On Solvability of One Class of Quasielliptic Systems

Abstract

We study the class of systems of differential equations defined by one class of matrix quasielliptic operators and establish solvability conditions for the systems and boundary value problems on ${𝕉}^{n}_{+}$ in the special scales of weighted Sobolev spaces $W^{l}_{p,\sigma}$. We construct the integral representations of solutions and obtain estimates for the solutions.

1. Introduction

We consider the class of quasielliptic systems in the whole space

$$ {\mathcal{L}}(D_{x})U=F(x),\quad x\in{𝕉}^{n}, $$

(1.1)

and the boundary value problems for them in the half-space

$$ \begin{cases}{\mathcal{L}}(D_{x})U=F(x),\quad x\in{𝕉}^{n}_{+},\\ {\mathcal{B}}(D_{x})U|_{x_{n}=0}=0.\end{cases} $$

(1.2)

These systems are determined by some class of matrix quasielliptic operators ${\mathcal{L}}(D_{x})$.

Let us specify the conditions on the operators ${\mathcal{L}}(D_{x})$ and ${\mathcal{B}}(D_{x})$. Denote by $l_{j,r}(i\xi)$ and $b_{j,r}(i\xi)$ the entries of the matrices ${\mathcal{L}}(i\xi)$ and ${\mathcal{B}}(i\xi)$ which are the symbols of the corresponding differential operators.

Condition 1

Let ${\mathcal{L}}(i\xi)$ be a $(\nu\times\nu)$-matrix. Suppose that there are vectors $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and ${\mathbf{t}}=(t_{1},\dots,t_{\nu})$, with $t_{r}>0$ and $t_{r}/\alpha_{j}\in{𝕅}$ such that

$$ l_{j,r}(c^{\alpha}i\xi)=c^{t_{r}}\,l_{j,r}(i\xi),\quad j,r=1,\dots,\nu, $$

for all $c>0$.

Condition 2

$\det\mathcal{L}(i\xi)=0$, $\xi\in{𝕉}^{n}$, if and only if $\xi=0$.

The matrix operators ${\mathcal{L}}(D_{x})$ satisfying Conditions 1 and 2 belong to the class of quasielliptic operators which was introduced by Volevich [1]. In particular, the following operators belong to the class of operators under consideration: homogeneous elliptic operators, Petrovskii’s elliptic and parabolic operators, Eidelman’s parabolic operators, backwards parabolic operators, and homogeneous quasielliptic operators (see [2]).

Condition 2 implies that the equation

$$ \det{\mathcal{L}}(is,i\lambda)=0,\quad s\in{𝕉}^{n-1}\backslash\{0\}, $$

(1.3)

has no real roots in $\lambda$. Denote the number of roots in the upper half-plane by $\mu$. We assume that $\mu$ is independent of $s\in{𝕉}^{n-1}\backslash\{0\}$.

Condition 3

Let ${\mathcal{B}}(i\xi)$ be a $(\mu\times\nu)$-matrix. Suppose that there is a vector $(m_{1},\dots,m_{\mu})$ satisfying $t_{r}-t_{\min}\leq t_{r}-m_{j}\leq t_{r}-\alpha_{n}$, $t_{\min}=\min(t_{1},\dots,t_{\nu})$ and such that

$$ b_{j,r}(c^{\alpha}i\xi)=c^{t_{r}-m_{j}}b_{j,r}(i\xi),\quad c>0,\ j=1,\dots,\mu,\ r=1,\dots,\nu. $$

Condition 4

Boundary value problem (1.2) satisfies the Lopatinskii condition; i.e., the boundary value problem on the half-axis

$$ \begin{cases}{\mathcal{L}}(is,D_{x_{n}})v=0,\quad x_{n}>0,\\ {\mathcal{B}}(is,D_{x_{n}})v|_{x_{n}=0}=\varphi,\\ \sup\limits_{x_{n}>0}|v|<\infty\end{cases} $$

(1.4)

is uniquely solvable for all $s\in{𝕉}^{n-1}\backslash\{0\}$ and $\varphi$.

The goal of this article is to obtain solvability conditions and construct solutions to the quasielliptic systems (1.1) in ${𝕉}^{n}$ and to their boundary value problems (1.2) on ${𝕉}^{n}_{+}$. We search solutions to the problems in the special scales of weighted Sobolev spaces $W^{l}_{p,\sigma}$ which were introduced in [3].

Recall the definition of $W^{\mathbf{k/\alpha}}_{p,\sigma}({𝕉}^{n})$, with ${\mathbf{k/\alpha}}=(k/\alpha_{1},\dots,k/\alpha_{n})$, $k/\alpha_{i}\in{𝕅}$, $1<p<\infty$, and $\sigma\geq 0$.

Definition

A locally summable function $u(x)$ belongs to $W^{\mathbf{k/\alpha}}_{p,\sigma}({𝕉}^{n})$ if $u(x)$ has the generalized derivatives $D^{\beta}_{x}u(x)$ on ${𝕉}^{n}$ for $\beta\alpha\leq k$, and

$$ \big{\|}(1+\langle x\rangle)^{-\sigma(k-\beta\alpha)}D_{x}^{\beta}u(x),L_{p}({𝕉}^{n})\big{\|}<\infty,\quad\langle x\rangle^{2}=\sum\limits_{i=1}^{n}x_{i}^{2/\alpha_{i}}. $$

The norm on $W^{\mathbf{k/\alpha}}_{p,\sigma}({𝕉}^{n})$ is defined as follows:

$$ \big{\|}u(x),W^{\mathbf{k/\alpha}}_{p,\sigma}({𝕉}^{n})\big{\|}=\sum\limits_{0\leq\beta\alpha\leq k}\big{\|}(1+\langle x\rangle)^{-\sigma(k-\beta\alpha)}D_{x}^{\beta}u(x),L_{p}({𝕉}^{n})\big{\|}. $$

(1.5)

In the isotropic case when $k/\alpha_{1}=\dots=k/\alpha_{n}=\overline{l}$, (1.5) is equivalent to

$$ \sum\limits_{0\leq|\beta|\leq\overline{l}}\big{\|}(1+|x|)^{-\sigma(\overline{l}-|\beta|)}D^{\beta}_{x}u(x),L_{p}({𝕉}^{n})\big{\|}. $$

In this case, the spaces under consideration coincide for $\sigma=1$ with the spaces often called the Nirenberg–Walker–Cantor spaces $M^{p}_{\overline{l},m}({𝕉}^{n})$, with $m=-\overline{l}$ (for instance, see [4,5,6]).

We consider the general case when the entries $k/\alpha_{i}$ of the vector of smoothness can differ from one another.

Recall that $C^{\infty}_{0}({𝕉}^{n})$ is everywhere dense in $W^{\mathbf{k/\alpha}}_{p,\sigma}({𝕉}^{n})$ for $\sigma\leq 1$ (see [3]). Henceforth, we assume that $1\geq\sigma\geq 0$.

Introduce the weighted Sobolev space ${\mathbf{W}}^{\mathbf{t/\alpha}}_{p,\sigma}({𝕉}^{n})$ for vector-functions, where the parameters of smoothness are determined by the vectors $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and ${\mathbf{t}}=(t_{1},\dots,t_{\nu})$ from Condition 1.

We say that a vector-function $U(x)=(U^{1}(x),\dots,U^{\nu}(x))^{T}$ belongs to

$$ {\mathbf{W}}^{\mathbf{t/\alpha}}_{p,\sigma}({𝕉}^{n})=\prod\limits_{k=1}^{\nu}W^{\mathbf{t_{k}/\alpha}}_{p,\sigma}({𝕉}^{n}),\quad{\mathbf{t_{k}/\alpha}}=(t_{k}/\alpha_{1},\dots,t_{k}/\alpha_{n}),\ 1<p<\infty,\ \sigma\geq 0, $$

if each of the entries $U^{k}(x)$ belongs to $W^{\mathbf{t_{k}/\alpha}}_{p,\sigma}({𝕉}^{n})$ and we set

$$ \big{\|}U(x),{\mathbf{W}}^{\mathbf{t/\alpha}}_{p,\sigma}({𝕉}^{n})\big{\|}=\sum\limits_{k=1}^{\nu}\big{\|}U^{k}(x),W_{p,\sigma}^{\mathbf{t_{k}/\alpha}}({𝕉}^{n})\big{\|}. $$

We also write

$$ {\mathbf{L}}_{p}({𝕉}^{n})=\prod\limits_{r=1}^{\nu}L_{p}({𝕉}^{n}). $$

Definition

Denote by $L_{p,\gamma}({𝕉}^{n})$ the weighted space with the norm

$$ \|f(x),L_{p,\gamma}({𝕉}^{n})\|=\|(1+\langle x\rangle)^{-\gamma}f(x),L_{p}({𝕉}^{n})\|. $$

We say that a vector-function $F(x)=(F^{1}(x),\dots,F^{\nu}(x))^{T}$ belongs to

$$ {\mathbf{L}}_{p,\gamma}({𝕉}^{n})=\prod\limits_{r=1}^{\nu}L_{p,\gamma}({𝕉}^{n}) $$

if each of the entries $F^{k}(x)$ belongs to $L_{p,\gamma}(𝕉^{n})$, and we set

$$ \|F(x),{\mathbf{L}}_{p,\gamma}({𝕉}^{n})\|=\sum\limits_{r=1}^{\nu}\|F^{r}(x),L_{p,\gamma}({𝕉}^{n})\|. $$

Similarly, we define ${\mathbf{L}}_{p}({𝕉}^{n}_{+})$, ${\mathbf{L}}_{p,\gamma}({𝕉}^{n}_{+})$, $W^{\mathbf{t_{k}/\alpha}}_{p,\sigma}({𝕉}^{n}_{+})$, and ${\mathbf{W}}^{\mathbf{t/\alpha}}_{p,\sigma}({𝕉}^{n}_{+})$.

2. Statement of the Main Results

We recall now that the first theorem about isomorphism for the class of quasielliptic operators ${\mathcal{L}}(D_{x})$ in the case when $t_{1}=\dots=t_{\nu}=1$ was proven in [7] by using the property of ${\mathbf{W}}^{\mathbf{t/\alpha}}_{p,\sigma}({𝕉}^{n})$. The unique solvability of “homogeneous” quasielliptic systems ensues from the theorem. Properties of quasielliptic operators for $t_{1}=\dots=t_{\nu}=1$ were further studied in [8]. Some theorems of isomorphism for wider classes of quasielliptic operators were proved in [2, 9,10,11].

Observe that the Sobolev spaces with power weights (for instance, see [5, 6, 12,13,14,15]) are also used in the theorems about isomorphic properties of elliptic operators.

Let us recall the theorem of isomorphism for the class of quasielliptic operators under consideration which was established in [2]. Put

$$ |\alpha|=\sum\limits^{n}_{i=1}\alpha_{i},\quad t_{\max}=\max\{t_{1},\dots,t_{\nu}\}. $$

Theorem 1

Let an operator ${\mathcal{L}}(D_{x})$ meet Conditions $1$ and $2$. If $|\alpha|/p>t_{\max}$, then

$$ {\mathcal{L}}(D_{x}):{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})\longrightarrow{\mathbf{L}}_{p}({𝕉}^{n}),\quad 1<p<\infty,\ \sigma=1, $$

is an isomorphism.

The unique solvability of (1.1) is straightforward from Theorem 1. Namely, if $|\alpha|/p>t_{\max}$ then for every $F(x)\in{\mathbf{L}}_{p}({𝕉}^{n})$ there is a unique solution $U(x)\in{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,1}({𝕉}^{n})$ to (1.1) and $\big{\|}U(x),{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,1}({𝕉}^{n})\big{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n})\|$ with a constant $c>0$ independent of $F(x)$.

The condition $|\alpha|/p>t_{\max}$ is essential for the unique solvability of (1.1) in ${\mathbf{W}}^{\mathbf{t}/\alpha}_{p,1}({𝕉}^{n})$. However, the condition imposes rather stringent constraints on the degree of summability $p$ and $\sigma=1$. In the following theorem, we establish unique solvability conditions for system (1.1) in the whole scale of weighted Sobolev spaces ${\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$ with significantly weaker constraints on the parameters $p$ and $\sigma$.

Theorem 2

Let an operator ${\mathcal{L}}(D_{x})$ meet Conditions 1 and 2. If

$$ |\alpha|/p>\sigma t_{\max},\quad|\alpha|/p^{\prime}>(1-\sigma)t_{\max},\quad 1/p+1/p^{\prime}=1, $$

(2.1)

then for every vector-function $F(x)\in{\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})$ there is a unique solution $U(x)\in{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$ to (1.1); moreover,

$$ \big{\|}U(x),{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})\big{\|}\leq C\|F(x),{\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})\| $$

(2.2)

with a constant $C>0$ independent of $F(x)$.

Theorem 3

Let the conditions of Theorem 2 be met. Then

$$ \big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}(x),L_{p}({𝕉}^{n})\big{\|}\leq c\|\langle x\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}{\mathcal{L}}(D_{x})U(x),{\mathbf{L}}_{p}({𝕉}^{n})\|, $$

(2.3)

$$ 0\leq\beta\alpha\leq t_{k},\quad k=1,\dots,\nu, $$

for all vector-functions $U(x)=(U^{1}(x),\dots,U^{\nu}(x))^{T}\in{\mathbf{C}}^{\infty}_{0}({𝕉}^{n})$, with a constant $c>0$ independent of $U(x)$.

Remark 1

Theorem 2 generalizes the unique solvability theorem for “homogeneous” quasielliptic systems $(t_{1}=\dots=t_{\nu}=1)$ from [7, 8].

Remark 2

As is demonstrated by examples, in some cases conditions (2.1) are necessary and sufficient for the unique solvability of systems (1.1) in ${\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$. For instance, it is easily seen for the polyharmonic equation

$$ \Delta^{m}u=f(x),\quad x\in{𝕉}^{n},\ n>2m, $$

by using the results from [16, Chapter 12]. Observe that, for solvability of the equation in $W^{2m}_{p}({𝕉}^{n})$ for $p\leq\frac{n}{n-2m}$, i.e., $|\alpha|/p^{\prime}\leq 1$, $\sigma=0$, the right-hand side $f(x)$ needs to satisfy the orthogonality conditions of the form

$$ \int\limits_{{𝕉}^{n}}f(x)x^{\beta}dx=0,\quad|\beta|\leq s(n,m,p). $$

Remark 3

The inequality of the form (2.3) for $\sigma=1$ is obtained in [4] for elliptic operators; and (2.3) is established in [7] for “homogeneous” quasielliptic operators.

Let state some theorem of unique solvability for (1.2).

Theorem 4

Let ${\mathcal{L}}(D_{x})$ and ${\mathcal{B}}(D_{x})$ meet Conditions 1–4. If (2.1) hold, then (1.2) has a unique solution $U(x)\in{\mathbf{W}}^{{\mathbf{t}/\alpha}}_{p,\sigma}({𝕉}^{n}_{+})$ for every vector-function $F(x)\in{\mathbf{L}}_{p}({𝕉}^{n}_{+})\cap{\mathbf{L}}_{1,-\sigma t_{\max}}({𝕉}^{n}_{+})$; moreover,

$$ \bigl{\|}U(x),{\mathbf{W}}^{{\mathbf{t}/\alpha}}_{p,\sigma}({𝕉}^{n}_{+})\bigr{\|}\leq c(\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|+\|F(x),{\mathbf{L}}_{1,-\sigma t_{\max}}({𝕉}^{n}_{+})\|) $$

(2.4)

with a constant $c>0$ independent of $F(x)$.

Remark 4

Theorem 4 generalizes the solvability theorems for boundary value problems for quasielliptic equations in [17, 18]; and for systems in [19,20,21].

Remark 5

Conditions (2.1) are important for the unique solvability of boundary value problems on ${𝕉}^{n}_{+}$ for quasielliptic equations and systems. In particular, for $|\alpha|/p<\sigma t_{\max}$, the homogeneous problem can have a nontrivial solution. If $|\alpha|/p^{\prime}\leq(1-\sigma)t_{\max}$, then there exist boundary value problems without solutions in ${\mathbf{W}}^{{\mathbf{t}/\alpha}}_{p,\sigma}({𝕉}^{n}_{+})$ even for compactly-supported infinitely differentiable $F(x)$ (see [18, 22,23,24]). For instance, if

$$ \int\limits_{\Gamma(s)}\mathcal{B}(is,i\lambda)\mathcal{L}^{-1}(is,i\lambda)\,d\lambda\neq 0,\quad s\in{𝕉}^{n-1}\setminus\{0\}, $$

for $\sigma=0$, $t_{1}=\dots=t_{\nu}=1$, where $\Gamma(s)$ is a contour in the complex plane encircling all roots of equation (1.3); then the condition

$$ \int\limits_{{𝕉}^{n}_{+}}F(x)\,dx=0 $$

is necessary for solvability of (1.2) in ${\mathbf{W}}^{{\mathbf{1}/\alpha}}_{p}({𝕉}^{n}_{+})$ for $1\geq|\alpha|/p^{\prime}>1-\alpha_{\min}$, $p\in(1,2]$.

3. Proof of Solvability of (1.1)

To prove Theorem 2, we follow the scheme of [2, 7]. Let us sketch the scheme of the proof and inspect the essential differences.

As in [2, 7], to prove the solvability of (1.1), we use the method of constructing approximate solutions which was described in [25] and based on Uspenskii’s integral representation [26] for summable functions (see also [25, Chapter 1]).

We suppose first that the entries $F^{j}(x)$ of $F(x)\in{\mathbf{L}}_{p}({𝕉}^{n})$ on the right-hand side of (1.1) have compact supports.

Consider the family of integral operators $P_{k,h}$, $k=1,\dots,\nu$, $0<h<1$, of the following form:

$$ P_{k,h}F(x)=(2\pi)^{-n}\int\limits^{h^{-1}}_{h}v^{-|\alpha|/t_{k}}\int\limits_{{𝕉}^{n}}\int\limits_{{𝕉}^{n}}\exp\bigg{(}i\frac{x-y}{v^{\alpha/t_{k}}}\xi\bigg{)}G_{k}(\xi)\bigg{(}\sum\limits^{\nu}_{r=1}l^{k,r}(\xi)F^{r}(y)\bigg{)}\,d\xi dydv, $$

(3.1)

where $l^{k,r}(\xi)$ are the entries of the inverse matrix $({\mathcal{L}}(i\xi))^{-1}$ and

$$ G_{k}(\xi)=2m\langle\xi\rangle^{2m}_{k}\exp\bigl{(}-\langle\xi\rangle^{2m}_{k}\bigr{)},\quad\langle\xi\rangle^{2}_{k}=\sum\limits^{n}_{i=1}\xi^{2t_{k}/{\alpha_{i}}}_{i},\ m\in{𝕅}. $$

(3.2)

It is demonstrated in [2] that the vector-function

$$ U_{h}(x)=P_{h}F(x)=(P_{1,h}F(x),\dots,P_{\nu,h}F(x))^{T} $$

(3.3)

is an approximate solution to (1.1).

Note that (3.1) and Conditions 1 and 2 imply that $U^{k}_{h}(x)=P_{k,h}F(x)$ are infinitely differentiable and, obviously, we can find a natural $m_{1}$ such that $U^{k}_{h}(x)$ in (3.2) are summable to every power $p\geq 1$ for $m\geq m_{1}$. Henceforth, we assume that $m\geq m_{1}$ in (3.2).

It ensues from the lemmas below that, under conditions (2.1),

$$ \bigl{\|}U_{h}(x),\ {\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})\bigr{\|}\leq C\|F(x),\ {\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})\| $$

(3.4)

with a constant $C>0$ independent of $F(x)$ and $h$; moreover,

$$ \bigl{\|}U_{h_{1}}(x)-U_{h_{2}}(x),\ {\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})\bigr{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(3.5)

First, we recall the estimates of [2] of the leading derivatives of the entries of (3.3).

Lemma 3.1

Let $\beta=(\beta_{1},\dots,\beta_{n})$, $\beta\alpha=t_{k}$, $k=1,\dots,\nu$. Then

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{h}(x),\ L_{p}({𝕉}^{n})\bigr{\|}\leq C_{\beta}\|F(x),\ {\mathbf{L}}_{p}({𝕉}^{n})\| $$

(3.6)

with a constant $C_{\beta}>0$ independent of $F(x)$ and $h$; moreover,

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{h_{1}}(x)-D^{\beta}_{x}U^{k}_{h_{2}}(x),\ L_{p}({𝕉}^{n})\bigr{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(3.7)

To estimate the norms of the derivatives

$$ D^{\beta}_{x}U^{k}_{h}(x),\quad 0\leq\beta\alpha<t_{k},\ k=1,\dots,\nu, $$

(3.8)

we need some estimates for the integrals

$$ {\mathcal{K}}^{k,r}_{\beta,h}(x)=\int\limits^{h^{-1}}_{h}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\int\limits_{{𝕉}^{n}}\exp\bigg{(}i\frac{x\xi}{v^{\alpha/t_{k}}}\bigg{)}G_{k}(\xi)(i\xi)^{\beta}l^{k,r}(\xi)\,d\xi dv, $$

(3.9)

$$ x\in{𝕉}^{n},\quad 0<h<1. $$

The following analog of Lemma 3.3 from [9] takes place:

Lemma 3.2

Let $|\alpha|+\beta\alpha>t_{k}$. Then there is $m_{2}$ such that for $m\geq m_{2}$ in (3.2) we have

$$ \langle x\rangle^{|\alpha|+\beta\alpha-t_{k}}\bigl{|}{\mathcal{K}}^{k,r}_{\beta,h}(x)\bigr{|}\leq c,\quad x\in{𝕉}^{n}, $$

(3.10)

with a constant $c>0$ independent of $h$ and $x$.

Proof

Consider the function

$$ J(x)=\langle x\rangle^{|\alpha|+\beta\alpha-t_{k}}{\mathcal{K}}^{k,r}_{\beta,h}(x),\quad x\in{𝕉}^{n}. $$

Making the change $v=\omega\langle x\rangle^{t_{k}}$ for $x\neq 0$ in (3.9), rewrite this function as

$$ J(x)=\int\limits^{h^{-1}/\langle x\rangle^{t_{k}}}_{h/\langle x\rangle^{t_{k}}}\omega^{-|\alpha|/{t_{k}}-\beta\alpha/{t_{k}}}\int\limits_{{𝕉}^{n}}\exp\bigg{(}i\frac{x\xi}{\langle x\rangle^{\alpha}\omega^{\alpha/t_{k}}}\bigg{)}G_{k}(\xi)(i\xi)^{\beta}l^{k,r}(\xi)\,d\xi d\omega. $$

By Conditions 1 and 2, $\mu(\xi)=(i\xi)^{\beta}l^{k,r}(\xi)$ is quasihomogeneous in $\alpha$. Therefore, by the definition of (3.2), we can find $m_{2}\in{𝕅}$ such that

$$ \bigg{|}\int\limits_{{𝕉}^{n}}e^{iz\xi}G_{k}(\xi)(i\xi)^{\beta}l^{k,r}(\xi)\,d\xi\bigg{|}\leq c^{\prime}(1+\langle z\rangle)^{-|\alpha|},\quad z\in{𝕉}^{n}, $$

with a constant independent of $z$. Hence,

$$ |J(x)|\leq c^{\prime}\int\limits^{\infty}_{0}\omega^{-|\alpha|/{t_{k}}-\beta\alpha/{t_{k}}}(1+\omega^{-1/t_{k}})^{-|\alpha|}\,d\omega. $$

Thus, since $|\alpha|+\beta\alpha>t_{k}$, $0\leq\beta\alpha<t_{k}$, we obtain $|J(x)|\leq c<\infty$; i.e., (3.10).

The lemma is proved.

Henceforth, we assume that $m\geq\max\{m_{1},m_{2}\}$ and turn to estimating the norms of (3.8).

Lemma 3.3

Let $\beta=(\beta_{1},\dots,\beta_{n})$, $0\leq\beta\alpha<t_{k}$ and $|\alpha|/p>\sigma(t_{k}-\beta\alpha)>t_{k}-\beta\alpha-|\alpha|/p^{\prime}$, $1/p+1/p^{\prime}=1$. Then

$$ \big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{h}(x),L_{p}({𝕉}^{n})\big{\|}\leq c\|\langle x\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}F(x),{\mathbf{L}}_{p}({𝕉}^{n})\|, $$

(3.11)

with a constant $c>0$ independent of $F(x)$ and $h$.

Proof

By (3.1), (3.2), and (3.9), $D^{\beta}_{x}U^{k}_{h}(x)$ can be written as

$$ D^{\beta}_{x}U^{k}_{h}(x)=\sum\limits^{\nu}_{r=1}(2\pi)^{-n}\int\limits_{{𝕉}^{n}}{\mathcal{K}}^{k,r}_{\beta,h}(x-y)F^{r}(y)\,dy. $$

By Lemma 3.2,

$$ \begin{gathered}\displaystyle\big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{h}(x),L_{p}({𝕉}^{n})\big{\|}\\ \displaystyle\leq c\sum\limits^{\nu}_{r=1}\bigg{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}\int\limits_{{𝕉}^{n}}\langle x-y\rangle^{-(|\alpha|+\beta\alpha-t_{k})}|F^{r}(y)|\,dy,L_{p}({𝕉}^{n})\bigg{\|}.\end{gathered} $$

The conditions imply that $|\alpha|+\beta\alpha-t_{k}>0$; therefore,

$$ \begin{gathered}\displaystyle\big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{h}(x),L_{p}({𝕉}^{n})\big{\|}\\ \displaystyle\leq c_{1}\sum\limits^{\nu}_{r=1}\bigg{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}\int\limits_{{𝕉}^{n}}\prod^{n}_{i=1}|x_{i}-y_{i}|^{(t_{k}-\beta\alpha)/|\alpha|-1}|F^{r}(y)|\,dy,L_{p}({𝕉}^{n})\bigg{\|}.\end{gathered} $$

Since $t_{k}-\beta\alpha>0$, $1\geq\sigma\geq 0$, we obtain

$$ \begin{gathered}\displaystyle\big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{h}(x),L_{p}({𝕉}^{n})\big{\|}\\ \displaystyle\leq c_{2}\sum\limits^{\nu}_{r=1}\bigg{\|}\int\limits_{{𝕉}^{n}}\prod^{n}_{i=1}|x_{i}|^{\sigma(\beta\alpha-t_{k})/|\alpha|}\,|x_{i}-y_{i}|^{(t_{k}-\beta\alpha)/|\alpha|-1}|y_{i}|^{(1-\sigma)(\beta\alpha-t_{k})/|\alpha|}\\ \displaystyle\times\langle y\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}|F^{r}(y)|dy,\ L_{p}({𝕉}^{n})\bigg{\|}.\end{gathered} $$

Put $a=\sigma(t_{k}-\beta\alpha)/|\alpha|$ and $b=(1-\sigma)(t_{k}-\beta\alpha)/|\alpha|$. By hypotheses, $a<1/p$, $b<1/p^{\prime}$, $a+b>0$. Applying the Hardy–Littlewood inequality [27], we arrive at (3.11).

The lemma is proved.

Lemma 3.4

Let the conditions of Theorem 2 be met and $\beta=(\beta_{1},\dots,\beta_{n})$, $0\leq\beta\alpha<t_{k}$. If $F(x)\in{\mathbf{L}}_{p}({𝕉}^{n})$ is compactly-supported then

$$ \big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\big{(}D^{\beta}_{x}U^{k}_{h_{1}}(x)-D^{\beta}_{x}U^{k}_{h_{2}}(x)\big{)},L_{p}({𝕉}^{n})\big{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0,\ k=1,\dots,\nu. $$

(3.12)

Proof

Put

$$ K^{k,r}_{\beta}(x)=\int\limits_{{𝕉}^{n}}\exp(ix\xi)G_{k}(\xi)(i\xi)^{\beta}l^{k,r}(\xi)\,d\xi. $$

(3.13)

From (3.1), (3.3), and (3.9) it follows that

$$ D^{\beta}_{x}U^{k}_{h}(x)=\sum\limits^{\nu}_{r=1}(2\pi)^{-n}\int\limits^{h^{-1}}_{h}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\int\limits_{{𝕉}^{n}}K^{k,r}_{\beta}\bigg{(}\frac{x-y}{v^{\alpha/t_{k}}}\bigg{)}F^{r}(y)\,dydv. $$

Using this representation and Minkowski’s inequality, for $h_{2}>h_{1}>0$ we obtain

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\big{(}D^{\beta}_{x}U^{k}_{h_{1}}(x)-D^{\beta}_{x}U^{k}_{h_{2}}(x)\big{)},L_{p}({𝕉}^{n})\big{\|}\\ \displaystyle\leq\sum\limits^{\nu}_{r=1}\int\limits^{h_{2}}_{h_{1}}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\bigg{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\int\limits_{{𝕉}^{n}}K^{k,r}_{\beta}\bigg{(}\frac{x-y}{v^{\alpha/t_{k}}}\bigg{)}F^{r}(y)\,dy,L_{p}({𝕉}^{n})\bigg{\|}\,dv\\ \displaystyle+\sum\limits^{\nu}_{r=1}\int\limits^{h^{-1}_{1}}_{h^{-1}_{2}}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\bigg{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\int\limits_{{𝕉}^{n}}K^{k,r}_{\beta}\bigg{(}\frac{x-y}{v^{\alpha/t_{k}}}\bigg{)}F^{r}(y)\,dy,L_{p}({𝕉}^{n})\bigg{\|}\,dv\\ \displaystyle=\sum\limits^{\nu}_{r=1}J^{1}_{r}(h_{1},h_{2})+\sum\limits^{\nu}_{r=1}J^{2}_{r}(h_{1},h_{2}).\end{gathered} $$

First, estimate $J^{1}_{r}(h_{1},h_{2})$. Applying Young’s inequality, we obviously get

$$ \begin{gathered}\displaystyle J^{1}_{r}(h_{1},h_{2})\leq\int\limits^{h_{2}}_{h_{1}}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\bigg{\|}K^{k,r}_{\beta}\bigg{(}\frac{x}{v^{\alpha/t_{k}}}\bigg{)},L_{1}({𝕉}^{n})\bigg{\|}\|F^{r}(y),L_{p}({𝕉}^{n})\|dv\\ \displaystyle=\int\limits^{h_{2}}_{h_{1}}v^{-\beta\alpha/t_{k}}\,dv\big{\|}K^{k,r}_{\beta}(z),L_{1}({𝕉}^{n})\big{\|}\|F^{r}(y),\ L_{p}({𝕉}^{n})\|.\end{gathered} $$

Hence, using (3.13) and the condition $\beta\alpha<t_{k}$, it follows that

$$ J^{1}_{r}(h_{1},h_{2})\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(3.14)

Estimate $J^{2}_{r}(h_{1},h_{2})$. Using the inequality

$$ (1+\langle x\rangle)^{-1}\langle x-y\rangle\leq a(1+\langle y\rangle),\quad a=\operatorname{const}, $$

(3.15)

we infer

$$ \begin{gathered}\displaystyle J^{2}_{r}(h_{1},h_{2})\leq\int\limits^{h^{-1}_{1}}_{h^{-1}_{2}}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\\ \displaystyle\times\bigg{\|}\int\limits_{{𝕉}^{n}}\langle x-y\rangle^{-\sigma(t_{k}-\beta\alpha)}K^{k,r}_{\beta}\bigg{(}\frac{x-y}{v^{\alpha/t_{k}}}\bigg{)}(1+\langle y\rangle)^{\sigma(t_{k}-\beta\alpha)}F^{r}(y)\,dy,L_{p}({𝕉}^{n})\bigg{\|}\,dv.\end{gathered} $$

By applying Young’s inequality, we obviously get

$$ \begin{gathered}\displaystyle J^{2}_{r}(h_{1},h_{2})\leq\int\limits^{h^{-1}_{1}}_{h^{-1}_{2}}v^{-|\alpha|/t_{k}-\beta\alpha/t_{k}}\bigg{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}K^{k,r}_{\beta}\bigg{(}\frac{x}{v^{\alpha/t_{k}}}\bigg{)},L_{p}({𝕉}^{n})\bigg{\|}\,dv\\ \displaystyle\times\|(1+\langle y\rangle)^{\sigma(t_{k}-\beta\alpha)}F^{r}(y),L_{1}({𝕉}^{n})\|\\ \displaystyle=\int\limits^{h^{-1}_{1}}_{h^{-1}_{2}}v^{-|\alpha|/p^{\prime}t_{k}-\beta\alpha/t_{k}-\sigma(1-\beta\alpha/t_{k})}\,dv\big{\|}\langle z\rangle^{-\sigma(t_{k}-\beta\alpha)}K^{k,r}_{\beta}(z),L_{p}({𝕉}^{n})\big{\|}\\ \displaystyle\times\|(1+\langle y\rangle)^{\sigma(t_{k}-\beta\alpha)}F^{r}(y),\ L_{1}({𝕉}^{n})\|.\end{gathered} $$

Since $|\alpha|/p>\sigma t_{\max}$, $t_{k}>\beta\alpha$, and $F^{r}(y)$ is compactly-supported; using (3.13), we infer that

$$ J^{2}_{r}(h_{1},h_{2})\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(3.16)

Recalling (3.14) and (3.16), we arrive at (3.12). The lemma is proved.

Proof of Theorem 2

Estimates (3.6) and (3.11) yield (3.4), while (3.7) and (3.12) imply (3.5). By completeness of ${\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$, it follows from (3.4) and (3.5) that there is a continuous linear operator

$$ P:{\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})\longrightarrow{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n}),\quad 1<p<\infty,\ 0\leq\sigma\leq 1, $$

defined on compactly-supported vector-functions $F(x)$ by the formula

$$ PF(x)=\lim\limits_{h\to 0}P_{h}F(x); $$

moreover, the vector-function $U(x)=PF(x)\in{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$ is a solution to (1.1). Owing to the density of the set of compactly-supported vector-functions in ${\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})$, the operator $P$ can be uniquely extended to the whole space ${\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})$ with the same norm. We preserve the notation $P$ for the extended operator.

By (3.4), the linear operators

$$ P_{h}:\ {\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})\longrightarrow{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n}),\quad 1<p<\infty,\ 0\leq\sigma\leq 1, $$

are continuous; and the collection of their norms is bounded: $\|P_{h}\|\leq C$. Consequently, by the Banach–Steinhaus Theorem,

$$ \big{\|}P_{h}F(x)-PF(x),{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})\big{\|}\to 0\quad\text{as }h\to 0 $$

for all $F(x)\in{\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})$.

By the above arguments, some solution $U(x)\in{\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n})$ to (1.1) exists for any right-hand side $F(x)\in{\mathbf{L}}_{p,(\sigma-1)t_{\max}}({𝕉}^{n})$; moreover, (2.2) holds.

The uniqueness of the solution to system (1.1) in the space under consideration is proven by analogy with [9].

Theorem 2 is proved.

Proof of Theorem 3

Estimate (2.3) is straightforward from Theorem 2 for $\beta\alpha=t_{k}$, $k=1,\dots,\nu$. Assume that $\beta\alpha<t_{k}$. Putting

$$ U^{k}_{h}(x)=P_{k,h}F(x),\quad F(x)={\mathcal{L}}(D_{x})U(x), $$

we derive

$$ \big{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}P_{k,h}({\mathcal{L}}(D)U)(x),L_{p}({𝕉}^{n})\big{\|}\leq c\|\langle x\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}{\mathcal{L}}(D_{x})U(x),\ {\mathbf{L}}_{p}({𝕉}^{n})\| $$

from Lemma 3.3 with a constant $c>0$ independent of $U(x)$ and $h$. Then, for every $\varepsilon>0$ we have

$$ \begin{gathered}\displaystyle J_{k,\varepsilon}=\bigl{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}(x),L_{p}(\{\langle x\rangle\geq\varepsilon\})\bigr{\|}\\ \displaystyle\leq\bigl{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}P_{k,h}({\mathcal{L}}(D)U)(x),\ L_{p}(\{\langle x\rangle\geq\varepsilon\})\bigr{\|}\\ \displaystyle+\bigl{\|}\langle x\rangle^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}(P_{k,h}({\mathcal{L}}(D)U)(x)-U^{k}(x)),\ L_{p}(\{\langle x\rangle\geq\varepsilon\})\bigr{\|}\\ \displaystyle\leq c\|\langle x\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}{\mathcal{L}}(D_{x})U(x),\ {\mathbf{L}}_{p}({𝕉}^{n})\|\\ \displaystyle+c(\varepsilon)\|(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}(P_{k,h}({\mathcal{L}}(D)U)(x)-U^{k}(x)),L_{p}(\{\langle x\rangle\geq\varepsilon\})\|,\end{gathered} $$

(3.17)

where $c(\varepsilon)=(1+1/\varepsilon)^{\sigma(t_{k}-\beta\alpha)}$. By the proof of Theorem 2,

$$ \bigl{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}(P_{k,h}({\mathcal{L}}(D)U)(x)-U^{k}(x)),\ L_{p}({𝕉}^{n})\bigr{\|}\to 0\quad\text{as }h\to 0. $$

Passing to the limit in (3.17) as $h\to 0$ yields

$$ J_{k,\varepsilon}\leq c\|\langle x\rangle^{(1-\sigma)(t_{k}-\beta\alpha)}{\mathcal{L}}(D_{x})U(x),\ {\mathbf{L}}_{p}({𝕉}^{n})\|. $$

Hence, passing to the limit as $\varepsilon\to 0$ implies (2.3).

Theorem 3 is proved.

4. Construction of Approximate Solutions to (1.2)

The proof of Theorem 4 is based on the construction of approximate solutions to boundary value problems for systems with constant coefficients (see [19]). We present the construction in this section.

We suppose that $F(x)\in{\mathbf{C}}^{\infty}(\overline{{𝕉}^{n}_{+}})$ and $F(x)\equiv 0$ for $|x|\gg 1$.

Consider the boundary value problem of the form (1.4) on the half-axis for the system of ordinary differential equations with parameter $s\in{𝕉}^{n-1}\backslash\{0\}$:

$$ \begin{cases}{\mathcal{L}}(is,D_{x_{n}})\omega=\widetilde{F}(s,x_{n}),\quad x_{n}>0,\\ {\mathcal{B}}(is,D_{x_{n}})\omega|_{x_{n}=0}=0,\\ \sup\limits_{x_{n}>0}|\omega|<\infty,\end{cases} $$

(4.1)

where $\widetilde{F}(s,x_{n})$ is the Fourier transform of the vector-function $F(x^{\prime},x_{n})$ with respect to $x^{\prime}$.

By the Lopatinskii condition, (4.1) is uniquely solvable. The solution of (4.1) can be represented as

$$ \omega(s,x_{n})=\omega_{0}(s,x_{n})+v(s,x_{n}), $$

(4.2)

where $\omega_{0}(s,x_{n})$ is a bounded solution to the system

$$ {\mathcal{L}}(is,D_{x_{n}})\omega=\widetilde{F}(s,x_{n}),\quad x_{n}>0, $$

and the vector-function $v(s,x_{n})$ is a solution to (1.4) with

$$ \varphi(s)=-{\mathcal{B}}(is,D_{x_{n}})\omega_{0}(s,x_{n})|_{x_{n}=0}. $$

Put $a(is,i\lambda)=\det{\mathcal{L}}(is,i\lambda)$. Denote by $\widetilde{\mathcal{L}}(is,i\lambda)$ the adjugate matrix to ${\mathcal{L}}(is,i\lambda)$. The identity

$$ {\mathcal{L}}(is,D_{x_{n}})\widetilde{\mathcal{L}}(is,D_{x_{n}})\omega(x_{n})\equiv a(is,D_{x_{n}})\omega(x_{n}) $$

is obviously valid for sufficiently smooth vector-functions $\omega(x_{n})$. Equation (1.3) has no real roots; therefore, the boundary value problem on the axis

$$ \begin{cases}a(is,D_{x_{n}})u=g(x_{n}),\quad-\infty<x_{n}<\infty,\\ \sup\limits_{-\infty<x_{n}<\infty}|u|<\infty\end{cases} $$

(4.3)

has a unique solution for $s\in{𝕉}^{n-1}\backslash\{0\}$ and every bounded $g(x_{n})\in C({𝕉})$. Consequently, using the formula of the solution to (4.3) (for instance, see [25, Chapter 1]), as $\omega_{0}(s,x_{n})$, we can take the bounded vector-function

$$ \omega_{0}(s,x_{n})=\widetilde{\mathcal{L}}(is,D_{x_{n}})R\widetilde{F}(s,x_{n}) $$

(4.4)

where

$$ R\widetilde{F}(s,x_{n})=\int\limits^{x_{n}}_{0}J_{+}(s,x_{n}-y_{n})\widetilde{F}(s,y_{n})\,dy_{n}+\int\limits^{\infty}_{x_{n}}J_{-}(s,x_{n}-y_{n})\widetilde{F}(s,y_{n})\,dy_{n}, $$

$$ J_{+}(s,x_{n})=\frac{1}{2\pi}\int\limits_{\Gamma^{+}}\frac{\exp(ix_{n}\lambda)}{a(is,i\lambda)}\,d\lambda,\quad J_{-}(s,x_{n})=-\frac{1}{2\pi}\int\limits_{\Gamma^{-}}\frac{\exp(ix_{n}\lambda)}{a(is,i\lambda)}\,d\lambda, $$

(4.5)

and the contour $\Gamma^{+}=\Gamma^{+}(s)$ encircles all roots of (1.3) in the upper half-plane, while $\Gamma^{-}=\Gamma^{-}(s)$ encircles the roots in the lower half-plane. Denote by $\{\omega_{1}(s,x_{n}),\dots,\omega_{\mu}(s,x_{n})\}$ the canonical basis of (1.4); i.e., each vector-function $\omega_{j}(s,x_{n})$ is a solution to (1.4) with the unit boundary vector $\varphi=e_{j}$ whose $j$th entry is $1$. Then the vector-function $v(s,x_{n})$ from (4.2) can be represented as

$$ v(s,x_{n})=\sum\limits^{\mu}_{j=1}\varphi^{j}(s)\omega_{j}(s,x_{n}), $$

(4.6)

where $\varphi(s)=-{\mathcal{B}}(is,D_{y_{n}})\omega_{0}(s,y_{n})|_{y_{n}=0}$.

By (4.2), (4.4), and (4.6), we derive the representation of the solution to (4.1) in the form

$$ \omega(s,x_{n})=\widetilde{\mathcal{L}}(is,D_{x_{n}})\,R\,\widetilde{F}(s,x_{n})+\sum^{\mu}_{j=1}\varphi^{j}(s)\omega_{j}(s,x_{n}). $$

(4.7)

Construct a solution to (1.2). By applying the inverse Fourier transform to (4.7) with respect to $s$, we can obtain a formal solution to (1.2). However, the contour integrals (4.5) and the components of the canonical basis of (1.4), in general, have nonintegrable singularities for $s=0$. Therefore, to obtain a formula for a solution to (1.2) we need to regularize the inverse Fourier transform. To this end, use Uspenskii’s integral representation for $f(x^{\prime})\in L_{p}({𝕉}^{n-1})$ (see [26]):

$$ f(x^{\prime})=\lim\limits_{h\to 0}(2\pi)^{1-n}\int\limits^{h^{-1}}_{h}v^{-1}\int\limits_{{𝕉}^{n-1}}\int\limits_{{𝕉}^{n-1}}\exp{(i(x^{\prime}-y^{\prime})s)}G(sv^{\alpha^{\prime}})f(y^{\prime})\,dsdy^{\prime}dv, $$

(4.8)

where

$$ G(s)=2m\langle s\rangle^{2m}\exp(-\langle s\rangle^{2m}),\quad\langle s\rangle^{2}=\sum\limits^{n-1}_{i=1}s^{2/\alpha_{i}}_{i}, $$

(4.9)

and the limit is understood in the sense of convergence in $L_{p}({𝕉}^{n-1})$ (see [25, Chapter 1]). The natural $m$ can be taken however large. By analogy with [19], introduce the vector-functions

$$ U_{h}(x)=(2\pi)^{(1-n)/2}\int\limits^{h^{-1}}_{h}v^{-1}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\omega(s,x_{n})\,dsdv, $$

(4.10)

where $\omega(s,x_{n})$ is defined in (4.7). It follows from the above that

$$ \begin{cases}{\mathcal{L}}(D_{x})U_{h}(x)=F_{h}(x),\quad x\in{𝕉}^{n}_{+},\\ {\mathcal{B}}(D_{x})U_{h}(x^{\prime},x_{n})|_{x_{n}=0}=0,\end{cases} $$

where

$$ F_{h}(x)=(2\pi)^{1-n}\int\limits^{h^{-1}}_{h}v^{-1}\int\limits_{{𝕉}^{n-1}}\int\limits_{{𝕉}^{n-1}}\exp{(i(x^{\prime}-y^{\prime})s)}G(sv^{\alpha^{\prime}})F(y^{\prime},x_{n})\,dsdy^{\prime}dv. $$

By (4.8), $U_{h}(x)$ can be considered as an approximate solution to (1.2), and the existence of a solution to (1.2) reduces to proving the convergence of $\{U_{h}(x)\}$ in ${\mathbf{W}}^{\mathbf{t}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$.

To obtain $L_{p}$-estimates for $U_{h}(x)$, we will use

Lemma 4.1

For all $x_{n}>0$, $s\in{𝕉}^{n-1}\backslash\{0\}$, $\gamma_{n}$, and $\kappa=(\kappa_{1},\dots,\kappa_{n-1})$, the following estimates are valid:

$$ \bigl{|}D^{\kappa}_{s}\bigl{(}D^{\gamma_{n}}_{x_{n}}\widetilde{\mathcal{L}}_{r,l}(is,D_{x_{n}})J_{+}(s,x_{n})\bigr{)}\bigr{|}\leq c\langle s\rangle^{(\gamma_{n}+1)\alpha_{n}-\kappa\alpha^{\prime}-t_{r}}\exp{(-\delta x_{n}\langle s\rangle^{\alpha_{n}})}, $$

$$ \bigl{|}D^{\kappa}_{s}\bigl{(}D^{\gamma_{n}}_{x_{n}}\widetilde{\mathcal{L}}_{r,l}(is,D_{x_{n}})J_{-}(s,-x_{n})\bigr{)}\bigr{|}\leq c\langle s\rangle^{(\gamma_{n}+1)\alpha_{n}-\kappa\alpha^{\prime}-t_{r}}\exp{(-\delta x_{n}\langle s\rangle^{\alpha_{n}})}, $$

$$ \bigl{|}D^{\gamma_{n}}_{x_{n}}D^{\kappa}_{s}\omega_{j}^{r}(s,x_{n})\bigr{|}\leq c\langle s\rangle^{\gamma_{n}\alpha_{n}-\kappa\alpha^{\prime}+m_{j}-t_{r}}\exp{(-\delta x_{n}\langle s\rangle^{\alpha_{n}})}, $$

$$ r,l=1,\dots,\nu,\quad j=1,\dots,\mu, $$

where $c$ and $\delta>0$ are constant, $\widetilde{\mathcal{L}}_{r,l}(is,i\xi_{n})$ are the entries of $\widetilde{\mathcal{L}}(is,i\xi_{n})$, and $\omega_{j}^{r}(s,x_{n})$ are the entries of $\omega_{j}(s,x_{n})$.

Proof. This repeats the proof of Lemma 4.2 from [25, Chapter 4].

5. Solvability of (1.2)

Consider the vector-function $U_{h}(x)$ from (4.10) and represent it, in accord with [19], as

$$ U_{h}(x)=U_{0,h}(x)+\sum\limits_{j=1}^{\mu}U_{j,h}(x), $$

(5.1)

where

$$ U_{0,h}(x)=\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}\mathcal{G}(s,h)\widetilde{\mathcal{L}}(is,D_{x_{n}})R\widetilde{F}(s,x_{n})\,ds, $$

(5.2)

$$ U_{j,h}(x)=\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}\mathcal{G}(s,h)\varphi^{j}(s)\omega_{j}(s,x_{n})\,ds, $$

(5.3)

$$ \mathcal{G}(s,h)=(2\pi)^{(1-n)/2}\int\limits_{h}^{h^{-1}}v^{-1}G(sv^{\alpha^{\prime}})\,dv, $$

(5.4)

$$ \varphi^{j}(s)=-{\mathcal{B}}_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})\,R\widetilde{F}(s,y_{n})|_{y_{n}=0}, $$

$$ {\mathcal{B}}_{j}(is,D_{y_{n}})=(b_{j,1}(is,D_{y_{n}}),\dots,b_{j,\nu}(is,D_{y_{n}})). $$

Denote the $k$th entry of $U_{0,h}(x)$ by $U^{k}_{0,h}(x)$.

Lemma 5.1

Let $\beta=(\beta_{1},\dots,\beta_{n})$ and $\beta\alpha=t_{k}$, $k=1,\dots,\nu$. Then

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{0,h}(x),\,L_{p}({𝕉}^{n}_{+})\bigr{\|}\leq c\bigl{\|}F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\bigr{\|},\quad 0<h<1, $$

(5.5)

with $c>0$ independent of $h$ and $F(x)$; moreover,

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{0,h_{1}}(x)-D^{\beta}_{x}U^{k}_{0,h_{2}}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(5.6)

Proof. This repeats the proof from [19]; we focus on some changes.

Extend the vector-function $F(x^{\prime},x_{n})$ by zero for $x_{n}<0$. Denote the Fourier transform of $F(x)$ in $x=(x^{\prime},x_{n})$ by $\widehat{F}(\xi)$. As demonstrated in [19],

$$ \widetilde{\mathcal{L}}(is,D_{x_{n}})R\widetilde{F}(s,x_{n})\equiv(2\pi)^{-1/2}\int\limits^{\infty}_{-\infty}e^{ix_{n}\xi_{n}}({\mathcal{L}}(is,i\xi_{n}))^{-1}\widehat{F}(s,\xi_{n})\,d\xi_{n} $$

(5.7)

for $s\in{𝕉}^{n-1}\backslash\{0\}$. By (5.7), for $\beta\alpha=t_{k}$ we obtain

$$ D^{\beta}_{x}U^{k}_{0,h}(x)=(2\pi)^{-n/2}\int\limits_{h}^{h^{-1}}v^{-1}\int\limits_{{𝕉}^{n}}\exp{(ix\xi)}G(sv^{\alpha^{\prime}})(i\xi)^{\beta}(({\mathcal{L}}(i\xi))^{-1})_{k}\widehat{F}(\xi)\,d\xi dv,\quad\xi=(s,\xi_{n}), $$

where $(({\mathcal{L}}(i\xi))^{-1})_{k}$ is the $k$th row of $({\mathcal{L}}(i\xi))^{-1}$. By the Lizorkin Theorem [28] and Conditions 1 and 2, the elements of the row $\xi^{\beta}(({\mathcal{L}}(i\xi))^{-1})_{k}$, $\beta\alpha=t_{k}$, are multipliers. Using this and the properties of Uspenskii’s representation [25], we come to (5.5) and (5.6).

Lemma 5.2

Let $\beta=(\beta_{1},\dots,\beta_{n})$, $\beta\alpha=t_{k}$, $k=1,\dots,\nu$. Then

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{j,h}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|,\quad 0<h<1, $$

(5.8)

with $c>0$ independent of $h$ and $F(x)$; moreover,

$$ \bigl{\|}D^{\beta}_{x}U^{k}_{j,h_{1}}(x)-D^{\beta}_{x}U^{k}_{j,h_{2}}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(5.9)

Proof

Consider the $k$th entry of the vector-function $U_{j,h}(x)$ from (5.3). Estimate $D^{\beta}_{x}U^{k}_{j,h}(x)$, where $\beta\alpha=t_{k}$. We follow the scheme of [19]. Represent $D^{\beta}_{x}U^{k}_{j,h}(x)$ as the sum of two summands:

$$ \begin{gathered}\displaystyle D^{\beta}_{x}U^{k}_{j,h}(x)=\int\limits_{{𝕉}^{n-1}}\int\limits_{0}^{\infty}\exp{(ix^{\prime}s)}(is)^{\beta^{\prime}}\mathcal{G}(s,h)\\ \displaystyle\times B_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})D^{\beta_{n}+1}_{y_{n}}\omega_{j}^{k}(s,x_{n}+y_{n})\,dy_{n}dsdv\\ \displaystyle+\int\limits_{{𝕉}^{n-1}}\int\limits_{0}^{\infty}\exp{(ix^{\prime}s)}(is)^{\beta^{\prime}}\mathcal{G}(s,h)\\ \displaystyle\times B_{j}(is,D_{y_{n}})D_{y_{n}}\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})D^{\beta_{n}}_{x_{n}}\omega_{j}^{k}(s,x_{n}+y_{n})\,dy_{n}ds\\ \displaystyle=\Phi^{1}_{j,h,k}(x)+\Phi^{2}_{j,h,k}(x),\end{gathered} $$

(5.10)

where $\mathcal{G}(s,h)$ is defined in (5.4). Consider the first summand. By the property of the Fourier transform, for $x_{n}>0$ we obtain

$$ \begin{gathered}\displaystyle\Phi^{1}_{j,h,k}(x)=\int\limits_{{𝕉}^{n}}\exp{(ix^{\prime}s)}(is)^{\beta^{\prime}}\mathcal{G}(s,h)\\ \displaystyle\times B_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})\theta(y_{n})D^{\beta_{n}+1}_{y_{n}}\omega_{j}^{k}(s,x_{n}+y_{n})\theta(x_{n}+y_{n})\,dy_{n}ds\\ \displaystyle=(2\pi)^{-1}\int\limits_{{𝕉}^{n}}\exp{(ix^{\prime}s-ix_{n}\xi_{n})}\langle s\rangle^{m_{j}-t_{k}+\beta_{n}\alpha_{n}}(is)^{\beta^{\prime}}\mu_{j,k}(s,\xi_{n})\\ \displaystyle\times\bigg{(}\int\limits_{R}\exp{(-it_{n}\xi_{n})}\mathcal{G}(s,h)B_{j}(is,D_{t_{n}})\widetilde{\mathcal{L}}(is,D_{t_{n}})R\widetilde{F}(s,t_{n})\theta(t_{n})\,dt_{n}\bigg{)}\,dsd\xi_{n},\end{gathered} $$

where

$$ \mu_{j,k}(s,\xi_{n})=\langle s\rangle^{t_{k}-m_{j}-\beta_{n}\alpha_{n}}\int\limits_{0}^{\infty}\exp{(iy_{n}\xi_{n})}D^{\beta_{n}+1}_{y_{n}}\omega_{j}^{k}(s,y_{n})\,dy_{n}. $$

Conditions 1–4 yield

$$ \omega_{j}^{k}(s,x_{n})=c^{t_{k}-m_{j}}\omega_{j}^{k}(c^{\alpha^{\prime}}s,c^{-\alpha_{n}}x_{n}),\quad c>0; $$

and so, by the Lizorkin Theorem [28], $\mu_{j,k}(s,\xi_{n})$ is a multiplier. Then

$$ \begin{gathered}\displaystyle\bigl{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}\leq c_{1}\bigg{\|}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}(is)^{\beta^{\prime}}\mathcal{G}(s,h)\\ \displaystyle\times\langle s\rangle^{m_{j}-t_{k}+\beta_{n}\alpha_{n}}B_{j}(is,D_{x_{n}})\widetilde{\mathcal{L}}(is,D_{x_{n}})R\widetilde{F}(s,x_{n})\theta(x_{n})\,ds,L_{p}({𝕉}^{n})\bigg{\|}\,dv.\end{gathered} $$

Since the elements

$$ (\langle s\rangle^{\alpha_{n}}+i\xi_{n})\langle s\rangle^{m_{j}-\alpha_{n}}B_{j}(is,i\xi_{n}){\mathcal{L}}^{-1}(is,i\xi_{n}) $$

(5.11)

are multipliers, we obtain

$$ \begin{gathered}\displaystyle\bigl{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}\leq c_{2}\bigg{\|}\int\limits_{{𝕉}^{n}}\exp{(ix^{\prime}s+ix_{n}\xi_{n})}\mathcal{G}(s,h)(is)^{\beta^{\prime}}\langle s\rangle^{-t_{k}+\beta_{n}\alpha_{n}}\\ \displaystyle\times\frac{\langle s\rangle^{\alpha_{n}}}{\langle s\rangle^{\alpha_{n}}+i\xi_{n}}\widehat{F}(s,\xi_{n})\,d\xi_{n}ds,{\mathbf{L}}_{p}({𝕉}^{n}_{+})\bigg{\|}.\end{gathered} $$

By applying

$$ \int\limits_{𝕉}\frac{\exp{(ix_{n}\xi_{n})}}{\langle s\rangle^{\alpha_{n}}+i\xi_{n}}\widehat{F}(s,\xi_{n})\,d\xi_{n}=(2\pi)^{-1/2}\int\limits_{0}^{x_{n}}\exp{(-(x_{n}-y_{n})\langle s\rangle^{\alpha_{n}})}\widetilde{F}(s,y_{n})\,dy_{n} $$

and Young’s inequality, we derive

$$ \begin{gathered}\displaystyle\big{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c_{4}\bigg{\|}\int\limits_{{𝕉}^{n}}\int\limits_{{𝕉}^{n-1}}\exp{(i(x^{\prime}-y^{\prime})s)}\exp{(-(x_{n}-y_{n})\langle s\rangle^{\alpha_{n}})}\theta(x_{n}-y_{n})\\ \displaystyle\times(is)^{\beta^{\prime}}\langle s\rangle^{\alpha_{n}-t_{k}+\beta_{n}\alpha_{n}}\mathcal{G}(s,h)F(y)\theta(y_{n})\,dsdy,{\mathbf{L}}_{p}({𝕉}^{n})\bigg{\|}\\ \displaystyle\leq c_{5}\bigg{\|}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}\exp{(-x_{n}\langle s\rangle^{\alpha_{n}})}\theta(x_{n})\\ \displaystyle\times\mathcal{G}(s,h)(is)^{\beta^{\prime}}\langle s\rangle^{\alpha_{n}-t_{k}+\beta_{n}\alpha_{n}}\,ds,L_{1}({𝕉}^{n})\bigg{\|}\|F(y),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|.\end{gathered} $$

Since

$$ \int\limits_{h}^{h^{-1}}v^{-1}G(sv^{\alpha^{\prime}})\,dv\equiv\exp{(-h^{2m}\langle s\rangle^{2m})}-\exp{(-h^{-2m}\langle s\rangle^{2m})}, $$

by Minkowski’s inequality and the fact that $\beta^{\prime}\alpha^{\prime}+\beta_{n}\alpha_{n}=t_{k}$, we obtain

$$ \begin{gathered}\displaystyle\big{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c_{6}\bigg{\|}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}\exp{(-x_{n}\langle s\rangle^{\alpha_{n}})}\theta(x_{n})\\ \displaystyle\times\exp{(-\langle s\rangle^{2m})}(is)^{\beta^{\prime}}\langle s\rangle^{\alpha_{n}-t_{k}+\beta_{n}\alpha_{n}}\,ds,L_{1}({𝕉}^{n})\bigg{\|}\|F(y),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|.\end{gathered} $$

Choosing $m$ in (4.9) sufficiently large, we arrive at the sought estimate for $\Phi^{1}_{j,h,k}(x)$. Analogous arguments for $\Phi^{2}_{j,h,k}(x)$ yield (5.8). Convergence (5.9) is proved similarly.

The lemma is proved.

Lemma 5.3

Let $\beta=(\beta_{1},\dots,\beta_{n})$, $0\leq\beta\alpha<t_{k}$, and $|\alpha|/p>\sigma(t_{k}-\beta\alpha)>t_{k}-\beta\alpha-|\alpha|/p^{\prime}$, $1/p+1/p^{\prime}=1$. Then

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{0,h}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c(\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|+\|(1+\langle x\rangle)^{\sigma(t_{k}-\beta\alpha)}F(x),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|),\quad 0<h<1,\end{gathered} $$

(5.12)

with a constant $c>0$ independent of $F(x)$ and $h$; moreover,

$$ \big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\big{(}D^{\beta}_{x}U^{k}_{0,h_{1}}(x)-D^{\beta}_{x}U^{k}_{0,h_{2}}(x)\big{)},L_{p}({𝕉}^{n}_{+})\big{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(5.13)

Proof

We will present the arguments for $|\beta|=0$.

Consider the $k$th entry of $U_{0,h}(x)$ from (5.2) and represent it as the sum of two summands:

$$ U^{k}_{0,h}(x)=U^{k}_{0,h,1}(x)+U^{k}_{0,h,2}(x), $$

(5.14)

where

$$ U^{k}_{0,h,1}(x)=(2\pi)^{(1-n)/2}\int\limits_{h}^{1}v^{-1}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\widetilde{\mathcal{L}}_{k}(is,D_{x_{n}})R\widetilde{F}(s,x_{n})\,dsdv, $$

$$ U^{k}_{0,h,2}(x)=(2\pi)^{(1-n)/2}\int\limits_{1}^{h^{-1}}v^{-1}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\widetilde{\mathcal{L}}_{k}(is,D_{x_{n}})R\widetilde{F}(s,x_{n})\,dsdv, $$

with $\widetilde{\mathcal{L}}_{k}(is,i\xi_{n})$ the $k$th row of $\widetilde{\mathcal{L}}(is,i\xi_{n})$. Using the homogeneity of the entries of ${\mathcal{L}}(is,i\xi_{n})$ and the definition of $R\widetilde{F}(s,x_{n})$, we derive

$$ \begin{gathered}\displaystyle U^{k}_{0,h,1}(x)=(2\pi)^{1-n}\int\limits_{h}^{1}v^{-1}\int\limits_{{𝕉}^{n}}K_{0,k}(v,x^{\prime}-y^{\prime},x_{n}-y_{n})\theta(y_{n})F(y)\,dydv,\\ \displaystyle U^{k}_{0,h,2}(x)=(2\pi)^{1-n}\int\limits_{1}^{h^{-1}}v^{-1}\int\limits_{{𝕉}^{n}}K_{0,k}(v,x^{\prime}-y^{\prime},x_{n}-y_{n})\theta(y_{n})F(y)\,dydv,\end{gathered} $$

(5.15)

where

$$ K_{0,k}(v,x^{\prime},x_{n})=\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\widetilde{\mathcal{L}}_{k}(is,D_{x_{n}})[J_{+}(s,x_{n})\theta(x_{n})+J_{-}(s,x_{n})\theta(-x_{n})]\,ds $$

and $\theta(x_{n})$ is the Heaviside function.

Consider $U^{k}_{0,h,1}(x)$. Since $\sigma t_{k}\geq 0$; therefore,

$$ \big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq\big{\|}U^{k}_{0,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}. $$

Conditions 1 and 2 imply that $K_{0,k}(v,x^{\prime},x_{n})=v^{t_{k}-|\alpha|}K_{0,k}(1,x^{\prime}v^{-\alpha^{\prime}},x_{n}v^{-\alpha_{n}})$, and, after the change $z_{j}=x_{j}v^{-\alpha_{j}}$, $j=1,\dots,n$, we obtain

$$ \big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c_{1}\int\limits_{h}^{1}v^{-1+t_{k}}\,dv\|K_{0,k}(1,z^{\prime},z_{n}),{\mathbf{L}}_{1}({𝕉}^{n})\|\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|. $$

In line with [17, 18], we can show that choosing a sufficiently large number $m$ in (4.9) in the definition of $G(s)$ yields

$$ \|K_{0,k}(1,z^{\prime},z_{n}),{\mathbf{L}}_{1}({𝕉}^{n})\|\leq c<\infty. $$

Then

$$ \big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|. $$

(5.16)

Consider the function $U^{k}_{0,h,2}(x)$ from (5.15). Applying (3.15), as well as Minkowski’s and Young’s inequalities, we obtain

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c\int\limits_{1}^{h^{-1}}v^{-1}\|\langle x\rangle^{-\sigma t_{k}}K_{0,k}(v,x^{\prime},x_{n}),L_{p}({𝕉}^{n})\|\,dv\|(1+\langle y\rangle)^{\sigma t_{k}}F(y),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|.\end{gathered} $$

By the arguments similar to estimation of $U^{k}_{0,h,1}(x)$, since $|\alpha|/p>\sigma t_{\max}$, we derive

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c_{1}\int\limits_{1}^{h^{-1}}v^{-1+(1-\sigma)t_{k}-|\alpha|/p^{\prime}}\,dv\|(1+\langle y\rangle)^{\sigma t_{k}}F(y),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|.\end{gathered} $$

Hence, the condition $|\alpha|/p^{\prime}>(1-\sigma)t_{k}$ implies

$$ \big{\|}(1+\langle x\rangle)^{-\sigma t_{k}}U^{k}_{0,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|(1+\langle y\rangle)^{\sigma t_{k}}F(y),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|, $$

(5.17)

with a constant $c>0$ independent of $F(x)$ and $h$.

We obtain (5.12) for $|\beta|=0$ from (5.14), (5.16), and (5.17). The proof of (5.12) for $t_{k}>\beta\alpha>0$, as well as (5.13), is carried out in exactly the same way.

The lemma is proved.

Lemma 5.4

Let $\beta=(\beta_{1},\dots,\beta_{n})$, $0\leq\beta\alpha<t_{k}$, and $|\alpha|/p>\sigma(t_{k}-\beta\alpha)>t_{k}-\beta\alpha-|\alpha|/p^{\prime}$, $1/p+1/p^{\prime}=1$. Then

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}D^{\beta}_{x}U^{k}_{j,h}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c(\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|+\|(1+\langle x\rangle)^{\sigma(t_{k}-\beta\alpha)}F(x),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|),\quad 0<h<1,\end{gathered} $$

(5.18)

with a constant $c>0$ independent of $F(x)$ and $h$; moreover,

$$ \big{\|}(1+\langle x\rangle)^{-\sigma(t_{k}-\beta\alpha)}\big{(}D^{\beta}_{x}U^{k}_{j,h_{1}}(x)-D^{\beta}_{x}U^{k}_{j,h_{2}}(x)\big{)},L_{p}({𝕉}^{n}_{+})\big{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

(5.19)

Proof

The main difficulty in establishing (5.18) is the case that $|\beta|=0$. To settle it, consider the $k$th entry of the vector-function $U_{j,h}(x)$ from (5.3) and represent the latter, in line with [19], as

$$ U^{k}_{j,h}(x)=U^{k}_{j,h,1}(x)+U^{k}_{j,h,2}(x), $$

(5.20)

where

$$ \begin{gathered}\displaystyle U^{k}_{j,h,1}(x)=(2\pi)^{(1-n)/2}\int\limits_{h}^{1}v^{-1}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\int\limits_{0}^{\infty}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\\ \displaystyle\times D_{y_{n}}(B_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})\omega_{j}^{k}(s,x_{n}+y_{n}))\,dy_{n}dsdv,\\ \displaystyle U^{k}_{j,h,2}(x)=(2\pi)^{(1-n)/2}\int\limits_{1}^{h^{-1}}v^{-1}\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\int\limits_{0}^{\infty}W_{j}(s,z_{n})\widetilde{F}(s,z_{n})\,dz_{n}\omega_{j}^{k}(s,x_{n})\,dsdv,\\ \displaystyle W_{j}(s,x_{n})=-{\mathcal{B}}_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})J_{-}(s,y_{n}-x_{n})|_{y_{n}=0}.\end{gathered} $$

Since $\sigma t_{k}\geq 0$; therefore,

$$ \big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq\bigl{\|}U^{k}_{j,h,1}(x),L_{p}({𝕉}^{n}_{+})\bigr{\|}. $$

(5.21)

Estimate the first summand $U^{k}_{j,h,1}(x)$ from (5.20). By analogy with [19], represent this function as

$$ \begin{gathered}\displaystyle U^{k}_{j,h,1}(x)=(2\pi)^{(1-n)/2}\int\limits_{h}^{1}v^{-1}\int\limits_{{𝕉}^{n-1}}\int\limits_{0}^{\infty}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\\ \displaystyle\times B_{j}(is,D_{y_{n}})\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})D_{y_{n}}\omega_{j}^{k}(s,x_{n}+y_{n})\,dy_{n}dsdv\\ \displaystyle+(2\pi)^{(1-n)/2}\int\limits_{h}^{1}v^{-1}\int\limits_{{𝕉}^{n-1}}\int\limits_{0}^{\infty}\exp{(ix^{\prime}s)}G(sv^{\alpha^{\prime}})\\ \displaystyle\times B_{j}(is,D_{y_{n}})D_{y_{n}}\widetilde{\mathcal{L}}(is,D_{y_{n}})R\widetilde{F}(s,y_{n})\omega_{j}^{k}(s,x_{n}+y_{n})\,dy_{n}dsdv\\ \displaystyle=\Phi^{1}_{j,h,k}(x)+\Phi^{2}_{j,h,k}(x).\end{gathered} $$

(5.22)

Arguing similarly to the estimation of $\Phi^{1}_{j,h,k}(x)$ in Lemma 5.2 and using the fact that the function

$$ \langle s\rangle^{t_{k}-m_{j}}\int\limits_{0}^{\infty}\exp{(iy_{n}\xi_{n})}D_{y_{n}}\omega_{j}^{k}(s,y_{n})\,dy_{n} $$

and the elements of (5.11) are multipliers, we obtain

$$ \big{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c_{5}\int\limits_{h}^{1}v^{-1}\big{\|}K_{k}^{\alpha_{n}}(v,x^{\prime},x_{n}),L_{1}({𝕉}^{n})\big{\|}\,dv\|F(y),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|, $$

where

$$ K_{k}^{\alpha_{n}}(v,x^{\prime},x_{n})=\int\limits_{{𝕉}^{n-1}}\exp{(ix^{\prime}s)}\exp{(-x_{n}\langle s\rangle^{\alpha_{n}})}\theta(x_{n})G(sv^{\alpha^{\prime}})\langle s\rangle^{\alpha_{n}-t_{k}}\,ds. $$

Since

$$ K_{k}^{\alpha_{n}}(v,x^{\prime},x_{n})=v^{t_{k}-|\alpha|}K_{k}^{\alpha_{n}}(1,x^{\prime}v^{-\alpha^{\prime}},x_{n}v^{-\alpha_{n}}), $$

making the change

$$ z_{i}=x_{i}v^{-\alpha_{i}},\quad i=1,\dots,n, $$

we derive

$$ \big{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c_{5}\int\limits_{h}^{1}v^{-1+t_{k}}\,dv\big{\|}K_{k}^{\alpha_{n}}(1,z^{\prime},z_{n}),L_{1}({𝕉}^{n})\big{\|}\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|. $$

By the definition of $K_{k}^{\alpha_{n}}(1,z^{\prime},z_{n})$, we obtain

$$ \big{\|}\Phi^{1}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|, $$

with a constant $c>0$ independent of $F(x)$ and $h$. By similar arguments for the second summand in (5.22), we arrive at the estimate

$$ \big{\|}\Phi^{2}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|. $$

From (5.21), (5.22), and the above inequalities we obtain that

$$ \big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,1}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|, $$

(5.23)

with a constant $c>0$ independent of $F(x)$ and $h$. Estimate $U^{k}_{j,h,2}(x)$ from (5.20). Using (3.15) and Minkowski’s inequality, we derive

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c_{2}\int\limits_{1}^{h^{-1}}v^{-1}\bigg{\|}\int\limits_{{𝕉}^{n}_{+}}\langle x-y\rangle^{-t_{k}\sigma}\\ \displaystyle\times\bigg{|}\int\limits_{{𝕉}^{n-1}}\exp{(i(x^{\prime}-y^{\prime})s)}G(sv^{\alpha^{\prime}})W_{j}(s,y_{n})\omega^{k}_{j}(s,x_{n})\,ds\bigg{|}\\ \displaystyle\times(1+\langle y\rangle)^{t_{k}\sigma}|F(y)|\,dy,L_{p}({𝕉}^{n}_{+})\bigg{\|}\,dv=c_{2}\int\limits_{1}^{h^{-1}}v^{-1}\bigg{\|}\int\limits_{{𝕉}^{n}_{+}}|K_{j,k}(v,x^{\prime}-y^{\prime},x_{n},y_{n})|\\ \displaystyle\times((1+\langle y\rangle)^{t_{k}\sigma}|F(y)|)^{(1/p+1/p^{\prime})}\,dy,{\mathbf{L}}_{p}({𝕉}^{n}_{+})\bigg{\|}\,dv,\end{gathered} $$

where

$$ K_{j,k}(v,x^{\prime}-y^{\prime},x_{n},y_{n})=\langle x-y\rangle^{-t_{k}\sigma}\int\limits_{{𝕉}^{n-1}}\exp{(i(x^{\prime}-y^{\prime})s)}G(sv^{\alpha^{\prime}})W_{j}(s,y_{n})\omega^{k}_{j}(s,x_{n})\,ds. $$

(5.24)

Applying Hölder’s inequality, we obtain

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c_{2}\int\limits_{1}^{h^{-1}}v^{-1}\bigg{\|}\int\limits_{{𝕉}^{n}_{+}}|K_{j,k}(v,x^{\prime}-y^{\prime},x_{n},y_{n})|^{p}(1+\langle y\rangle)^{t_{k}\sigma}|F(y)|\,dy,L_{1}({𝕉}^{n}_{+})\bigg{\|}^{1/p}\,dv\\ \displaystyle\times\|(1+\langle y\rangle)^{t_{k}\sigma}|F(y)|,{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|^{1/p^{\prime}}.\end{gathered} $$

Using the Tonelli Theorem, rewrite this inequality as

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}\Phi^{2}_{j,h,k}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c_{2}\int\limits_{1}^{h^{-1}}v^{-1}\bigg{\|}\int\limits_{{𝕉}^{n}_{+}}|K_{j,k}(v,x^{\prime}-y^{\prime},x_{n},y_{n})|^{p}\,dx(1+\langle y\rangle)^{t_{k}\sigma}|F(y)|,L_{1}({𝕉}^{n}_{+})\bigg{\|}^{1/p}\,dv\\ \displaystyle\times\|(1+\langle y\rangle)^{t_{k}\sigma}F(y),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|^{1/p^{\prime}}.\end{gathered} $$

(5.25)

Put

$$ A_{j,k}(v,y)=\int\limits_{{𝕉}^{n}}|K_{j,k}(v,x^{\prime}-y^{\prime},x_{n},y_{n})|^{p}\theta{(x_{n})}\theta{(y_{n})}\,dx. $$

Repeating the arguments from Lemma 2 in [18, § 4] and using Lemma 4.1 and the inequality $|\alpha|/p>t_{k}\sigma$, we derive

$$ A_{j,k}(v,y)\leq cv^{p((1-\sigma)t_{k}-|\alpha|/p^{\prime})} $$

(5.26)

with a constant $c>0$ independent of $v$ and $y$. Inserting (5.26) in (5.25), we get

$$ \begin{gathered}\displaystyle\big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\\ \displaystyle\leq c\int\limits_{1}^{h^{-1}}\!v^{-1+(1-\sigma)t_{k}-|\alpha|/p^{\prime}}\,dv\|(1+\langle x\rangle)^{t_{k}\sigma}F(x),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|.\end{gathered} $$

By the hypotheses of the lemma, $(1-\sigma)t_{k}-|\alpha|/p^{\prime}<0$, and so

$$ \big{\|}(1+\langle x\rangle)^{-t_{k}\sigma}U^{k}_{j,h,2}(x),L_{p}({𝕉}^{n}_{+})\big{\|}\leq c\|(1+\langle x\rangle)^{t_{k}\sigma}F(x),{\mathbf{L}}_{1}({𝕉}^{n}_{+})\|, $$

(5.27)

with a constant $c>0$ independent of $F(x)$ and $h$.

Since

$$ U^{k}_{j,h}(x)=U^{k}_{j,h,1}(x)+U^{k}_{j,h,2}(x), $$

by (5.23) and (5.27), we arrive at (5.18) for $|\beta|=0$. Obtaining (5.18) for $t_{k}>\beta\alpha>0$, as well as (5.19), is carried out similarly.

The lemma is proved.

Proof of Theorem 4. This follows from Lemmas 5.1–5.4.

Indeed, by the above lemmas and under conditions (2.1), the vector-function $U_{h}(x)$ defined in (4.10) and (5.1) belongs to ${\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$ for every $F(x)\in{\mathbf{C}}^{\infty}(\overline{{𝕉}^{n}_{+}})$ such that $F(x)\equiv 0$ for $|x|\gg 1$ and the following estimate holds:

$$ \big{\|}U_{h}(x),{\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})\big{\|}\leq c(\|F(x),{\mathbf{L}}_{p}({𝕉}^{n}_{+})\|+\|F(x),{\mathbf{L}}_{1,-\sigma t_{\max}}({𝕉}^{n}_{+})\|) $$

with a constant $c>0$ independent of $F(x)$ and $h$; moreover,

$$ \big{\|}U_{h_{1}}(x)-U_{h_{2}}(x),{\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})\big{\|}\to 0\quad\text{as }h_{1},h_{2}\to 0. $$

By completeness of ${\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$, there exists $U(x)\in{\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$ such that

$$ \big{\|}U_{h}(x)-U(x),{\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})\big{\|}\to 0\quad\text{as }h\to 0. $$

$U_{x}$ is a solution to (1.2) and estimate (2.4) holds for it. Since the set of the functions in $C^{\infty}(\overline{{𝕉}^{n}_{+}})$ vanishing for large $|x|$ is everywhere dense in $L_{p}({𝕉}^{n}_{+})\cap L_{1,-\sigma t_{\max}}({𝕉}^{n}_{+})$, boundary value problem (1.2) is solvable in ${\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$ for every vector-function $F(x)\in{\mathbf{L}}_{p}({𝕉}^{n}_{+})\cap{\mathbf{L}}_{1,-\sigma t_{\max}}({𝕉}^{n}_{+})$, and (2.4) holds for the solution.

The proof of uniqueness of the solution to (1.2) in ${\mathbf{W}}^{{\mathbf{t}}/\alpha}_{p,\sigma}({𝕉}^{n}_{+})$, $\sigma\in[0,1]$, follows the proof of uniqueness of solutions to the boundary value problem for quasielliptic equations (see [18]).

Theorem 4 is proved.