overline{\Omega}$), and choose functions $\psi_{i}\in
C_{0}^{\infty}(B_{r_{i}/2}(x_{i})$ that are 1 on $B_{r_{i}/4}(x_{i})$. Denote
by $P_{i}$ and $\Sigma_{i}$ the associated operator and domain for which

$\varphi(P_{0}-\lambda)u=\varphi(P_{i}-\lambda)u$ (5.22)

holds when $u\in D_{q}(\overline{\Omega})$ with $\operatorname{supp}u\subset
B_{r_{i}/4}(x_{i})$, and $\operatorname{supp}\varphi\subset
B_{r_{i}/2}(x_{i})$, $\lambda\in V_{\delta,K_{i}}$, according to Propositions
5.4 and 5.7.

We want to construct an approximate inverse of $P_{0,D}-\lambda$ by use of
these identities in the local coordinate patches.

For a given $f\in L_{q}(\Omega)$, let $u=u(\lambda)\in
D_{q}(\overline{\Omega})$ be the family of functions satisfying

$(P_{0}-\lambda)u(\lambda)=f\text{ on }\Omega,\text{ for }\lambda\in
V_{\delta,K}.$

By multiplication by $\varrho_{i}$, we find
$\varrho_{i}(P_{0}-\lambda)u=\varrho_{i}f$, and hence

$(P_{0}-\lambda)\varrho_{i}u=\varrho_{i}f+[\varrho_{i},P_{0}]u\quad\text{on
}\Omega.$ (5.23)

Multiplication by $\psi_{i}$ gives

$\psi_{i}(P_{0}-\lambda)\varrho_{i}u=\psi_{i}\varrho_{i}f+\psi_{i}[\varrho_{i},P_{0}]u=\varrho_{i}f+\psi_{i}[\varrho_{i},P_{0}]u\quad\text{on
}\Omega.$

By Propositions 5.4 and 5.7, the left-hand side equals
$\psi_{i}(P_{i}-\lambda)\varrho_{i}u$, hence

$\psi_{i}(P_{i}u-\lambda)\varrho_{i}u=\varrho_{i}f+\psi_{i}[\varrho_{i},P_{0}]u\quad\text{on
}\Omega,\text{ supported in }B_{r_{i}/2}(x_{i}).$

In particular,

$1_{\Sigma_{i}}\psi_{i}(P_{i}-\lambda)\varrho_{i}u=1_{\Sigma_{i}}(\varrho_{i}f+\psi_{i}[\varrho_{i},P_{0}]u)\quad\text{on
}\Omega.$ (5.24)

Here we observe that

$1_{\Sigma_{i}}\varrho_{i}f=1_{\Omega}\varrho_{i}f,$

when $f$ is considered as extended by 0 outside $\Omega$.

For the left-hand side of (5.24), we note that by a commutation with
$\psi_{i}$,

$\psi_{i}(P_{i}-\lambda)\varrho_{i}u=(P_{i}-\lambda)\varrho_{i}u-[P_{i},\psi_{i}]\varrho_{i}\psi_{i}u,$

since $\psi_{i}\varrho_{i}=\varrho_{i}$, and for the right-hand side,

$\psi_{i}[\varrho_{i},P_{0}]u=[\varrho_{i},P_{0}]\psi_{i}u+[\psi_{i},[\varrho_{i},P_{0}]]u;$

this leads to the formula

$\begin{split}1_{\Sigma_{i}}(P_{i}-\lambda)\varrho_{i}u&=1_{\Sigma_{i}}\varrho_{i}f+1_{\Sigma_{i}}S_{i}\psi_{i}u+1_{\Sigma_{i}}S_{i}^{\prime}u,\text{
with}\\\ S_{i}&=[P_{i},\psi_{i}]\varrho_{i}+[\varrho_{i},P_{0}],\\\
S_{i}^{\prime}&=[\psi_{i},[\varrho_{i},P_{0}]].\end{split}$ (5.25)

Here $S_{i}$ is a $\psi$do of order $2a-1$ and $S_{i}^{\prime}$ is of order
$2a-2$; the latter order is $\leq 0$ and the former is so when
$a\leq\frac{1}{2}$.

Now compose all this with $(P_{i,D}-\lambda)^{-1}\colon L_{q}(\Sigma_{i})\to
D_{q}(\overline{\Sigma}_{i})$, arriving at

$\varrho_{i}u=(P_{i,D}-\lambda)^{-1}1_{\Sigma_{i}}\varrho_{i}f+(P_{i,D}-\lambda)^{-1}1_{\Sigma_{i}}(S_{i}\psi_{i}u+S_{i}^{\prime}u).$
(5.26)

This has the form of an $\mathcal{R}$-controlled operator family acting on $f$
and two operators acting on $u$ with lower order factors, one of them applied
to the global $u$. Summation over $i$ gives a representation of
$u=R_{\lambda}f$ as an $\mathcal{R}$-controlled sum and a remainder term that
should behave better for $|\lambda|\to\infty$:

$\begin{split}R_{\lambda}f&=u=R_{0,\lambda}f+T_{\lambda}u,\text{ where }\\\
R_{0,\lambda}f&=\sum_{i\leq
N}(P_{i,D}-\lambda)^{-1}1_{\Sigma_{i}}\varrho_{i}f,\\\
T_{\lambda}u&=\sum_{i\leq
N}(P_{i,D}-\lambda)^{-1}1_{\Sigma_{i}}(S_{i}\psi_{i}u+S_{i}^{\prime}u).\end{split}$
(5.27)

Here we let $\lambda\in V_{\delta,\overline{K}}$, where
$\overline{K}=\max_{i\leq N}\\{K_{i}\\}$. The first line shows:

$(1-T_{\lambda})R_{\lambda}=R_{0,\lambda}\quad\text{on $\Omega$ for
}\lambda\in V_{\delta,\overline{K}}.$ (5.28)

To obtain a useful formula for $R_{\lambda}$ from $R_{0,\lambda}$ and
$T_{\lambda}$ is easiest when $a\leq\frac{1}{2}$, since all the $S_{i}$ and
$S_{i}^{\prime}$ are then bounded in $L_{q}$-norm. However, we shall give just
one formulation of the proof that works for all $0<a<1$.

Consider

$H_{\lambda}=\sum_{k=0}^{\infty}T_{\lambda}^{k}R_{0,\lambda}.$ (5.29)

If the series converges, then

$H_{\lambda}-T_{\lambda}H_{\lambda}=\sum_{k=0}^{\infty}T_{\lambda}^{k}R_{0,\lambda}-\sum_{k=1}^{\infty}T_{\lambda}^{k}R_{0,\lambda}=R_{0,\lambda},$

so

$(1-T_{\lambda})H_{\lambda}=R_{0,\lambda}.$

This is the equation, $R_{\lambda}$ should solve, cf. (5.28). If
$1-T_{\lambda}$ is invertible in a suitable sense, we can conclude that
$R_{\lambda}=H_{\lambda}$.

Let us first investigate the invertibility of $1-T_{\lambda}$. We have for
$R_{i,\lambda}=(P_{i,D}-\lambda)^{-1}$ the standard resolvent estimates when
$\lambda\in V_{\delta,\overline{K}}$:

$\|\lambda R_{i,\lambda}f\|_{L_{q}(\Sigma_{i})}\leq
c\|f\|_{L_{q}(\Sigma_{i})},\quad\|R_{i,\lambda}f\|_{\dot{H}_{q}^{a}(\overline{\Sigma}_{i})}\leq
c_{1}\|R_{i,\lambda}f\|_{D_{q}(\overline{\Sigma}_{i})}\leq
c_{2}\|f\|_{L_{q}(\Sigma_{i})},$

when $f\in L_{q}(\overline{\Sigma}_{1})$. Since $(2a-1)_{+}\in\,[0,a)$, there
is an interpolation inequality (as in (4.11))

$\|v\|_{\dot{H}_{q}^{2a-1}(\overline{\Sigma}_{i})}\leq
c_{3}\|v\|^{\theta}_{L_{q}(\Sigma_{i})}\|v\|^{1-\theta}_{\dot{H}_{q}^{a}(\overline{\Sigma}_{i})},$
(5.30)

where $\theta=1-(2a-1)_{+}/a$, equal to $(1-a)/a$ if $a>\frac{1}{2}$ and $1$
if $a\leq\frac{1}{2}$. Then for
$u\in\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega})$,

$\displaystyle\|R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}$
$\displaystyle+S_{i}^{\prime})u\|_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Sigma}_{i})}\leq
c_{3}\|R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})u\|^{\theta}_{L_{q}(\Sigma_{i})}\|R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})u\|^{1-\theta}_{\dot{H}_{q}^{a}(\overline{\Sigma}_{i})}$
$\displaystyle\leq
c_{3}|\lambda|^{-\theta}(c_{1}\|1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})u\|_{L_{q}(\Sigma_{i})})^{\theta}(c_{2}\|1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})u\|_{L_{q}(\Sigma_{i})})^{1-\theta}$
$\displaystyle\leq
c_{4}|\lambda|^{-\theta}(\|\psi_{i}u\|_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Sigma}_{i})}+\|u\|_{L_{q}(\Omega)})\leq
c_{5}|\lambda|^{-\theta}\|u\|_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega})}.$

It follows that

$\|T_{\lambda}u\|_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega})}=\Big{\|}\sum_{i=1}^{N}R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})u\Big{\|}_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega})}\leq
c_{6}|\lambda|^{-\theta}\|u\|_{\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega})}.$

Thus for $|\lambda|$ sufficiently large, $\sum_{k\geq 0}T_{\lambda}^{k}$
converges in $\mathcal{L}(\dot{H}_{q}^{(2a-1)_{+}}(\overline{\Omega}))$, so
$1-T_{\lambda}$ is bijective there, and since $H_{\lambda}$ and $R_{\lambda}$
range in the subspace $D_{q}(\overline{\Omega})$, $R_{\lambda}$ identifies
with $H_{\lambda}$.

Now let us show $\mathcal{R}$-boundedness for large $|\lambda|$. The $k$’th
term in the series is

$T_{\lambda}^{k}R_{0,\lambda}=T_{\lambda}^{k}\sum_{j=1}^{N}R_{j,\lambda}1_{\Sigma_{j}}\varrho_{j}.$

For $k=1$, $\lambda
T_{\lambda}R_{0,\lambda}=\lambda\sum_{i,j=1}^{N}R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})R_{j,\lambda}1_{\Sigma_{j}}\varrho_{j}$
has an $\mathcal{R}$-bound estimated by

$\displaystyle\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\Big{\\{}\lambda\sum_{i,j=1}^{N}R_{i,\lambda}1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})R_{j,\lambda}1_{\Sigma_{j}}\varrho_{j}\big{|}\lambda\in
V_{\delta,K_{1}}\Big{\\}}$
$\displaystyle\leq\sum_{i,j=1}^{N}\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{\lambda
R_{i,\lambda}|\lambda\in
V_{\delta,K_{1}}\\}\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})R_{j,\lambda}1_{\Sigma_{j}}\varrho_{j}|\lambda\in
V_{\delta,K_{1}}\\}$

by the sum and product rules. Since $S_{i}\psi_{i}+S_{i}^{\prime}$ is of order
$(2a-1)_{+}$, we can use Theorem 4.8 $3^{\circ}$, (5.30) and the fact that
$D_{q}(\overline{\Sigma}_{1})\subset\dot{H}_{q}^{a}(\overline{\Sigma}_{1})$ to
show that for $K_{1}\geq\overline{K}$, the $\mathcal{R}$-bound of the second
factor in each term is $\leq cK_{1}^{-\theta}$ when $\lambda\in
V_{\delta,K_{1}}$. Denote

$\max_{i\leq N}\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{\lambda
R_{i,\lambda}\mid\lambda\in V_{\delta,\overline{K}}\\}=C_{0}.$

For a given $0<\varepsilon<1$, take $K_{1}$ so large that for all
$i,j=1,\dots,N$,

$\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{1_{\Sigma_{i}}(S_{i}\psi_{i}+S_{i}^{\prime})R_{j,\lambda}1_{\Sigma_{j}}\varrho_{j}\mid\lambda\in
V_{\delta,K_{1}}\\}\leq\varepsilon$ (5.31)

Then by summation over $i,j$,

$\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{\lambda
T_{\lambda}R_{0,\lambda}\mid\lambda\in V_{\delta,K_{1}}\\}\leq
C_{0}N^{2}\varepsilon.$

For $T^{k}R_{0,\lambda}$ there are similar formulas with $k$ factors of the
second type:

$T_{\lambda}^{k}R_{0,\lambda}=\sum_{i_{1},\dots,i_{k+1}=1}^{N}R_{i_{1},\lambda}1_{\Sigma_{i_{1}}}(S_{i_{1}}\psi_{i_{1}}+S_{i_{1}}^{\prime})\dots
R_{i_{k},\lambda}1_{\Sigma_{i_{k}}}(S_{i_{k}}\psi_{i_{k}}+S_{i_{k}}^{\prime})R_{i_{k+1},\lambda}1_{\Sigma_{i_{k+1}}}\varrho_{i_{k+1}}.$

Here we find the estimate

$\mathcal{R}_{\mathcal{L}(L_{q}(\Omega))}\\{\lambda
T^{k}_{\lambda}R_{0,\lambda}\mid\lambda\in V_{\delta,K_{1}}\\}\leq
C_{0}N^{k+1}\varepsilon^{k}.$

Then, if we adapt the choice of $K_{1}$ such that (5.31) holds with
$\varepsilon<1/N$, the series (5.29) converges with respect to
$\mathcal{R}$-bounds (by [11, Proposition 4.8]). Then
$R_{\lambda}=H_{\lambda}$ has been determined and is $\mathcal{R}$-sectorial
on $V_{\delta,K_{1}}$. ∎

###### Remark 5.9

It is seen from the proof that the evenness of the symbol of $P$ is only
needed in a small neighborhood of the boundary; away from this, strong
ellipticity suffices.

##  6 Results for linear evolution equations

We now turn to the consequences for heat problems.

Thanks to the results in Section 5, we can now obtain maximal regularity
results in much more general cases than the one in Proposition 5.2.

###### Theorem 6.1

Let $\Omega$ be bounded with $C^{1+\tau}$-boundary for some $\tau>2a$, and let
$1<p,q<\infty$. Let $P=\operatorname{Op}(p)$ satisfy Hypothesis 3.1, and
assume that the principal symbol $p_{0}(x,\xi)$ is real positive at each
boundary point $x\in\partial\Omega$. Let $I=(0,T)$ for some $T\in(0,\infty)$.
Then for any $f\in L_{p}(I;L_{q}(\Omega))$, the heat equation (1.1) has a
unique solution $u\in C^{0}(\overline{I};L_{q}(\Omega))$ satisfying

$u\in L_{p}(I;D_{q}(\overline{\Omega}))\cap H^{1}_{p}(I;L_{q}(\Omega)).$ (6.1)

###### Proof.

Because of Theorem 5.8, the shifted operator $P_{D,q}+k\colon
D(P_{D,q})=H^{a(2a)}_{q}(\overline{\Omega})\subset L_{q}(\Omega)\to
L_{q}(\Omega)$ satisfies the second statement of Theorem 4.5 for some $k>0$
sufficiently large. Hence $P_{D,q}+k$ has maximal $L_{p}$-regularity on
$I=\mathbb{R}_{+}$. This implies that $P_{D,q}$ has maximal $L_{p}$-regularity
on $I=(0,T)$ for any $T\in(0,\infty)$. ∎

Note that the theorem allows $p\neq q$.

Nonhomogeneous boundary problems can also be considered. There is a local
Dirichlet boundary condition associated with $P$, namely the assignment of
$\gamma_{0}(u/d_{0}^{a-1})$; recall
$d_{0}(x)=\operatorname{dist}(x,\partial\Omega)$ near $\partial\Omega$,
extended smoothly to $\Omega$. As shown in earlier works (cf. [22], [26]), it
is natural to study the problem

$Pu=f\text{ in
}\Omega,\quad\gamma_{0}(u/d_{0}^{a-1})=\varphi,\quad\operatorname{supp}u\subset\overline{\Omega},$
(6.2)

for $u$ in the $(a-1)$-transmission space
$H_{q}^{(a-1)(2a)}(\overline{\Omega})$ (cf. (2.14)ff.), which is mapped by
$r^{+}P$ into $L_{q}(\Omega)$ by [26, Theorem 3.5]. This is a larger space
than $D_{q}(\overline{\Omega})=H_{q}^{a(2a)}(\overline{\Omega})$, satisfying

$H_{q}^{a(2a)}(\overline{\Omega})=\\{u\in
H_{q}^{(a-1)(2a)}(\overline{\Omega})\mid\gamma_{0}(u/d_{0}^{a-1})=0\\}.$ (6.3)

The problem (6.2) is Fredholm solvable with $u\in
H_{q}^{(a-1)(2a)}(\overline{\Omega})$ for $f,\varphi$ given in $L_{q}(\Omega)$
resp. $B_{q,q}^{a+1-1/q}(\partial\Omega)$, when $\tau>2a+1$ [26, Theorem 5.1].

Note that the case $\varphi=0$ in (6.2) is the homogeneous Dirichlet problem.
There is the notation for the boundary mapping, provided with a normalizing
constant,

$\gamma_{0}^{a-1}\colon u\mapsto\Gamma(a+1)\gamma_{0}(u/d_{0}^{a-1}).$

By [26, Theorem 2.3] with $\mu=a-1$, there holds:

###### Proposition 6.2

When $\tau\geq 1$ and $a-1+\frac{1}{q}<s<\tau$ with $s<\tau+a-1$, the mapping
$\gamma_{0}^{a-1}$ is continuous from $H_{q}^{(a-1)(s)}(\overline{\Omega})$ to
$B_{q,q}^{s-a+1-\frac{1}{q}}(\partial\Omega)$ and has a right inverse
$K_{(0)}^{a-1}$ that maps continuously

$K_{(0)}^{a-1}\colon B_{q,q}^{s-a+1-\frac{1}{q}}(\partial\Omega)\to
H_{q}^{(a-1)(s)}(\overline{\Omega}).$

In particular,

$K^{a-1}_{(0)}\colon B_{q,q}^{a+1-\frac{1}{q}}(\partial\Omega)\to
H_{q}^{(a-1)(2a)}(\overline{\Omega}),\;K^{a-1}_{(0)}\colon
B_{q,q}^{\varepsilon}(\partial\Omega)\to
H_{q}^{(a-1)(a-1+\frac{1}{q}+\varepsilon)}(\overline{\Omega}),$ (6.4)

for $\varepsilon>0$ (subject to $s=a-1+\frac{1}{q}+\varepsilon<\tau+a-1$).

By Lemma 5.3 in [26], $H_{q}^{(a-1)(s)}(\overline{\Omega})\subset
L_{q}(\Omega)$ for $s\geq 0$, when $q<\frac{1}{1-a}$. We assume this for the
nonhomogeneous heat problem:

$\begin{split}\partial_{t}u+Pu&=f\text{ on }\Omega\times I,\\\
\gamma_{0}(u/d_{0}^{a-1})&=\psi\text{ on }\partial\Omega\times I,\\\
u&=0\text{ on }(\mathbb{R}^{n}\setminus\Omega)\times I,\\\
u|_{t=0}&=0.\end{split}$ (6.5)

Here we can show:

###### Theorem 6.3

In addition to the assumptions of Theorem 6.1, assume that $\tau>2a+1$ and
$q<\frac{1}{1-a}$. Then (6.5) has for $f\in L_{p}(I;L_{q}(\Omega))$, $\psi\in
L_{p}(I;B_{q,q}^{a+1-1/q}(\partial\Omega))\cap
H_{p}^{1}(I;B_{q,q}^{\varepsilon}(\partial\Omega))$ with $\psi(x,0)=0$
($\varepsilon>0$) a unique solution $u$ satisfying

$u\in L_{p}(I;H_{q}^{(a-1)(2a)}(\overline{\Omega}))\cap
H_{p}^{1}(I;L_{q}(\Omega)).$ (6.6)

###### Proof.

Considering the boundary mapping and its right inverse as constant in $t$, we
can add a time-parameter $t$, and have in view of Propostion 6.2 and (6.4) for
any $p\in(1,\infty)$ that with $I=(0,T)$,

$\displaystyle\gamma_{0}^{a-1}$ $\displaystyle\colon
L_{p}(I;H_{q}^{(a-1)(2a)}(\overline{\Omega}))\to
L_{p}(I;B_{q,q}^{a+1-\frac{1}{q}}(\partial\Omega)),$
$\displaystyle\gamma_{0}^{a-1}$ $\displaystyle\colon
H^{1}_{p}(I;H_{q}^{(a-1)(a-1+\frac{1}{q}+\varepsilon)}(\overline{\Omega}))\to
H^{1}_{p}(I;B_{q,q}^{\varepsilon}(\partial\Omega)),$

with right inverses $K^{a-1}_{(0)}$ continuous in the opposite direction.

For the given $\psi$ as in the assumptions, let
$v(x,t)=K^{a-1}_{(0)}\psi(x,t)$; it lies in
$L_{p}(I;H_{q}^{(a-1)(2a)}(\overline{\Omega}))$ and in
$H^{1}_{p}(I;L_{q}(\Omega))$ (since
$H_{q}^{(a-1)(a-1+\frac{1}{q}+\varepsilon)}(\overline{\Omega})\subset
L_{q}(\Omega)$), and satisfies

$\gamma_{0}^{a-1}v=\psi,\quad v|_{t=0}=0,\quad r^{+}Pv\in
L_{p}(I;L_{q}(\Omega)),\quad\partial_{t}v\in L_{p}(I;L_{q}(\Omega)).$

Then $w=u-v$ is in $L_{p}(I;H_{q}^{(a-1)(2a)}(\overline{\Omega}))$ with
$\gamma_{0}^{a-1}w=0$, hence in $L_{p}(I;H_{q}^{a(2a)}(\overline{\Omega}))$ by
(6.3). Moreover, $(r^{+}P+\partial_{t})(u-v)\in L_{p}(I;L_{q}(\Omega))$. Thus
in order for $u$ to solve (6.5), $w$ must solve a problem (1.1) with
homogeneous boundary condition and $f$ replaced by $f-(r^{+}P+\partial_{t})v$.
Here Theorem 6.1 assures that there is a unique solution $w\in
L_{p}(I;H_{q}^{a(2a)}(\overline{\Omega}))\cap H_{p}^{1}(I;L_{q}(\Omega))$.
Then $u=v+w$ is the unique solution of (6.5), satisfying (6.6). ∎

Let us also mention that one can use the resolvent estimates (just in uniform
norms) to show results for other function spaces. For example, by a strategy
of Amann [3]:

###### Theorem 6.4

Assumptions as in Theorem 6.1. Let $s$ be noninteger $>0$. For any
$f\in\dot{C}^{s}(\overline{\mathbb{R}}_{+};L_{q}(\Omega))$ there is a unique
solution
$u\in\dot{C}^{s}(\overline{\mathbb{R}}_{+};D_{q}(\overline{\Omega}))$, and
there holds

$f\in\dot{C}^{s}(\overline{\mathbb{R}}_{+};L_{q}(\Omega))\iff
u\in\dot{C}^{s}(\overline{\mathbb{R}}_{+};D_{q}(\overline{\Omega}))\cap\dot{C}^{s+1}(\overline{\mathbb{R}}_{+};L_{q}(\Omega)).$
(6.7)

###### Proof.

The proof goes exactly as in [24, Theorem 5.14]. The notation
$\dot{C}^{s}(\overline{\mathbb{R}}_{+};X)$ indicates the functions in
$C^{s}({\mathbb{R}};X)$ vanishing for $t<0$. ∎

As in Remark 5.9 we observe that the evenness of the symbol $p(x,\xi)$ is only
needed in a small neighborhood of the boundary.

##  7 Applications to nonlinear evolution equations

In this last section we present an application of the result on maximal
regularity established in Theorem 6.1 to existence of strong solutions of the
nonlinear nonlocal parabolic equation

$\displaystyle\partial_{t}u+a_{0}(x,u)Pu$ $\displaystyle=f(x,u)$
$\displaystyle\text{in }\Omega\times(0,T),$ (7.1) $\displaystyle u$
$\displaystyle=0$ $\displaystyle\text{on
}(\mathbb{R}^{n}\setminus\Omega)\times(0,T),$ $\displaystyle u|_{t=0}$
$\displaystyle=u_{1}$ $\displaystyle\text{in }\Omega,$

for some $T>0$.

###### Theorem 7.1

Let $\Omega$ be a bounded domain with $C^{1+\tau}$-boundary for some
$\tau>2a$, and let $1<p,q<\infty$ be such that

$(a+\tfrac{1}{q})(1-\tfrac{1}{p})-\tfrac{n}{q}>0.$ (7.2)

If $n=1$, assume moreover $\frac{1}{q}<a$. Let $P$ satisfy Hypothesis 3.1, and
assume that the principal symbol $p_{0}(x,\xi)$ is real positive at each
boundary point $x\in\partial\Omega$. Moreover, for an open set
$U\subset\mathbb{R}$ with $0\in U$, let $a_{0}\in
C^{\max(1,\tau)}(\mathbb{R}^{n}\times U,\mathbb{R})$ with $a_{0}(x,s)>0$ for
all $s\in U$ and $x\in\mathbb{R}^{n}$, let $f\colon\mathbb{R}^{n}\times
U\to\mathbb{R}\colon(x,u)\mapsto f(x,u)$ be continuous and locally Lipschitz
with respect to $u\in U$, and let
$u_{0}\in(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}\cap
C^{\tau}(\overline{\Omega})$ with $\overline{u_{0}(\Omega)}\subset U$. Then
there are $\varepsilon_{0},T>0$ such that for every $u_{1}\in
X_{\gamma,1}:=(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}$ with
$\|u_{0}-u_{1}\|_{X_{\gamma},1}\leq\varepsilon_{0}$, the system (7.1)
possesses a unique solution

$u\in L_{p}((0,T);D_{q}(\overline{\Omega}))\cap
H^{1}_{p}((0,T);L_{q}(\Omega)).$

###### Proof.

We prove the result by applying a local existence result for an abstract
evolution equation by Köhne et al. [31, Theorem 2.1], which can also be found
in [35, Theorem 5.1.1]. Alternatively, one could also use a result by Clément
and Li [8, Theorem 2.1]. To this end we choose $X_{0}=L_{q}(\Omega)$,
$X_{1}=D_{q}(\overline{\Omega})$. Note that (7.2) implies $\frac{1}{q}<a$ when
$n\geq 2$, so that
$D_{q}(\overline{\Omega})\hookrightarrow\dot{H}_{q}^{a+\frac{1}{q}-\varepsilon}(\overline{\Omega})$
by (3.6) for all $n\geq 1$. Here
$\dot{H}_{q}^{a+\frac{1}{q}-\varepsilon}(\overline{\Omega})\hookrightarrow\overline{H}_{q}^{a+\frac{1}{q}-\varepsilon}(\Omega)$.

Then in the notation of [31] (with $\mu=1$)

$\displaystyle X_{\gamma,1}$
$\displaystyle:=(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}$
$\displaystyle\hookrightarrow(L_{q}(\Omega),\overline{H}_{q}^{a+\frac{1}{q}-\varepsilon}(\Omega))_{1-\frac{1}{p},p}=\overline{B}_{q,p}^{(a+\frac{1}{q}-\varepsilon)(1-\frac{1}{p})}(\Omega)\hookrightarrow
C^{0}(\overline{\Omega})$ (7.3)

for $\varepsilon>0$ sufficiently small, in view of (7.2). Moreover, let

$V_{1}:=\\{u\in X_{\gamma,1}\mid u(x)\in U\text{ for all
}x\in\overline{\Omega}\\}.$

Then $V_{1}\subset X_{\gamma,1}$ is open due to (7.3) and the fact that
$U\subset\mathbb{R}$ is open. Moreover, since $a_{0},f\colon U\to\mathbb{R}$
are locally Lipschitz continuous, we have that

$u\mapsto a_{0}(\cdot,u(\cdot)),u\mapsto f(\cdot,u(\cdot))\in
C^{0,1}(V_{1},C^{0}(\overline{\Omega})).$

Now we define $A\colon V_{1}\to\mathcal{L}(X_{1},X_{0})$ and $F\colon V_{1}\to
X_{0}$ by

$A(u)=a_{0}(\cdot,u(\cdot))P,\quad F(u)=f(\cdot,u(\cdot))\qquad\text{for all
}u\in V_{1}.$

Because of $P\in\mathcal{L}(D_{q}(\overline{\Omega}),L_{q}(\Omega))$, this
yields

$\displaystyle A\in C^{0,1}(V_{1},\mathcal{L}(X_{1},X_{0})),\ F\in
C^{0,1}(V_{1},X_{0}).$

Finally, we note that, since $u_{0}\in X_{\gamma,1}\cap
C^{\tau}(\overline{\Omega})$, we have $a_{0}(\cdot,u_{0}(\cdot))\in
C^{\tau}(\overline{\Omega})$. Thus $A(u_{0})=\operatorname{Op}(\tilde{p})$,
with $\tilde{p}(x,\xi)=a_{0}(x,u_{0}(x))p(x,\xi)$ for all
$x,\xi\in\mathbb{R}^{n}$, satisfies again Hypothesis 3.1. Therefore $A(u_{0})$
has maximal $L_{p}$-regularity on every finite time interval $I=(0,T)$,
$0<T<\infty$ due to Theorem 6.1. Hence all assumptions of [31, Theorem 2.1]
with $\mu=1$ are satisfied. This yields the statement of the theorem. ∎

###### Remark 7.2

Actually, the uniqueness statement in Theorem 7.1 holds in a slightly stronger
local sense: If $u,\tilde{u}\in
L_{p}((0,T^{\prime});D_{q}(\overline{\Omega}))\cap
H^{1}_{p}((0,T^{\prime});L_{q}(\Omega))$ are solutions of (7.1) with $(0,T)$
replaced by $(0,T^{\prime})$ for some $T^{\prime}\in(0,T]$ and initial value
as before, then $u\equiv\tilde{u}$. This follows immediately from the proof of
[31, Theorem 2.1], which is based on the contraction mapping principle and
uses that $T$ is sufficiently small.

Finally, we apply the previous result to a fractional nonlinear diffusion
equation with a nonzero exterior condition, of the form

$\displaystyle\partial_{t}w+P\varphi(w)$ $\displaystyle=0$
$\displaystyle\text{in }\Omega\times(0,T),$ (7.4) $\displaystyle w$
$\displaystyle=w_{b}$ $\displaystyle\text{on
}(\mathbb{R}^{n}\setminus\Omega)\times(0,T),$ $\displaystyle w|_{t=0}$
$\displaystyle=w_{1}$ $\displaystyle\text{in }\Omega,$

for some function $\varphi\in C^{2}(\mathbb{R}_{+},\mathbb{R})$ with
$\varphi^{\prime}(s)>0$ for all $s\in\mathbb{R}_{+}$.

###### Corollary 7.3

Let $\Omega$ be bounded with $C^{1+\tau}$-boundary for some $\tau>2a$, and let
$1<p,q<\infty$ be such that (7.2) holds, assuming also $\frac{1}{q}<a$ if
$n=1$. Let $P$ satisfy Hypothesis 3.1, and assume that the principal symbol
$p_{0}(x,\xi)$ is real positive at each boundary point $x\in\partial\Omega$,
and that $P$ maps real functions to real functions. Moreover, let $\varphi\in
C^{2}(\mathbb{R}_{+})$ be real with $\varphi^{\prime}(s)>0$ for all
$s\in\mathbb{R}_{+}$, let $w_{b}\in{H}^{2a}_{q}(\mathbb{R}^{n})\cap
C^{\tau}(\mathbb{R}^{n})$ be real with $\inf_{x\in\mathbb{R}^{n}}w_{b}(x)>0$
for all $x\in\mathbb{R}^{n}$, and let
$w_{0}\colon\overline{\Omega}\to\mathbb{R}_{+}$ be such that
$\varphi(w_{0})-\varphi(w_{b})\in(L_{q}(\Omega),D_{q}(\overline{\Omega}))_{1-\frac{1}{p},p}\cap
C^{\tau}(\overline{\Omega})$. Then there is some $\varepsilon_{0}>0$ such that
for every $w_{1}\colon\Omega\to\mathbb{R}_{+}$ with
$\varphi(w_{1})-\varphi(w_{0})\in X_{\gamma,1}$ (cf. (7.3)) and
$\|\varphi(w_{0})-\varphi(w_{1})\|_{X_{\gamma,1}}\leq\varepsilon_{0}$, the
system (7.4) possesses a unique solution $w\in\bigcap_{0\leq
s<\frac{1}{q}}L_{p}((0,T);\overline{H}^{a+s}_{q}(\Omega))\cap
H^{1}_{p}((0,T);L_{q}(\Omega))$ with $\varphi(w)-\varphi(w_{b})\in
L_{p}((0,T);D_{q}(\overline{\Omega}))$ for some $T>0$.

###### Proof.

We use a reformulation of (7.4) in the form (7.1). First of all, in view of
(7.2)ff., we have $w_{0}\in C^{0}(\overline{\Omega})$ and therefore

$\delta:=\min\big{\\{}\inf_{x\in\overline{\Omega}}w_{0}(x),\inf_{x\in\mathbb{R}^{n}}w_{b}(x)\big{\\}}>0.$

Hence there is some $\tilde{\varphi}\in C^{2}(\mathbb{R},\mathbb{R})$ with
$\tilde{\varphi}^{\prime}(s)>0$ for all $s\in\mathbb{R}$ and
$\tilde{\varphi}(s)=\varphi(s)$ for all $s\geq\frac{\delta}{2}$. Moreover, we
define

$\displaystyle a_{0}(x,s)$
$\displaystyle=\tilde{\varphi}^{\prime}(\tilde{\varphi}^{-1}(s+\varphi(w_{b}(x))))$
$\displaystyle\text{for all }s\in U:=\mathbb{R},x\in\mathbb{R}^{n},$
$\displaystyle f(x,s)$ $\displaystyle=-a_{0}(x,s)P(\varphi(w_{b}(x)))(x)$
$\displaystyle\text{for all }s\in\mathbb{R},x\in\Omega,$

and
$u_{0}:=\tilde{\varphi}(w_{0})-\tilde{\varphi}(w_{b})=\varphi(w_{0})-\varphi(w_{b})$.
Hence we can apply Theorem 7.1 and get the existence of some
$\varepsilon_{0}>0$ and $T>0$ such that for every
$w_{1}\colon\Omega\to\mathbb{R}_{+}$ with
$\tilde{\varphi}(w_{1})-\varphi(w_{0})\in X_{\gamma,1}$ (cf. (7.3)) and
$\|\tilde{\varphi}(w_{0})-\varphi(w_{1})\|_{X_{\gamma,1}}\leq\varepsilon_{0}$
there is a unique solution $u\in L_{p}((0,T);D_{q}(\overline{\Omega}))\cap
H^{1}_{p}((0,T);L_{q}(\Omega))$ of (7.1). Moreover, by choosing
$\varepsilon_{0}>0$ sufficiently small, we can achieve that
$\|\tilde{\varphi}(w_{0})-\varphi(w_{1})\|_{X_{\gamma,1}}\leq\varepsilon_{0}$
implies $\|w_{0}-w_{1}\|_{C^{0}(\overline{\Omega})}<\delta/2$, since
$\tilde{\varphi}\colon\mathbb{R}\to\mathbb{R}$ is strictly monotone. Hence
$\inf_{x\in\overline{\Omega}}w_{0}>\delta/2$ and
$\tilde{\varphi}(w_{0})=\varphi(w_{0})$ in that case. Now let us define
$w:=\tilde{\varphi}^{-1}(u+\varphi(w_{b}))$.

Then $w\in
L_{p}((0,T);\overline{H}^{a+\frac{1}{q}-\varepsilon}_{q}(\Omega))\cap
H^{1}_{p}((0,T);L_{q}(\Omega))$ since

$u+\varphi(w_{b})\in
L_{p}((0,T);\overline{H}^{a+\frac{1}{q}-\varepsilon}_{q}(\Omega))\cap
H^{1}_{p}((0,T);L_{q}(\Omega)),$

$\tilde{\varphi}\in C^{2}(\mathbb{R})$ and by well-known results on
composition operators on Sobolev and Bessel potential spaces and

$\displaystyle\partial_{t}w$
$\displaystyle=(\tilde{\varphi}^{-1})^{\prime}(u+\varphi(w_{b}))\partial_{t}u=-(\tilde{\varphi}^{-1})^{\prime}(u+\varphi(w_{b}))a_{0}(\cdot,u(\cdot))P(u+\varphi(w_{b}))=-P(\tilde{\varphi}(w)).$

Moreover, since

$\displaystyle
L_{p}((0,T);\overline{H}^{a+\frac{1}{q}-\varepsilon}_{q}(\Omega))\cap
H^{1}_{p}((0,T);L_{q}(\Omega))$ $\displaystyle\hookrightarrow
BUC([0,T];\overline{H}^{(a+\frac{1}{q}-\varepsilon)(1-\frac{1}{p})}_{q}(\Omega))\hookrightarrow
C^{0}([0,T]\times\overline{\Omega})$

for $\varepsilon>0$ sufficiently small due to (7.2)ff. and
$\inf_{x\in\overline{\Omega}}w_{0}>\delta$, we can achieve

$\inf_{x\in\overline{\Omega},t\in[0,T]}w(x,t)>\delta/2$

by choosing $T>0$ sufficiently small. Hence $\tilde{\varphi}(w)=\varphi(w)$.
Finally, $\tilde{\varphi}(w)-\varphi(w_{b})=u\in
L_{p}((0,T);D_{q}(\overline{\Omega}))$ by definition. This shows existence of
a solution.

It remains to show uniqueness of the constructed solution $w$. To this end let
$\tilde{w}\in
L_{p}((0,T);\overline{H}^{a+\frac{1}{q}-\varepsilon}_{q}(\Omega))\cap
H^{1}_{p}((0,T);L_{q}(\Omega))$ with $\varphi(\tilde{w})-\varphi(w_{b})\in
L_{p}((0,T);D_{q}(\overline{\Omega}))$ be another solution of (7.4) and
consider

$t_{0}:=\sup\left\\{T^{\prime}\in[0,T]\mid
w|_{[0,T^{\prime}]}\equiv\tilde{w}|_{[0,T^{\prime}]}\right\\}.$

We show by contradiction that $t_{0}=T$, which implies the uniqueness. Hence
assume $t_{0}<T$. Since $w,\tilde{w}\in C^{0}([0,T]\times\overline{\Omega})$,
we have

$\inf_{x\in\overline{\Omega}}\tilde{w}(x,t_{0})=\inf_{x\in\overline{\Omega}}w(x,t_{0})>\delta/2.$

Hence there is some $T^{\prime}\in(t_{0},T)$ such that

$\inf_{x\in\overline{\Omega},t\in[t_{0},T^{\prime}]}\tilde{w}(x,t)>\delta.$

Therefore
$\tilde{u}:=\varphi(\tilde{w})|_{[0,T^{\prime}]}=\tilde{\varphi}(\tilde{w})|_{[0,T^{\prime}]}\in
L_{p}((0,T^{\prime});D_{q}(\overline{\Omega}))\cap
H^{1}_{p}((0,T^{\prime});L_{q}(\Omega))$ is a solution of (7.1) with $(0,T)$
replaced by $(0,T^{\prime})$. Since $u|_{[0,T^{\prime}]}$ solves the same
system, the improved uniqueness statement of Remark 7.2 implies that
$\tilde{u}|_{[0,T^{\prime}]}=u|_{0,T^{\prime}]}$. This yields
$\tilde{w}|_{[0,T^{\prime}]}=w|_{[0,T^{\prime}]}$, which is a contradiction to
the definition of $t_{0}$. Hence $t_{0}=T$, and uniqueness is shown. ∎

###### Example 7.4

Choosing $\varphi(w)=w^{m}$ for $m>1$ in (7.4) yields a case including the
fractional porous medium equation; in the latter,
$P\varphi(w)=(-\Delta)^{a}w^{m}$.

The problem with $\varphi$ was studied e.g. in Hölder spaces in the case
$\Omega=\mathbb{R}^{n}$ and $P=(-\Delta)^{a}$ by Vázques, de Pablo, Quirós and
Rodríguez in [44], which lists a number of applications including the
fractional porous medium equation. Roidos and Shao obtained maximal
$L_{p}$-regularity results in [36] in cases like
$P=(-\nabla\cdot\mathfrak{a}(x)\nabla)^{a}$, with $\Omega$ replaced by a
smooth closed $n$-dimensional Riemannian manifold; they applied it in their
Section 6.1 to porous medium equations for $P=(-\Delta)^{a}$. The present
study achieves these types of results for the first time on domains $\Omega$
with boundary; examples include $P=L^{a}$ where $L$ is as in (7.6) below.

Corollary 7.3 applies moreover to pseudodifferential operators $P$ satisfying
Hypothesis 3.1 with $p(x,\xi)$ real and vanishing odd-numbered symbol terms
$p_{2k+1},k\in\mathbb{N}_{0}$, so that $p(x,-\xi)=p(x,\xi)$; cf. Remark 7.5
below.

For completeness, we give some details on when operators in complex function
spaces map real functions to real functions:

###### Remark 7.5

A function $u\in\mathcal{S}({\mathbb{R}}^{n})$ is real if and only if
$\hat{u}(-\xi)=\overline{\hat{u}(\xi)}$ for all $\xi\in{\mathbb{R}}^{n}$. It
follows from (2.8) that $P=\operatorname{Op}(p)$ maps real functions to real
functions if and only if $p(x,-\xi)=\overline{p(x,\xi)}$ for all
$x,\xi\in{\mathbb{R}}^{n}$. This gives one criterion for preserving real
functions.

For operators arising from functional calculus, another criterion may be
convenient: When $A$ is a linear operator in
$L_{q}({\mathbb{R}}^{n},\mathbb{C})$, define $\overline{A}$ by
$\overline{A}u=\overline{A\overline{u}}$, with $D(\overline{A})=D(A)$. Then
$A$ maps real functions to real functions if and only if $\overline{A}=A$.
Assume this, and let $f$ be a function on $\mathbb{C}$, holomorphic on the
resolvent set of $A$, satisfying

$\overline{f(\lambda)}=f(\overline{\lambda}).$ (7.5)

Let the operator $f(A)$ be defined by a Dunford integral
$f(A)u=\tfrac{i}{2\pi}\int_{\mathcal{C}}f(\lambda)(A-\lambda)^{-1}u\,d\lambda$,
where $\mathcal{C}$ is a curve encircling the spectrum of $A$
counterclockwise. Note that from $\overline{(A-\lambda)}=A-\overline{\lambda}$
follows $\overline{(A-\lambda)^{-1}}=(A-\overline{\lambda})^{-1}$ when
$\lambda$ is in the resolvent set. Hence, in view of (7.5),

$\displaystyle\overline{f(A)}u$
$\displaystyle=\overline{\tfrac{i}{2\pi}\int_{\mathcal{C}}f(\lambda)(A-\lambda)^{-1}\overline{u}\,d\lambda}=\tfrac{-i}{2\pi}\int_{\mathcal{C}}f(\overline{\lambda})(\overline{A}-\overline{\lambda})^{-1}u\,d\overline{\lambda}$
$\displaystyle=\tfrac{i}{2\pi}\int_{\mathcal{C}^{\prime}}f(\mu)(\overline{A}-\mu)^{-1}u\,d\mu=f(A)u,$

since $\overline{A}=A$ (here $\mathcal{C}^{\prime}$ is the curve obtained by
conjugation of $\mathcal{C}$ and oriented counterclockwise). Thus $f(A)$
preserves real functions.

This can be used for example when $P=L^{a}$, where

$Lu=-\sum_{j,k=1}^{n}\partial_{j}a_{jk}(x)\partial_{k}u+b(x)u,$ (7.6)

with $(a_{jk}(x))_{1\leq j,k\leq n}$ being a real, symmetric, $x$-dependent
matrix with a positive lower bound for $x\in{\mathbb{R}}^{n}$, and $b(x)\geq
0$. $L$ preserves real functions. The fractional powers $L^{a}=LL^{a-1}$
($0<a<1$) can be defined under mild smoothness hypotheses on the coefficients;
then they also preserve real functions. When all coefficients are in
$C^{\infty}_{b}({\mathbb{R}}^{n})$, the construction of Seeeley [42] shows
that $P$ has a smooth symbol satisfying Hypothesis 3.1. When coefficients are
just $C^{\tau}$, there is a principal symbol $p_{0}=\bigl{(}\sum
a_{jk}(x)\xi_{j}\xi_{k}\bigr{)}^{a}$ satisfying Hypothesis 3.1 , but the
remainder term would need further analysis.

## References

  * [1] H. Abels, Pseudodifferential boundary value problems with non-smooth coefficients, Comm. Part. Diff. Eq., 30 (2005), pp. 1463–1503.
  * [2] H. Abels and G. Grubb, Fractional-order operators on nonsmooth domains, J. Lond. Math. Soc. (2), 107 (2023), pp. 1297–1350.
  * [3] H. Amann, Operator-valued fourier multipliers, vector-valued besov spaces, and applications, Math. Nachr, 186 (1997), pp. 5–56.
  * [4] U. Biccari, M. Warma, and E. Zuazua, Local regularity for fractional heat equations, in Recent advances in PDEs: analysis, numerics and control, vol. 17 of SEMA SIMAI Springer Ser., Springer, Cham, 2018, pp. 233–249.
  * [5] D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, in Probability and analysis (Varenna, 1985), vol. 1206 of Lecture Notes in Math., Springer, Berlin, 1986, pp. 61–108.
  * [6] H. Chang-Lara and G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), pp. 139–172.
  * [7] J.-H. Choi, K.-H. Kim, and J. Ryu, Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets, Discrete Contin. Dyn. Syst., 43 (2023), pp. 3338–3377.
  * [8] P. Clément and S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem, Adv. Math. Sci. Appl., 3 (1993/94), pp. 17–32.
  * [9] E. B. Davies, Heat kernels and spectral theory, vol. 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989.
  * [10] R. Denk, An introduction to maximal regularity for parabolic evolution equations, in Nonlinear partial differential equations for future applications, vol. 346 of Springer Proc. Math. Stat., Springer, Singapore, [2021] ©2021, pp. 1–70.
  * [11] R. Denk, M. Hieber, and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), pp. viii+114.
  * [12] R. Denk and J. Seiler, Maximal $L_{p}$-regularity of non-local boundary value problems, Monatsh. Math., 176 (2015), pp. 53–80.
  * [13] H. Dong, P. Jung, and D. Kim, Boundedness of non-local operators with spatially dependent coefficients and $L_{p}$-estimates for non-local equations, Calc. Var. Partial Differential Equations, 62 (2023), pp. Paper No. 62, 28.
  * [14] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013), pp. 1539–1573.
  * [15] X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: parabolic equations, J. Funct. Anal., 272 (2017), pp. 4165–4221.
  * [16] M. Fukushima, Y. Ōshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, vol. 19 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1994.
  * [17] G. Grubb, Pseudo-differential boundary problems in $L_{p}$ spaces, Comm. Part. Diff. Eq. 15, 289-340, 1990.
  * [18] , Parameter-elliptic and parabolic pseudodifferential boundary problems in global $L^{p}$ sobolev spaces, Math. Z., 218 (1995), pp. 43–90.
  * [19] , Functional Calculus of Pseudodifferential Boundary Problems, 2nd Edition, Birkhäuser, Basel - Boston - Berlin, 1996.
  * [20] , Distributions and operators, vol. 252 of Graduate Texts in Mathematics, Springer, New York, 2009.
  * [21] , Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators, Analysis and P.D.E., 7 (2014), pp. 1649–1682.
  * [22] , Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$-transmission pseudodifferential operators, Adv. Math., 268 (2015), pp. 478–528.
  * [23] , Regularity in $L_{p}$ Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), pp. 2634–2660.
  * [24] , Fractional-order operators: boundary problems, heat equations, in Mathematical analysis and applications—plenary lectures, vol. 262 of Springer Proc. Math. Stat., Springer, Cham, 2018, pp. 51–81.
  * [25] , Limited regularity of solutions to fractional heat and Schrödinger equations, Discrete Contin. Dyn. Syst., 39 (2019), pp. 3609–3634.
  * [26] , Resolvents for fractional-order operators with nonhomogeneous local boundary conditions, J. Funct. Anal., 284 (2023), pp. Paper No. 109815, 55.
  * [27] G. Grubb and N. J. Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in $L_{p}$ Sobolev spaces, Acta Math., Vol. 171, No. 2, 165-229, 1993.
  * [28] L. Hörmander,  Seminar notes on pseudo-differential operators and boundary problems.  Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/search/, 1966.
  * [29] , The Analysis of Linear Partial Differential Operators III, Springer, Berlin-Heidelberg-New York, 1985.
  * [30] T. Jin and J. Xiong, Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., 35 (2015), pp. 5977–5998.
  * [31] M. Köhne, J. Prüss, and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_{p}$-spaces, J. Evol. Equ., 10 (2010), pp. 443–463.
  * [32] D. Lamberton, Équations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces $L^{p}$, J. Funct. Anal., 72 (1987), pp. 252–262.
  * [33] T. Leonori, I. Peral, A. Primo, and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), pp. 6031–6068.
  * [34] J. Marschall, Parametrices for nonregular elliptic pseudodifferential operators, Math. Nachr., 159 (1992), pp. 175–188.
  * [35] J. Prüss and G. Simonett, Moving interfaces and quasilinear parabolic evolution equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016.
  * [36] N. Roidos and Y. Shao, Maximal $L_{q}$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces, Pure Appl. Funct. Anal., 7 (2022), pp. 1037–1063.
  * [37] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), pp. 3–26.
  * [38] X. Ros-Oton, J. Serra, and E. Valdinoci, Pohozaev identities for anisotropic integrodifferential operators, Comm. Partial Differential Equations, 42 (2017), pp. 1290–1321.
  * [39] X. Ros-Oton and H. Vivas, Higher-order boundary regularity estimates for nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), pp. Paper No. 111, 20.
  * [40] R. T. Seeley, Refinement of the functional calculus of Calderón and Zygmund, Indag. Math., 27 (1965), pp. 521–531.  Nederl. Akad. Wetensch. Proc. Ser. A 68.
  * [41] , Integro-differential operators on vector bundles, Trans. Amer. Math. Soc., 117 (1965), pp. 167–204.
  * [42] , Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10 (1967), pp. 288–307. Amer. Math. Soc., R. I.
  * [43] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978\.
  * [44] J. L. Vázquez, A. de Pablo, F. Quirós, and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations, J. Eur. Math. Soc. (JEMS), 19 (2017), pp. 1949–1975.