Parabolic problems in generalized Sobolev spaces

TL;DR

This paper studies the solvability and regularity of inhomogeneous parabolic boundary value problems within generalized anisotropic Sobolev spaces, introducing a function parameter to characterize subordinate regularity and establishing isomorphism results.

Contribution

It introduces a framework for analyzing parabolic problems in generalized Sobolev spaces with a function parameter, proving operator isomorphism and regularity results.

Findings

Operator is an isomorphism on generalized Sobolev spaces

Established local regularity of solutions

Derived conditions for continuity of generalized derivatives

Abstract

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $\varphi:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $\varphi$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.

Equations

$$\mathcal{B}_{p,\mu}:=\bigl{\{}w\in\mathcal{S}^{\prime}(\mathbb{R}^{k}):\mu\widehat{w}\in L_{p}(\mathbb{R}^{k})\bigr{\}}$$

$$\mu (\xi ):=(1+|\xi {|}^2{)}^{s/2}\phi ((1+|\xi {|}^2{)}^{1/2}),$$

$$\mu(\xi^{\prime},\xi_{k})=\bigl{(}1+|\xi^{\prime}|^{2}+|\xi_{k}|^{1/b}\bigr{)}^{s/2}\varphi\bigl{(}(1+|\xi^{\prime}|^{2}+|\xi_{k}|^{1/b})^{1/2}\bigr{)}$$

$$\begin{array}{c}A(x,t,D_x,{\partial }_t)u(x,t)\equiv \sum _{|\alpha |+2b\beta \le 2m}{a}^{\alpha ,\beta }(x,t)D_x^{\alpha }{\partial }_t^{\beta }u(x,t)=f(x,t)\\ \text{mbox}forallx\in G\text{mbox}andt\in (0,\tau );\end{array}$$

$$\begin{array}{c}{B}_j(x,t,D_x,{\partial }_t)u(x,t)\equiv \sum _{|\alpha |+2b\beta \le m_j}{b}_j^{\alpha ,\beta }(x,t)D_x^{\alpha }{\partial }_t^{\beta }u(x,t)\upharpoonright S=g_j(x,t)\\ \text{mbox}forallx\in \Gamma ,t\in (0,\tau )\text{mbox}andj\in \{1,\dots ,m\};\end{array}$$

$$({\partial }_t^{k}u)(x,0)=h_k(x)\text{mbox}forallx\in G\text{mbox}andk\in \{0,\dots ,\varkappa -1\}.$$

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

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

$$A^{\circ }(x,t,\xi ,p)\equiv \sum _{|\alpha |+2b\beta =2m}{a}^{\alpha ,\beta }(x,t){\xi }^{\alpha }{p}^{\beta }\ne 0\text{mbox}whenever|\xi |+|p|\ne 0.$$

$$B_j^{\circ }(x,t,\xi +\zeta \nu (x),p)\equiv \sum _{|\alpha |+2b\beta =m_j}{b}_j^{\alpha ,\beta }(x,t)(\xi +\zeta \nu (x){)}^{\alpha }{p}^{\beta },j=1,\dots ,m,$$

$$\prod _{j=1}^m(\zeta -{\zeta }_j^{+}(x,t,\xi ,p)).$$

$$\frac{\mu (\xi )}{\mu (\eta )}\le c(1+|\xi -\eta |{)}^l\text{mbox}whenever\xi ,\eta \in {\mathbb{R}}^k.$$

$$\underset{{\mathbb{R}}^k}{\int }{\mu }^2(\xi )|\hat{w}(\xi ){|}^{2}d\xi <\infty .$$

$$(w_1,w_2{)}_{H^{\mu }({\mathbb{R}}^k)}=\underset{{\mathbb{R}}^k}{\int }{\mu }^2(\xi )\widehat{w_1}(\xi )\overline{\widehat{w_2}(\xi )}d\xi ,$$

$$\Vert w{\Vert }_{H^{\mu }({\mathbb{R}}^k)}:=(w,w{)}_{H^{\mu }({\mathbb{R}}^k)}^{1/2}.$$

$$\displaystyle H^{\mu}(V):=\bigl{\{}w\!\upharpoonright\!V:\,w\in H^{\mu}(\mathbb{R}^{k})\bigr{\}},$$

$$\displaystyle\|u\|_{H^{\mu}(V)}:=\inf\bigl{\{}\|w\|_{H^{\mu}(\mathbb{R}^{k})}:\,w\in H^{\mu}(\mathbb{R}^{k}),\;u=w\!\upharpoonright\!V\bigr{\}},$$

$$H^{\mu}_{Q}(\mathbb{R}^{k}):=\bigl{\{}w\in H^{\mu}(\mathbb{R}^{k}):\,\mathrm{supp}\,w\subseteq Q\bigr{\}},$$

$$(u_1,u_2{)}_{H^{\mu }(V)}:=(w_1-{\rm Y}{w}_1,w_2-{\rm Y}{w}_2{)}_{H^{\mu }({\mathbb{R}}^k)},$$

$$\begin{gathered}\mu(\xi^{\prime},\xi_{k})=\bigl{(}1+|\xi^{\prime}|^{2}+|\xi_{k}|^{2\gamma}\bigr{)}^{s/2}\varphi\bigl{(}(1+|\xi^{\prime}|^{2}+|\xi_{k}|^{2\gamma})^{1/2}\bigr{)}\\ \mbox{for all}\;\;\xi^{\prime}\in\mathbb{R}^{k-1}\;\;\mbox{and}\;\;\xi_{k}\in\mathbb{R}.\end{gathered}$$

$$\phi (r):=(\log r{)}^{{\theta }_1}(\log \log r{)}^{{\theta }_2}\dots (\underset{k\text{mbox}times}{\underbrace{\log \dots \log }}r{)}^{{\theta }_k}\text{mbox}ofr\gg 1,$$

$$H^{s_1,s_1\gamma }({\mathbb{R}}^k)\hookrightarrow H^{s,s\gamma ;\phi }({\mathbb{R}}^k)\hookrightarrow H^{s_0,s_0\gamma }({\mathbb{R}}^k)\text{mbox}whenever{s}_0<s<s_1.$$

$$\displaystyle c_{0}\bigl{(}1+|\xi^{\prime}|^{2}+|\xi_{k}|^{2\gamma}\bigr{)}^{s_{0}/2}$$

$$\displaystyle\leq c_{1}\bigl{(}1+|\xi^{\prime}|^{2}+|\xi_{k}|^{2\gamma}\bigr{)}^{s_{1}/2}$$

$$\bigl{\{}H^{s,s\gamma;\varphi}(\mathbb{R}^{k}):\,s\in\mathbb{R},\,\varphi\in\mathcal{M}\,\bigr{\}}.$$

$$v_j(y,t):={\chi }_j({\theta }_j(y))v({\theta }_j(y),t)\text{mbox}ofy\in {\mathbb{R}}^{n-1}\text{mbox}andt\in (0,\tau )$$

$$(v,v^{\circ }{)}_{H^{s,s\gamma ;\phi }(S)}:=\sum _{j=1}^{\lambda }(v_j,v_j^{\circ }{)}_{H^{s,s\gamma ;\phi }(\Pi )},$$

$$\Vert v{\Vert }_{H^{s,s\gamma ;\phi }(S)}:=(v,v{)}_{H^{s,s\gamma ;\phi }(S)}^{1/2}.$$

$${\theta }_j^{\ast }:\Pi ={\mathbb{R}}^{n-1}\times (0,\tau )\leftrightarrow {\Gamma }_j\times (0,\tau ),j=1,\dots ,\lambda ,$$

$$\mu(\xi)=\bigl{(}1+|\xi|^{2}\bigr{)}^{s/2}\varphi\bigl{(}(1+|\xi|^{2})^{1/2}\bigr{)}\quad\mbox{of}\;\;\xi\in\mathbb{R}^{k}.$$

$$(\omega ,{\omega }^{\circ }{)}_{H^{s;\phi }(\Gamma )}:=\sum _{j=1}^{\lambda }({\omega }_j,{\omega }_j^{\circ }{)}_{H^{s;\phi }({\mathbb{R}}^{n-1})},$$

Full text

The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686.

1. Introduction

Isomorphism theorems form a core of the modern theory of parabolic initial-boundary value problems. These theorems assert that the parabolic problems are well-posed (in the sense of Hadamard) on appropriate pairs of Sobolev or Hölder anisotropic normed spaces [1, 9, 15, 10, 11, 12, 25, 27]. Otherwise speaking, the bounded operators corresponding to the mentioned problems set isomorphisms on these pairs. The isomorphism theorems play a key role in the investigations of regularity of solutions to parabolic problems, their Green functions, control problems for systems governed by parabolic equations, and others (see., e.g., [10, 11, 27, 20, 35]). Note that the case of Hilbert spaces is of a special interest for applications of these theorems [11, 27]). This case deals with anisotropic Sobolev spaces [48] based on the Lebesgue spaces of square integrable functions.

Certainly, the more finely a scale of function spaces is calibrated, the more precise results may be obtained with the help of this scale. Number parameters, which is used for Sobolev or Hölder spaces, provide a calibration that proved to be rough for various problems of mathematical analysis [13, 22, 50, 53], theory of partial differential equations [16, 17, 38, 40, 41], theory of stochastic processes [21], and others. In 1963, motivated by applications to partial differential equations, Hörmander [16] introduced and investigated the normed distribution spaces

[TABLE]

parametrized with a sufficiently general function parameter $\mu:\mathbb{R}^{k}\to(0,\infty)$. Here, the number parameter $p$ satisfies $1\leq p\leq\infty$, and $\widehat{w}$ denotes the Fourier transform of the tempered distribution $w$. Hörmander gave important applications of his spaces to the investigation of the existence and regularity of solutions to partial differential equations (see also his monograph [17]). The most complete results were obtained for the class of hypoelliptic equations, to which parabolic equations pertain. If $p=2$, the Hörmander spaces become Hilbert ones and turn out to be a broad generalization of the inner product Sobolev spaces.

Hörmander’s monograph [16] attracted a great attention to generalized Sobolev spaces and stimulated various investigations concerning these spaces and their applications, mostly to mathematical analysis (see, e.g., [13, 22, 53, 54] and references therein). However, these spaces were applied rarely to multidimensional boundary-value problems up to the recent time (we may refer only to [47]). This was caused by the absence of reasonable notion of generalized Sobolev spaces over manifolds (such spaces should be independent of local charts on the manifold) and by the lack of analytical tools to work with these spaces.

Recently the situation has essentially changed. Mikhailets and Murach [36, 38] have built a theory of solvability of general elliptic boundary-value problems in generalized Sobolev spaces of the form $H^{s;\varphi}:=\mathcal{B}_{2,\mu}$, where

[TABLE]

$s\in\mathbb{R}$, and the function $\varphi:[1,\infty)\to(0,\infty)$ varies slowly at infinity in the sense of Karamata. Note that these spaces are isotropic because the function $\mu$ depends only on $|\xi|$. The function parameter $\varphi$ defines a subordinate regularity of the distributions $w\in H^{s;\varphi}$ with respect to the basic power regularity given by the number $s$. If $\varphi(\cdot)\equiv 1$, the space $H^{s;\varphi}$ will become the inner product Sobolev space of order $s$. The main research method of this theory is the interpolation with a function parameter of Hilbert spaces and linear operators acting on these spaces. The Hörmander spaces used in the theory are obtained by this interpolation applied to pairs of inner product Sobolev spaces. This allows the authors to define the corresponding spaces over smooth manifolds and facilitates the application of these spaces to elliptic problems. Of late years this theory was extended to a wider class of generalized Sobolev spaces [2, 3], namely to all Hilbert spaces that are interpolation ones between inner product Sobolev spaces [37, 39].

The above-mentioned interpolation method proved to be useful in the theory of parabolic initial-boundary value problems as well. This was shown in papers [28, 30, 32, 33, 34] for some classes of parabolic problems. These papers deal with the generalized anisotropic Sobolev spaces $H^{s,s/(2b);\varphi}:=\mathcal{B}_{2,\mu}$, where

[TABLE]

for all $\xi^{\prime}\in\mathbb{R}^{k-1}$ and $\xi_{k}\in\mathbb{R}$. Here, $s\in\mathbb{R}$, $\varphi$ is the above-mentioned function, and $\nobreak{1\leq b\in\mathbb{Z}}$, with the even integer $2b$ characterizing the parabolicity of the partial differential equation investigated. These papers present isomorphism theorems for the parabolic problems considered in the indicated spaces and give some applications of these theorems to the study of the regularity of generalized solutions to the parabolic problems.

Note that various methods of the interpolation with a number parameter between normed spaces are used in the theory of multidimensional boundary-value problems [4, 26, 27, 45, 51, 52]. However, the application of these methods to Sobolev spaces (or other classical function spaces depending on number parameters only) does not give spaces parametrized with function parameters.

The purpose of this paper is to prove an isomorphism theorem for a general parabolic initial-boundary value problem given in a finite multidimensional cylinder and considered in the generalized Sobolev spaces $H^{s,s/(2b);\varphi}$. We use this theorem to prove some results on the local regularity of generalized solutions to the problem under investigation. Our main method is the interpolation with a function parameter between Hilbert spaces. This method allows us to deduce the isomorphism theorem from the known theorem [1, 11, 12, 27, 56] on the well-posedness of the general parabolic problem in anisotropic Sobolev spaces. Note that the transition from the case [32] of homogeneous initial conditions to the general case is not easy even for the Sobolev spaces of integer orders (see [1, Sections 10 and 11]). Specifically, this transition uses the description of the spaces in terms of the spacial variables and time variable. To avoid this difficulty and other obstacles, we prefer to resort to the interpolation of spaces and operators that correspond to the parabolic problem with inhomogeneous initial conditions.

The paper consists of six sections and Appendix. Section 1 is Introduction. Section 2 contains the statement of the general parabolic initial-boundary value problem. Section 3 discusses generalized anisotropic Sobolev spaces in which we investigate this problem. Our main results are formulated in Section 4. The basic result is Isomorphism Theorem 4.1 for the parabolic problem considered in the above-mentioned generalized Sobolev spaces. As applications of this theorem, we give Theorems 4.3 and 4.4. Theorem 4.3 deals with the local regularity of the generalized solution to the problem. Theorem 4.4 yields sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set. These conditions are essentially finer than their versions obtained in the framework of the Sobolev spaces [19, 18] and are sharp. Section 5 is devoted to our basic research method, the interpolation with a function parameter between Hilbert spaces. The main results are proved in Section 6. Appendix discusses the equivalence of the compatibility conditions imposed on the right-hand sides of the parabolic problem to those considered in the cited papers [1, 12, 56].

2. Statement of the problem

We arbitrarily choose an integer $n\geq 2$ and a real number $\tau>0$. Suppose that $G$ is a bounded domain in $\mathbb{R}^{n}$ and that its boundary $\Gamma:=\partial G$ is an infinitely smooth closed manifold of dimension $n-1$. (Of course, the $C^{\infty}$-structure on $\Gamma$ is induced by $\mathbb{R}^{n}$.) 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 area, with their closures $\overline{\Omega}=\overline{G}\times[0,\tau]$ and $\overline{S}=\Gamma\times[0,\tau]$. We naturally identify $\overline{G}$ with the lower base $\{(x,0):x\in\overline{G}\}$ of the closed cylinder $\overline{\Omega}$.

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

[TABLE]

[TABLE]

[TABLE]

Here, $b$, $m$, and all $m_{j}$ are arbitrarily choosen integers that satisfy the conditions $m\geq b\geq 1$, $\varkappa:=m/b\in\mathbb{Z}$, and $m_{j}\geq 0$. All the coefficients of the linear partial differential expressions $A:=A(x,t,D_{x},\partial_{t})$ and $B_{j}:=B_{j}(x,t,D_{x},\partial_{t})$, with $j\in\{1,\dots,m\}$, are supposed to be infinitely smooth complex-valued functions given on $\overline{\Omega}$ and $\overline{S}$ respectively; i.e., each

[TABLE]

and each

[TABLE]

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 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 $\alpha_{1},...,\alpha_{n}$ and $\beta$ that satisfy the condition written under the integral sign. As usual, $\xi^{\alpha}:=\xi_{1}^{\alpha_{1}}\ldots\xi_{n}^{\alpha_{n}}$ for $\xi:=(\xi_{1},\ldots,\xi_{n})\in\mathbb{C}^{n}$.

We recall [1, Section 9, Subsection 1] that the initial-boundary value problem (2.1)–(2.3) is called parabolic in $\Omega$ if the following Conditions 2.1 and 2.2 are satisfied.

Condition 2.1**.**

If $x\in\overline{G}$, $t\in[0,\tau]$, $\xi\in\mathbb{R}^{n}$, and $p\in\mathbb{C}$ with $\mathrm{Re}\,p\geq 0$, then

[TABLE]

To formulate the next condition, we 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 $p\in\mathbb{C}$ with $\mathrm{Re}\,p\geq 0$ such that $|\xi|+|p|\neq 0$. Let $\nu(x)$ be the unit vector of the inward normal to $\Gamma$ at $x$. It follows from Condition 2.1 and the inequality $n\geq 2$ that the polynomial $A^{\circ}(x,t,\xi+\zeta\nu(x),p)$ 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 provided that each root is taken the number of times equal to its multiplicity.

Condition 2.2**.**

The polynomials

[TABLE]

in $\zeta\in\mathbb{C}$ are linearly independent modulo

[TABLE]

Note Condition 2.1 is that under which the partial differential equation $Au=\nobreak f$ is $2b$-parabolic in $\overline{\Omega}$ in the sense of I. G. Petrovskii [44]. Besides, Condition 2.2 means that the system of boundary partial differential expressions $\{B_{1},\ldots,B_{m}\}$ covers $A$ on $\overline{S}$. This condition is introduced by Zagorskii [55], as noticed in [1, § 9, Subsection 1].

We investigate parabolic problem (2.1)–(2.3) in appropriate generalized Sobolev spaces considered in the next section.

3. Generalized Sobolev spaces related to the problem

Throughout the paper, we use complex distribution spaces and interpret distributions as antilinear functionals. Among the normed distribution spaces $\mathcal{B}_{p,\mu}$ introduced and investigated by Hörmander in [16, Section 2.2], we need the inner product spaces $H^{\mu}(\mathbb{R}^{k}):=\mathcal{B}_{2,\mu}$, which give a broad generalization of the concept of Sobolev spaces (in the framework of Hilbert spaces). Here, $1\leq k\in\mathbb{Z}$, and $\nobreak{\mu:\mathbb{R}^{k}\rightarrow(0,\infty)}$ is an arbitrary Borel measurable function for which there exist positive numbers $c$ and $l$ such that

[TABLE]

By definition, the linear space $H^{\mu}(\mathbb{R}^{k})$ consists of all distributions $w\in\mathcal{S}^{\prime}(\mathbb{R}^{k})$ whose Fourier transform $\widehat{w}$ is a locally Lebesgue integrable function such that

[TABLE]

The inner product in $H^{\mu}(\mathbb{R}^{k})$ is defined by the formula

[TABLE]

where $w_{1},w_{2}\in H^{\mu}(\mathbb{R}^{k})$; this inner product induces the norm

[TABLE]

As usual, $\mathcal{S}^{\prime}(\mathbb{R}^{k})$ stands for the linear topological space of all tempered distributions on $\mathbb{R}^{k}$; this space is the antidual of the Schwartz space $\mathcal{S}(\mathbb{R}^{k})$ of rapidly decreasing functions on $\mathbb{R}^{k}$.

According to [16, Section 2.2], the space $H^{\mu}(\mathbb{R}^{k})$ is Hilbert and separable with respect to this inner product. Besides, this space is continuously embedded in $\mathcal{S}^{\prime}(\mathbb{R}^{k})$, and the set $\mathcal{S}(\mathbb{R}^{k})$ is dense in $H^{\mu}(\mathbb{R}^{k})$, as well as the set $C^{\infty}_{0}(\mathbb{R}^{k})$ of all compactly supported $C^{\infty}$-functions on $\mathbb{R}^{k}$ (see also [17, Section 10.1]). We will say that the function parameter $\mu$ is the regularity index for the space $H^{\mu}(\mathbb{R}^{k})$ and its versions $H^{\mu}(\cdot)$.

A version of $H^{\mu}(\mathbb{R}^{k})$ for an arbitrary nonempty open set $V\subset\mathbb{R}^{k}$ is introduced in the standard way. Namely,

[TABLE]

where $u\in H^{\mu}(V)$. Here, as usual, $w\!\upharpoonright\!V$ stands for the restriction of the distribution $w$ to the open set $V$. In other words, $H^{\mu}(V)$ is the factor space of the separable Hilbert space $H^{\mu}(\mathbb{R}^{k})$ by its subspace

[TABLE]

with $Q:=\mathbb{R}^{k}\backslash V$. Thus, $H^{\mu}(V)$ is also Hilbert and separable. The norm (3.2) is induced by the inner product

[TABLE]

where $w_{j}\in H^{\mu}(\mathbb{R}^{k})$, $w_{j}=u_{j}$ in $V$ for each $j\in\{1,\,2\}$, and $\Upsilon$ is the orthogonal projector of the space $H^{\mu}(\mathbb{R}^{k})$ onto its subspace (3.3). The spaces $H^{\mu}(V)$ and $H^{\mu}_{Q}(\mathbb{R}^{k})$ were introduced and investigated by Volevich and Paneah [54, Section 3].

It follows directly from the definition of $H^{\mu}(V)$ and properties of $H^{\mu}(\mathbb{R}^{k})$ that the space $H^{\mu}(V)$ is continuously embedded in the linear topological space $\mathcal{D}^{\prime}(V)$ of all distributions on $V$ and that the set $\{w\!\upharpoonright\!\overline{V}:w\in C^{\infty}_{0}(\mathbb{R}^{k})\}$ is dense in $H^{\mu}(V)$.

Suppose that the integer $k\geq 2$, and arbitrarily choose a real number $\gamma>0$. We need the Hörmander spaces $H^{\mu}(\mathbb{R}^{k})$ and their versions in the case where the regularity index $\mu$ takes the form

[TABLE]

Here, the number parameter $s$ is real, whereas the function parameter $\varphi$ runs over a certain class $\mathcal{M}$.

By definition, the class $\mathcal{M}$ consists of all Borel measurable functions $\varphi:[1,\infty)\rightarrow(0,\infty)$ such that

• a)

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

• b)

the function $\varphi$ varies slowly at infinity in the sense of Karamata [23], i.e. $\varphi(\lambda r)/\varphi(r)\rightarrow\nobreak 1$ as $r\rightarrow\infty$ for every $\lambda>0$.

The theory of slowly varying functions (at infinity) is set forth in [7, 8, 46]. Their standard examples are the functions

[TABLE]

where the parameters $1\leq k\in\mathbb{Z}$ and $\theta_{1},\theta_{2},\ldots,\theta_{k}\in\mathbb{R}$ are arbitrarily chosen.

Note that the regularity index (3.4) satisfies condition (3.1) (see [32, Appendix]). Dealing with the above-stated parabolic problem, we need the Hörmander spaces $H^{\mu}(\mathbb{R}^{k})$ with the regularity index (3.4) only in the case where $\gamma=1/(2b)$. However, it is naturally to introduce these spaces for arbitrary $\gamma>0$.

Let $s\in\mathbb{R}$ and $\varphi\in\mathcal{M}$. We put $H^{s,s\gamma;\varphi}(\mathbb{R}^{k}):=H^{\mu}(\mathbb{R}^{k})$ in the case where $\mu$ is of the form (3.4). Specifically, if $\varphi(r)\equiv 1$, then $H^{s,s\gamma;\varphi}(\mathbb{R}^{k})$ becomes the anisotropic Sobolev inner product space $H^{s,s\gamma}(\mathbb{R}^{k})$ of order $(s,s\gamma)$ [6, 48]. Generally, if $\varphi\in\mathcal{M}$, we have the dense continuous embeddings:

[TABLE]

Indeed, let $s_{0}<s<s_{1}$; since $\varphi\in\mathcal{M}$, there exist positive numbers $c_{0}$ and $c_{1}$ such that $c_{0}\,r^{s_{0}-s}\leq\varphi(r)\leq c_{1}\,r^{s_{1}-s}$ for every $r\geq 1$ (see e.g., [46, Section 1.5, Property $1^{\circ}$]). Then

[TABLE]

for arbitrary $\xi^{\prime}\in\mathbb{R}^{k-1}$ and $\xi_{k}\in\mathbb{R}$. This directly entails the continuous embeddings (3.5). They are dense because the set $C^{\infty}_{0}(\mathbb{R}^{k})$ is dense in all the spaces in (3.5).

Consider the class of Hörmander inner product spaces

[TABLE]

The embeddings (3.5) show, that in (3.6) the function parameter $\varphi$ defines subordinate regularity with respect to the basic anisotropic $(s,s\gamma)$-regularity. Specifically, if $\varphi(r)\rightarrow\infty$ [or $\varphi(r)\rightarrow\nobreak 0$] as $r\rightarrow\infty$, then $\varphi$ defines supplementary positive [or negative] regularity. In other words, $\varphi$ refines the basic regularity $(s,s\gamma)$.

We need versions of the function spaces (3.6) for the cylinder $\Omega=G\times(0,\tau)$ and its lateral boundary $S=\Gamma\times(0,\tau)$. We put $H^{s,s\gamma;\varphi}(\Omega):=H^{\mu}(\Omega)$ in the case where $\mu$ is of the form (3.4) with $k:=n+1$. For the function space $H^{s,s\gamma;\varphi}(\Omega)$, the numbers $s$ and $s\gamma$ serve as the regularity indices of distributions $u(x,t)$ with respect to the spatial variable $x\in G$ and to the time variable $t\in(0,\tau)$ respectively.

Following [29, Section 1], we will define the function space $H^{s,s\gamma;\varphi}(S)$ with the help of special local charts on $S$. Let $s>0$ and $\varphi\in\mathcal{M}$. We put $H^{s,s\gamma;\varphi}(\Pi):=H^{\mu}(\Pi)$ for the strip $\Pi:=\mathbb{R}^{n-1}\times(0,\tau)$ in the case where $\mu$ is defined by formula (3.4) with $k:=n$. Recall that, according to our assumption, $\Gamma=\partial\Omega$ is an infinitely smooth closed manifold of dimension $n-1$, the $C^{\infty}$-structure on $\Gamma$ being induced by $\mathbb{R}^{n}$. From this structure we arbitrarily choose a finite atlas formed by local charts $\nobreak{\theta_{j}:\mathbb{R}^{n-1}\leftrightarrow\Gamma_{j}}$ with $j=1,\ldots,\lambda$. Here, the open sets $\Gamma_{1},\ldots,\Gamma_{\lambda}$ make up a covering of $\Gamma$. 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 $\Gamma$.

By definition, the linear space $H^{s,s\gamma;\varphi}(S)$ consists of all square integrable functions $\nobreak{v:S\to\mathbb{C}}$ that the function

[TABLE]

belongs to $H^{s,s\gamma;\varphi}(\Pi)$ for each $j\in\{1,\ldots,\lambda\}$. The inner product in $H^{s,s\gamma;\varphi}(S)$ is defined by the formula

[TABLE]

where $v,v^{\circ}\in H^{s,s\gamma;\varphi}(S)$. This inner product induces the norm

[TABLE]

The space $H^{s,s\gamma;\varphi}(S)$ is separable Hilbert one and does not depend up to equivalence of norms on the choice of local charts and partition of unity on $\Gamma$ [29, Theorem 1]. (The proof in [29] is done in the $\gamma\in\mathbb{Q}$ case we really need; the general case is treated similarly to [32, Lemma 3.1]). Note that this space is actually defined with the help of the following special local charts on $S$:

[TABLE]

where $\theta_{j}^{*}(y,t):=(\theta_{j}(y),t)$ for all $y\in\mathbb{R}^{n-1}$ and $t\in(0,\tau)$.

Let $s\in\mathbb{R}$ and $\varphi\in\mathcal{M}$. We also need the isotropic space $H^{s;\varphi}(V)$ over an arbitrary open nonempty set $V\subseteq\mathbb{R}^{k}$, with $k\geq 1$. We put $H^{s;\varphi}(V):=H^{\mu}(V)$ in the case where

[TABLE]

Since the function (3.8) is radial (i.e., depends only on $|\xi|$), the space $H^{s;\varphi}(V)$ is isotropic. We will use the spaces $H^{s;\varphi}(V)$ given over the whole Euclidean space $V:=\mathbb{R}^{k}$ or over the domain $V:=G$ in $\mathbb{R}^{n}$.

Besides, we will use the space $H^{s;\varphi}(\Gamma)$ over $\Gamma=\partial\Omega$. It is defined with the help of the above-mentioned collection of local charts $\{\theta_{j}\}$ and partition of unity $\{\chi_{j}\}$ on $\Gamma$ similarly to the spaces over $S$. By definition, the linear space $H^{s;\varphi}(\Gamma)$ consists of all distributions $\omega$ on $\Gamma$ that for each number $j\in\{1,\ldots,\lambda\}$ the distribution $\omega_{j}(y):=\chi_{j}(\theta_{j}(y))\,\omega(\theta_{j}(y))$ of $y\in\mathbb{R}^{n-1}$ belongs to $H^{s;\varphi}(\mathbb{R}^{n-1})$. The inner product in $H^{s;\varphi}(\Gamma)$ is defined by the formula

[TABLE]

where $\omega,\omega^{\circ}\in H^{s;\varphi}(\Gamma)$. It induces the norm

[TABLE]

The space $H^{s;\varphi}(\Gamma)$ is separable Hilbert one and does not depend up to equivalence of norms on our choice of local charts and partition of unity on $\Gamma$ [38, Theorem 2.1].

Note that the classes of isotropic inner product spaces

[TABLE]

were selected, investigated, and systematically applied to elliptic differential operators and elliptic boundary-value problems by Mikhailets and Murach [38, 36].

If $\varphi\equiv 1$, then the considered spaces $H^{s,s\gamma;\varphi}(\cdot)$ and $H^{s;\varphi}(\cdot)$ become the inner product Sobolev spaces $H^{s,s\gamma}(\cdot)$, anisotropic, and $H^{s}(\cdot)$, isotropic, respectively. It follows directly from (3.5) that

[TABLE]

Analogously,

[TABLE]

see [38, Theorems 2.3(iii) and 3.3(iii)]. These embeddings are continuous and dense. Certainly, if $s=0$, then $H^{s}(\cdot)=H^{s,s\gamma}(\cdot)$ is the Hilbert space $L_{2}(\cdot)$ of all square integrable functions given on the corresponding measurable set.

In the Sobolev case of $\varphi\equiv 1$, we will omit the index $\varphi$ in designations of distribution spaces that will be introduced on the base of the spaces $H^{s,s\gamma;\varphi}(\cdot)$ and $H^{s;\varphi}(\cdot)$.

4. Main results

We will formulate an isomorphism theorem for the parabolic problem (2.1)–(2.3) in the generalized Sobolev spaces introduced and then apply it to the investigation of regularity of solutions to the problem.

In order that a regular solution $u(x,t)$ to this problem exist, the right-hand sides of the problem should satisfy certain compatibility conditions (see, e.g., [25, Chapter 4, Section 5]). These conditions consist in that the partial derivatives $(\partial^{k}_{t}u)(x,0)$, which could be found from the parabolic equation (2.1) and initial conditions (2.3), should satisfy the boundary conditions (2.2) and some relations that are obtained by the differentiation of the boundary conditions with respect to $t$. To write these compatibility conditions, we previously consider the problem in appropriate anisotropic Sobolev spaces.

We associate the linear mapping

[TABLE]

with the problem (2.1)–(2.3). Put

[TABLE]

(Specifficaly, if $m_{j}\leq 2m-1$ for each $j\in\{1,\ldots,m\}$, then $\sigma_{0}=2m$.) Let real $s\geq\sigma_{0}$; the mapping (4.1) extends uniquely (by continuity) to a bounded linear operator

[TABLE]

with

[TABLE]

This follows directly from the known properties of partial differential operators and trace operators on anisotropic Sobolev spaces (see, e.g., [48, Chapter I, Lemma 4, and Chapter II, Theorems 3 and 7]). Choosing any function $u(x,t)$ from the space $H^{s,s/(2b)}(\Omega)$, we define the right-hand sides

[TABLE]

of the problem by the formula

[TABLE]

where $\Lambda$ is the operator (4.2).

The compatibility conditions for the functions $f$, $g_{j}$, and $h_{k}$ arise naturally in such a way. According to [48, Chapter II, Theorem 7], the traces $(\partial^{k}_{t}u)(\cdot,0)\in H^{s-2bk-b}(G)$ are well defined by closure for all $k\in\mathbb{Z}$ such that $0\leq k<s/(2b)-1/2$ (and only for these $k$). These traces are expressed from (2.1) and (2.3) in terms of $f$ and $h_{k}$ as follows.

The parabolicity Condition 2.1 in the case of $\xi=0$ and $p=1$ means that the coefficient $a^{(0,\ldots,0),\varkappa}(x,t)\neq 0$ for all $x\in\overline{G}$ and $t\in[0,\tau]$. We can therefore solve the parabolic equation (2.1) with respect to $\partial^{\varkappa}_{t}u(x,t)$; namely,

[TABLE]

with $a_{0}^{\alpha,\beta}:=-a^{\alpha,\beta}/a^{(0,\ldots,0),\varkappa}\in C^{\infty}(\overline{\Omega})$. Let $k\in\mathbb{Z}$ satisfy $0\leq k<s/(2b)-1/2$. It follows from the initial conditions (2.3), equality (4.5), and the equalities obtained by the differentiation of (4.5) $k-\varkappa$ times with respect to $t$ that

[TABLE]

These equalities hold for almost all $x\in G$, and partial derivatives are interpreted in the sense of the theory of distributions.

Besides, according to [48, Chapter II, Theorem 7], for each $j\in\{1,\dots,m\}$ the traces $\partial^{\,k}_{t}g_{j}(\cdot,0)\in H^{s-m_{j}-1/2-2bk-b}(\Gamma)$ are well defined by closure for all $k\in\mathbb{Z}$ such that $0\leq k<(s-m_{j}-1/2-b)/(2b)$ (and only for these $k$). We can express these traces in terms of the function $u(x,t)$ and its time derivatives; namely,

[TABLE]

for almost all $x\in\Gamma$. Here, all the functions

[TABLE]

of $x\in G$ are expressed in terms of the functions $f(x,t)$ and $h_{k}(x)$ by the recurrent formula (4.6). (As usual, $[m_{j}/(2b)]$ denotes the integral part of $m_{j}/(2b)$.)

Substituting (4.6) in the right-hand side of (4.7), we obtain the compatibility conditions

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

due to (4.4). The right-hand side of the equality in (4.8) is well defined because the function $B_{j,k}(v_{0},\dots,v_{[m_{j}/(2b)]+k})$ belongs to $H^{s-m_{j}-2bk-b}(G)$ and the trace

[TABLE]

is therefore defined by closure whenever $s-m_{j}-2bk-b-1/2>0$.

The number of the compatibility conditions (4.8) is a function of $s\geq\sigma_{0}$. This function is discontinuous at $s$ if and only if $(s-m_{j}-1/2-b)/(2b)\in\mathbb{Z}$. Thus, the set of all its discontinuities coincides with

[TABLE]

Note, if $s\leq\min\{m_{1},\ldots,m_{m}\}+b+1/2$, there are no compatibility conditions.

Our main 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 $s>\sigma_{0}$ and function parameter $\varphi\in\mathcal{M}$. We also consider the Sobolev case where $s=\sigma_{0}$ and $\varphi\equiv 1$ (we need it to formulate our results). We take $H^{s,s/(2b);\varphi}(\Omega)$ as the domain of this isomorphism. Its range is imbedded in the Hilbert space

[TABLE]

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

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

[TABLE]

that satisfy the compatibility conditions (4.8). These conditions are well defined for every indicated $F$ because they are well defined whenever $F\in\mathcal{H}^{s-\varepsilon-2m,(s-\varepsilon-2m)/(2b)}$ and $0<\varepsilon\ll 1$ and because

[TABLE]

[This continuous embedding follows directly from (3.9) and (3.10).] We endow the linear space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ with the inner product and norm in the Hilbert space $\mathcal{H}^{s-2m,(s-2m)/(2b);\varphi}$. The space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ is complete, i.e. Hilbert. Indeed,

[TABLE]

whenever $0<\varepsilon\ll 1$. Here, the space $\mathcal{Q}^{s-\varepsilon-2m,(s-\varepsilon-2m)/(2b)}$ 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 right-hand side of this equality is complete with respect to the sum of the norms in the components of the intersection, this sum being equivalent to the norm in $\mathcal{H}^{s-2m,(s-2m)/(2b);\varphi}$ due to (4.13). Thus, the space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ is complete (with respect to the latter norm).

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

[TABLE]

Here, the number $\varepsilon\in(0,1/2)$ is arbitrarily chosen, and the right-hand side of the equality is the result of the interpolation with the parameter $1/2$ of the written pair of Hilbert spaces. We will recall the definition of the interpolation between Hilbert spaces in Section 5. The Hilbert space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ defined by formula (4.14) does not depend on our choice of $\varepsilon$ up to equivalence of norms and is continuously embedded in $\mathcal{H}^{s-2m,(s-2m)/(2b);\varphi}$. This will be shown in Remark 6.5.

Theorem 4.1**.**

For arbitrary $s>\sigma_{0}$ and $\varphi\in\nobreak\mathcal{M}$, the mapping (4.1) extends uniquely (by continuity) to an isomorphism

[TABLE]

This theorem is known in the Sobolev case where $\varphi\equiv 1$. It is proved in this case by Agranovich and Vishik [1, Theorem 12.1] under the restriction $s,s/(2b)\in\mathbb{Z}$. This restriction can be removed; see, e.g., Lions and Magenes’ monograph [27, Theorem 6.2] in the case of $b=1$ and the normal boundary conditions, and Zhitarashu’s paper [56, Theorem 9.1] in the general case. Their results include the limiting case of $s=\sigma_{0}$. Note that these papers deal with another equivalent form of the compatibility conditions (4.8), which will be discussed in Appendix.

We will deduce Theorem 4.1 from the Sobolev case with the help of the interpolation with a function parameter between Hilbert spaces. This will be done in Section 6 after we investigate the necessary interpolation properties of the spaces used in (4.15).

Note that we have to define the range of the isomorphism (4.15) by the interpolation formula (4.14) in the $s\in E$ case because this isomorphism can cease holding if we define $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ in the way used in the $s\notin E$ case. This is suggested by Solonnikov’s result [49, Section 6] concerning the heat equation in Sobolev spaces; see also [27, Remark 6.4].

Let us discuss the regularity properties of the generalized solution to the parabolic problem (2.1)–(2.3). We assume further in this section that the right-hand sides $f$, $g_{j}$, and $h_{k}$ of the problem are arbitrary distributions given respectively on $\Omega$, $S$ and $G$. A function $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ is said to be a (strong) generalized solution to this problem if

[TABLE]

here, $\Lambda$ is the bounded operator (4.2) for $s:=\sigma_{0}$. It follows from this condition that

[TABLE]

Moreover [56, Theorem 9.1], the problem has a unique solution $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ for every vector (4.16). We see now that the following result is a direct consequence of Theorem 4.1:

Corollary 4.2**.**

Assume that a function $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ is a generalized solution to the parabolic problem (2.1)–(2.3) whose right-hand sides satisfy the condition

[TABLE]

for some $s>\sigma_{0}$ and $\varphi\in\mathcal{M}$. Then $u\in H^{s,s/(2b);\varphi}(\Omega)$.

Let us formulate a local version of this result. Let $U$ be an open subset of $\mathbb{R}^{n+1}$ such that $\Omega_{0}:=U\cap\Omega\neq\emptyset$ and $U\cap\Gamma=\emptyset$. Put $\Omega^{\prime}:=U\cap\partial\overline{\Omega}$, $S_{0}:=U\cap S$, $S^{\prime}:=U\cap\{(x,\tau);x\in\Gamma\}$, and $G_{0}:=U\cap G$. We need to introduce local versions of the spaces $H^{s,s/(2b);\varphi}(\Omega)$, $H^{s,s/(2b);\varphi}(S)$ and $H^{s;\varphi}(G)$ with $s>0$ and $\varphi\in\mathcal{M}$.

We let $H^{s,s/(2b);\varphi}_{\mathrm{loc}}(\Omega_{0},\Omega^{\prime})$ denote the linear space of all distributions $u$ in $\Omega$ such that $\chi u\in H^{s,s/(2b);\varphi}(\Omega)$ for every function $\chi\in C^{\infty}(\overline{\Omega})$ subject to $\mathrm{supp}\,\chi\subset\Omega_{0}\cup\Omega^{\prime}$. Analogously, $H^{s,s/(2b);\varphi}_{\mathrm{loc}}(S_{0},S^{\prime})$ denotes the linear space of all distributions $v$ on $S$ such that $\chi v\in H^{s,s/(2b);\varphi}(S)$ for every function $\chi\in C^{\infty}(\overline{S})$ subject to $\mathrm{supp}\,\chi\subset S_{0}\cup S^{\prime}$. Finally, $H^{s;\varphi}_{\mathrm{loc}}(G_{0})$ stands for the linear space of all distributions $w$ in $G$ such that $\chi w\in H^{s;\varphi}(G)$ for every function $\chi\in C^{\infty}(\overline{G})$ satisfying $\mathrm{supp}\,\chi\subset G_{0}$.

Theorem 4.3**.**

Let $s>\sigma_{0}$ and $\varphi\in\mathcal{M}$. Assume that a function $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ is a generalized solution to the parabolic problem (2.1)–(2.3) whose right-hand sides satisfy the following conditions:

[TABLE]

Then $u\in H^{s,s/(2b);\varphi}_{\mathrm{loc}}(\Omega_{0},\Omega^{\prime})$.

If $\Omega^{\prime}=\emptyset$, Theorem 4.3 asserts that the regularity of $u$ increases on neighbourhoods of internal points of $\overline{\Omega}$. If $G_{0}=\emptyset$, this theorem states that the regularity of $u(x,t)$ increases whenever $t>0$. In this case, the theorem follows directly from [32, Theorem 4.3] provided $\sigma_{0}/(2b)\in\mathbb{Z}$. Remark that we restrict ourselves to the case $U\cap\Gamma=\emptyset$ because the conclusion of Theorem 4.3 is not true in the general case.

Using the spaces introduced, we obtain sufficient conditions under which the generalized solution $u$ and its generalized derivatives of a prescribed order are continuous on $\Omega_{0}\cup\Omega^{\prime}$.

Theorem 4.4**.**

Let an integer $p\geq 0$ satisfy $p+b+n/2>\sigma_{0}$. Assume that a function $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ is a generalized solution to the parabolic problem (2.1)–(2.3) whose right-hand sides satisfy conditions (4.17)–(4.19) for $s:=p+b+n/2$ and some function parameter $\varphi\in\mathcal{M}$ subject to

[TABLE]

Then the solution $u(x,t)$ and all its generalized derivatives $D_{x}^{\alpha}\partial_{t}^{\beta}u(x,t)$ with $|\alpha|+2b\beta\leq p$ are continuous on $\Omega_{0}\cup\Omega^{\prime}$.

As to the conclusion of this theorem, note that a distribution $v$ in $\Omega$ is called continuous on the set $\Omega_{0}\cup\Omega^{\prime}$ if there exists a continuous function $v_{0}$ on $\Omega_{0}\cup\Omega^{\prime}$ such that

[TABLE]

for every test function $\omega\in C^{\infty}(\Omega)$ subject to $\mathrm{supp}\,\omega\subset\Omega_{0}$. Here, $v(\omega)$ stands for the value of the functional $v$ at $\omega$. (It is not difficult to show that this definition is equivalent to the following: $\chi v\in C(\overline{\Omega})$ for every function $\chi\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\chi\subset\Omega_{0}\cup\Omega^{\prime}$.)

Remark 4.5**.**

Condition (4.20) in Theorem 4.4 is sharp. Namely, let $s:=p+b+n/2$ and $\varphi\in\mathcal{M}$, and assume that for every function $u\in\nobreak H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ the following implication holds:

[TABLE]

Then $\varphi$ satisfies condition (4.20).

Note also that the use of generalized Sobolev spaces allows us to obtain a finer result then it is possible in the framework of Sobolev spaces. Namely, if we formulate an analog of Theorem 4.4 for the Sobolev case of $\varphi\equiv\nobreak 1$, we have to replace the condition of this theorem with a stronger one. Thus, we have to claim that the right-hand sides of the problem (2.1)–(2.3) satisfy conditions (4.17)–(4.19) for certain $s>p+b+n/2$. This claim is stronger than the condition of Theorem 4.4 due to the left-hand embeddings in (3.9) and (3.10). This theorem can be used to obtain sufficient conditions under which the generalized solution $u$ to the parabolic problem is classical (see [31]).

We will prove Theorems 4.3 and 4.4 at the end of Section 6 and then substantiate Remark 4.5.

5. Interpolation with a function parameter between Hilbert spaces

This method of interpolation is a natural generalization of the classical interpolation method by S. Krein and J.-L. Lions (see their monographs [24, Chapter IV, Section 1, Subsection 10] and [26, Chapter 1, Sections 2 and 5]) to the case where a general enough function is used instead of a number as an interpolation parameter. We restrict ourselves to the case of separable complex Hilbert spaces and mainly follow the monograph [38, 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. For $X$ there is a positive-definite self-adjoint operator $J$ in $X_{0}$ with the domain $X_{1}$ such that $\|Jv\|_{X_{0}}=\|v\|_{X_{1}}$ for every $v\in X_{1}$. This operator is uniquely determined by $X$ and is called the generating operator for $X$; see, e.g., [24, Chapter IV, Theorem 1.12]. The operator sets an isometric isomorphism between $X_{1}$ and $X_{0}$.

Let $\mathcal{B}$ denote the set of all Borel measurable functions $\psi:(0,\infty)\rightarrow(0,\infty)$ such that $\psi$ is bounded on each compact interval $[a,b]$, with $0<a<b<\infty$, and that $1/\psi$ is bounded on every semiaxis $[a,\infty)$, with $a>0$.

Choosing a function $\psi\in\mathcal{B}$ arbitrarily, we consider the (generally, unbounded) operator $\psi(J)$ in $X_{0}$ as the Borel function $\psi$ of $J$. This operator is built with the help of Spectral Theorem applied to the self-adjoint operator $J$. Let $[X_{0},X_{1}]_{\psi}$ or, simply, $X_{\psi}$ denote the domain of $\psi(J)$ endowed with the inner product $(v_{1},v_{2})_{X_{\psi}}:=(\psi(J)v_{1},\psi(J)v_{2})_{X_{0}}$ and the corresponding norm $\|v\|_{X_{\psi}}:=\|\psi(J)v\|_{X_{0}}$. The linear space $X_{\psi}$ is Hilbert and separable with respect to this norm.

A function $\psi\in\mathcal{B}$ is called an interpolation parameter if the following condition is satisfied for all regular pairs $X=[X_{0},X_{1}]$ and $Y=[Y_{0},Y_{1}]$ of Hilbert spaces and for an arbitrary linear mapping $T$ given on $X_{0}$: if the restriction of $T$ to $X_{j}$ is a bounded operator $T:X_{j}\rightarrow Y_{j}$ for each $j\in\{0,1\}$, then the restriction of $T$ to $X_{\psi}$ is also a bounded operator $T:X_{\psi}\rightarrow Y_{\psi}$.

If $\psi$ is an interpolation parameter, we will say that the Hilbert space $X_{\psi}$ is obtained by the interpolation with the function parameter $\psi$ of the pair $X=\nobreak[X_{0},X_{1}]$ or, otherwise speaking, between the spaces $X_{0}$ and $X_{1}$. In this case, the dense and continuous embeddings $X_{1}\hookrightarrow X_{\psi}\hookrightarrow X_{0}$ hold.

The class of all interpolation parameters (in the sense of the given definition) admits a constructive description. Namely, a function $\psi\in\mathcal{B}$ is an interpolation parameter if and only if $\psi$ is pseudoconcave in a neighbourhood of infinity. The latter property means that there exists a concave positive function $\psi_{1}(r)$ of $r\gg 1$ that both the functions $\psi/\psi_{1}$ and $\psi_{1}/\psi$ are bounded in some neighbourhood of infinity. This criterion follows from Peetre’s description of all interpolation functions for the weighted Lebesgue spaces [42, 43] (this result of Peetre is set forth in the monograph [5, Theorem 5.4.4]). The proof of the criterion is given, e.g., in [38, Section 1.1.9].

The application of this criterion to power functions gives the classical result by Krein and Lions. Namely, the function $\psi(r)\equiv r^{\theta}$ is an interpolation parameter if and only if $\nobreak{0\leq\theta\leq 1}$. In this case, the exponent $\theta$ serves as a number parameter of the interpolation, and the interpolation space $X_{\psi}$ is also denoted by $X_{\theta}$. We used this interpolation in our definition (4.14), with $\theta=1/2$.

For the readers’ convenience, we formulate the general interpolation properties used systematically below. The first of them enables us to reduce the interpolation of subspaces to the interpolation of the whole spaces (see [38, Theorem 1.6] or [52, Section 1.17.1, Theorem 1]). As usual, subspaces of normed spaces are supposed to be closed. Generally, we consider nonorthogonal projectors onto subspaces of a Hilbert space.

Proposition 5.1**.**

Let $X=[X_{0},X_{1}]$ be a regular pair of Hilbert spaces, and let $Y_{0}$ be a subspace of $X_{0}$. Then $Y_{1}:=X_{1}\cap Y_{0}$ is a subspace of $X_{1}$. Suppose that there exists a linear mapping $P$ on $X_{0}$ such that $P$ is a projector of the space $X_{j}$ onto its subspace $Y_{j}$ for each $j\in\{0,\,1\}$. Then the pair $[Y_{0},Y_{1}]$ is regular, and $[Y_{0},Y_{1}]_{\psi}=X_{\psi}\cap Y_{0}$ with equivalence of norms for an arbitrary interpolation parameter $\psi\in\mathcal{B}$. Here, $X_{\psi}\cap Y_{0}$ is a subspace of $X_{\psi}$.

The second property reduces the interpolation of orthogonal sums of Hilbert spaces to the interpolation of their summands (see [38, Theorem 1.8]).

Proposition 5.2**.**

Let $[X_{0}^{(j)},X_{1}^{(j)}]$, with $j=1,\ldots,q$, be a finite collection of regular pairs of Hilbert spaces. Then

[TABLE]

with equality of norms for every function $\psi\in\mathcal{B}$.

The third property is Reiteration Theorem for the interpolation [38, Theorem 1.3].

Proposition 5.3**.**

Let $\alpha,\beta,\psi\in\mathcal{B}$, and suppose that the function $\alpha/\beta$ is bounded in a neighbourhood of infinity. Define the function $\omega\in\mathcal{B}$ by the formula $\omega(r):=\alpha(r)\psi(\beta(r)/\alpha(r))$ for $r>0$. Then $\omega\in\mathcal{B}$, and $[X_{\alpha},X_{\beta}]_{\psi}=X_{\omega}$ with equality of norms for every regular pair $X$ of Hilbert spaces. Besides, if $\alpha,\beta,\psi$ are interpolation parameters, then $\omega$ is also an interpolation parameter.

Our proof of Theorem 4.1 is based on the key fact that the interpolation with an appropriate function parameter between marginal Sobolev spaces in (3.9) and (3.10) gives the intermediate spaces $H^{s,s\gamma;\varphi}(\cdot)$ and $H^{s;\varphi}(\cdot)$ respectively. Let us formulate this property separately for isotropic and for anisotropic spaces.

Proposition 5.4**.**

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

[TABLE]

Then the function $\psi\in\mathcal{B}$ is an interpolation parameter, and the equality of spaces

[TABLE]

holds true with equivalence of norms for arbitrary $\lambda\in\mathbb{R}$ provided that $W=G$ or $W=\Gamma$. If $W=\mathbb{R}^{k}$ with $1\leq k\in\mathbb{Z}$, then (5.2) holds true with equality of norms.

The proof of this proposition is given in [38, Theorems 1.14, 2.2, and 3.2] for the cases where $W=\mathbb{R}^{k}$, $W=\Gamma$, and $W=G$ respectively.

Proposition 5.5**.**

Let $s_{0},s,s_{1}\in\mathbb{R}$ satisfy $0\leq s_{0}<s<s_{1}$, and let $\varphi\in\mathcal{M}$. Define the interpolation parameter $\psi\in\mathcal{B}$ by formula (5.1). Then the equality of spaces

[TABLE]

holds true with equivalence of norms for arbitrary real $\lambda\leq s_{0}$ provided that $W=\Omega$ or $W=S$. If $W=\mathbb{R}^{k}$ with $2\leq k\in\mathbb{Z}$, then (5.3) holds true with equality of norms without the assumption that $0\leq s_{0}$.

This result is proved in [29, Theorem 2 and Lemma 1] for the cases where $W=S$ and $W=\mathbb{R}^{k}$ respectively. In the $W=\Omega$ case, the proof of the result is the same as the proof of its analog for a strip [29, Lemma 2]. Note that we represent the indexes as $s-\lambda$ etc. for the sake of convenience of our application of Propositions 5.4 and 5.5 to the spaces used in (4.3).

6. Proofs

To deduce Theorem 4.1 from its known counterpart in the Sobolev case, we need to prove a version of Proposition 5.5 for the range of isomorphism (4.15). This proof is based on the following lemma about properties of the operator that assigns the Cauchy data to an arbitrary function $g\in H^{s,s/(2b);\varphi}(S)$.

Lemma 6.1**.**

Choose an integer $r\geq 1$, and consider the linear mapping

[TABLE]

This mapping extends uniquely (by continuity) to a bounded linear operator

[TABLE]

for arbitrary $s>2br-b$ and $\varphi\in\mathcal{M}$. This operator is right invertible; moreover, there exists a continuous linear mapping $T:(L_{2}(\Gamma))^{r}\to L_{2}(S)$ that the restriction of $T$ to the space $\mathbb{H}^{s;\varphi}(\Gamma)$ is a bounded linear operator

[TABLE]

for all $s>2br-b$ and $\varphi\in\mathcal{M}$ and that $RTv=v$ for every $v\in\mathbb{H}^{s;\varphi}(\Gamma)$.

Proof.

We first prove an analog of this lemma for Hörmander spaces defined on $\mathbb{R}^{n}$ and $\mathbb{R}^{n-1}$ instead of $S$ and $\Gamma$. Then we deduce the lemma with the help of the special local charts on $S$.

Consider the linear mapping

[TABLE]

Here, we interpret $w$ as a function $w(x,t)$ of $x\in\mathbb{R}^{n-1}$ and $t\in\mathbb{R}$ so that $R_{0}w\in(\mathcal{S}(\mathbb{R}^{n-1}))^{r}$. Choose $s>2br-b$ and $\varphi\in\mathcal{M}$ arbitrarily, and prove that the mapping (6.4) extends uniquely (by continuity) to a bounded linear operator

[TABLE]

This fact is known in the Sobolev case of $\varphi\equiv 1$ due to [48, Chapter II, Theorem 7]. Using the interpolation with a function parameter between Sobolev spaces, we can deduce this fact in the general case of arbitrary $\varphi\in\mathcal{M}$.

Namely, choose $s_{0},s_{1}\in\mathbb{R}$ such that $2br-b<s_{0}<s<s_{1}$, and consider the bounded linear operators

[TABLE]

Let $\psi$ be the interpolation parameter (5.1). Then the restriction of the mapping (6.6) with $j=0$ to the space

[TABLE]

is a bounded operator

[TABLE]

The latter equality is due to Proposition 5.5. This operator is an extension by continuity of the mapping (6.4) because the set $\mathcal{S}(\mathbb{R}^{n})$ is dense in $H^{s,s/(2b);\varphi}(\mathbb{R}^{n})$. Owing to Propositions 5.2 and 5.4, we get

[TABLE]

Hence, the bounded linear operator (6.8) is the required operator (6.5).

Let us now build a continuous linear mapping

[TABLE]

such that its restriction to every space $\mathbb{H}^{s;\varphi}(\mathbb{R}^{n-1})$, with $s>2br-b$ and $\varphi\in\mathcal{M}$, is a bounded operator between the spaces $\mathbb{H}^{s;\varphi}(\mathbb{R}^{n-1})$ and $H^{s,s/(2b);\varphi}(\mathbb{R}^{n})$ and that this operator is a right inverse of (6.5).

Similarly to [16, Proof of Theorem 2.5.7], we define the linear mapping

[TABLE]

on the linear topological space of all vectors

[TABLE]

We consider $T_{0}v$ as a distribution on the Euclidean space $\mathbb{R}^{n}$ of points $(x,t)$, with $x=(x_{1},\ldots,x_{n-1})\in\mathbb{R}^{n-1}$ and $t\in\mathbb{R}$. In (6.11), the function $\beta\in C^{\infty}_{0}(\mathbb{R})$ is chosen so that $\beta=1$ in a certain neighbourhood of zero. As usual, $F_{\xi\mapsto x}^{-1}$ denotes the inverse Fourier transform with respect to $\xi=(\xi_{1},\ldots,\xi_{n-1})\in\mathbb{R}^{n-1}$, and $\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$. The variable $\xi$ is dual to $x$ relative to the direct Fourier transform $\widehat{w}(\xi)=(Fw)(\xi)$ of a function $w(x)$.

Obviously, the mapping (6.11) is well defined and acts continuously between $(\mathcal{S}^{\prime}(\mathbb{R}^{n-1}))^{r}$ and $\mathcal{S}^{\prime}(\mathbb{R}^{n})$. It is also evident that the restriction of this mapping to the space $(L_{2}(\mathbb{R}^{n-1}))^{r}$ is a continuous operator from $(L_{2}(\mathbb{R}^{n-1}))^{r}$ to $L_{2}(\mathbb{R}^{n})$.

We assert that

[TABLE]

Since $v\in(\mathcal{S}(\mathbb{R}^{n-1}))^{r}$ implies $T_{0}v\in\mathcal{S}(\mathbb{R}^{n-1})$, the left-hand side of the equality (6.12) is well defined. Let us prove this equality.

Choosing $j\in\{0,\dots,r-1\}$ and $v=(v_{0},\dots,v_{r-1})\in(\mathcal{S}(\mathbb{R}^{n-1}))^{r}$ arbitrarily, we get

[TABLE]

for every $\xi\in\mathbb{R}^{n-1}$. In the third equality, we have used the fact that $\beta=1$ in a neighbourhood of zero. Thus, the Fourier transforms of all the corresponding components of the vectors $R_{0}T_{0}v$ and $v$ coincide, which is equivalent to (6.12).

Let us now prove that the restriction of the mapping (6.11) to each space

[TABLE]

with $0\leq m\in\mathbb{Z}$, is a bounded operator between $\mathbb{H}^{2bm}(\mathbb{R}^{n-1})$ and $H^{2bm,m}(\mathbb{R}^{n})$. Note that the integers $2bm-2bk-b$ may be negative in (6.13).

Let an integer $m\geq 0$. We use the fact that the norm in the space $H^{2bm,m}(\mathbb{R}^{n})$ is equivalent to the norm

[TABLE]

(see, e.g., [6, Section 9.1]). Here and below in this proof, $\|\cdot\|$ stands for the norm in the Hilbert space $L_{2}(\mathbb{R}^{n})$. Of course, $\partial_{x_{j}}$ and $\partial_{t}$ denote the operators of generalized differentiation with respect to $x_{j}$ and $t$ respectively. Choosing $v=(v_{0},\dots,v_{r-1})\in(\mathcal{S}(\mathbb{R}^{n-1}))^{r}$ arbitrarily and using the Parseval equality, we obtain the following:

[TABLE]

Let us estimate each of these three integrals separately. We begin with the third integral. Changing the variable $\tau=\langle\xi\rangle^{2b}t$ in the interior integral with respect to $t$, we get the equalities

[TABLE]

Hence,

[TABLE]

with

[TABLE]

Using the same changing of $t$ in the second integral, we obtain the following:

[TABLE]

Hence,

[TABLE]

with

[TABLE]

The first integral is estimated analogously:

[TABLE]

Thus, we conclude that

[TABLE]

for any $v\in(\mathcal{S}(\mathbb{R}^{n-1}))^{r}$, with the number $c>0$ being independent of $v$. Since the set \bigl{(}S(\mathbb{R}^{n-1})\bigr{)}^{r} is dense in $\mathbb{H}^{2bm}(\mathbb{R}^{n-1})$, it follows from the latter estimate that the mapping (6.11) sets a bounded linear operator

[TABLE]

Let us deduce from this fact that the mapping (6.11) acts continuously between the spaces $\mathbb{H}^{s;\varphi}(\mathbb{R}^{n-1})$ and $H^{s,s/(2b);\varphi}(\mathbb{R}^{n})$ for every $s>2br-b$ and $\varphi\in\mathcal{M}$. Put $s_{0}=0$, and choose an integer $s_{1}>s$ such that $s_{1}/(2b)\in\mathbb{Z}$, and consider the linear bounded operators

[TABLE]

Let, as above, $\psi$ be the interpolation parameter (5.1). Then the restriction of the mapping (6.14) with $j=0$ to the space

[TABLE]

is a bounded operator

[TABLE]

Here, we have used formulas (6.7) and (6.9), which remain true for the considered $s_{0}$ and $s_{1}$.

Now the equality (6.12) extends by continuity over all vectors $v\in\mathbb{H}^{s;\varphi}(\mathbb{R}^{n-1})$. Hence, the operator (6.15) is right inverse to (6.5). Thus, the required mapping (6.10) is built.

We need to introduce analogs of the operators (6.5) and (6.15) for the strip

[TABLE]

Let $s>2br-b$ and $\varphi\in\mathcal{M}$. Given $u\in H^{s,s/(2b);\varphi}(\Pi)$, we put $R_{1}u:=R_{0}w$, where a function $w\in H^{s,s/(2b);\varphi}(\mathbb{R}^{n})$ satisfies the condition $w\!\upharpoonright\!\Pi=u$. Evidently, this definition does not depend on the choice of $w$. The linear mapping $u\mapsto R_{1}u$ is a bounded operator

[TABLE]

This follows immediately from the boundedness of the operator (6.5) and from the definition of $H^{s,s/(2b);\varphi}(\Pi)$.

Let us build a right inverse of (6.16) on the base of the mapping (6.11). We put $T_{1}v:=(T_{0}v)\!\upharpoonright\!\Pi$ for arbitrary $v\in(L_{2}(\mathbb{R}^{n-1}))^{r}$. Owing to (6.10), the linear mapping $v\mapsto T_{1}v$ acts continuously between $(L_{2}(\mathbb{R}^{n-1}))^{r}$ and $L_{2}(\Pi)$. Moreover, its restriction to $\mathbb{H}^{s;\varphi}(\mathbb{R}^{n-1})$ is a bounded operator

[TABLE]

This follows directly from the boundedness of the operator (6.15). Besides,

[TABLE]

Thus, the operator (6.17) is a right inverse of (6.16).

Using operators (6.16) and (6.17), we can now prove our lemma with the help of the special local charts (3.7) on $S$. As above, let $s>2br-b$ and $\varphi\in\mathcal{M}$. Choosing $g\in C^{\infty}(\overline{S})$ arbitrarily, we get the following:

[TABLE]

Here, $c$ denotes the norm of the bounded operator (6.16), and, as usual, the symbol "$\circ$" stands for a composition of functions. Recall that $\{\theta_{j}\}$ is a collection of local charts on $\Gamma$ and that $\{\chi_{j}\}$ is an infinitely smooth partition of unity on $\Gamma$. Thus,

[TABLE]

This implies that the mapping (6.1) extends by continuity to the bounded linear operator (6.2).

Let us now build a continuous linear mapping $T:(L_{2}(\Gamma))^{r}\to L_{2}(S)$ whose restriction to $\mathbb{H}^{s;\varphi}(\Gamma)$ is a right inverse of (6.2). Consider the linear mapping of flattening of $\Gamma$

[TABLE]

This mapping acts continuously between $L_{2}(\Gamma)$ and $(L_{2}(\mathbb{R}^{n-1}))^{\lambda}$. Moreover, its restriction to $H^{\sigma;\varphi}(\Gamma)$ is an isometric operator

[TABLE]

Besides, consider the linear mapping of sewing of $\Gamma$

[TABLE]

Here, each function $\eta_{j}\in C_{0}^{\infty}(\mathbb{R}^{n-1})$ is chosen so that $\eta_{j}=1$ on the set $\theta^{-1}_{j}(\mathrm{supp}\,\chi_{j})$, whereas $O_{j}$ denotes the operator of the extension by zero to $\Gamma$ of a function given on $\Gamma_{j}$. The restriction of this mapping to $(H^{\sigma;\varphi}(\mathbb{R}^{n-1}))^{\lambda}$ is a bounded operator

[TABLE]

and this operator is left inverse to (6.18) (see [38, the proof of Theorem 2.2]).

The mapping $K$ induces the operator $K_{1}$ of the sewing of the manifold $S=\Gamma\times(0,\tau)$ by the formula

[TABLE]

for arbitrary functions $u_{1},\dots,u_{\lambda}\in L_{2}(\Pi)$ and almost all $x\in\Gamma$ and $t\in(0,\tau)$. The linear operator $K_{1}$ acts continuously between $(L_{2}(\Pi))^{\lambda}$ and $L_{2}(S)$. Moreover, its restriction to $(H^{\sigma,\sigma/(2b);\varphi}(\Pi))^{\lambda}$ is a bounded operator

[TABLE]

(see [29, the proof of Theorem 2]).

Given $v:=(v_{0},v_{1},\dots,v_{r-1})\in(L_{2}(\Gamma))^{r}$, we set

[TABLE]

where

[TABLE]

for each integer $k\in\{0,\ldots,r-1\}$. The linear mapping $v\mapsto Tv$ acts continuously between $(L_{2}(\Gamma))^{r}$ and $L_{2}(S)$, which follows directly from the corresponding properties of $L$, $T_{1}$, and $K_{1}$. The restriction of this mapping to $\mathbb{H}^{s;\varphi}(\Gamma)$ is the required bounded operator (6.3). Indeed, its boundedness follows immediately from the boundedness of the operators (6.17), (6.18), and (6.19). Besides, the operator (6.3) is a right inverse of (6.2) because

[TABLE]

for an arbitrary vector $v=(v_{0},v_{1},\dots,v_{r-1})\in\mathbb{H}^{s;\varphi}(\Gamma)$. Here, the index $k$ runs over the set $\{0,\dots,r-1\}$ and denotes the $k$-th component of a vector. ∎

Using this lemma, we will now prove a version of Proposition 5.5 for the range of isomorphism (4.15). It is sufficient to restrict ourselves to the $s\notin E$ case; recall that $E$ is denoted by (4.12). Let $\{J_{l}:1\leq l\in\mathbb{Z}\}$ stand for the collection of all connected components of the set $(\sigma_{0},\infty)\setminus E$. Each component $J_{l}$ is a certain finite subinterval of $(\sigma_{0},\infty)$.

Lemma 6.2**.**

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

[TABLE]

holds true up to equivalence of norms.

Proof.

It relies on Proposition 5.1 and the interpolation formula for the pair of Sobolev spaces $\mathcal{H}^{s_{0}-2m,(s_{0}-2m)/(2b)}$ and $\mathcal{H}^{s_{1}-2m,(s_{1}-2m)/(2b)}$. Bearing in mind the compatibility conditions (4.8), consider the set

[TABLE]

for each $j\in\{1,\dots,m\}$. This set does not depend on $s\in J_{l}$. Let $q_{l,j}^{\star}$ denote the number of all elements of (6.21), and put $q_{l,j}:=q_{l,j}^{\star}-1$ for convenience.

Let us build a linear mapping $P_{l}$ on

[TABLE]

that its restriction to the space $\mathcal{H}^{\sigma-2m,(\sigma-2m)/(2b)}$ is a projector of this space on its subspace $\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b)}$ for every $\sigma\in J_{l}$. Choosing a vector

[TABLE]

arbitrarily, we put

[TABLE]

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

[TABLE]

with the functions $v_{0},v_{1}\dots,v_{[m_{j}/(2b)]+q_{l,j}}$ and the differential operators $B_{j,0},\ldots,B_{j,q_{l,j}}$ being defined by (4.9) and (4.10) respectively and with $T_{l,j}$ denoting the linear mapping $T$ from Lemma 6.1 in the $r=q_{l,j}^{\star}$ case.

The linear mapping

[TABLE]

given on the space (6.22) is required. Indeed, its restriction to the space $\mathcal{H}^{\sigma-2m,(\sigma-2m)/(2b)}$ is a bounded operator on this space for every $\sigma\in J_{l}$, which follows from (4.9)–(4.11) and (6.3). Note that we use the boundedness of the operators (6.3) in the case where $r=q_{l,j}^{\star}$, $s=\sigma-m_{j}-1/2$, and $\varphi(\cdot)\equiv 1$, with the condition $s>2br-b$ being satisfied because

[TABLE]

Besides, it follows from the definition of $P_{l}$ and the compatibility conditions (4.8) that $P_{l}F\in\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b)}$ for every $F\in\mathcal{H}^{\sigma-2m,(\sigma-2m)/(2b)}$. Indeed, these conditions for the vector $P_{l}F=(f,g^{*}_{1},\dots,g^{*}_{m},h_{0},\dots,h_{\varkappa-1})$ become

[TABLE]

for all $j$ and $k$ indicated in (4.8). However,

[TABLE]

provided that $q_{l,j}\geq 0$. If $q_{l,j}=-1$, there will not be the compatibility conditions involving $g_{j}$. Thus, the vector $P_{l}F$ satisfies the compatibility conditions, i.e. $P_{l}F\in\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b)}$. Moreover, $F\in\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b)}$ implies that $P_{l}F=F$. Namely, if $F\in\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b)}$, then (4.8) holds, which entails that all $w_{j,k}=0$, i.e. $g_{1}^{*}=g_{1}$, …, $g_{m}^{*}=g_{m}$.

Now we use Proposition 5.1 in which $X_{j}:=\mathcal{H}^{s_{j}-2m,(s_{j}-2m)/(2b)}$ and $Y_{j}:=\mathcal{Q}^{s_{j}-2m,(s_{j}-2m)/(2b)}$ for each $j\in\{0,1\}$, and $P:=P_{l}$. According to this proposition, the pair

[TABLE]

is regular, and

[TABLE]

The right-hand side of this equality is a subspace of

[TABLE]

Owing to Propositions 5.2, 5.4, and 5.5, we obtain the following equalities:

[TABLE]

Thus,

[TABLE]

up to equivalence of norms. Formulas (6.24) and (6.25) give

[TABLE]

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

Lemma 6.2 just proved will be used in the proof of Theorem 4.1 in the $s\notin E$ case. Examining the opposite case, we need the following two results.

Lemma 6.3**.**

Let numbers $s,\varepsilon\in\mathbb{R}$ satisfy $s>\varepsilon>0$, and let $\varphi\in\mathcal{M}$. Then the equality of spaces

[TABLE]

holds true up to equivalence of norms provided that $W=\Omega$ or $W=S$.

Proof.

Choose a number $\delta>0$ such that $s-\varepsilon-\delta>0$. According to Proposition 5.5 for $\lambda=0$, we have the equalities

[TABLE]

and

[TABLE]

Here, the interpolation parameters $\alpha$ and $\beta$ are defined by the formulas

[TABLE]

and $\alpha(r)=\beta(r):=1$ if $0<r<1$. Owing to Propositions 5.3 and 5.5, we then get

[TABLE]

Here, the interpolation parameter $\omega$ is defined by the formulas

[TABLE]

and $\omega(r):=1$ if $0<r<1$. As to Proposition 5.5, note that the interpolation parameter $\omega$ equals the right-hand side of (5.1) if we put $s_{0}:=s-\varepsilon-\delta$ and $s_{1}:=s+\varepsilon+\delta$. Thus, (6.26) is valid. ∎

Lemma 6.4**.**

Let numbers $s\in\mathbb{R}$ and $\varepsilon>0$ satisfy $s-\varepsilon>\sigma_{0}$, and let $\varphi\in\mathcal{M}$. Then the equality of spaces

[TABLE]

holds true up to equivalence of norms.

Proof.

Relation (6.27) follows from Proposition 5.2 and formula (6.26) and its analog for the isotropic space $H^{\sigma;\varphi}(G)$, with $\sigma>0$ (see [38, Lemma 4.3]). Indeed,

[TABLE]

∎

Now we are in position to prove Theorem 4.1.

Proof of Theorem 4.1..

Let $s>\sigma_{0}$ and $\varphi\in\mathcal{M}$. We first consider the case where $s\notin E$. Then $s\in J_{l}$ for a certain 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 $s_{j}/(2b)+1/2\notin\mathbb{Z}$ whenever $j\in\{0,1\}$. According to Zhitarashu [56, Theorem 9.1], the mapping (4.1) extends uniquely (by continuity) to an isomorphism

[TABLE]

(see also the book [12, Theorem 5.7]). Let $\psi$ be the interpolation parameter (5.1). Then the restriction of the operator (6.28) with $j=0$ to the space

[TABLE]

is an isomorphism

[TABLE]

Here, the equalities of spaces hold true up to equivalence of norms due to Proposition 5.5 and Lemma 6.2. The operator (6.29) is an extension by continuity of the mapping (4.1) because $C^{\infty}(\overline{\Omega})$ is dense in $H^{s,s/(2b);\varphi}(\Omega)$. Thus, Theorem 4.1 is proved in the case considered.

Examine now the $s\in E$ case. Choose $\varepsilon\in(0,1/2)$ arbitrarily. Since $s\pm\varepsilon\notin E$ and $s-\varepsilon>\sigma_{0}$, we have the isomorphisms

[TABLE]

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

[TABLE]

Recall that the last equality is the definition of the space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$. To complete the proof, it remains to apply Lemma 6.3 for $W=\Omega$ to (6.31). ∎

Remark 6.5**.**

Let $s\in E$. The space $\mathcal{Q}^{s-2m,(s-2m)/(2b);\varphi}$ defined by formula (4.14) is independent of the choice of the number $\varepsilon\in(0,1/2)$ up to equivalence of norms. Indeed, according to Theorem 4.1 we have the isomorphism

[TABLE]

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

[TABLE]

in view of $s\mp\varepsilon\in(0,\sigma_{0})\setminus E$ and the definition of the left-hand space. It follows from this that the embedding operator acts continuously from (4.14) to (6.27), as stated.

Proof of Theorem 4.3..

We will first prove that, under its hypotheses (4.17)–(4.19), the implication

[TABLE]

holds for each integer $\lambda\geq 1$ subject to $s-\lambda+1>\sigma_{0}$.

We arbitrarily choose a function $\chi\in C^{\infty}(\overline{\Omega})$ with $\mbox{supp}\,\chi\subset\Omega_{0}\cup\Omega^{\prime}$. For $\chi$ there exists a function $\eta\in C^{\infty}(\overline{\Omega})$ such that $\mbox{supp}\,\eta\subset\Omega_{0}\cup\Omega^{\prime}$ and $\eta=1$ in a neighbourhood of $\mbox{supp}\,\chi$. Interchanging each of the differential operators $A$, $B_{j}$ and $\partial^{k}_{t}$ with the operator of the multiplication by $\chi$, we can write

[TABLE]

Here, $\Lambda^{\prime}:=(A^{\prime},B^{\prime}_{1},\ldots,B^{\prime}_{m},C^{\prime}_{0},\dots C^{\prime}_{\varkappa-1})$ is an operator whose components act on every function $w(x,t)$ from $H^{\sigma_{0}-1,(\sigma_{0}-1)/(2b)}(\Omega)$ as follows:

[TABLE]

and

[TABLE]

where all $a^{\alpha,\beta}_{1}\in C^{\infty}(\overline{\Omega})$, $b_{j,1}^{\alpha,\beta}\in C^{\infty}(\overline{S})$ and $c_{l,\,k}\in C^{\infty}(\overline{G})$. This operator acts continuously between the spaces

[TABLE]

for every $\sigma>\sigma_{0}-1$. In the $\varphi(\cdot)\equiv 1$ case, this follows directly from (6.34), (6.35), (6.36), and the known properties of partial differential operators and trace operators on anisotropic Sobolev spaces (see, e.g., [48, Chapter I, Lemma 4, and Chapter II, Theorems 3 and 7]). Remark only that each operator (6.36), with $1\leq k\leq\varkappa-1$, acts continuously between the spaces $H^{\sigma,\sigma/(2b)}(\Omega)$ and $H^{\sigma-2b(k-1)-b}(G)\hookrightarrow H^{\sigma+1-2bk-b}(G)$. The boundedness of the operator (6.37) in the general situation is plainly deduced from this case with the help of the Propositions 5.4 and 5.5 (see also (6.25) for $s:=\sigma+1$).

Owing to (4.17), (4.18), and (4.19), we obtain the inclusion

[TABLE]

Besides, according to (6.37) with $\sigma:=s-\lambda$, we have the implication

[TABLE]

Hence, using (6.33), we conclude that

[TABLE]

To deduce the required property (6.32) from (6.38) let us prove that

[TABLE]

for every $\sigma>\sigma_{0}$. Assume that the premise of this implication is true for some $\sigma>\sigma_{0}$. Since $\mathrm{dist}(\mathrm{supp}\,\chi,\Gamma)>0$, we have the equality $\Lambda(\chi u)=0$ near $\Gamma$. Hence, the vector $\Lambda(\chi u)$ satisfies the compatibility conditions (4.8) in which $(f,g_{1},\dots,g_{m},h_{0},\dots,h_{\varkappa-1})$ means $\Lambda(\chi u)$ and $\sigma$ is taken instead of $s$. This yields (6.39) in the $\sigma\notin E$ case due to the definition of the space $\mathcal{Q}^{\sigma-2m,(\sigma-2m)/(2b);\varphi}$.

In the opposite case of $\sigma\in E$, this space is defined by the interpolation. Considering this case, we choose a function $\chi_{1}\in C^{\infty}(\overline{\Omega})$ such that $\chi_{1}=0$ in a neighbourhood of $\Gamma$ and that $\chi_{1}=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. The mapping $M_{\chi_{1}}:F\mapsto\chi_{1}F$ acts continuously between the spaces

[TABLE]

whenever $0<\varepsilon<1/2$ because the vector $\chi_{1}F$ satisfies the compatibility conditions (4.8) in which $(f,g_{1},\dots,g_{m},h_{0},\dots,h_{\varkappa-1})$ means $\chi_{1}F$ and the numbers $\sigma\pm\varepsilon\notin E$ are taken instead of $s$. Applying the interpolation with the number parameter $1/2$ to (6.40), we obtain a bounded operator

[TABLE]

According to the interpolation formulas (6.27) and (4.14), this operator acts between the spaces

[TABLE]

Owing to our choice of $\chi_{1}$, we have $\chi_{1}\Lambda(\chi u)=\Lambda(\chi u)$. Since $\Lambda(\chi u)\in\mathcal{H}^{\sigma-2m,(\sigma-2m)/(2b);\varphi}$ by our assumption, we conclude that

[TABLE]

due to (6.42). Thus, the implication (6.39) is proved.

Now, using properties (6.38), (6.39) with $\sigma:=s-\lambda+1$, and Corollary 4.2, we conclude that

[TABLE]

for every $\chi\in C^{\infty}(\overline{\Omega})$ subject to $\mbox{supp}\,\chi\subset\Omega_{0}\cup\Omega^{\prime}$. Note that Corollary 4.2 is applicable here because $\chi u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ by the hypothesis of the theorem and because $s-\lambda+1>\sigma_{0}$. Thus, we have proved the required implication (6.32).

Let us use this implication to prove the theorem, i.e. to show that $u\in H^{s,s/(2b);\varphi}_{\mathrm{loc}}(\Omega_{0},\Omega^{\prime})$. We separately examine the case of $s\notin\mathbb{Z}$ and the case of $s\in\mathbb{Z}$.

Consider first the case of $s\notin\mathbb{Z}$. In this case, there exists an integer $\lambda_{0}\geq 1$ such that

[TABLE]

Using the implication (6.32) successively for $\lambda:=\lambda_{0}$, $\lambda:=\lambda_{0}-1$,…, $\lambda:=1$, we conclude that

[TABLE]

Note that $u\in H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ by the hypothesis of the theorem.

Consider now the case of $s\in\mathbb{Z}$. In this case, there is no integer $\lambda_{0}$ that satisfies (6.43). Nevertheless, since $s-\varepsilon\notin\mathbb{Z}$ and $s-\varepsilon>\sigma_{0}$ whenever $0<\varepsilon<1$, the inclusion $u\in H^{s-\varepsilon,(s-\varepsilon)/(2b);\varphi}_{\mathrm{loc}}(\Omega_{0},\Omega^{\prime})$ holds true as we have just proved. Hence, using (6.32) with $\lambda:=1$, we conclude that

[TABLE]

∎

Proof of Theorem 4.4..

Choosing a sufficiently small number $\varepsilon>0$, we put $U_{\varepsilon}:=\{x\in U:\mathrm{dist}(x,\partial U)>\varepsilon\}$, $\Omega_{\varepsilon}:=U_{\varepsilon}\cap\Omega$, and $\Omega^{\prime}_{\varepsilon}:=U_{\varepsilon}\cap\partial\overline{\Omega}$. Consider a function $\chi_{\varepsilon}\in C^{\infty}(\overline{\Omega})$ such that $\mbox{supp}\,\chi_{\varepsilon}\subset\Omega_{0}\cup\Omega^{\prime}$ and that $\chi_{\varepsilon}=1$ on $\Omega_{\varepsilon}\cup\Omega^{\prime}_{\varepsilon}$. Owing to Theorem 4.3, we have the inclusion $\chi_{\varepsilon}u\in H^{s,s/(2b);\varphi}(\Omega)$ were $s=p+b+n/2$ and $\varphi$ satisfies condition (4.20). Hence, there exists a distribution $w_{\varepsilon}\in H^{s,s/(2b);\varphi}(\mathbb{R}^{n+1})$ such that $w_{\varepsilon}=\chi_{\varepsilon}u=u$ on $\Omega_{\varepsilon}$. Let the indices $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and $\beta$ satisfy the condition $|\alpha|+2b\beta\leq p$. Then, according to [32, Lemma 8.1(i)], the generalized partial derivative $D_{x}^{\alpha}\partial_{t}^{\beta}w_{\varepsilon}(x,t)$ is continuous on $\mathbb{R}^{n+1}$. Hence, the distribution $v(x,t):=D_{x}^{\alpha}\partial_{t}^{\beta}u(x,t)$ in $\Omega$ is continuous on $\Omega_{\varepsilon}\cup\Omega^{\prime}_{\varepsilon}$; i.e.,

[TABLE]

for every test function $\omega\in C^{\infty}(\Omega)$ subject to $\mathrm{supp}\,\omega\subset\Omega_{\varepsilon}$, with $v_{\varepsilon}$ denoting the continuous function $v_{\varepsilon}(x,t):=D_{x}^{\alpha}\partial_{t}^{\beta}w_{\varepsilon}(x,t)$ of $(x,t)\in\Omega_{\varepsilon}\cup\Omega^{\prime}_{\varepsilon}$. We define the continuous function $v_{0}$ on $\Omega_{0}\cup\Omega^{\prime}_{0}$ by the formula $v_{0}:=v_{\varepsilon}$ on $\Omega_{\varepsilon}\cup\Omega^{\prime}_{\varepsilon}$ whenever $0<\varepsilon\ll 1$. This function is well defined because $0<\delta<\varepsilon$ implies that $v_{\delta}=v_{\varepsilon}$ on $\Omega_{\varepsilon}\cup\Omega^{\prime}_{\varepsilon}$. Then $v$ satisfies (4.21) for every test function $\omega\in C^{\infty}(\Omega)$ with $\mathrm{supp}\,\omega\subset\Omega_{0}$ because $\mathrm{supp}\,\omega\subset\Omega_{\varepsilon}$ for a sufficiently small number $\varepsilon>0$ depending on $\omega$. ∎

Ending this section, we substantiate Remark 4.5. Let $\varphi\in\mathcal{M}$, and let an integer $p\geq 0$ be subject to the condition $s:=p+b+n/2>\sigma_{0}$. Assume that every function $u\in\nobreak H^{\sigma_{0},\sigma_{0}/(2b)}(\Omega)$ satisfies (4.22) and show that $\varphi$ then satisfies (4.20). Let $V$ be a nonempty open subset of $\mathbb{R}^{n+1}$ such that $\overline{V}\subset\Omega_{0}$. We arbitrarily choose a function $w\in H^{s,s/(2b);\varphi}(\mathbb{R}^{n+1})$ such that $\mathrm{supp}\,w\subset V$. Put $u:=w\!\upharpoonright\!\Omega\in H^{s,s/(2b);\varphi}(\Omega)$ and

[TABLE]

The function $u$ satisfies the premise of the implication (4.22). Hence, $u$ satisfies the conclusion of Theorem 4.4 due to our assumption. Thus, the generalized derivative $D_{x}^{\alpha}\partial_{t}^{\beta}u(x,t)$ is continuous on $\Omega_{0}\cup\Omega^{\prime}$ whenever $|\alpha|+2b\beta\leq p$. Specifically, each derivative $D_{1}^{j}u$, with $0\leq j\leq p$, is continuous on $V$. Therefore, each derivative $D_{1}^{j}w$, with $0\leq j\leq p$, is continuous on $\mathbb{R}^{n+1}$. Hence, $\varphi$ satisfies (4.20) due to [32, Lemma 8.1(ii)]. Remark 4.5 is substantiated.

Appendix

Along with the explicit compatibility conditions (4.8), other (and less explicit) forms of them are often used in the theory of general parabolic initial-boundary value problems (see, e.g., [1, 27, 12]). Our proof of Theorem 4.1 is based on the isomorphism theorem obtained in works [56, 12]. They refer to the compatibility conditions introduced in [1, § 11]. These conditions are equivalent to (4.8) on some assumptions about $s$, which is considered known. But we have not found the proof of this fact in the literature. Therefore, we prefer to give the proof for the sake of completeness of the presentation.

Let us formulate the compatibility conditions given in [1, § 11]. They use some function spaces, which we introduce now. Let $V$ be an open nonempty subset of $\mathbb{R}^{k}$, with $2\leq k\in\mathbb{Z}$, and let $s>0$. The linear space $H^{s,s/(2b)}_{+}(V)$ is defined to consist of the restrictions $u=w\!\upharpoonright\!V$ of all functions $w\in H^{s,s/(2b)}(\mathbb{R}^{k})$ which vanish whenever $t<0$, we considering $w$ as a function $w(x,t)$ of $x=(x_{1},\ldots,x_{k-1})\in\mathbb{R}^{k-1}$ and $t\in\mathbb{R}$. The space is endowed with the norm

[TABLE]

We need this Hilbert space in the case were $V=\Omega$ and $k=n+1$ or in the case were $V=\Pi$ and $k=n$, with $\Pi:=\mathbb{R}^{n-1}\times(0,\tau)$. Changing $H^{s,s\gamma;\varphi}(\Pi)$ for $H^{s,s\gamma}_{+}(\Pi)$ in the definition of $H^{s,s\gamma;\varphi}(S)$, we define the Hilbert space $H^{s,s\gamma}_{+}(S)$. This space does not depend up to equivalence of norms on our choice of local charts and partition of unity on $\Gamma$ [32, Lemma 3.1].

Assume that $s\geq\sigma_{0}$, $s\notin E$, and $s/(2b)+1/2\notin\mathbb{Z}$. A vector

[TABLE]

satisfies the compatibility conditions in the sense of [1, § 11] if there exists a function $v=v(x,t)$ from $H^{s,s/(2b)}(\Omega)$ such that

[TABLE]

(Note that $s+1/2\notin\mathbb{Z}\Rightarrow s\notin E$. Hence, the restrictions put on $s_{j}$ at the beginning of our proof of Theorem 4.1 and caused by the use of [56, Theorem 9.1] are somewhat stronger than the assumptions just made about $s$.)

Let us prove that the collection of these compatibility conditions is equivalent to (4.8) for every vector (6.44). Owing to [32, Lemma 5.1], we rewrite conditions (6.45) and (6.46) in the following equivalent form:

[TABLE]

and

[TABLE]

The collection of conditions (6.47) and (6.48) is equivalent to (4.9) where $0\leq k<s/(2b)-1/2$ provided that we put

[TABLE]

Indeed, note first that (6.48) is equivalent to the collection of conditions

[TABLE]

If $\varkappa\leq k<s/(2b)-1/2$, then

[TABLE]

for almost all $x\in G$. Note that $v_{k}$ presents only in the summand corresponding to $\nobreak{\alpha=(0,\ldots,0)}$, $\beta=\varkappa$, and $q=k-\varkappa$ and that this summand equals $-v_{k}$. Hence,

[TABLE]

It is evident now that (6.48) is equivalent to the collection of relations (4.9) where $\varkappa\leq k<s/(2b)-1/2$. This gives the required equivalence of (6.47) and (6.48) to (4.9).

Suppose now that a vector (6.44) satisfies the compatibility conditions (4.8) in which $v_{k}$ and $B_{j,k}$ are defined by (4.9) and (4.10). Let us define a vector

[TABLE]

by formulas (4.9), with $r:=[s/(2b)-1/2]$. Owing to [48, Chapter 2, Theorem 10] there exists a function $v\in H^{s,s/(2b)}(\Omega)$ that satisfies (6.50). It follows from this and (2.2) and (4.10) that

[TABLE]

Hence, (4.8) implies (6.49). Besides, (4.9) implies (6.47) and (6.48), as we have mentioned. Thus, the vector (6.44) satisfies conditions (6.45)–(6.47).

Conversely, suppose that a vector (6.44) satisfies relations (6.45)–(6.47) for a certain function $v\in H^{s,s/(2b)}(\Omega)$. This implies (4.9) and (6.49) provided that we define the functions $v_{k}$ by (6.50). Then the compatibility conditions (4.8) follow from (4.9) and (6.49) in view of (6.51).