Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces

Abstract

In this chapter an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation, and we study the Fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is also considered one spectral problem for a discontinuous second-order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

1. Introduction

Differential (also elliptic) equations play an especial role in the study of various processes and phenomena in natural science. Solvability problems of elliptic equations have a very rich history and remarkable monographs by various famous mathematicians are devoted to them. The theory of elliptic equations was developed in an comprehensive way in Hölder classes (solution in the classical sense) and in Hilbertian Sobolev spaces $W_2^k\left(\mathrm{\Omega }\right)$. In the above mentioned case, depending on the nature of the problem, there are various methods of solution (for instance, the method of potentials, the periodic case, the method of the theory of functions, spectral method, etc.), which cannot be said for the non-Hilbert case $W_p^k\left(\mathrm{\Omega }\right),p\ne 2$, in which each method faces certain difficulties. All considered spaces are separable Banach spaces and infinitely differentiable and finite functions are dense in them. In the study of solvability of differential equations these facts are significant. Note that one of the methods to solve differential equations is a spectral method. To justify the solution by this method, one should study the basis properties of the root vectors of the considered spectral problem in the appropriate Banach function space.

In connection with applications in problems of mechanics, mathematical physics and pure mathematics, the so-called non-standard spaces of functions have greatly increased and the list of such spaces includes Lebesgue spaces with a variable summability index, Morrey spaces, grand Lebesgue spaces, Orlicz spaces, etc. For more details one can see the monographs [1, 2, 3, 4, 5, 6]. Compared with other areas of mathematics, the apparatus of harmonic analysis has been fairly well studied in relation to these spaces. The problems of analysis and approximation theory have been relatively well studied in Lebesgue spaces with variable summability index and Morrey spaces (see [7, 8, 9, 10, 11, 12, 13, 14]). The above mentioned problems have begun to be studied in Grand Lebesgue spaces, and valuable results have been obtained in this direction (see [15, 16]). The solvability problems of partial differential equations have also begun to be studied in the Sobolev spaces generated by these spaces (see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]). Morrey-Sobolev and grand-Sobolev spaces are not separable and therefore infinitely differentiable and finite functions are not dense in them, in this reason the study the problems of solvability of differential equations in these spaces are of special scientific interest. Therefore, it is necessary to extract reasonable subspaces dictated by differential equations and develop an instruments for studying the solvability of differential equations in these subspaces.

An $m$-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for $m$-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation and we study the fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is considered one spectral problem for a discontinuous second order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

2. Needful information

2.1 Standard notation

$Z_{+}$ will be the set of non-negative integers. $\overline{1,n}=1,2,\dots ,n$. $\mathbb{C}$ is the set of complex numbers. $B_r\left(x_0\right)=\left\{x\in R^n:\left|x-x_0\right|<r\right\}$ will denote the open ball in $R^n$ centered at $x_0$, where $\left|x\right|=\sqrt{x_1^2+\dots +x_n^2},x=\left(x_1,\dots ,x_n\right)$. $⟨·;·⟩$ is a scalar product in $R^n$. $\mathit{mes}\left(M\right)$ will stand for the Lebesgue measure of the set $M$; $\mathrm{\partial \Omega }$ will be the boundary of the domain $\mathrm{\Omega }$; $\overline{\mathrm{\Omega }}=\mathrm{\Omega }\bigcup ‍\mathrm{\partial \Omega }$; $\mathit{\text{diam}}\mathrm{\Omega }$ will stand for the diameter of the set $\mathrm{\Omega }$; $f{/}_M$ denotes the restriction of $f$ to $M$. $C_0^{\infty }\left(\mathrm{\Omega }\right)-$ will denote the space of infinitely differentiable and finite functions in $\mathrm{\Omega }$ and $C^{\left(m\right)}\left(\mathrm{\Omega }\right)-$will stand for the space of $m$-th order continuously differentiable functions in the domain $\mathrm{\Omega }$. $C_0^m\left(\mathrm{\Omega }\right)-$will stand for the space of $m$-th order continuously differentiable and finite functions in the domain $\mathrm{\Omega }$. $D_L-$will stand for the domain of the operator $L$; $R_T-$will stand for the range of the operator $T$; $\mathit{\text{KerT}}-$ is the kernel of the operator $T$; $T^{\ast }$ is the adjoint of $T$; $\left[X;Y\right]$ is a Banach space of bounded operators acting from $X$ to $Y$; $\left[X\right]=\left[X;X\right]$. Throughout this paper, $q\prime$ will denote the conjugate of a number, i.e. $\frac{1}{q}+\frac{1}{q\prime }=1$.

2.2 Elliptic operator of $m$-th order and some necessary facts

Let $\mathrm{\Omega }\subset R^n$ be some bounded domain with the rectifiable boundary $\mathrm{\partial \Omega }$. We will use the notations of [19]. $\alpha =\left({\alpha }_1,\dots ,{\alpha }_n\right)$ will be the multiindex with the coordinates ${\alpha }_k\in Z_{+},\forall k=\overline{1,n}$; ${\partial }_i=\frac{\partial }{x_i}$ will denote the differentiation operator, ${\partial }^{\alpha }={\partial }_1^{{\alpha }_1}{\partial }_2^{{\alpha }_2}\dots {\partial }_n^{{\alpha }_n}$. For every $\xi =\left({\xi }_1,\dots ,{\xi }_n\right)$ we assume ${\xi }^{\alpha }={\xi }_1^{{\alpha }_1}{\xi }_2^{{\alpha }_2}\dots {\xi }_n^{{\alpha }_n}$. Let $L$ be an elliptic differential operator of $m$-th order

$$L=\sum _{\left|p\right|\le m}‍a_p\left(x\right){\partial }^p,$$E1

where $p=\left(p_1,\dots ,p_n\right)$, $p_k\in Z_{+},\forall k=\overline{1,n},a_p\left(·\right)\in L_{\infty }\left(\mathrm{\Omega }\right)$ are real functions. Consider the elliptic operator $L_0$:

$$L_0=\sum _{\left|p\right|=m}‍a_p^0{\partial }^p,$$E2

with the constant coefficients $a_p^0$ and denote by $J\left(·\right)$ a fundamental solution of Eq. (2) [28].

Let $L$ be an elliptic operator and consider a “tangential operator”

$$L_{x_0}=\sum _{\left|p\right|=m}‍a_p\left(x_0\right){\partial }^p,$$E3

at every point $x_0\in \mathrm{\Omega }$. Denote by $J_{x_0}\left(·\right)$ the fundamental solution of the equation $L_{x_0}\phi =0$. The function $J_{x_0}\left(·\right)$ is called a parametrics for the equation $\mathit{L\phi }=0$ with a singularity at the point $x_0$. Let

$$S_{x_0}\phi =\psi \left(x\right)=\int ‍J_{x_0}\left(x-y\right)\phi \left(y\right)\mathit{dy},$$E4

and

$$T_{x_0}=S_{x_0}\left(L_{x_0}-L\right).$$E5

Denote the operators $S_{x_0},L_{x_0}$ and $T_{x_0}$, corresponding to the point $x_0=0$, by $S_0,L_0$ and $T_0$, respectively.

Let us give the definition of smooth boundary.

Definition 1.1 We will say that the boundary $\partial \Omega$ of a domain $\Omega \subset R^n$ belongs to class $C^{\left(k\right)}$ if each sufficiently small piece of it can be mapped onto a segment of the hyperplane $x_n=0$ using a coordinate transformation $y\left(x\right)=\left(y_1\left(x\right);\dots ;y_n\left(x\right)\right)$ with a positive Jacobian so that $y_i\in C^{\left(k\right)},\forall i=\overline{1,n}$.

2.3 Grand-Sobolev spaces $W_{q)}^m\left(\mathrm{\Omega }\right)$and $W{N}_{q)}^m\left(\mathrm{\Omega }\right)$

Define the grand-Lebesgue space $L_{q)}\left(\mathrm{\Omega }\right)$, $1<q<+\infty$ (throughout this paper we will assume that this condition holds on $q$). Grand-Lebesgue space $L_{q)}\left(\mathrm{\Omega }\right)$is a Banach space of (Lebesgue) measurable functions $f$ on $\mathrm{\Omega }$ with the norm

$${‖f‖}_{q)}=\underset{0<\epsilon <q-1}{sup}{\left(\epsilon {\int }_{\mathrm{\Omega }}‍{\left|f\right|}^{q-\epsilon }\mathit{dx}\right)}^{\frac{1}{q-\epsilon }}.$$E6

The following continuous embeddings hold

$$L_q\left(\mathrm{\Omega }\right)\subset L_{q)}\left(\mathrm{\Omega }\right)\subset L_{q-\epsilon }\left(\mathrm{\Omega }\right),$$E7

where $\epsilon \in \left(0,q-1\right)$ is an arbitrary number. The space $L_{q)}\left(\mathrm{\Omega }\right)$ is not separable.

Below in this section we will assume that every function defined on $\mathrm{\Omega }$ is extended by zero to $R^n\backslash \overline{\mathrm{\Omega }}$. Let $T_{\delta }$ be a shift operator, i.e. $\left(T_{\delta }f\right)\left(x\right)=f\left(\delta +x\right),\forall x\in \mathrm{\Omega }$ and $\forall \delta \in R^n$. Let

$$N_{q)}\left(\mathrm{\Omega }\right)=\left\{f\in L_{q)}\left(\mathrm{\Omega }\right):{‖T_{\delta }f-f‖}_{q)}\to 0,\delta \to 0\right\}.$$E8

The space $N_{q)}\left(\mathrm{\Omega }\right)$ is a Banach space with the norm ${‖·‖}_{q)}$, (i.e. is the subspace of $L_{q)}\left(\mathrm{\Omega }\right)$.)

The following lemma is true (see [17]).

Lemma 1.2 $\overline{C_0^{\infty }\left(\Omega \right)}=$ $N_{q)}\left(\Omega \right)$ (the closure is taken in $L_{q)}\left(\Omega \right)$).

Let us include the following lemma without proof.

Lemma 1.3 The embeddings

$$L_q\left(\mathrm{\Omega }\right)\subset N_{q)}\left(\mathrm{\Omega }\right)\subset L_{q)}\left(\mathrm{\Omega }\right)\subset L_1\left(\mathrm{\Omega }\right),$$E9

hold and every inclusion is strict.

Denote by $W_{q)}^m\left(\mathrm{\Omega }\right)$ the grand-Sobolev space generated by the norm

$${‖f‖}_{W_{q)}^m}=\sum _{k=0}^m‍{‖f^{\left(k\right)}‖}_{q)}.$$E10

Let

$$W{N}_{q)}^m\left(\mathrm{\Omega }\right)=\left\{f\in W_{q)}^m\left(\mathrm{\Omega }\right):{‖T_{\delta }f-f‖}_{W_{q)}^m}\to 0,\delta \to 0\right\}.$$E11

Consider the following singular kernel

$$k\left(x\right)=\frac{\omega \left(x\right)}{{\left|x\right|}^n},$$E12

where $\omega \left(x\right)$ is a positive homogeneous function of degree zero, which is infinitely differentiable and satisfies

$${\int }_{\left|x\right|=1}‍\omega \left(x\right)\mathit{d\sigma }=0,$$E13

$\mathit{d\sigma }$ being a surface element on the unit sphere. By $K$ we will denote the corresponding singular integral

$$\left(\mathit{Kf}\right)\left(x\right)=k\ast f\left(x\right)={\int }_{\mathrm{\Omega }}‍f\left(y\right)k\left(x-y\right)\mathit{dy}.$$E14

The following theorem is valid for the operator $K$ (see [4]).

Theorem 1.4 [4] The inclusion $K\in \left[L_{q)}\left(\Omega \right)\right],1<q<+\infty$ is valid, i.e. $\exists c>0$:

$${‖\mathit{Kf}‖}_{q)}\le c{‖f‖}_{q)},\forall f\in L_{q)}\left(\mathrm{\Omega }\right)$$E15

The validity of the following lemma is given in [17].

Lemma 1.5 [17] $N_{q)}\left(\Omega \right),1<q<+\infty$, is an invariant subspace of the singular operator $K$ in $L_{q)}\left(\Omega \right)$.

Considering the expression for the norm $N_{q)}$, it is not hard to prove the following.

Proposition 1.6 Let $\Omega \subset R^n-$be a bounded domain and $L$ be an elliptic operator with coefficients $a_p\in L_{\infty }\left(\Omega \right),\forall \left|p\right|\le m$. Then it is valid $L\in \left[W{N}_{q)}^m\left(\Omega \right);N_{q)}\left(\Omega \right)\right]$, i.e. the following inequality

$${‖\mathit{Lu}‖}_{N_{q)}\left(\mathrm{\Omega }\right)}\le C{‖u‖}_{W{N}_{q)}^m\left(\mathrm{\Omega }\right)},\forall u\in W{N}_{q)}^m\left(\mathrm{\Omega }\right),$$E16

holds, where $C>0$ is a constant independent of $u$.

In the sequel, when $\mathrm{\Omega }=B_r$ the spaces $L_{q)}\left(\mathrm{\Omega }\right),N_{q)}\left(\mathrm{\Omega }\right),W_{q)}\left(\mathrm{\Omega }\right)$ and $W{N}_{q)}\left(\mathrm{\Omega }\right)$ will be redenoted by $L_{q)}\left(r\right),N_{q)}\left(r\right),W_{q)}^m\left(r\right)$ and $W{N}_{q)}^m\left(r\right)$, respectively. Along with $W{N}_{q)}^m\left(\mathrm{\Omega }\right)$, consider the following space of functions $N_{q)}^m\left(\mathrm{\Omega }\right)$ equipped with the norm

$${‖f‖}_{N_{q)}^m\left(\mathrm{\Omega }\right)}=\sum _{\left|p\right|\le m}‍d_{\mathrm{\Omega }}^{\left|p\right|-\frac{n}{q}}{‖{\partial }^{p}f‖}_{L_{q)\left(\mathrm{\Omega }\right)}},$$E17

where $d_{\mathrm{\Omega }}=\mathit{\text{diam}}\mathrm{\Omega }$, and we will assume $N_{q)}^0\left(\mathrm{\Omega }\right)=N_{q)}\left(\mathrm{\Omega }\right)$. The closure of $C_0^{\infty }\left(\mathrm{\Omega }\right)$ in $N_{q)}^m\left(\mathrm{\Omega }\right)$ ($N_q^m\left(\mathrm{\Omega }\right)$) we will denote by ${\stackrel{\circ }{N}}_{q)}^m\left(\mathrm{\Omega }\right)$ (${\stackrel{\circ }{N}}_q^m\left(\mathrm{\Omega }\right)$).

3. Main lemma

3.1 Solvability in the small

Introduce the following

Definition 1.7 We will say that the operator $L$ has the property $P_{x_0})$ if its coefficients satisfy the conditions: i) $a_p\in L_{\infty }\left(B_r\left(x_0\right)\right)$, $\forall \left|p\right|\le m$, for some $r>0$; ii) $\exists r>0:$ for $\left|p\right|=m$ the coefficient $a_p\left(·\right)$ coincides a.e. in $B_r\left(x_0\right)$ with some function bounded and continuous at the point $x_0$.

It is absolutely clear that if $a_p\in C\left(\mathrm{\Omega }\right)$, $\forall \left|p\right|\le m$, then $L$ has the property $P_{x_0})$ for $\forall x_0\in \mathrm{\Omega }$.

In establishing the interior Schauder-type estimate for grand-Sobolev spaces $N_{q)}^m\left(\mathrm{\Omega }\right)$ the following Main Lemma, proved in [18] plays a key role.

Main Lemma. Let the $m$-th order elliptic operator $L$ have the property $P_{x_0})$ at the point $x_0$. Let $\phi \in N_{q)}^m\left(B_r\left(x_0\right)\right)$ and $\phi$ vanish in a neighborhood of $\left|x-x_0\right|=r$. Then for $q>1$ it holds

$${‖T_{x_0}\phi ‖}_{N_{q)}^m\left(B_r\left(x_0\right)\right)}\le \sigma \left(r\right){‖\phi ‖}_{N_{q)}^m\left(B_r\left(x_0\right)\right)},$$E18

where the function $\sigma \left(r\right)\to 0,r\to 0$, depends only on the ellipticity constant $L_{x_0}$, on the coefficients of $L$ and their moduli of continuity.

Let us consider the $m$-th order elliptic operator $L$ with the coefficients $a_p\left(x\right)$ defined by (1), and the corresponding operator $T_{x_0}$ defined by (5). Using Main Lemma, it is proved the following local existence theorem.

Theorem 1.8 Let $L$ be an $m$-th order elliptic operator which has the property $P_{x_0})$ at some point $x_0\in \Omega$ and $f\in G_{q)}\left(\Omega \right),1<q<+\infty$. Then, for sufficiently small $r$, there exists a solution of the equation $\mathit{Lu}=f$ belonging to the class $N_{q)}\left(B_r\left(x_0\right)\right)$.

4. Interior Schauder type estimates

Let $\omega \left(·\right)$ be an infinitely differentiable function on $\left[0,1\right]$ such that for $0\le t<\frac{1}{3}$, $\omega \left(t\right)\equiv 1$ and for $\frac{2}{3}<t\le 1$, $\omega \left(t\right)\equiv 0$. For $0<R_1<R_2$ we put

$$\xi \left(x\right)=\left\{\begin{array}{l}1,\left|x\right|\le R_1,\\ \omega \left(\frac{\left|x\right|-R_1}{R_2-R_1}\right),R_1<\left|x\right|\le R_2.\end{array}\right.$$E19

Regarding this function it holds the following.

Lemma 1.9 There is a constant $C>0$ depending only on $R_2$ and $\omega \left(·\right)$, such that for $\forall R_1:0<R_1<R_2$, there is

$${‖\xi ‖}_{C^m\left(R_2\right)}\le C{\left(1-\frac{R_1}{R_2}\right)}^{-m}.$$E20

Accept the following property with respect to the domain $\mathrm{\Omega }$.

Property $\mathit{\alpha })$.We say that the domain $\Omega$ admits the continuation of functions of the space $N_{q)}^k\left(\Omega \right)$ if there exists a domain $\Omega \prime \supset \overline{\Omega }$ and a linear mapping $\theta$ of the space $N_{q)}^k\left(\Omega \right)$ into $N_{q)}^k\left(\Omega \prime \right)$ such that

$$\mathit{\theta u}=u\mathit{\text{in}}\mathrm{\Omega },$$

$${‖\mathit{\theta u}‖}_{N_{q)}^k\left(\mathrm{\Omega }\prime \right)}\le \mathit{\text{const}}{‖u‖}_{N_{q)}^k\left(\mathrm{\Omega }\right)},$$E21

holds.

So, the following lemma is true.

Lemma 1.10 Let the domain $\Omega$ have the Property $\alpha )$ with respect to space $N_{q)}^k\left(\Omega \right)$. Then $\exists C>0$ depending only on $n,q$ and on a constant from (21), which holds

$${‖\phi ‖}_{N_{q)}^k\left(\mathrm{\Omega }\right)}\le \epsilon {‖\phi ‖}_{N_{q)}^{k+1}\left(\mathrm{\Omega }\right)}+C{\epsilon }^{-k}{‖\phi ‖}_{L_{q)}\left(\mathrm{\Omega }\right)},$$E22

for $\forall k=\overline{1,m-1}$ and $\forall \epsilon >0$.

The main result of this section is the following Schauder type estimate.

Theorem 1.11 Let the coefficients of $m$-th order elliptic operator $L$ satisfy the following conditions: i) $a_p\left(·\right)\in C\left(\overline{\Omega }\right)$, $\forall p:\left|p\right|=m$; ii) $a_p\left(·\right)\in L_{\infty }\left(\Omega \right)$, $\forall p:\left|p\right|<m$; where $\Omega \subset R^n-$ bounded domain with boundary $\partial \Omega$. Let ${\Omega }_0\subset \Omega$ be an arbitrary compact. Then for $\forall u\in W{G}_{q)}^m\left(\Omega \right)$, $1<q<+\infty$, the following a priori estimate holds

$${‖u‖}_{N_{q)}^m\left({\mathrm{\Omega }}_0\right)}\le C\left({‖\mathit{Lu}‖}_{L_{q)}\left(\mathrm{\Omega }\right)}+{‖u‖}_{L_{q)}\left(\mathrm{\Omega }\right)}\right),$$E23

where the constant $C$ depends only on the ellipticity constant $m,\mathrm{\Omega },{\mathrm{\Omega }}_0$ of $L$, on the coefficients of the operator $L$.

5. Extension of functions from $N_{\left.q\right)}^m\left(\mathrm{\Omega }\right)$Compactness

Consider the question of the possibility of extension of a function $f$ from class $N_{\left.q\right)}^m\left(\mathrm{\Omega }\right)$ to a wider class $N_{\left.q\right)}^m\left(\mathrm{\Omega }\prime \right)$ with $\mathrm{\Omega }\prime \supset \overline{\mathrm{\Omega }}$. Following the classical case (see, monograph [29]), first consider the case when $\mathrm{\Omega }\prime$ is a cube with an edge $2a>0:K_a=\left\{\left|y_i\right|<a,i=\overline{1,n}\right\}$, and $\mathrm{\Omega }$ is a parallelepiped $K_a^{+}=K_a\bigcap ‍\left\{y_n>0\right\}$ .

The following lemma is true.

Lemma 1.12 For $\forall f\in W{N}_{q)}^k\left(K_a^{+}\right)$ there exists an extension $F\in W{N}_{q)}^k\left(K_a\right)$ and, in addition, inequality

$${‖F‖}_{N_{\left.q\right)}^m\left(K_a\right)}\le C{‖f‖}_{N_{\left.q\right)}^m\left(K_a^{+}\right)},$$E24

holds.

It is completely analogous to the monograph [29, p. 129], it is proved the following

Lemma 1.13 Let $f\in W{N}_{q)}^k\left(\Omega \right)$ and for $\forall \xi \in \partial \Omega$ there exists a function $F_{\xi }\left(x\right)$, defined in a ball $B_r\left(\xi \right)$ of some radius $r=r\left(\xi \right)>0$, such that $F_{\xi }\left(x\right)=f\left(x\right),\forall x\in \Omega \cap B_r\left(\xi \right)$ and $F_{\xi }\in W{N}_{q)}^k\left(B_r\left(\xi \right)\right)$. Besides

$${‖F_{\xi }‖}_{W{N}_{q)}^k\left(B_r\left(\xi \right)\right)}\le C{‖f‖}_{W{N}_{q)}^k\left(\mathrm{\Omega }\right)},$$E25

is true, where $C>0$ is a constant independent of $f$. Then, for any $\rho >0$, the function $f$ has an extension $F$ to the domain ${\mathrm{\Omega }}_{\rho }=\bigcup _{x\in \mathrm{\Omega }}‍B_{\rho }\left(x\right)$ with the properties $F\in W{N}_{q)}^k\left({\mathrm{\Omega }}_{\rho }\right),F\left(x\right)=0,\forall x\in {\mathrm{\Omega }}_{\rho }\backslash {\mathrm{\Omega }}_{\rho /2}$: and the inequality

$${‖F‖}_{W{N}_{q)}^k\left({\mathrm{\Omega }}_{\rho }\right)}\le C{‖f‖}_{W{N}_{q)}^k\left(\mathrm{\Omega }\right)},$$E26

holds, where the constant $C>0$, is dependent only on domain $\mathrm{\Omega }$ and $\rho$.

Using Main Lemma and Lemmas 1.12, 1.13, similarly to [29, p. 130] it is proved the following extension.

Theorem 1.14 Let $\Omega$, $\Omega \prime$ be bounded domains in $R^n$, $\overline{\Omega }\subset \Omega \prime$ and $\partial \Omega \in C^{\left(m\right)}$. Then for $\forall f\in N_{\left.q\right)}^m\left(\Omega \right)$ there exists a finite extension $F\in N_{\left.q\right)}^m\left(\Omega \prime \right)$ in $\Omega \prime$ and the following estimate

$${‖F‖}_{N_{q)}^m\left(\mathrm{\Omega }\prime \right)}\le C{‖f‖}_{N_{q)}^m\left(\mathrm{\Omega }\right)},$$E27

is valid, where $C>0$ is a constant independent of $f$.

Consider the compactness question of the family in $N_{\left.q\right)}\left(\mathrm{\Omega }\right)$. The following theorem is true.

Theorem 1.15 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(1\right)}$. Then a set, bounded in $N_{\left.q\right)}^1\left(\Omega \right)$, is compact in $N_{\left.q\right)}\left(\Omega \right)$.

Analogously to Theorem 1.15, the following theorem is also proved.

Theorem 1.16 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(k\right)}$. If the set of functions is bounded in $N_{\left.q\right)}^k\left(\Omega \right)$, then the set of their traces on $\left(n-1\right)$-dimensional surface $\Gamma \subset \overline{\Omega }$ from the class $C^{\left(k\right)}$ is compact in $W_{\left.q\right)}^r\left(\Omega \right)$, $\forall r=\overline{0,k-1}$.

6. Trace of functions from the grand-Sobolev space ${\mathit{N}}_{\mathit{q})}^{\mathit{m}}\left(\mathbf{\Omega }\right)$

In this section, we will define a concept of the trace for functions from the grand-Sobolev space $N_{q)}^m\left(\mathrm{\Omega }\right)$ on an $\left(n-1\right)$-dimensional differentiable surface.

Based on the embedding $N_{q)}^m\left(\mathrm{\Omega }\right)\subset N_{q)}^1\left(\mathrm{\Omega }\right),\forall m\ge 2$, it is sufficient to define this concept regarding the functions from $N_{q)}^1\left(\mathrm{\Omega }\right)$. So, assume $S\subset \overline{\mathrm{\Omega }}$: $S\in C^{\left(1\right)}$ is some $\left(n-1\right)$-dimensional surface. Let $x_0\in S$ be an arbitrary point. Then it is obvious that there exists a sufficiently small neighborhood of this point $S_{x_0}\subset S$, such that uniquely projected onto some domain $D$ of the plane $x_n=0$ in $R^n$ and it has the equation

$$x_n=\phi \left(x\prime \right)\in C^{\left(1\right)}\left(\overline{D}\right),x\prime =\left(x_1,\dots ,x_{n-1}\right)\in D.$$E28

$\mathrm{\Omega }$ is bounded domain and we will consider that it is placed inside a cube $\left\{0<x_i<a,i=\overline{1,n}\right\}$, with an edge $a>0$. Let $f\in {\stackrel{\circ }{C}}^{\infty }\left(\mathrm{\Omega }\right)$ be some finite function in $\mathrm{\Omega }$. For $\forall \left(x\prime ;\phi \left(x\prime \right)\right)\in S_{x_0}$ we have

$$f\left(x\right){/}_{S_{x_0}}=f\left(x\prime ;\phi \left(x\prime \right)\right)={\int }_0^{\phi \left(x\prime \right)}‍\frac{\partial f\left(x\prime ;{\xi }_n\right)}{\partial {\xi }_n}d{\xi }_n.$$E29

Let $\epsilon \in \left(0,q-1\right)-$be an arbitrary number, $q_{\epsilon }=q-\epsilon$ and $\frac{1}{q_{\epsilon }}+\frac{1}{q{\prime }_{\epsilon }}=1$. Applying Hölder’s inequality from (29), we obtain

$${\left|f{/}_{S_{x_0}}\right|}^{q_{\epsilon }}\le {\left|\phi \left(x\prime \right)\right|}^{\frac{q_{\epsilon }}{q{\prime }_{\epsilon }}}{\int }_0^{\phi \left(x\prime \right)}‍{\left|\frac{\partial f\left(x\prime ;{\xi }_n\right)}{\partial {\xi }_n}\right|}^{q_{\epsilon }}d{\xi }_n\le a^{\frac{q_{\epsilon }}{q{\prime }_{\epsilon }}}{\int }_0^{\phi \left(x\prime \right)}‍{\left|\frac{\partial f\left(x\prime ;{\xi }_n\right)}{\partial {\xi }_n}\right|}^{q_{\epsilon }}d{\xi }_n.$$E30

Let $C=max\left\{a;1\right\}$. Consequently

$$\epsilon {\left|f{/}_{S_{x_0}}\right|}^{q_{\epsilon }}\le C^{q_{\epsilon }}\epsilon {\int }_0^{\phi \left(x\prime \right)}‍{\left|\frac{\partial f\left(x\prime ;{\xi }_n\right)}{\partial {\xi }_n}\right|}^{q_{\epsilon }}d{\xi }_n,$$E31

where $C$ is a constant independent of $f$ and $\epsilon$. Multiplying by $\sqrt{1+{\phi }_{x_1}^2+\dots +{\phi }_{x_{n-1}}^2}$ and integrating over $D$ we obtain

$${‖{\epsilon }^{\frac{1}{q_{\epsilon }}}f‖}_{L_{q_{\epsilon }}\left(S_{x_0}\right)}^{q_{\epsilon }}\le C{‖{\epsilon }^{\frac{1}{q_{\epsilon }}}\frac{\partial f}{\partial x_n}‖}_{L_{q_{\epsilon }}\left(\mathrm{\Omega }\right)}^{q_{\epsilon }}.$$E32

Since the surface $S$ can be covered by a finite number of surfaces of type $S_{x_0}$, then summing the corresponding inequalities (from (32)) we establish

$${‖{\epsilon }^{\frac{1}{q_{\epsilon }}}f‖}_{L_{q_{\epsilon }}\left(S\right)}^{q_{\epsilon }}\le C{‖{\epsilon }^{\frac{1}{q_{\epsilon }}}\frac{\partial f}{\partial x_n}‖}_{L_{q_{\epsilon }}\left(\mathrm{\Omega }\right)}^{q_{\epsilon }},$$E33

where $C>0$ is a constant independent of $f$ and $\epsilon$. This immediately implies

$${‖{\epsilon }^{\frac{1}{q_{\epsilon }}}f‖}_{L_{q_{\epsilon }}\left(S\right)}\le C\sum _{\left|p\right|=1}‍d_{\mathrm{\Omega }}{‖{\epsilon }^{\frac{1}{q_{\epsilon }}}{\partial }^{p}f‖}_{L_{q_{\epsilon }}\left(\mathrm{\Omega }\right)}\le C{‖{\epsilon }^{\frac{1}{q_{\epsilon }}}{\partial }^{p}f‖}_{N_{q_{\epsilon }}^1\left(\mathrm{\Omega }\right)}.$$E34

Taking first $\underset{0<\epsilon <q-1}{sup}$ on the right and then the same $sup$ on the left, from this estimate for $\forall f\in {\stackrel{0}{C}}^{\infty }\left(\mathrm{\Omega }\right)$, we have

$${‖f‖}_{L_{\left.q\right)}\left(S\right)}\le C{‖f‖}_{N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)}.$$E35

If $\mathrm{\partial \Omega }\in C^{\left(1\right)}$, then Theorem 1.14 implies that the inequality (35) holds for $\forall f\in C^{\left(1\right)}\left(\overline{\mathrm{\Omega }}\right)$.

Let $f\in N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)$ be an arbitrary function. Then $\exists \left\{f_n\right\}\subset C^{\infty }\left(\overline{\mathrm{\Omega }}\right)$:

$${‖f_n-f‖}_{N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)}\to 0,n\to \infty .$$E36

It follows directly from (35) that the sequence $\left\{f_n\right\}$ is fundamental in $L_{\left.q\right)}\left(S\right)$:

$${‖f_n-f_m‖}_{L_{\left.q\right)}\left(S\right)}\to 0,n,m\to \infty .$$E37

From the completeness of $L_{\left.q\right)}\left(S\right)$ it follows that $\exists f_S\in L_{\left.q\right)}\left(S\right)$:

$${‖f_n-f_S‖}_{L_{\left.q\right)}\left(S\right)}\to 0,n\to \infty .$$E38

Similarly the classical case, it is proved that $f_S$ does not depend on the choice of the sequence $\left\{f_n\right\}$.

$f_S$ is called the trace of the function $f\in N_{\left.q\right)}^1\left(\Omega \right)$ on $S$ and we will denote it by the operator ${\Gamma }_S:$ $\mathit{\Gamma f}=f{/}_S$.

Based on the concept of ${\mathrm{\Gamma }}_S$, we define the following linear space

$$N_{\left.q\right)}^1\left(S\right)=N_{\left.q\right)}^1\left(\mathrm{\Omega }\right){/}_S=\left\{f\in L_{\left.q\right)}\left(S\right):\exists u\in N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)\Rightarrow f={\mathrm{\Gamma }}_{S}u=u{/}_S\right\}.$$E39

For the case of $S=\mathrm{\partial \Omega }$, the operator ${\mathrm{\Gamma }}_S$ will be simply denoted by $\mathrm{\Gamma }:{\mathrm{\Gamma }}_{\mathrm{\partial \Omega }}=\mathrm{\Gamma }.$

The following lemma is true.

Lemma 1.17 Let $\Omega \subset R^n$ be an bounded domain and $\partial \Omega \in C^{\left(1\right)}$. Then the linear spaces ${\mathcal{F}}_{q)}^1$ and $N_{\left.q\right)}^1\left(\partial \Omega \right)$ are isomorphic, where

$$N_{\left.q\right)}^1\left(\mathrm{\partial \Omega }\right)\equiv N_{\left.q\right)}^1\left(\mathrm{\Omega }\right){/}_{\mathrm{\partial \Omega }}=\left\{f\in L_{\left.q\right)}\left(\mathrm{\partial \Omega }\right):\exists u\in N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)\Rightarrow f=\mathrm{\Gamma }u=u{/}_{\mathrm{\partial \Omega }}\right\}.$$E40

Based on this lemma, we define the norm in $N_{\left.q\right)}^1\left(\mathrm{\partial \Omega }\right)$

$${‖f‖}_{N_{\left.q\right)}^1\left(\mathrm{\partial \Omega }\right)}={‖{\mathrm{\Gamma }}^{-1}f‖}_{{\mathcal{F}}_{\left.q\right)}^1},\forall f\in N_{\left.q\right)}^1\left(\mathrm{\partial \Omega }\right).$$E41

Since ${\mathcal{F}}_{\left.q\right)}^1$ is a Banach space with respect to the norm ${‖·‖}_{{\mathcal{F}}_{\left.q\right)}^1}$, then this lemma immediately implies that $N_{\left.q\right)}^1\left(\mathrm{\partial \Omega }\right)$ is also Banach with respect to the norm (41).

The space $N_{\left.q\right)}^m\left(\mathrm{\partial \Omega }\right)$ is defined similarly and the corresponding lemma is true for the spaces ${\mathcal{F}}_{q)}^m$, where

$$N_{\left.q\right)}^m\left(\mathrm{\partial \Omega }\right)\equiv N_{\left.q\right)}^m\left(\mathrm{\Omega }\right){/}_{\mathrm{\partial \Omega }}=\left\{f\in L_{q)}\left(\mathrm{\partial \Omega }\right):\exists u\in N_{\left.q\right)}^m\left(\mathrm{\Omega }\right)\Rightarrow f=\mathrm{\Gamma }u=u{/}_{\mathrm{\partial \Omega }}\right\}.$$E42

The following theorem is true (regarding the proof see [30]).

Theorem 1.18 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(m\right)}$. If the set of functions is bounded in $N_{q)}^m\left(\Omega \right),m\ge 1$, then the set of their traces on the $\left(n-1\right)$-dimensional surface $S\subset \overline{\Omega }$ from the class $C^{\left(m\right)}$ is compact in $L_{q)}\left(S\right)$.

7. Schauder-type estimate up to the boundary

Using the results obtained in the previous sections, it is established a Schauder-type estimate up to the boundary for a second-order elliptic operator with nonsmooth coefficients. The following theorem is true.

Theorem 1.19 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$ and $L$ be a second-order elliptic operator (i.e. $m=2$) with coefficients $a_p\in C\left(\overline{\Omega }\right),\forall p:\left|p\right|=m$ and $a_p\in L_{\infty }\left(\Omega \right),\forall p:\left|p\right|<m$. Then for $\forall u\in N_{\left.q\right)}^2\left(\Omega \right)$ the following estimate

$${‖u‖}_{N_{\left.q\right)}^2\left(\mathrm{\Omega }\right)}\le C\left({‖\mathit{Lu}‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)}+{‖u‖}_{N_{\left.q\right)}^2\left(\mathrm{\partial \Omega }\right)}+{‖u‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)}\right),$$E43

holds true, where $C>0$ is a constant independent of $u$, but depends on the norms of the coefficients of $L$ in $L_{\infty }\left(\mathrm{\Omega }\right)$.

8. Solvability of the Dirichlet problem for a second-order elliptic operator

Let us apply the estimates established in the previous sections to the solvability question (in the strong sense) of the Dirichlet problem for a second-order elliptic type equation in classes $N_{\left.q\right)}^2\left(\mathrm{\Omega }\right)$. So, let $\mathrm{\Omega }\subset R^n$ be a domain with a boundary $\mathrm{\Omega }\in C^{\left(2\right)}$ . Assume that $f\in N_{\left.q\right)}\left(\mathrm{\Omega }\right)$ is a given function and $a_{\mathit{ij}}$ $\in C\left(\overline{\mathrm{\Omega }}\right)$; $a_i;a\in L_{\infty }\left(\mathrm{\Omega }\right)$ $i$;$j=\overline{1,n}$. Consider the equation

$$\mathit{Lu}=\sum _{i;j=1}^n‍a_{\mathit{ij}}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+\sum _{i=1}^n‍a_i\left(x\right)\frac{\partial u}{\partial x_i}+a\left(x\right)u=f\left(x\right),x\in \mathrm{\Omega }.$$E44

In the sequel we will assume that the following uniformly ellipticity condition holds a.e. in $\mathrm{\Omega }$

$$\nu {\left|\xi \right|}^2\le \sum _{i,j=1}^n‍a_{\mathit{ij}}\left(x\right){\xi }_i{\xi }_j\le {\nu }^{-1}{\left|\xi \right|}^2,\forall \xi \in R^n,$$E45

where $\nu \in \left(0,1\right]$ is some constant.

Under the solution of the Eq. (44) we mean a function $u\in N_{\left.q\right)}^2\left(\mathrm{\Omega }\right)$ for which equality (44) holds a.e. $x\in \mathrm{\Omega }$. Let $\phi \in N_{\left.q\right)}^2\left(\mathrm{\partial \Omega }\right)$ be a given function. Let us define the boundary condition

$$\mathrm{\Gamma }u=u{/}_{\mathrm{\partial \Omega }}=\phi$$E46

where $\mathrm{\Gamma }:N_{\left.q\right)}^2\left(\mathrm{\Omega }\right)\to N_{\left.q\right)}^2\left(\mathrm{\partial \Omega }\right)-$ is a trace operator.

We will say that the domain $\mathrm{\Omega }$ has a property $\left({\delta }_q\right)$ is class $\stackrel{\circ }{N_{q)}^2}\left(\mathrm{\Omega }\right)$, if the Dirichlet problem (46) is correctly solvable for the Poisson equation, i.e. the problem

$$\left\{\begin{array}{l}\mathrm{\Delta }u=f,\mathit{\text{in}}\mathrm{\Omega },\\ \mathrm{\Gamma }u=0,\mathit{on}\mathrm{\partial \Omega },\end{array}\right.$$E47

has a unique solution for $\forall f\in N_{q)}\left(\mathrm{\Omega }\right)$ in class $\stackrel{\circ }{N_{q)}^2}\left(\mathrm{\Omega }\right)$.

In order to solve the problem (44), (46) we apply the parameter continuation method (see e.g. [28, p. 247]).

Furthermore, assume that the operator $L$ satisfies the following inequality

$${‖u‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)}\le C{‖\mathit{Lu}‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)},\forall u\in \stackrel{\circ }{N_{q)}^2}\left(\mathrm{\Omega }\right),$$E48

where the constant $C$ depends only on the ellipticity constants of the operator $L$, on the sup norms of the coefficients $L$, on domain $\mathrm{\Omega }$ and is independent of the function $u\in \stackrel{\circ }{N_{q)}^2}\left(\mathrm{\Omega }\right)$.

We will say that the operator $\mathrm{L}$ has property (A) if an inequality (48) holds for an operator $\mathrm{L}$.

The question of whether inequality (48) holds (i.e. property (A)) we will consider later.

Thus, the following main theorem is true.

Theorem 1.20 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$ and $L$ is a second-order elliptic differential operator defined by expression (44) with coefficients $a_{\mathit{ij}}\in C\left(\overline{\Omega }\right)$; $a_i$; $a\in L_{\infty }\left(\Omega \right)$, $i;j=\overline{1,n}$. Assume that the domain $\Omega$ has property $\left({\Delta }_{q)}\right)$ in class $\stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$ and the operator $L$ has property (A). Then the equation $\mathit{Lu}=f$ is uniquely solvable for $\forall f\in N_{\left.q\right)}\left(\Omega \right)$ in class $\stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$, i.e. $\stackrel{\circ }{L:N_{q)}^2}\left(\Omega \right)\leftrightarrow N_{\left.q\right)}\left(\Omega \right)$ is an isomorphism and it is obvious that the estimate

$${‖u‖}_{N_{q)}^2\left(\mathrm{\Omega }\right)}\le C{‖f‖}_{N_{q)}\left(\mathrm{\Omega }\right)},\forall f\in N_{q)}\left(\mathrm{\Omega }\right),$$E49

holds true, where $C>0$ is a constant independent of $f$.

Now consider a homogeneous equation $\mathit{Lu}=0$ in $\mathrm{\Omega }$ with a nonhomogeneous boundary condition $\mathrm{\Gamma }u=u{/}_{\mathrm{\partial \Omega }}=\phi$, where $\phi \in N_{q)}^2\left(\mathrm{\Omega }\right)-$is given function. From the results of Section 6 it follows that $\exists \mathrm{\Phi }\in N_{q)}^2\left(\mathrm{\Omega }\right):\mathrm{\Gamma \Phi }=\phi .$ Suppose $\upsilon =u-\mathrm{\Phi }$ and let $f=-L\mathrm{\Phi }$. It is clear that $\mathrm{\Gamma }\upsilon =\upsilon {/}_{\mathrm{\partial \Omega }}=0$ and $\mathit{L\upsilon }=f$ in $\mathrm{\Omega }$. If $a_{\mathit{ij}};a_i$; $a\in L_{\infty }\left(\mathrm{\Omega }\right)$, $i;j=\overline{1,n}$, then, as follows from Proposition 1.6 that $f\in N_{q)}\left(\mathrm{\Omega }\right)$. Therefore, we can apply the Theorem 1.20 to the problem

$$\left\{\begin{array}{l}\mathit{L\upsilon }=f,a.e.\mathit{\text{in}}\mathrm{\Omega },\\ \upsilon {/}_{\mathrm{\partial \Omega }}=0.\end{array}\right.$$E50

If all the conditions of Theorem 1.20 are satisfied, then this problem is uniquely solvable in class $N_{q)}^2\left(\mathrm{\Omega }\right)$ and for the solution it is valid the following estimate

$${‖\upsilon ‖}_{N_{q)}^2\left(\mathrm{\Omega }\right)}\le C{‖f‖}_{N_{q)}\left(\mathrm{\Omega }\right)},$$E51

where $C>0$ is a constant independent of $f$. It is quite obvious that then the problem

$$\left\{\begin{array}{l}\mathit{Lu}=0,a.e.\mathit{\text{in}}\mathrm{\Omega },\\ u{/}_{\mathrm{\Gamma }}=\phi ,\end{array}\right.$$E52

is also uniquely solvable in $N_{q)}^2\left(\mathrm{\Omega }\right)$.

Taking into account expression (41) for the norm in $N_{q)}^2\left(\mathrm{\partial \Omega }\right)$, we obtain

$${‖u‖}_{N_{q)}^2\left(\mathrm{\Omega }\right)}\le C{‖\phi ‖}_{N_{q)}^2\left(\mathrm{\partial \Omega }\right)}.$$E53

Consider a nonhomogeneous equation with a nonhomogeneous boundary condition

$$\left\{\begin{array}{l}\mathit{Lu}=fa.e.\mathit{\text{in}}\mathrm{\Omega },\\ u{/}_{\mathrm{\partial \Omega }}=\phi ,\end{array}\right.$$E54

where $f\in N_{q)}\left(\mathrm{\Omega }\right)$ and $\phi \in N_{q)}^2\left(\mathrm{\partial \Omega }\right)-$are given functions. Representing the function $u$ in the form $u=v+w$, where

$$\left\{\begin{array}{l}\mathit{Lv}=f,\\ v{/}_{\mathrm{\partial \Omega }}=0,\end{array}\right.\left\{\begin{array}{l}\mathit{Lw}=0,\\ w{/}_{\mathrm{\partial \Omega }}=\phi ,\end{array}\right.$$E55

from Theorem 1.20 and taking into account estimate (53), we arrive at the following conclusion.

Theorem 1.21 Let the domain $\Omega$ and the operator $L$ satisfy all the conditions of Theorem 1.20. Then for $\forall f\in N_{q)}\left(\Omega \right)$ and $\forall \phi \in N_{q)}^2\left(\partial \Omega \right)$ the problem (54) is uniquely solvable in the space $N_{q)}^2\left(\partial \Omega \right)$and regarding the solution the estimate

$${‖u‖}_{N_{q)}^2\left(\mathrm{\Omega }\right)}\le C\left({‖f‖}_{N_{q)}\left(\mathrm{\Omega }\right)}+{‖\phi ‖}_{L_{q)}\left(\mathrm{\partial \Omega }\right)}\right),$$E56

is fulfilled, where $C>0$ is a constant independent of $f$ and $\phi$.

9. Some properties of a Riesz potential

For obtaining main results we need some properties of a Riesz potential and embedding theorems regarding the spaces $N_{\mathrm{q})}^m$. In this section, we will give some properties of an integral operator with a weak singularity. These properties are used to study the properties of functions from class $W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$. Let us remember the Sobolev integral identity

$$u\left(x\right)=\sum _{\left|\alpha \right|=0}^{k-1}‍x^{\alpha }{\int }_{\mathrm{\Omega }}‍b_{\alpha }\left(y\right)u\left(y\right)\mathit{dy}+\sum _{\left|\alpha \right|=k}‍{\int }_{\mathrm{\Omega }}‍\frac{A_{\alpha }\left(x;y\right)}{r^{n-k}}{\partial }^{\alpha }u\left(y\right)\mathit{dy},\forall u\in C^k\left(\overline{\mathrm{\Omega }}\right),$$E57

where $b_{\alpha }\in C\left(\overline{\mathrm{\Omega }}\right)$, $A_{\propto }\in L_{\infty }\left(\mathrm{\Omega }\times \mathrm{\Omega }\right)$ (generally speaking, for $x\ne y$: $A_{\propto }\left(x;y\right)$ is infinitely differentiable). In establishing many properties of a function from Sobolev classes the representation (57) plays a key role. In accordance with (57) consider the integral operator (Riesz potential).

$$\left(\mathit{K\rho }\right)\left(x\right)=V\left(x\right)={\int }_{\mathrm{\Omega }}‍\frac{A\left(x,y\right)}{r^{\lambda }}\rho \left(y\right)\mathit{dy},$$E58

where $r=\left|y-x\right|$; $x\in \mathrm{\Omega }\subset R^n$ is a bounded domain, $0\le \lambda <n$; $A\in L_{\infty }\left(\mathrm{\Omega }\times \mathrm{\Omega }\right)$. The following theorem is true.

Theorem 1.22 Let $\mathrm{\Omega }\subset R^n$ be a bounded domain, $\rho \in L_{\left.q\right)}\left(\mathrm{\Omega }\right)$, $A\in C\left(\overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}\right)$ and $\lambda q^{\prime }<n$. Then operator (58) acts compactly from $L_{\mathrm{q})}\left(\mathrm{\Omega }\right)$ to $C\left(\overline{\mathrm{\Omega }}\right)$.

It is true the following classical analogue

Theorem 1.23 Let $\lambda q^{\prime }\ge n$ and an integer $s$ satisfy $n-\left(n-\lambda \right)<s\le n$. Then the integral (58) defines a function that, on any intersection ${\Omega }_s$ of the set $\Omega$ by a plane of dimension $s$, is defined almost everywhere in the sense of the Lebesgue measure in $R^s$. The operator $K$ defined by formula (58) is bounded as an operator from $L_{\left.q\right)}\left(\Omega \right)$ to $L_r\left({\Omega }_s\right)$ (also from $L_{\left.q\right)}\left(\Omega \right)$ to $L_{\left.r\right)}\left({\Omega }_s\right)$), for $\forall r$: $1<r<r_0=\frac{\mathit{sq}}{n-\left(n-\lambda \right)q}$.

It is valid the following

Theorem 1.24 If $\lambda q^{\prime }\ge n$, then the operator $K$, defined by expression (58), acts compactly from $L_{\left.q\right)}\left(\Omega \right)$ to $L_r\left(\Omega \right)$ (also from $L_{q)}\left(\Omega \right)$ to $L_{\left.r\right)}\left(\Omega \right)$), for $\forall r:1<r<r_0=\frac{\mathit{nq}}{n-\left(n-\lambda \right)q}$.

10. Embedding theorems

To obtain Sobolev-type embedding theorems in spaces $W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$ we will use the results obtained in the previous section. Throughout this section, we assume that the domain $\mathrm{\Omega }\subset R^n-$ is bounded and stellar relative to some sphere. Remember that a domain is called stellar relative to some point if any ray outgoing from this point has one and only one common point with the boundary of this domain. A domain is stellar with respect to some set if it is stellar at every point of this set. The following theorem is true.

Theorem 1.25 If $\mathit{qk}>n$, then $W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$ compactly embedded in $C\left(\overline{\mathrm{\Omega }}\right)$.

The following theorem is true.

Theorem 1.26 Let $\mathit{qk}\le n$ and ${\Omega }_s\subset \Omega -$be a piecewise smooth manifold of $s$ dimensions, where $n-\mathit{kq}<s\le n$. Then $W_{\left.q\right)}^k\left(\Omega \right)$ continuously embedded in $L_r\left({\Omega }_s\right)$ (also in $L_{r)}\left({\Omega }_s\right)$), where $1<r<r_0=\frac{\mathit{sq}}{n-\mathit{kq}}$.

The following theorems are proved in a completely similar way.

Theorem 1.27 If $\mathit{qk}\le n$, then $W_{\left.q\right)}^k\left(\Omega \right)$ compactly embedded in $L_r\left(\Omega \right)$ (also in $L_{r)}\left(\Omega \right)$), where $1\le r<\frac{\mathit{nq}}{n-\mathit{kq}}$ .

The following theorem is also true.

Theorem 1.28 Let $u\in W_{\left.q\right)}^k\left(\Omega \right)$. Then it has all possible generalized derivatives of any order $l<k$ in $\Omega$. At the same time $W_{\left.q\right)}^k\left(\Omega \right)$ compactly embedded in $C^l\left(\overline{\Omega }\right)$, if $\left(k-l\right)q>n$ and in $W_r^l\left(\Omega \right)$(also in $W_{r)}^l\left(\Omega \right)$), if $\left(k-l\right)q\le n$ and $1\le r<\frac{\mathit{nq}}{n-\mathit{kq}}$.

Let us give some equivalent norms in the grand-Sobolev spaces $W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$ (then in $N_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$). Let a function $f\left(·\right)$ continuous in $R^r$ have the following properties

$$\alpha )f\left(t\right)\ge 0\wedge f\left(t\right)=0\iff t=0;$$

$$\beta )f\left(\mathit{\lambda t}\right)=\left|\lambda \right|f\left(t\right),\forall \lambda \in R\wedge \forall t\in R^r;$$E59

$$\gamma )f\left(t+\tau \right)\le f\left(t\right)+f\left(\tau \right),\forall t;\tau \in R^r.$$

In a completely analogous way to the classical case, the following theorem is proved.

Theorem 1.29 Let $r$ denote the number of distinct monomials of degree $\le k-1$ and let $l_1,\dots ,l_r$ be linear functionals bounded on $W_{\left.q\right)}^k\left(\Omega \right)$ that do not simultaneously vanish on any polynomial of degree $\le k-1$, except for the identically zero. Let $f\left(·\right)$ be a continuous function in $R^r$, having the properties of a norm $\alpha )-\gamma )$. Then the norm

$${‖u‖}_{\left.q\right),k}^{\ast }=f\left(l_{1}u;l_{2}u;\dots ;l_{r}u\right)+\sum _{\left|\alpha \right|=k}‍{‖{\partial }^{\alpha }u‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)},$$E60

is equivalent to the norm ${‖·‖}_{W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)}$.

The following lemma is true.

Lemma 1.30 Let $\Omega \subset R^n$ be a bounded domain with sufficiently smooth boundary $\partial \Omega$. Then the norm ${‖·‖}_{\left.q\right),k}^{\ast }$, defined by expression

$${‖u‖}_{\left.q\right),k}^{\ast }=\sum _{\left|\alpha \right|=0}^{k-1}‍\left|{\int }_{\mathrm{\partial \Omega }}‍{\partial }^{\alpha }\mathit{ud\sigma }\right|+\sum _{\left|\alpha \right|=k}‍{‖{\partial }^{\alpha }u‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)},$$E61

is equivalent to ${‖·‖}_{W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)}$ in $W_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$. In this case for $u\in \stackrel{\circ }{W_{q)}^k}\left(\mathrm{\Omega }\right)$ the following inequality

$${‖u‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)}\le C\sum _{\left|\alpha \right|=k}‍{‖{\partial }^{\alpha }u‖}_{L_{\left.q\right)}\left(\mathrm{\Omega }\right)},$$E62

holds, $C$ is a constant independent of $u$.

11. About the property (a). Fredholmness

Let us get back to the question of whether property (A) is satisfied. Let $\mathrm{\Omega }\subset R^n$ be a bounded domain. Let $S\subset \overline{\mathrm{\Omega }}$ be some $\left(n-1\right)$-dimensional surface. Define the following class of functions. Let $S\subset \overline{\mathrm{\Omega }}$ belong to class $C^{\left(k\right)}$ and $\epsilon >0$ be some number. Put ${\mathrm{\Omega }}_{\epsilon }\left(S\right)=\left\{x\in \mathrm{\Omega }:\rho \left(x;S\right)>\epsilon \right\}$.

We say that the function $f$ belongs to class $C_0^k\left(S\right)$ (i.e. $f$ vanishes in some neighborhood $S$), if $f\in C^k\left(\overline{\Omega }\right)$ and $\exists \epsilon >0:f\left(x\right)=0$, $\forall x\in \Omega \backslash {\Omega }_{\epsilon }\left(S\right)$. Denote by $\stackrel{\circ }{N_{q)}^k}\left(\Omega ;S\right)$ the closure $C_0^k\left(S\right)$ in $N_{\left.q\right)}^k\left(\Omega \right)$.

Thus, it is clear that $\stackrel{\circ }{N_{q)}^k}\left(\mathrm{\Omega };\mathrm{\partial \Omega }\right)=\stackrel{\circ }{N_{q)}^k}\left(\mathrm{\Omega }\right)$. Denote by ${\mathcal{F}}_{\left.q\right)}^m\left(\mathrm{\Omega };S\right)$ the factor space $N_{\left.q\right)}^m\left(\mathrm{\Omega }\right)/\stackrel{\circ }{N_{q)}^m}\left(\mathrm{\Omega };S\right)$. Thus, ${\mathcal{F}}_{\left.q\right)}^m\left(\mathrm{\Omega };\mathrm{\partial \Omega }\right)={\mathcal{F}}_{\left.q\right)}^m$. Each function $f\in N_{\left.q\right)}^1\left(\mathrm{\Omega }\right)$ (also $f\in N_{\left.q\right)}^k\left(\mathrm{\Omega }\right)$) has a (unique) trace $f{/}_S$ on $S$, and $f{/}_S\in L_{\left.q\right)}\left(S\right)$. Consider the following class of functions

$$N_{\left.q\right)}^k\left(S\right)=N_{\left.q\right)}^k\left(\mathrm{\Omega }\right){/}_S=\left\{f\in L_{\left.q\right)}\left(S\right):\exists u\in N_{\left.q\right)}^k\left(\mathrm{\Omega }\right)\Rightarrow f=u{/}_S\right\}.$$E63

The following theorem is true.

Lemma 1.31 Let $S\subset \overline{\Omega }\wedge S\in C^{\left(k\right)}$ be $\left(n-1\right)$-dimensional surface. Then the linear spaces ${\mathcal{F}}_{\left.q\right)}^k\left(\Omega ;S\right)$ and $N_{\left.q\right)}^k\left(S\right)$, $k\ge 1$, are isomorphic.

It is not hard to see that if $f\in \stackrel{\circ }{N_{q)}^k}\left(\mathrm{\Omega };S\right)$, then $f{/}_S=0$, $\forall k\ge 1$. Applying this lemma, completely similar to Theorem 1.19, we can prove the following.

Theorem 1.32 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$. Let $L$ be a second-order elliptic operator with coefficients $a_{\mathit{ij}}\in C\left(\overline{\Omega }\right)$, $a_i,a$ $\in L_{\infty }\left(\Omega \right)$, $\forall i$, $j=\overline{1,n}$ defined by expression (44). Let $S\subset \overline{\Omega }\wedge S\in C^{\left(2\right)}$ be some $\left(n-1\right)$-dimensional surface and $\Omega \prime \subset \subset \Omega \bigcup ‍S$ (i.e. $\overline{\Omega \prime }\subset \Omega \bigcup ‍S$) be an arbitrary domain. Then the following estimate holds true for $\forall u\in \stackrel{\circ }{N_{q)}^2}\left(\Omega ;S\right)$:

$${‖u‖}_{N_{\left.q\right)}^2\left(\mathrm{\Omega }\prime \right)}\le C\left({‖\mathit{Lu}‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)}+{‖u‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)}\right),$$E64

where the constant $C$ depends only on the ellipticity constants of operator $L$, on the norms of the coefficients of $L$ in $L_{\infty }\left(\mathrm{\Omega }\right)$, on $S$ and $\mathrm{\Omega }\prime$ (is independent of $u$).

It is not hard to see that Theorem 1.19 is a particular case of this theorem, for this it is sufficient to take $S=\mathrm{\partial \Omega }$. By Theorem 1.32 completely analogous to Theorem 9.14 of the monograph [31, p. 240] the following theorem is proved.

Theorem 1.33 Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$ and $L$ be an elliptic operator (44) with coefficients $a_{\mathit{ij}}\in C\left(\overline{\Omega }\right)$, $a_i,a$ $\in L_{\infty }\left(\Omega \right)$, $i,j=\overline{1,n}$ . Then the following estimate holds for $\forall u\in \stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$:

$${‖u‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)}\le C{‖\mathit{Lu}-\mathit{\sigma u}‖}_{N_{\left.q\right)}\left(\mathrm{\Omega }\right)},$$E65

for $\forall \sigma \ge {\sigma }_0$, where $C$; ${\sigma }_0>0$ are some constants that independent of $u$.

The following theorem is the result of Theorems 1.20 and 1.33.

Theorem 1.34 Let $L$ be an elliptic operator (44) with coefficients $a_{\mathit{ij}}\in C\left(\overline{\Omega }\right)$, $a_i$, $a\in L_{\infty }\left(\Omega \right)$, $i,j=\overline{1,n}$. Let $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$, which has a property $\left({\Delta }_{q)}\right)$. Then $\exists {\sigma }_0>0$: the equation $\mathit{Lu}-\mathit{\sigma u}=f$ is uniquely solvable for $\forall f\in N_{\left.q\right)}\left(\Omega \right)$ in class $\stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$, $\forall \sigma \ge {\sigma }_0$.

The Fredholm alternatives hold for the equation $\mathrm{Lu}=\mathrm{f}$, i.e. the following main theorem is true.

Theorem 1.35 Let $L$ be an elliptic operator (44) with coefficients $a_{\mathit{ij}}\in C\left(\overline{\Omega }\right)$, $a_i$, $a\in L_{\infty }\left(\Omega \right)$, $i,j=\overline{1,n}$ and $\Omega \subset R^n$ be a bounded domain with a boundary $\partial \Omega \in C^{\left(2\right)}$, which has a property $\left({\Delta }_{q)}\right)$. Then: i) if $\mathit{\text{KerL}}=0$ in $\stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$, then the boundary value problem $\mathit{Lu}=f$, $u{/}_{\Gamma }=\phi$, has a unique solution for $\forall \phi \in {\mathcal{F}}_{\left.q\right)}^2\left(\Omega \right)$ and $\forall f\in N_{\left.q\right)}\left(\Omega \right)$; ii) $\mathit{\text{KerL}}$ is a finite-dimensional subspace in $\stackrel{\circ }{N_{q)}^2}\left(\Omega \right)$.

Regarding the proof of all these facts one can see the works [4, 6].

12. On one spectral problem in Morrey-Smirnov space

In this section we consider one spectral problem in Morrey-Smirnov space. Such spectral problems arise in the problem of vibrations of a loaded string with fixed ends is solved by applying the Fourier method (see [32, 33, 34]). Morrey space is also non separable space and we define its subspace in which the infinitely differentiable functions are dense. We prove that eigenfunctions of the considered spectral problem form a basis in this subspace after eliminating an arbitrary term from them.

We need some facts from the theory of Morrey-type spaces. Let $\mathrm{\Gamma }$ be some rectifiable Jordan curve on the complex plane $C.$ By ${\left|M\right|}_{\mathrm{\Gamma }}$ we denote the linear Lebesgue measure of the set $M\subset \mathrm{\Gamma }$. By the Morrey- Lebesgue space $L^{p,\alpha }\left(\mathrm{\Gamma }\right),0\le \alpha \le 1,p\ge 1$, we mean a normed space of all functions $f\left(·\right)$ measurable on $\mathrm{\Gamma }$ equipped with a finite norm ${‖f‖}_{L^{p,\alpha }\left(\mathrm{\Gamma }\right)}$:

$${‖f‖}_{L^{p,\alpha }\left(\mathrm{\Gamma }\right)}=\underset{B}{sup}{\left({\left|B\cap \mathrm{\Gamma }\right|}_{\mathrm{\Gamma }}^{\alpha -1}{\int }_{B\cap \mathrm{\Gamma }}‍{\left|f\left(\xi \right)\right|}^p\left|\mathit{d\xi }\right|\right)}^{\frac{1}{p}}<+\infty .$$E66

$L^{p,\alpha }\left(\mathrm{\Gamma }\right)$ is a Banach space and $L^{p,1}\left(\mathrm{\Gamma }\right)=L_p\left(\mathrm{\Gamma }\right),L^{p,0}\left(\mathrm{\Gamma }\right)=L_{\infty }\left(\mathrm{\Gamma }\right)$. The embedding $L^{p,{\alpha }_1}\left(\mathrm{\Gamma }\right)\subset L^{p,{\alpha }_2}\left(\mathrm{\Gamma }\right)$ is valid for $0\le {\alpha }_1\le {\alpha }_2\le 1$ . Thus $L^{p,\alpha }\left(\mathrm{\Gamma }\right)\subset L_p\left(\mathrm{\Gamma }\right),\forall \alpha \in \left[0,1\right],\forall p\ge 1$. The case of $\mathrm{\Gamma }\equiv \left[a,b\right]$ will be denoted by $L^{p,\alpha }\left(a,b\right)$.

Denote by ${\stackrel{\sim }{L}}^{p,\alpha }\left(a,b\right)$ the linear subspace of $L^{p,\alpha }\left(a,b\right)$ consisting of functions whose shifts are continuous in $L^{p,\alpha }\left(a,b\right)$, i.e. ${‖f\left(·+\delta \right)-f\left(·\right)‖}_{L^{p,\alpha }\left(a,b\right)}\to 0$ as $\delta \to 0.$ The closure of ${\stackrel{\sim }{L}}^{p,\alpha }\left(a,b\right)$ in $L^{p,\alpha }\left(a,b\right)$ will be denoted by $M^{p,\alpha }\left(a,b\right)$. In [35] the following theorem is proved.

Theorem 1.36 The exponential system ${\left\{e^{i\mathit{nt}}\right\}}_{n\in Z}$ is the basis in $M^{p,\alpha }\left(-\pi ,\pi \right),1<p<+\infty ,0<\alpha \le 1.$

Using this theorem, it is easy to obtain the following

Theorem 1.37 Each of the trigonometric systems ${\left\{\mathit{\text{sinnx}}\right\}}_{n=1}^{\infty }$ and ${\left\{\mathit{\text{cosnx}}\right\}}_{n=0}^{\infty }$ forms the basis for $M^{p,\alpha }\left(0,\pi \right),1<p<+\infty ,0<\alpha \le 1$.

Consider a sample eigenvalue problem for the discontinuous second-order differential operator

$$-y\prime \prime \left(x\right)=\mathit{\lambda y}\left(x\right),x\in \left(0,\frac{1}{3}\right)\cup \left(\frac{1}{3},1\right),$$E67

with the boundary conditions

$$\left.\begin{array}{l}y\left(0\right)=y\left(1\right)=0,\\ y\left(\frac{1}{3}-0\right)=y\left(\frac{1}{3}+0\right),\\ y\prime \left(\frac{1}{3}-0\right)-y\prime \left(\frac{1}{3}+0\right)=\mathit{\lambda my}\left(\frac{1}{3}\right),\end{array}\right\}$$E68

where $\lambda$ is the spectral parameter, $m$ is a non-zero complex number.

Let us give some results from [36], which we will need throughout the paper.

Lemma 1.38 [36] The spectral problem (67), (68) has two series of eigenfunctions which are given by the following expressions

$$y_{1,n}\left(x\right)=sin3\mathit{\pi nx},x\in \left[0,1\right],n=1,2,\dots ,$$E69

$$y_{2,n}\left(x\right)=\left\{\begin{array}{l}sin{\rho }_{2,n}\left(x-\frac{1}{3}\right)+sin{\rho }_{2,n}\left(x+\frac{1}{3}\right),x\in \left[0,\frac{1}{3}\right],\\ sin{\rho }_{2,n}\left(1-x\right),x\in \left[\frac{1}{3},1\right],n=0,1,2,\dots .\end{array}\right.$$E70

Let us construct the operator $L,$ which linearizes the problem (67), (68) in the direct sum $L_p\left(0,1\right)\oplus C$. Denote by $W_p^2\left(0,\frac{1}{3}\right)\oplus W_p^2\left(\frac{1}{3},1\right)$ the space of functions whose restrictions to intervals $\left(0,\frac{1}{3}\right)$ and $\left(\frac{1}{3},1\right)$ belong to Sobolev spaces $W_p^2\left(0,\frac{1}{3}\right)$ and $W_p^2\left(\frac{1}{3},1\right)$, respectively, where $1<p<\infty .$ Let us define the operator $L$ in the following way. As its domain $D_L$ we take the manifold

$$\begin{array}{l}{D}_L=\left\{\widehat{y}=\left(y\left(x\right),\mathit{my}\left(\frac{1}{3}\right)\right):y\left(x\right)\right.\in W_p^2\left(0,\frac{1}{3}\right)\oplus W_p^2\left(\frac{1}{3},1\right),\\ y\left(0\right)=y\left(1\right)=0,y\left(\frac{1}{3}-0\right)=\left.y\left(\frac{1}{3}+0\right)\right\},\end{array}$$E71

and for $\widehat{y}\in D_L$ the operator $L$ is defined by the relation

$$L\widehat{y}=\left(-y\prime \prime ;y\prime \left(\frac{1}{3}-0\right)-y\prime \left(\frac{1}{3}+0\right)\right).$$E72

The following lemma holds true.

Lemma 1.39 The operator $L$ defined by expressions (71), (72) is a densely defined closed operator with a completely continuous resolvent. The eigenvalues of the operator $L$ and the problem (67), (68) coincide. If $y\left(x\right)$ is the eigenfunction (associated function) of problem (67), (68), then $\widehat{y}=\left(y\left(x\right);\mathit{my}\left(\frac{1}{3}\right)\right)$ is the eigenvector (associated vector) of the operator $L.$

In order to obtain the main results, we need some concepts and facts from the theory of bases in a Banach space.

Recall the following definition.

Definition 1.40 The basis ${\left\{u_n\right\}}_{n\in N}$ of Banach space $X$ is called a $p$-basis, if for any $x\in X$ one has the inequality

$${\left(\sum _{n=1}^{\infty }‍{\left|⟨x,{\vartheta }_n⟩\right|}^p\right)}^{\frac{1}{p}}\le M‖x‖,$$E73

where ${\left\{{\vartheta }_n\right\}}_{n\in N}$ is the biorthogonal system for ${\left\{u_n\right\}}_{n\in N}$.

Definition 1.41 The sequences ${\left\{u_n\right\}}_{n\in N}$ and ${\left\{{\phi }_n\right\}}_{n\in N}$ of Banach space $X$ are called $p$- close, if

$$\sum _{n=1}^{\infty }‍{‖u_n-{\phi }_n‖}^p<\infty .$$E74

We will also use the following results from [37, 38] (see also [39, 40]).

Theorem 1.42 [37] Let ${\left\{x_n\right\}}_{n\in N}$ form a $q$-basis for the space $X$, and the system ${\left\{y_n\right\}}_{n\in N}$ is $p$- close to ${\left\{x_n\right\}}_{n\in N}$, where $\frac{1}{p}+\frac{1}{q}=1$. Then the following properties are equivalent:

1. ${\left\{y_n\right\}}_{n\in N}$ is complete in $X$;

2. ${\left\{y_n\right\}}_{n\in N}$ is minimal in $X$;

3. ${\left\{y_n\right\}}_{n\in N}$ forms an isomorphic basis to ${\left\{x_n\right\}}_{n\in N}$ for $X$.

Let $X_1=X\oplus C^m$ and ${\left\{{\widehat{u}}_n\right\}}_{n\in N}\subset X_1$ be some minimal system, and ${\left\{{\widehat{\vartheta }}_n\right\}}_{n\in N}\subset X_1^{\ast }=X^{\ast }\oplus C^m$ be its biorthogonal system:

$${\widehat{u}}_n=\left(u_n;{\alpha }_{n1},\dots ,{\alpha }_{\mathit{nm}}\right);{\widehat{\vartheta }}_n=\left({\vartheta }_n;{\beta }_{n1},\dots ,{\beta }_{\mathit{nm}}\right).$$E75

Let $J=\left\{n_1,\dots ,n_m\right\}$ be some set of $m$ natural numbers. Suppose

$$\delta =det{‖{\beta }_{n_{i}j}‖}_{i,j=\overline{1,m}}.$$E76

The following theorem holds true.

Theorem 1.43 [38] Let the system ${\left\{{\widehat{u}}_n\right\}}_{n\in N}$ form a basis for $X_1$. In order to the system ${\left\{u_n\right\}}_{n\in N_J}$, where $N_J=N\backslash J$, form a basis for $X$ it is necessary and sufficient that the condition $\delta \ne 0$ is satisfied. In this case the biorthogonal system to ${\left\{u_n\right\}}_{n\in N_J}$ is defined by

$${\vartheta }_n^{\ast }=\frac{1}{\delta }\left|\begin{array}{llll}{\vartheta }_n& {\vartheta }_{n1}& \dots & {\vartheta }_{\mathit{nm}}\\ {\beta }_{n1}& {\beta }_{n_{1}1}& \dots & {\beta }_{n_{m}1}\\ \dots & \dots & \dots & \dots \\ {\beta }_{\mathit{nm}}& {\beta }_{n_{1}m}& \dots & {\beta }_{n_{m}m}\end{array}\right|.$$E77

For $\delta =0$ the system ${\left\{u_n\right\}}_{n\in N_J}$ is not complete and is not minimal in $X$.

Let $X$ be a Banach space and ${\left\{u_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}$ be some system in $X$.

Let $a_{\mathit{ik}}^{\left(n\right)},i,k=\overline{1,m},n\in N,$ be some complex numbers. Put

$$A_n={\left(a_{\mathit{ik}}^{\left(n\right)}\right)}_{i,k=\overline{1,m}}\mathit{\text{and}}{\mathrm{\Delta }}_n=det{A}_n,n\in N.$$E78

Let us consider the following system in space $X$

$${\widehat{u}}_{\mathit{kn}}=\sum _{i=1}^m‍a_{\mathit{ik}}^{\left(n\right)}{u}_{\mathit{\text{in}}},k=\overline{1,m};n\in N.$$E79

Theorem 1.44 If the system ${\left\{u_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}$ forms a basis for $X$ and

$${\mathrm{\Delta }}_n\ne 0,\forall n\in N,$$E80

then the system ${\left\{{\widehat{u}}_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}$ forms a basis with parentheses for $X$. If in addition the conditions

$$\underset{n}{sup\left\{\right.}‖A_n‖,‖A_n^{-1}‖\left.\right\}<\infty ,\underset{n}{sup}\left\{‖u_{\mathit{kn}}‖,‖{\vartheta }_{\mathit{kn}}‖\right\}<\infty ,$$E81

hold, where ${\left\{{\vartheta }_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}\subset X^{\ast }$ is biorthogonal system to ${\left\{u_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}$, then the system ${\left\{{\widehat{u}}_{\mathit{kn}}\right\}}_{k=\overline{1,m};n\in N}$ forms the usual basis for $X$.

The following theorem holds true.

Theorem 1.45 The system of eigen and associated vectors of the operator $L$ forms the basis for space $M^{p,\alpha }\left(0,1\right)\oplus C,1<p<\infty ,0<\alpha \le 1.$

Now, let us consider the basicity of the system $\left\{y_0\right\}\cup {\left\{y_{i,n}\right\}}_{i=1,2;n\in N}^{\infty }$ with a removed function in space $M^{p,\alpha }\left(0,1\right).$

Theorem 1.46 If from the system of eigen and associated functions of problem (67), (68) $\left\{y_0\right\}\cup {\left\{y_{i,n}\right\}}_{i=1,2;n\in N}^{\infty }$ we eliminate any function $y_{2,n_0}\left(x\right)$, corresponding to a simple eigenvalue, then the new system forms a basis for $M^{p,\alpha }\left(0,1\right)$, $1<p<\infty ,$ $0<\alpha \le 1$. And if we eliminate any function $y_{1,n_0}\left(x\right)$ from this system, then the obtained system does not form a basis in $M^{p,\alpha }\left(0,1\right)$; moreover, in this case this system is not complete and is not minimal in this space.

Proof. For the eigenfunctions $\left\{z_0\right\}\cup {\left\{z_{i,n}\right\}}_{i=1,2;n\in N}^{\infty }$ of the adjoint problem we have $z_{1,n}\left(\frac{1}{3}\right)=0$ for any $n\in N$ and $z_{2,n}\left(\frac{1}{3}\right)\ne 0$. On the other hand, the eigenvectors of the adjoint operator $L^{\ast }$ are defined by ${\widehat{z}}_n=\left(z_n,\overline{m}{z}_n\left(\frac{1}{3}\right)\right),n=0,1,\dots ,$. Applying Theorem 1.43 to the system $\left\{{\widehat{y}}_0\right\}\cup {\left\{{\widehat{y}}_{i,n}\right\}}_{i=1,2;n\in N}^{\infty }$, we notice that $\delta =\overline{m}{z}_{1,n}\left(\frac{1}{3}\right)=0$ for any $n\in N$ and $\delta =\overline{m}{z}_{2,n}\left(\frac{1}{3}\right)\ne 0$ for any eigenfunction corresponding to a simple eigenvalue, and the statements of the theorem follow from the corresponding statements of Theorem 1.43.

Theorem is proved.

Conflict of interest

The authors declare no conflict of interest.