# Some problems for Petrovskii parabolic systems
in generalized Sobolev spaces

Aleksandr Dyachenko, Valerii Los  Aleksandr Dyachenko
National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic
Institute, Prospect Peremohy 37, 03056, Kyiv-56, Ukraine
<EMAIL_ADDRESS>Valerii Los
National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic
Institute, Prospect Peremohy 37, 03056, Kyiv-56, Ukraine
v$\underline{\phantom{k}}$<EMAIL_ADDRESS>
(Date: 29/03/2021)

###### Abstract.

We consider an inhomogeneous initial-boundary value problem for a Petrovskii
parabolic system of second order PDEs. We prove that this problem induces
isomorphisms between appropriate anisotropic generalized Sobolev spaces. The
regularity of these spaces are given by a pair of real numbers and by a
function parameter. The latter allows us to characterize the regularity of
solutions to the problem more finely as compared with anisotropic Sobolev
spaces.

###### Key words and phrases:

Parabolic initial-boundary value problem, generalized Sobolev space, slowly
varying function, isomorphism property, interpolation with a function
parameter.

###### 2010 Mathematics Subject Classification:

35K20, 46E35

##  1\. Introduction

The central result of the theory of parabolic initial-boundary value problems
states that these problems are well posed in the sense of Hadamard on
appropriate pairs of some normed function spaces. This theory generally uses
anisotropic versions of classical Hölder spaces or Sobolev spaces (see, e.g.,
[1, 32, 14, 16, 7, 8]). In recent years, other classes of function spaces are
applied more and more actively to parabolic problems; namely, spaces with
mixed norms, Triebel–Lizorkin spaces, weighted spaces, spaces of generalized
smoothness (see, e.g., [33, 11, 5, 15, 6, 23] and reference therein). Our
paper continues the series of works [23, 18, 25, 19, 20, 24] aimed to build a
theory of parabolic initial-boundary value problems in function spaces of
generalized anisotropic smoothness. This smoothness is given by a pair of real
numbers and by a radial function which varies slowly at infinity and
characterizes supplementary smoothness with respect to that given by the
numbers. Such spaces give a broad generalization of anisotropic versions of
inner product Sobolev spaces applied usually to parabolic equations. These
works have elaborated a relevant theory of solvability and maximal regularity
for scalar parabolic problems (involving one PDE). The present paper considers
a general enough inhomogeneous initial-boundary volume problem for a
Petrovskii parabolic system of second-order PDEs. We prove that this problem
is well posed on appropriate pairs of generalized anisotropic Sobolev spaces
just mentioned. This result is formulated in terms of an isomorphism theorem
and is proved with the help of the quadratic interpolation (with function
parameter) between some anisotropic Sobolev spaces. The special case of
homogeneous Cauchy data was studied in [21, 22]. Note that various spaces of
generalized smoothness proved to be useful in the theory of PDEs [9, 10, 27,
28, 29] and the theory of stochastic processes [12]. Specifically, the
monograph [27] presents a theory of elliptic boundary-value problems for
isotropic analogs of the spaces used in the present paper (see also [3, 2] for
elliptic problems in broader classes of generalized Sobolev spaces).

##  2\. Statement of the problem

We arbitrarily choose an integer $n\geq 2$ and a real number $\tau>0$. Let $G$
be a bounded domain in $\mathbb{R}^{n}$ with an infinitely smooth boundary
$\Gamma:=\partial G$. We put $\Omega:=G\times(0,\tau)$ and
$S:=\Gamma\times(0,\tau)$; thus, $\Omega$ is an open cylinder in
$\mathbb{R}^{n+1}$, and $S$ is its lateral boundary. Then
$\overline{\Omega}:=\overline{G}\times[0,\tau]$ and
$\overline{S}:=\Gamma\times[0,\tau]$ are the closures of $\Omega$ and $S$
respectively.

Consider the following parabolic initial-boundary-value problem in $\Omega$:

$\begin{split}\partial_{t}u_{j}(x,t)+\sum_{k=1}^{N}\sum_{|\alpha|\leq
2}&a^{\alpha}_{j,k}(x,t)\,D^{\alpha}_{x}u_{k}(x,t)=f_{j}(x,t)\\\ &\mbox{for
all}\quad(x,t)\in\Omega\quad\mbox{and}\quad j\in\\{1,\dots,N\\};\end{split}$
(2.1) $\begin{split}\sum_{k=1}^{N}\sum_{|\alpha|\leq
l_{j}}&b^{\alpha}_{j,k}(x,t)\,D^{\alpha}_{x}u_{k}(x,t)\big{|}_{S}=g_{j}(x,t)\\\
&\mbox{for all}\quad(x,t)\in S\quad\mbox{and}\quad
j\in\\{1,\dots,N\\};\end{split}$ (2.2)
$u_{j}(x,t)\big{|}_{t=0}=h_{j}(x)\quad\mbox{for all}\quad x\in
G\quad\mbox{and}\quad j\in\\{1,\ldots,N\\}.$ (2.3)

Here, the integer $N\geq 2$ is arbitrarily chosen, and
$l_{1},\ldots,l_{N}\in\\{0,\,1\\}$. All coefficients of the partial
differential expressions in formulas (2.1) and (2.2) are supposed to be
infinitely smooth complex-valued functions given on $\overline{\Omega}$ and
$\overline{S}$ respectively; i.e., each

$a_{j,k}^{\alpha,\beta}\in
C^{\infty}(\overline{\Omega}):=\bigl{\\{}w\\!\upharpoonright\overline{\Omega}\\!:\,w\in
C^{\infty}(\mathbb{R}^{n+1})\bigr{\\}}$

and each

$b_{j,k}^{\alpha,\beta}\in
C^{\infty}(\overline{S}):=\bigl{\\{}v\\!\upharpoonright\overline{S}\\!:\,v\in
C^{\infty}(\Gamma\times\mathbb{R})\bigr{\\}}.$

We use the notation $D^{\alpha}_{x}:=D^{\alpha_{1}}_{1}\dots
D^{\alpha_{n}}_{n}$, with $D_{k}:=i\,\partial/\partial{x_{k}}$, and
$\partial_{t}:=\partial/\partial t$ for the partial derivatives of functions
depending on $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$.
Here, $i$ is imaginary unit, and $\alpha=(\alpha_{1},...,\alpha_{n})$ is a
multi-index, with $|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$. In formulas (2.1)
and (2.2) and their analogs, we take summation over the integer-valued
nonnegative indices $\alpha_{1},...,\alpha_{n}$ that satisfy the condition
written under the integral sign.

We assume that the initial-boundary value problem (2.1)–(2.3) is Petrovskii
parabolic in the cylinder $\Omega$. Let us recall the corresponding definition
[32, Section 1, § 1]. Let

$A_{j,k}(x,t,D_{x},\partial_{t}):=\delta_{j,k}\partial_{t}+\sum_{|\alpha|\leq
2}a^{\alpha}_{j,k}(x,t)\,D^{\alpha}_{x}$ (2.4)

and

$B_{j,k}(x,t,D_{x}):=\sum_{|\alpha|\leq
l_{j}}b^{\alpha}_{j,k}(x,t)\,D^{\alpha}_{x}$ (2.5)

for all $j,k\in\\{1,\ldots,N\\}$. Here, $\delta_{j,k}$ is the Kronecker delta.
Using notation (2.4) and (2.5) we may rewrite all equalities in (2.1) and
(2.2) as follows:

$\sum_{k=1}^{N}A_{j,k}(x,t,D_{x},\partial_{t})u_{k}(x,t)=f_{j}(x,t)$

and

$\sum_{k=1}^{N}B_{j,k}(x,t,D_{x})u_{k}(x,t)\big{|}_{S}=g_{j}(x,t).$

Define the principal symbols of the linear partial differential operators
(2.4) and (2.5) by the formulas

$A^{(0)}_{j,k}(x,t,\xi,p):=\delta_{j,k}p+\sum_{|\alpha|=2}a^{\alpha}_{j,k}(x,t)\,\xi^{\alpha}$

and

$B^{(0)}_{j,k}(x,t,\xi):=\sum_{|\alpha|=l_{j}}b^{\alpha}_{j,k}(x,t)\,\xi^{\alpha}.$

These symbols are homogeneous polynomials in
$\xi:=(\xi_{1},\ldots,\xi_{n})\in\mathbb{C}^{n}$ and $p\in\mathbb{C}$ jointly
(as usual, $\xi^{\alpha}:=\xi_{1}^{\alpha_{1}}\ldots\xi_{n}^{\alpha_{n}}$).
Consider the matrices

$A^{(0)}(x,t,\xi,p):=\bigl{(}A^{(0)}_{j,k}(x,t,\xi,p)\bigr{)}_{j,k=1}^{N}\\\ $

and

$B^{(0)}(x,t,\xi):=\bigl{(}B^{(0)}_{j,k}(x,t,\xi)\bigr{)}_{j,k=1}^{N}.$

The problem (2.1)–(2.3) is said to be Petrovskii parabolic in $\Omega$ if it
satisfies the following two conditions (i) and (ii):

  * (i)

For arbitrary points $x\in\overline{G}$ and $t\in[0,\tau]$ and every vector
$\xi\in\mathbb{R}^{n}$, all the roots $p(x,t,\xi)$ of the polynomial $\det
A^{(0)}(x,t,\xi,p)$ in $p\in\mathbb{C}$ satisfy the inequality
$\mathrm{Re}\,p(x,t,\xi)\leq-\delta\,|\xi|^{2b}$ for some number $\delta>0$
that does not depend on $x$, $t$, and $\xi$.

To formulate Condition (ii), we fix a number $\delta_{1}\in(0,\delta)$, where
$\delta$ has appeared in Condition (i), and then arbitrarily choose a point
$x\in\Gamma$, real number $t\in[0,\tau]$, vector $\xi\in\mathbb{R}^{n}$
tangent to the boundary $\Gamma$ at $x$, and number $p\in\mathbb{C}$ such that
$\mathrm{Re}\,p\geq-\delta_{1}|\xi|^{2b}$ and $|\xi|+|p|\neq 0$. Let $\nu(x)$
denote the unit vector of the inward normal to $\Gamma$ at $x$. It follows
from Condition (i) and the inequality $n\geq 2$ that the polynomial $\det
A^{(0)}(x,t,\xi+\zeta\nu(x),p)$ in $\zeta\in\mathbb{C}$ has $m$ roots
$\zeta^{+}_{j}(x,t,\xi,p)$, $j=\nobreak 1,\ldots,m$, with positive imaginary
part and $m$ roots with negative imaginary part, taking into account
multiplicity of roots.

The second condition is formulated as follows:

  * (ii)

For some positive number $\delta_{1}<\delta$ and for every choice of the
parameters $x$, $t$, $\xi$ and $p$ indicated above, the rows of the matrix

$B^{(0)}(x,t,\xi+\zeta\nu(x),p)\cdot\widetilde{A}^{(0)}(x,t,\xi+\zeta\nu(x),p)$

are linearly independent modulo the polynomial
$\prod_{j=1}^{m}(\zeta-\zeta^{+}_{j}(x,t,\xi,p))$. Here, $\widetilde{A}^{(0)}$
is the transposed matrix of the cofactors of entries of $A^{(0)}$.

We investigate parabolic problem (2.1)–(2.3) in appropriate generalized
Sobolev inner product spaces considered in the next section. All functions
(and distributions) are supposed to be complex-valued.

##  3\. Generalized Sobolev spaces related to the problem

Following [25, Section 3] (see also [23, Section 3]), we consider generalized
Sobolev spaces used for the investigation of the parabolic problem
(2.1)–(2.3). They are parameterized with two numbers $s$ and $s/2$, where
$s\in\mathbb{R}$, and with a function $\varphi\in\mathcal{M}$. The class
$\mathcal{M}$ is defined to consist of all Borel measurable functions
$\varphi:[1,\infty)\rightarrow(0,\infty)$ such that

  * ($\ast$)

both the functions $\varphi$ and $1/\varphi$ are bounded on each compact
interval $[1,d]$, with $1<d<\infty$;

  * ($\ast\ast$)

$\varphi$ is a slowly varying function at infinity in the sense of J. Karamata
[13]; i.e.,

$\lim_{r\rightarrow\infty}\frac{\varphi(\lambda
r)}{\varphi(r)}=1\quad\mbox{for every}\quad\lambda>0.$ (3.1)

The theory of slowly varying functions is set forth in [4]. We give an
important and standard example of functions satisfying (3.1) by putting

$\varphi(r):=(\log r)^{\theta_{1}}\,(\log\log
r)^{\theta_{2}}\ldots(\,\underbrace{\log\ldots\log}_{k\;\mbox{\small{times}}}r\,)^{\theta_{k}}\quad\mbox{for}\quad
r\gg 1,$ (3.2)

where the parameters $k\in\mathbb{N}$ and
$\theta_{1},\theta_{2},\ldots,\theta_{k}\in\mathbb{R}$ are chosen arbitrarily.
The functions (3.2) form the logarithmic multiscale, which has a number of
applications in the theory of function spaces. Some other examples of slowly
varying functions can be found in [4, Sec. 1.3.3] and [27, Sec. 1.2.1].

By definition, the (complex) linear space $H^{s,s/2;\varphi}(\mathbb{R}^{k})$,
where $2\leq k\in\mathbb{Z}$, consists of all tempered distributions $w$ on
$\mathbb{R}^{k}$ whose (complete) Fourier transform $\widetilde{w}$ is locally
Lebesgue integrable over $\mathbb{R}^{k}$ and satisfies the condition

$\|w\|_{H^{s,s/2;\varphi}(\mathbb{R}^{k})}:=\biggl{(}\;\int\limits_{\mathbb{R}^{k-1}}\int\limits_{\mathbb{R}}r^{2s}(\xi,\eta)\,\varphi^{2}(r(\xi,\eta))\,|\widetilde{w}(\xi,\eta)|^{2}\,d\xi\,d\eta\biggr{)}^{1/2}<\infty,$
(3.3)

where

$r(\xi,\eta):=\bigl{(}1+|\xi|^{2}+|\eta|\bigr{)}^{1/2}\quad\mbox{for
all}\;\;\xi\in\mathbb{R}^{k-1}\;\;\mbox{and}\;\;\eta\in\mathbb{R}.$

This space is Hilbert and separable with respect to the norm (3.3).

It is a special case of the spaces $\mathcal{B}_{p,\mu}$ introduced by
Hörmander [9, Section 2.2]; namely,
$H^{s,s/2;\varphi}(\mathbb{R}^{k})=\mathcal{B}_{p,\mu}$ provided that $p=2$
and $\mu(\xi,\eta)\equiv r^{s}(\xi,\eta)\varphi(r(\xi,\eta))$. The space
$H^{s,s/2;\varphi}(\mathbb{R}^{k})$ give a broad generalization of the concept
of Sobolev spaces (in the framework of Hilbert spaces). If
$\varphi(\cdot)\equiv 1$, the space $H^{s,s/2;\varphi}(\mathbb{R}^{k})$
becomes the anisotropic Sobolev space $H^{s,s/2}(\mathbb{R}^{k})$. Generally,
we have the dense continuous embeddings

$H^{s_{1},s_{1}/2}(\mathbb{R}^{k})\hookrightarrow
H^{s,s/2;\varphi}(\mathbb{R}^{k})\hookrightarrow
H^{s_{0},s_{0}/2}(\mathbb{R}^{k})\quad\mbox{whenever}\quad s_{0}<s<s_{1}.$
(3.4)

Basing on $H^{s,s/2;\varphi}(\mathbb{R}^{k})$, consider some Hilbert function
spaces relating to the problem (2.1)–(2.3). Let $V$ be an open nonempty set in
$\mathbb{R}^{k}$. (Specifically, we need the case where $V=\Omega$, with
$k=n+1$.) Put

$H^{s,s/2;\varphi}(V):=\bigl{\\{}w\\!\upharpoonright\\!V:\,w\in
H^{s,s/2;\varphi}(\mathbb{R}^{k})\bigl{\\}}.$ (3.5)

The linear space (3.5) is endowed with the norm

$\|u\|_{H^{s,s/2;\varphi}(V)}:=\inf\bigl{\\{}\,\|w\|_{H^{s,s/2;\varphi}(\mathbb{R}^{k})}:w\in
H^{s,s/2;\varphi}(\mathbb{R}^{k}),\;\,u=w\\!\upharpoonright\\!V\bigl{\\}},$
(3.6)

where $u\in H^{s,s/2;\varphi}(V)$. This space is Hilbert and separable with
respect to this norm. The set $C^{\infty}(\overline{\Omega})$ is dense in
$H^{s,s/2;\varphi}(\Omega)$.

We also need the space $H^{s,s/2;\varphi}(S)$ on the lateral boundary $S$ of
the cylinder $\Omega$ (see [17, Section 1]). We restricting ourselves to the
$s>0$ case. Briefly saying, this space consists of all functions $v\in
L_{2}(S)$ that yield functions from the space $H^{s,s/2;\varphi}(\Pi)$ on
$\Pi:=\mathbb{R}^{n-1}\times(0,\tau)$ with the help of some local coordinates
on $\overline{S}$. Let us recall the detailed definition.

We arbitrarily choose a finite atlas on $\Gamma$ of class $C^{\infty}$. Let
this atlas be formed by some local charts
$\theta_{j}:\mathbb{R}^{n-1}\leftrightarrow\Gamma_{j}$, with
$j=1,\ldots,\lambda$. Here, $\Gamma_{1},\ldots,\Gamma_{\lambda}$ are open
nonempty subsets of $\Gamma$ such that
$\Gamma:=\Gamma_{1}\cup\cdots\cup\Gamma_{\lambda}$. We also arbitrarily choose
functions $\chi_{j}\in C^{\infty}(\Gamma)$, with $j=1,\ldots,\lambda$, such
that $\mathrm{supp}\,\chi_{j}\subset\Gamma_{j}$ and
$\chi_{1}+\cdots+\chi_{\lambda}=1$ on $\Gamma$. Thus, these functions form a
partition of unity on $\Gamma$.

By definition, the complex linear space $H^{s,s/2;\varphi}(S)$ consists of all
functions $v\in L_{2}(S)$ such that the function
$v_{j}(y,t):=\chi_{j}(\theta_{j}(y))v(\theta_{j}(y),t)$ of
$y\in\mathbb{R}^{n-1}$ and $t>0$ belongs to $H^{s,s/2;\varphi}(\Pi)$ for each
$j\in\\{1,\ldots,\lambda\\}$. (As usual, $L_{2}(S)$ denotes the space of all
square integrable functions on the surface $S$.) The space
$H^{s,s/2;\varphi}(S)$ is endowed with the norm

$\|v\|_{H^{s,s/2;\varphi}(S)}:=\bigl{(}\|v_{1}\|_{H^{s,s/2;\varphi}(\Pi)}^{2}+\cdots+\|v_{\lambda}\|_{H^{s,s/2;\varphi}(\Pi)}^{2}\bigr{)}^{1/2}.$

This space is Hilbert and separable with respect to this norm and does not
depend up to equivalence of norms on the indicated choice of an atlas and
partition of unity on $\Gamma$ [17, Theorem 1].

We also need the isotropic generalized Sobolev spaces
$H^{s;\varphi}(\mathbb{R}^{k})$, $H^{s;\varphi}(G)$, and
$H^{s;\varphi}(\Gamma)$ investigated by Mikhailets and Murach [27, 26]. Let
$s\in\mathbb{R}$ and $\varphi\in\mathcal{M}$. By definition, the complex
linear space $H^{s;\varphi}(\mathbb{R}^{k})$, where $1\leq k\in\mathbb{Z}$,
consists of all tempered distributions $w$ on $\mathbb{R}^{k}$ whose
(complete) Fourier transform $\widetilde{w}$ is locally Lebesgue integrable
over $\mathbb{R}^{k}$ and satisfies the condition

$\|w\|_{H^{s;\varphi}(\mathbb{R}^{k})}:=\biggl{(}\;\int\limits_{\mathbb{R}^{k}}\langle\xi\rangle^{2s}\,\varphi^{2}(\langle\xi\rangle)\,|\widetilde{w}(\xi)|^{2}\,d\xi\biggr{)}^{1/2}<\infty.$
(3.7)

Here, as usual, $\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$ is the smooth modulus
of $\xi\in\mathbb{R}^{k}$. This space is Hilbert and separable with respect to
the norm (3.7). Notice that $H^{s;\varphi}(\mathbb{R}^{k})$ is the inner
product space $\mathcal{B}_{2,\mu}(\mathbb{R}^{k})$ corresponding to the
function parameter $\mu(\xi):=\langle\xi\rangle^{s}\varphi(\langle\xi\rangle)$
of $\xi\in\mathbb{R}^{k}$. The space $H^{s;\varphi}(\mathbb{R}^{k})$ is
isotropic because the function $\mu(\xi)$ depends only on $|\xi|$. If
$\varphi(\cdot)\equiv 1$, the space $H^{s;\varphi}(\mathbb{R}^{k})$ becomes
the isotropic Sobolev space $H^{s}(\mathbb{R}^{k})$.

Basing on $H^{s;\varphi}(\mathbb{R}^{k})$, consider some Hilbert function
spaces on $G$ and $\Gamma$. Let $V$ be an open nonempty set in
$\mathbb{R}^{k}$. (Specifically, we need the case where $V=G$, with $k=n$.)
Put

$H^{s;\varphi}(V):=\bigl{\\{}w\\!\upharpoonright\\!V:\,w\in
H^{s;\varphi}(\mathbb{R}^{k})\bigl{\\}}.$ (3.8)

The norm in the linear space (3.8) is defined by the formula

$\|h\|_{H^{s;\varphi}(V)}:=\,\inf\bigl{\\{}\,\|w\|_{H^{s;\varphi}(\mathbb{R}^{k})}:w\in
H^{s;\varphi}(\mathbb{R}^{k}),\;\,h=w\\!\upharpoonright\\!V\bigl{\\}},$ (3.9)

with $h\in H^{s;\varphi}(V)$. This space is Hilbert and separable with respect
to this norm.

The linear space $H^{s;\varphi}(\Gamma)$ is defined to consist of all
distributions $\omega$ on $\Gamma$ such that every distribution
$\omega_{j}(y):=\chi_{j}(\theta_{j}(y))\,\omega(\theta_{j}(y))$ of
$y\in\mathbb{R}^{n-1}$ belongs to $H^{s;\varphi}(\mathbb{R}^{n-1})$, where
$j\in\\{1,\ldots,\lambda\\}$. The space $H^{s;\varphi}(\Gamma)$ is equipped
with the norm

$\|\omega\|_{H^{s;\varphi}(\Gamma)}:=\bigl{(}\|\omega_{1}\|_{H^{s;\varphi}(\mathbb{R}^{n-1})}^{2}+\cdots+\|\omega_{\lambda}\|_{H^{s;\varphi}(\mathbb{R}^{n-1})}^{2}\bigr{)}^{1/2}.$

This space is Hilbert and separable and is independent (up to equivalence of
norms) of our choice of local charts and partition of the unit on $\Gamma$
(see [27, Theorem 2.1]).

If $\varphi\equiv 1$, the spaces $H^{s,s/2;\varphi}(\cdot)$ and
$H^{s;\varphi}(\cdot)$ become the inner product Sobolev spaces
$H^{s,s/2}(\cdot)$ and $H^{s}(\cdot)$, respectively. Owing to (3.4), we have
the embeddings

$H^{s_{1},s_{1}/2}(\cdot)\hookrightarrow
H^{s,s/2;\varphi}(\cdot)\hookrightarrow
H^{s_{0},s_{0}/2}(\cdot)\quad\mbox{whenever}\quad s_{0}<s<s_{1}.$ (3.10)

Besides,

$H^{s_{1}}(\cdot)\hookrightarrow H^{s;\varphi}(\cdot)\hookrightarrow
H^{s_{0}}(\cdot)\quad\mbox{whenever}\quad s_{0}<s<s_{1},$ (3.11)

as is noted in [27, Theorems 2.3(iii) and 3.3(iii)]. All these embeddings are
continuous and dense.

Generally, if $\varphi(\cdot)\equiv 1$, we will suppress the index $\varphi$
in the designations of relevant spaces.

##  4\. Main result

We will formulate an isomorphism theorem for the parabolic problem (2.1)–(2.3)
in the generalized Sobolev spaces introduced in pervious section. Considering
this problem, we put $u:=(u_{1},\ldots,u_{N})$, $f:=(f_{1},\ldots,f_{N})$,
$g:=(g_{1},\ldots,g_{N})$, and $h:=(h_{1},\ldots,h_{N})$. We write the system
(2.1) and the boundary conditions (2.2) in the matrix form $Au=f$ and
$Bu|_{S}=g$; here

$A:=(A_{j,k}(x,t,D_{x},\partial_{t}))_{j,k=1}^{N}\quad\mbox{and}\quad
B:=\bigl{(}B_{j,k}(x,t,D_{x})\bigr{)}_{j,k=1}^{N}$

are matrix differential operators. We associate the linear mapping

$u\mapsto\Lambda
u:=\bigl{(}Au,Bu,u\\!\upharpoonright\\!G\bigr{)},\quad\mbox{where}\;\,u\in\bigl{(}C^{\infty}(\overline{\Omega})\bigr{)}^{N},$
(4.1)

with the problem (2.1)–(2.3).

In order that a regular enough solution $u$ to the problem (2.1)–(2.3) exist,
the right-hand sides to the problem should satisfy certain compatibility
conditions (see, e.g., [32, Section 14]). These conditions consist in that the
partial derivatives $\partial^{r}_{t}u_{j}(x,t)\big{|}_{t=0}$, which can be
found from the parabolic system (2.1) and initial conditions (2.3), should
satisfy the boundary conditions (2.2) and some relations obtained by the
differentiation of the boundary conditions with respect to $t$. To write these
compatibility conditions, we consider problem (2.1)–(2.3) in the corresponding
pairs of Sobolev spaces.

Let real $s\geq 2$. The mapping (4.1) extends uniquely (by continuity) to a
bounded linear operator

$\begin{split}\Lambda:\>&\bigl{(}H^{s,s/2}(\Omega)\bigr{)}^{N}\\\
&\leftrightarrow\bigl{(}H^{s-2,s/2-1}(\Omega)\bigr{)}^{N}\oplus\bigoplus_{j=1}^{N}H^{s-l_{j}-1/2,(s-l_{j}-1/2)/2}(S)\oplus\bigl{(}H^{s-1}(G)\bigr{)}^{N}.\end{split}$
(4.2)

This follows directly from [30, Chapter I, Lemma 4, and Chapter II, Theorems 3
and 7].

Choosing any vector-function $u(x,t)$ from the space
$(H^{s,s/2}(\Omega))^{N}$, we define the right-hand sides

$\begin{gathered}f\in\bigl{(}H^{s-2,s/2-1}(\Omega)\bigr{)}^{N},\quad\\\
g\in\bigoplus_{j=1}^{N}H^{s-l_{j}-1/2,(s-l_{j}-1/2)/2}(S),\quad\mbox{and}\quad
h\in\bigl{(}H^{s-1}(G)\bigr{)}^{N}\end{gathered}$ (4.3)

of the problem by the formula $(f,g,h):=\Lambda u$ with the help of the
bounded operator (4.2).

Compatibility conditions for functions $f_{j}$, $g_{j}$, and $h_{j}$ naturally
arise in such a way. According to [30, Chapter II, Theorem 7], the traces
$\partial^{\,r}_{t}u_{j}(\cdot,0)\in H^{s-2r-1}(G)$ (for all
$j\in\\{1,\dots,N\\}$) are well defined by closure for all $r\in\mathbb{Z}$
such that $0\leq r<s/2-1/2$ (and only for these $r$). These traces are
expressed from the equations (2.1) and initial conditions (2.3) in terms of
the functions $f_{j}$ and $h_{j}$ by recurrent formula:

$\begin{split}u_{j}(x,0)&=h_{j}(x)\quad\mbox{for all}\quad
j\in\\{1,\dots,N\\},\\\
(\partial^{\,r}_{t}u_{j})(x,0)&=-\sum_{k=1}^{N}\sum_{|\alpha|\leq
2}\sum\limits_{q=0}^{r-1}\binom{r-1}{q}(\partial^{\,r-1-q}_{t}a_{j,k}^{\alpha})(x,0)\,D^{\alpha}_{x}(\partial^{q}_{t}u_{k})(x,0)+\\\
&+\partial^{\,r-1}_{t}f_{j}(x,0)\\\ &\qquad\mbox{for all}\quad
j\in\\{1,\dots,N\\}\quad\mbox{and}\\\ &\qquad\mbox{for each}\quad
r\in\mathbb{Z}\quad\mbox{such that}\quad 1\leq r<s/2-1/2.\end{split}$ (4.4)

These equalities hold true for almost all $x\in G$.

Besides, according to [30, Chapter II, Theorem 7], for each
$j\in\\{1,\dots,N\\}$ the traces $\partial^{\,r}_{t}g_{j}(\cdot,0)\in
H^{s-l_{j}-3/2-2r}(\Gamma)$ are well defined by closure for all
$r\in\mathbb{Z}$ such that $0\leq r<(s-l_{j}-3/2)/2$ (and only for these $r$).
We express these traces in terms of the functions $u_{j}(x,t)$ and its time
derivatives by the formula

$\displaystyle(\partial^{r}_{t}g_{j})(x,0)$
$\displaystyle=\partial^{r}_{t}\biggl{(}\sum_{k=1}^{N}\sum_{|\alpha|\leq
l_{j}}b^{\alpha}_{j,k}(x,t)\,D^{\alpha}_{x}u_{k}(x,t)\biggr{)}|_{t=0}$ (4.5)
$\displaystyle=\sum_{k=1}^{N}\sum_{|\alpha|\leq
l_{j}}\,\sum_{q=0}^{r}\binom{r}{q}(\partial^{r-q}_{t}b^{\alpha}_{j,k})(x,0)\,D^{\alpha}_{x}(\partial^{q}_{t}u_{k})(x,0)$

for almost all $x\in\Gamma$. Here, all the functions
$(\partial^{\,q}_{t}u_{k})(x,0)$ of $x\in G$ are expressed in terms of the
functions $f_{j}(x,t)$ and $h_{j}(x)$ by the recurrent formula (4.4).

Substituting (4.4) in the right-hand side of formula (4.5), we obtain the
compatibility conditions

$\begin{gathered}\partial^{r}_{t}g_{j}\\!\upharpoonright\\!\Gamma=\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})\\!\upharpoonright\\!\Gamma\\\
\mbox{for each}\;\;j\in\\{1,\dots,N\\}\;\;\mbox{and}\;\;r\in\mathbb{Z}\\\
\mbox{such that}\;\;0\leq r<(s-l_{j}-3/2)/2.\end{gathered}$ (4.6)

Here, the functions $v_{1,0},v_{2,0},\ldots$ are defined almost everywhere on
$G$ by the formulas

$\displaystyle v_{j,0}(x)$ $\displaystyle=h_{j}(x),$ (4.7) $\displaystyle
v_{j,r}(x)$ $\displaystyle=-\sum_{k=1}^{N}\sum_{|\alpha|\leq
2}\sum\limits_{q=0}^{r-1}\binom{r-1}{q}(\partial^{\,r-1-q}_{t}a_{j,k}^{\alpha})(x,0)\,D^{\alpha}_{x}v_{k,q}(x,0)+$
$\displaystyle\qquad\quad+\partial^{\,r-1}_{t}f_{j}(x,0)\quad\mbox{if}\quad
r\geq 1,$

and

$\displaystyle\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})(x)$
(4.8) $\displaystyle=\sum_{k=1}^{N}\sum_{|\alpha|\leq
l_{j}}\,\sum_{q=0}^{r}\binom{r}{q}(\partial^{r-q}_{t}b^{\alpha}_{j,k})(x,0)\,D^{\alpha}_{x}v_{k,q}(x,0)$

for almost all $x\in G$. Note, that

$v_{j,r}\in H^{s-2r-1}(G)\quad\mbox{for each}\quad
r\in\mathbb{Z}\cap[0,s/2-1/2)$

due to (4.3). The right-hand side of the equality in (4.6) is well defined
because the function
$\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})$ belongs
to $H^{s-l_{j}-2r-1}(G)$ and the trace

$\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})\\!\upharpoonright\\!\Gamma\in
H^{s-l_{j}-2r-3/2}(\Gamma)$ (4.9)

is therefore defined by closure whenever $s-l_{j}-2r-3/2>0$.

The number of the compatibility conditions (4.6) is a function of $s\geq 2$.
This function is discontinuous at $s$ if and only if
$(s-l_{j}-3/2)/2\in\mathbb{Z}$. Thus, the set of all its discontinuities
coincides with

$E:=\\{2l+l_{j}+3/2:j,l\in\mathbb{Z},\;1\leq j\leq N,\;l\geq
0\\}\cap(2,\infty).$ (4.10)

Note, if $s\leq 5/2$ and $l_{j}=1$ for some $j$, there are no compatibility
conditions involving $g_{j}$.

Our result on the parabolic problem (2.1)–(2.3) consists in that the linear
mapping (4.1) extends uniquely to an isomorphism between appropriate pairs of
generalized Sobolev spaces introduced in the previous section. Let us indicate
these pairs. We arbitrarily choose a real number $s>2$ and function parameter
$\varphi\in\mathcal{M}$. We take $(H^{s,s/2;\varphi}(\Omega))^{N}$ as the
domain of this isomorphism. Its range is imbedded in the Hilbert space

$\displaystyle\mathcal{H}^{s-2,s/2-1;\varphi}:=$
$\displaystyle\bigl{(}H^{s-2,s/2-1;\varphi}(\Omega)\bigr{)}^{N}$
$\displaystyle\oplus\bigoplus_{j=1}^{N}H^{s-l_{j}-1/2,(s-l_{j}-1/2)/2;\varphi}(S)\oplus\bigl{(}H^{s-1;\varphi}(G)\bigr{)}^{N}$

and is denoted by $\mathcal{Q}^{s-2,s/2-1;\varphi}$. [If $\varphi\equiv 1$,
then $\mathcal{H}^{s-2,s/2-1;\varphi}$ is the target space of (4.2).] We
separately define $\mathcal{Q}^{s-2,s/2-1;\varphi}$ in the cases where
$s\notin E$ and where $s\in E$.

Suppose first that $s\notin E$. By definition, the linear space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ consists of all vectors

$F:=\bigl{(}f_{1},\dots,f_{N},g_{1},\dots,g_{N},h_{1},\dots,h_{N}\bigr{)}\in\mathcal{H}^{s-2,s/2-1;\varphi}$

that satisfy the compatibility conditions (4.6). These conditions are well
defined for every indicated $F$ because they are well defined whenever
$F\in\mathcal{H}^{s-\varepsilon-2,s/2-\varepsilon/2-1}$ and $0<\varepsilon\ll
1$ and because

$\mathcal{H}^{s-2,s/2-1;\varphi}\hookrightarrow\mathcal{H}^{s-\varepsilon-2,s/2-\varepsilon/2-1}.$
(4.11)

This continuous embedding follows directly from (3.10) and (3.11). We endow
the linear space $\mathcal{Q}^{s-2,s/2-1;\varphi}$ with the norm in the
Hilbert space $\mathcal{H}^{s-2,s/2-1;\varphi}$. The space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ is complete, i.e. Hilbert. Indeed,

$\mathcal{Q}^{s-2,s/2-1;\varphi}=\mathcal{H}^{s-2,s/2-1;\varphi}\cap\mathcal{Q}^{s-\varepsilon-2,s/2-\varepsilon/2-1}$

whenever $0<\varepsilon\ll 1$. Here, the space
$\mathcal{Q}^{s-\varepsilon-2,s/2-\varepsilon/2-1}$ is complete because the
differential operators and trace operators used in the compatibility
conditions are bounded on the corresponding pairs of Sobolev spaces.
Therefore, the space

$\mathcal{H}^{s-2,s/2-1;\varphi}\cap\mathcal{Q}^{s-\varepsilon-2,s/2-\varepsilon/2-1}$

is complete with respect to the sum of the norms in the components of the
intersection, this sum being equivalent to the norm in
$\mathcal{H}^{s-2,s/2-1;\varphi}$ due to (4.11). Thus, the space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ is complete (with respect to the latter
norm).

If $s\in E$, then we define the Hilbert space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ by means of the quadratic interpolation
between its analogs just introduced. Namely, we put

$\mathcal{Q}^{s-2,s/2-1;\varphi}:=\bigl{[}\mathcal{Q}^{s-\varepsilon-2,s/2-\varepsilon/2-1;\varphi},\mathcal{Q}^{s+\varepsilon-2,s/2+\varepsilon/2-1;\varphi}\bigr{]}_{1/2}.$
(4.12)

Here, the number $\varepsilon\in(0,1/2)$ is arbitrarily chosen, and the right-
hand side of the equality is the result of the quadratic interpolation with
the parameter $1/2$ of the written pair of Hilbert spaces. The Hilbert space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ defined by formula (4.12) does not depend on
our choice of $\varepsilon$ up to equivalence of norms and is continuously
embedded in $\mathcal{H}^{s-2,s/2-1;\varphi}$. This will be shown in Remark
5.1 at the end of Section 5.

Now, we may formulate the main result of the paper.

######  Theorem 4.1.

For arbitrary $s>2$ and $\varphi\in\nobreak\mathcal{M}$ the mapping (4.1)
extends uniquely (by continuity) to an isomorphism

$\Lambda:\,\bigl{(}H^{s,s/2;\varphi}(\Omega)\bigr{)}^{N}\leftrightarrow\mathcal{Q}^{s-2,s/2-1;\varphi}.$
(4.13)

This Theorem is known in the Sobolev case where $\varphi\equiv 1$, which is
proved in this case by Solonnikov [32, Theorem 5.4] for general parabolic
systems under the restriction $s,s/2\in\mathbb{Z}$. This restriction can be
removed, as is shown in Eidel’man and Zhitarashu’s monograph [8, Theorem 5.7];
their result includes the limiting case of $s=2$. Theorem 4.1 on isomorphisms
has many applications.

Using this theorem, we can investigate global and local regularity of
generalized solutions to the problem under consideration. This theorem also
allows us to obtain new sufficient conditions for the continuity of
generalized solutions, specifically to find conditions under which the
solutions are classical (c.f. [23, 24, 19, 20, 21, 22]). These applications
will be given in another paper.

Note that the necessity to define the target space
$\mathcal{Q}^{s-2,s/2-1;\varphi}$ separately in the $s\in E$ case is caused by
the following: if we defined this space for $s\in E$ in the way used in the
$s\notin E$ case, then the isomorphism (4.13) would not be valid at least for
$\varphi\equiv 1$. This follows from a result by Solonnikov [31, Section 6]
(see also [16, Remark 6.4]).

##  5\. Proof of the main result

We deduce Theorem 4.1 from its known counterpart in the Sobolev case [8,
Theorem 5.7] with the help of the quadratic interpolation with a suitable
function parameter. As to this interpolation, we mainly use definitions,
notations, and properties given in [27, Section 1.1]. Let $X:=[X_{0},X_{1}]$
be an ordered pair of separable complex Hilbert spaces such that $X_{1}$ is a
dense linear manifold in $X_{0}$ and that the embedding $X_{1}\subseteq X_{0}$
is continuous. This pair is called regular. Let $[X_{0},X_{1}]_{\psi}$ denotes
the Hilbert space, obtained by the quadratic interpolation with the function
parameter $\psi$ of the pair $X=\nobreak[X_{0},X_{1}]$ or (in other words)
between $X_{0}$ and $X_{1}$. We define the interpolation parameter $\psi$ by
the formula

$\psi(r):=\begin{cases}\;r^{(s-s_{0})/(s_{1}-s_{0})}\,\varphi(r^{1/(s_{1}-s_{0})})&\text{if}\quad
r\geq 1,\\\ \;\varphi(1)&\text{if}\quad 0<r<1,\end{cases}$ (5.1)

where $s_{0},s,s_{1}\in\mathbb{R}$ satisfy $s_{0}<s<s_{1}$, and
$\varphi\in\mathcal{M}$.

It follows from the definition of the space $\mathcal{Q}^{s-2,s/2-1;\varphi}$
that we need a relevant interpolation formula for this space in the case where
$s\notin E$, with $E$ being denoted by (4.10). Let $\\{J_{l}:1\leq
l\in\mathbb{Z}\\}$ stand for the collection of all connected components of the
set $(2,\infty)\setminus E$. Each component $J_{l}$ is a certain finite
subinterval of $(2,\infty)$.

######  Lemma 5.1.

Let $1\leq l\in\mathbb{Z}$. Assume that real numbers $s_{0},s,s_{1}\in J_{l}$
satisfy $s_{0}<s<s_{1}$ and that $\varphi\in\mathcal{M}$. Define an
interpolation parameter $\psi$ by (5.1). Then

$\mathcal{Q}^{s-2,s/2-1;\varphi}=\,\bigl{[}\mathcal{Q}^{s_{0}-2,s_{0}/2-1},\mathcal{Q}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}$
(5.2)

up to equivalence of norms.

###### Proof.

We first prove an analog of formula (5.2) for the space
$\mathcal{H}^{s-2,s/2-1;\varphi}$. According to [17, Theorem 2], [17, Lemma 2]
(with $\Omega$ instead $\Pi$) and [27, Theorems 1.5, 1.14(i), 3.2], we have

$\displaystyle\bigl{[}$
$\displaystyle\mathcal{H}^{s_{0}-2,s_{0}/2-1},\mathcal{H}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}$
$\displaystyle=\bigl{(}\bigl{[}H^{s_{0}-2,s_{0}/2-1}(\Omega),H^{s_{1}-2,s_{1}/2-1}(\Omega)\bigr{]}_{\psi}\bigr{)}^{N}$
$\displaystyle\qquad\oplus\bigoplus_{j=1}^{N}\bigl{[}H^{s_{0}-l_{j}-1/2,\,(s_{0}-l_{j}-1/2)/2}(S),H^{s_{1}-l_{j}-1/2,\,(s_{1}-l_{j}-1/2)/2}(S)\bigr{]}_{\psi}$
$\displaystyle\qquad\oplus\bigl{(}\bigl{[}H^{s_{0}-1}(G),H^{s_{1}-1}(G)\bigr{]}_{\psi}\bigr{)}^{N}$
$\displaystyle=\bigl{(}H^{s-2,s/2-1;\varphi}(\Omega)\bigr{)}^{N}\oplus\bigoplus_{j=1}^{N}H^{s-l_{j}-1/2,(s-l_{j}-1/2)/2;\varphi}(S)\oplus\bigl{(}H^{s-1;\varphi}(G)\bigr{)}^{N}$
$\displaystyle=\mathcal{H}^{s-2,s/2-1;\varphi}.$

Thus,

$\bigl{[}\mathcal{H}^{s_{0}-2,s_{0}/2-1},\mathcal{H}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}=\mathcal{H}^{s-2,s/2-1;\varphi}$
(5.3)

up to equivalence of norms.

Consider the set

$\biggl{\\{}k\in\mathbb{Z}:0\leq
k<\frac{1}{2}\,\biggl{(}s-l_{j}-\frac{3}{2}\biggr{)}\biggr{\\}}$ (5.4)

for each $j\in\\{1,\dots,N\\}$. This set relates to the compatibility
conditions (4.6) and does not depend on $s$ whenever $s$ ranges over an
arbitrary chosen set $J_{l}$. Let $q_{l,j}$ denote the number of all elements
of (5.4).

Letting $1\leq l\in\mathbb{Z}$, we define a linear mapping $P_{l}$ on

$\bigcup_{\sigma\in J_{l}}\mathcal{H}^{\sigma-2,\sigma/2-1}$ (5.5)

such that the restriction of $P_{l}$ to the space
$\mathcal{H}^{\sigma-2,\sigma/2-1}$ is a projector of this space on its
subspace $\mathcal{Q}^{\sigma-2,\sigma/2-1}$ for every $\sigma\in J_{l}$. For
any vector

$F:=\bigl{(}f_{1},\dots,f_{N},g_{1},\dots,g_{N},h_{1},\dots,h_{N}\bigr{)}\in\bigcup_{\sigma\in
J_{l}}\mathcal{H}^{\sigma-2,\sigma/2-1},$

we put

$\left\\{\begin{array}[]{ll}g^{*}_{j}:=g_{j}&\hbox{whenever}\;q_{l,j}=0,\\\
g^{*}_{j}:=g_{j}+T_{l,j}(w_{j,0},\dots,w_{j,q_{l,j}-1})&\hbox{whenever}\;q_{l,j}\geq
1\end{array}\right.$ (5.6)

for each $j\in\\{1,\dots,N\\}$. Here,

$\displaystyle w_{j,0}$
$\displaystyle:=\mathcal{B}_{j,0}(v_{1,0},\dots,v_{N,0})\\!\upharpoonright\\!\Gamma-
g_{j}\\!\upharpoonright\\!\Gamma,$ $\displaystyle\dots$ $\displaystyle
w_{j,q_{l,j}-1}$
$\displaystyle:=\mathcal{B}_{j,q_{l,j}-1}(v_{1,0},\dots,v_{N,0},\dots,v_{1,q_{l,j}-1},\dots,v_{N,q_{l,j}-1})\\!\upharpoonright\\!\Gamma$
$\displaystyle\,-\partial_{t}^{q_{l,j}-1}g_{j}\\!\upharpoonright\\!\Gamma,$

with the functions $v_{1,0},\dots,v_{N,q_{l,j}-1}$ and the differential
operators $\mathcal{B}_{j,0},\ldots,\mathcal{B}_{j,q_{l,j}-1}$ being defined
by (4.7) and (4.8) respectively and with $T_{l,j}$ denoting the linear mapping
$T$ from [25, Lemma 6.1] in the $r=q_{l,j}$ case.

The linear mapping

$\displaystyle P_{l}$
$\displaystyle:\,\bigl{(}f_{1},\dots,f_{N},g_{1},\dots,g_{N},h_{1},\dots,h_{N}\bigr{)}$
$\displaystyle\mapsto\bigl{(}f_{1},\dots,f_{N},g^{*}_{1},\dots,g^{*}_{N},h_{1},\dots,h_{N}\bigr{)}$

given on (5.5) is required. Indeed, its restriction to the space
$\mathcal{H}^{\sigma-2,\sigma/2-1}$ is a bounded operator on this space for
every $\sigma\in J_{l}$, which follows from (4.7)–(4.9) and [25, Lemma 6.1].
We use the boundedness of the operators $T$ from [25, Lemma 6.1] in the case
where $r=q_{l,j}$, $s=\sigma-l_{j}-1/2$, and $\varphi(\cdot)\equiv 1$, with
the condition $s>2r-1$ being satisfied because

$q_{l,j}<\frac{1}{2}\biggl{(}\sigma-
l_{j}-\frac{3}{2}\biggr{)}+1\quad\mbox{whenever}\quad\sigma\in J_{l}.$

Besides, it follows from the definitions of $P_{l}$ and the space
$\mathcal{Q}^{\sigma-2,\sigma/2-1}$ that

$P_{l}F=(f_{1},\dots,f_{N},g^{*}_{1},\dots,g^{*}_{N},h_{1},\dots,h_{N})\in\mathcal{Q}^{\sigma-2,\sigma/2-1}$

for every $F\in\mathcal{H}^{\sigma-2,\sigma/2-1}$. Indeed,

$\displaystyle\partial^{r}_{t}g_{j}^{*}\\!\upharpoonright\\!\Gamma=$
$\displaystyle\partial^{r}_{t}g_{j}\\!\upharpoonright\\!\Gamma+\partial^{r}_{t}T_{l,j}(w_{j,0},\dots,w_{j,q_{l,j}-1})\\!\upharpoonright\\!\Gamma$
$\displaystyle=$
$\displaystyle\partial^{r}_{t}g_{j}\\!\upharpoonright\\!\Gamma+w_{j,r}$
$\displaystyle=$
$\displaystyle\partial^{r}_{t}g_{j}\\!\upharpoonright\\!\Gamma+\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})\\!\upharpoonright\\!\Gamma-\partial^{r}_{t}g_{j}\\!\upharpoonright\\!\Gamma$
$\displaystyle=$
$\displaystyle\mathcal{B}_{j,r}(v_{1,0},\dots,v_{N,0},\dots,v_{1,r},\dots,v_{N,r})\\!\upharpoonright\\!\Gamma$

for all $j$ and $r$ indicated in (4.6). Thus, the vector $P_{l}F$ satisfies
the compatibility conditions, i.e.
$P_{l}F\in\mathcal{Q}^{\sigma-2,\sigma/2-1}$. Moreover,
$F\in\mathcal{Q}^{\sigma-2,\sigma/2-1}$ implies that $P_{l}F=F$. Namely, if
$F\in\mathcal{Q}^{\sigma-2,\sigma/2-1}$, then (4.6) holds, which entails that
all $w_{j,r}=0$, i.e. $g_{1}^{*}=g_{1}$, …, $g_{N}^{*}=g_{N}$.

According to [27, Theorem 1.6], the pair

$\bigl{[}\mathcal{Q}^{s_{0}-2,s_{0}/2-1},\mathcal{Q}^{s_{1}-2,s_{1}/2-1}\bigr{]}$

is regular, and

$\displaystyle\bigl{[}\mathcal{Q}^{s_{0}-2,s_{0}/2-1},\mathcal{Q}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}$
(5.7)
$\displaystyle=\bigl{[}\mathcal{H}^{s_{0}-2,s_{0}/2-1},\mathcal{H}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}\cap\mathcal{Q}^{s_{0}-2,s_{0}/2-1}$

up to equivalence of norms. Formulas (5.7) and (5.3) give

$\displaystyle\bigl{[}\mathcal{Q}^{s_{0}-2,s_{0}/2-1},\mathcal{Q}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}$
$\displaystyle=\mathcal{H}^{s-2,s/2-1;\varphi}\cap\mathcal{Q}^{s_{0}-2,s_{0}/2-1}=\mathcal{Q}^{s-2,s/2-1;\varphi}.$

The latter equality holds true because $s,s_{0}\in J_{l}$; i.e., all elements
of the subspace $\mathcal{Q}^{s-2,s/2-1;\varphi}$ satisfy the same
compatibility conditions as elements of $\mathcal{Q}^{s_{0}-2,s_{0}/2-1}$.

∎

###### The proof of Theorem 4.1..

Let $s>2$ and $\varphi\in\mathcal{M}$. We first consider the case where
$s\notin E$. Then $s\in J_{l}$ for some integer $l\geq 1$. Choose numbers
$s_{0},s_{1}\in J_{l}$ such that $s_{0}<s<s_{1}$ and that
$s_{j}+1/2\notin\mathbb{Z}$ and $s_{j}/2+1/2\notin\mathbb{Z}$ whenever
$j\in\\{0,1\\}$. According to Zhitarashu and Eidelman [8, Theorem 5.7], the
mapping (4.1) extends uniquely (by continuity) to an isomorphism

$\Lambda:\,\bigl{(}H^{s_{j},s_{j}/2}(\Omega)\bigr{)}^{N}\leftrightarrow\mathcal{Q}^{s_{j}-2,s_{j}/2-1}\quad\mbox{for
each}\quad j\in\\{0,1\\}.$ (5.8)

We define the interpolation parameter $\psi$ by (5.1). It follows from [27,
Theorem 1.5], [17, Lemma 2] (with $\Omega$ instead $\Pi$) and Lemma 5.1 that
that the restriction of the operator (5.8) with $j=0$ to the space

$\bigl{[}\bigl{(}H^{s_{0},s_{0}/2}(\Omega)\bigr{)}^{N},\bigl{(}H^{s_{1},s_{1}/2}(\Omega)\bigr{)}^{N}\bigr{]}_{\psi}=\bigl{(}H^{s,s/2;\varphi}(\Omega)\bigr{)}^{N}$

sets an isomorphism

$\Lambda:\,\bigl{(}H^{s,s/2;\varphi}(\Omega)\bigr{)}^{N}\leftrightarrow\bigl{[}\mathcal{Q}^{s_{0}-2,s_{0}/2-1},\mathcal{Q}^{s_{1}-2,s_{1}/2-1}\bigr{]}_{\psi}=\mathcal{Q}^{{s-2,s/2-1};\varphi}.$

Since the set $(C^{\infty}(\overline{\Omega}))^{N}$ is dense in
$(H^{s,s/2;\varphi}(\Omega))^{N}$, the operator (5) is an extension by
continuity of (4.1). Hence, Theorem 4.1 is proved in the $s\notin E$ case.

Now we consider the case $s\in E$. Choose $\varepsilon\in(0,1/2)$ arbitrarily.
Since $s\pm\varepsilon\notin E$ and $s-\varepsilon>2$, we have the
isomorphisms

$\Lambda:\bigl{(}H^{s\pm\varepsilon,(s\pm\varepsilon)/2;\varphi}(\Omega)\bigr{)}^{N}\leftrightarrow\mathcal{Q}^{s\pm\varepsilon-2,(s\pm\varepsilon)/2-1;\varphi}$
(5.9)

as has just been proved. They imply that the mapping (4.1) extends uniquely
(by continuity) to an isomorphism

$\displaystyle\Lambda:\bigl{[}\bigl{(}H^{s-\varepsilon,(s-\varepsilon)/2;\varphi}(\Omega)\bigr{)}^{N},\bigl{(}H^{s+\varepsilon,(s+\varepsilon)/2;\varphi}(\Omega)\bigr{)}^{N}\bigr{]}_{1/2}$
(5.10)
$\displaystyle\leftrightarrow\bigl{[}\mathcal{Q}^{s-\varepsilon-2,(s-\varepsilon)/2-1;\varphi},\mathcal{Q}^{s+\varepsilon-2,(s+\varepsilon)/2-1;\varphi}\bigr{]}_{1/2}=\mathcal{Q}^{s-2,s/2-1;\varphi}.$

The last equality is the definition of the space
$\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$. Owing to [25, Formula (61)], we have

$\bigl{[}\bigl{(}H^{s-\varepsilon,(s-\varepsilon)/2;\varphi}(\Omega)\bigr{)}^{N},\bigl{(}H^{s+\varepsilon,(s+\varepsilon)/2;\varphi}(\Omega)\bigr{)}^{N}\bigr{]}_{1/2}\\\
=\bigl{(}H^{s,s/2;\varphi}(\Omega)\bigr{)}^{N}.$ (5.11)

It remains to apply (5.11) to (5.10). ∎

######  Remark 5.1.

Let $s\in E$. The space $\mathcal{Q}^{s-2,s/2-1;\varphi}$ defined by (4.12)
does not depend on $\varepsilon\in(0,1/2)$ up to equivalence of norms. Indeed,
according to Theorem 4.1, the isomorphism

$\Lambda:\bigl{(}H^{s,s/2;\varphi}(\Omega)\bigr{)}^{N}\leftrightarrow\bigl{[}\mathcal{Q}^{s-\varepsilon-2,(s-\varepsilon)/2-1;\varphi},\mathcal{Q}^{s+\varepsilon-2,(s+\varepsilon)/2-1;\varphi}\bigr{]}_{1/2}$

holds true whenever $0<\varepsilon<1/2$. This directly implies the mentioned
independence. Besides, the space $\mathcal{Q}^{s-2,s/2-1;\varphi}$ is embedded
continuously in $\mathcal{H}^{s-2,s/2-1;\varphi}$. Indeed, choosing
$\varepsilon\in(0,1/2)$, we get the continuous embeddings

$\mathcal{Q}^{s\mp\varepsilon-2,(s\mp\varepsilon)/2-1;\varphi}\hookrightarrow\mathcal{H}^{s\mp\varepsilon-2,(s\mp\varepsilon)/2-1;\varphi}$
(5.12)

in view of $s\mp\varepsilon\in(2,\infty)\setminus E$ and the definition of the
left-hand space. Owing to the interpolation formula (5.11) and its analogs for
the spaces on $S$ and $G$, we obtain

$\mathcal{H}^{s-2,s/2-1;\varphi}=\bigl{[}\mathcal{H}^{s-\varepsilon-2,(s-\varepsilon)/2-1;\varphi},\mathcal{H}^{s+\varepsilon-2,(s+\varepsilon)/2-1;\varphi}\bigr{]}_{1/2}$
(5.13)

(see also [24, Lemma 6.4]). It follows from (5.12) that the embedding operator
acts continuously from (4.12) to (5.13), as was stated.

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