UDC 517.947.42 : 513.881

ON INTEGRALS OF POTENTIAL TYPE AND EMBEDDING THEOREMS IN SPACES WITH MIXED NORM

A. Kh. Gudiev

Introduction. This article is devoted to the study of properties of functions of many variables belonging to the Sobolev space with mixed norm

\[ W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k), \]

and also to the study of properties of abstract additive functions of sets \(\varphi(E)\) belonging to the space

\[ \psi^{(l)}_{(p_1,p_2,\ldots,p_k)}(X;\Omega_1\times\Omega_2\times\cdots\times\Omega_k). \]

Definitions of these spaces are given below.

In order to state more concretely the questions that are solved in this article, we shall say that a given function space is an ordinary function space if the space \(L_p\) is used in its definition; if, however, the space
\[ L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k) \]
is used in its definition, then we shall say that this is a space with mixed norm.

Definition. Embedding theorems in which an embedding of an ordinary function space \(M_1\) into a space \(M_2\) of the same type is established will be called embedding theorems of type I. This type of embedding theorem will be denoted by

\[ M_1\to M_2. \]

Definition. Embedding theorems in which an embedding of an ordinary function space \(M_1\) into a space \(N_2\) with mixed norm is established will be called embedding theorems of type II. This type of embedding theorem will be denoted by

\[ M_1\to N_2. \]

Embedding theorems of type I originate with the works of S. L. Sobolev and at present have been studied rather well (see the survey article by S. M. Nikol’skii [7]).

Embedding theorems of type II arose in connection with a hypothesis of S. L. Sobolev and S. M. Nikol’skii [4]. As is known, theorems of type II are more general than theorems of type I, in the sense that the corresponding theorems of type I follow from theorems of type II as a special case. In addition, with the appearance of embedding theorems of type II it became possible to speak, in a definite sense, about traces of functions from \(W_p^{(l)}\), \(B_p^{(l)}\), \(H_p^r\), \(\psi_p^{(l)}\) on hyperplanes or smooth manifolds of any number of dimensions.

Research on embedding theorems of type II has only just begun. However, a third type of embedding theorem is also possible, more general than types I and II. We shall denote this type by

\[ N_1 \to N_2, \]

where \(N_i\) are spaces with mixed norm.

In this article it is proved that, under certain conditions, embedding theorems of type III hold.

These results are the first results establishing an embedding of a space with mixed norm into another space of the same type.

We note that the corresponding theorems of types I and II follow from a theorem of type III as special cases.

In all embedding theorems of type II [5, 15–17], restrictions are imposed on the parameters \(p_1, p_2, \ldots, p_k\) of the spaces with mixed norm:

\[ p_1 \leq p_2 \leq \cdots \leq p_k. \]

The results of the present work make it possible to remove these restrictions. As it turns out, these restrictions were not essential, but followed from the method of proof.

One of the main results of the article is Theorem 1, which characterizes the action of an operator of potential type in spaces with mixed norm and is, it seems to us, the most general theorem of this kind. Special cases of this theorem were considered at various times in the works [1, 2, 6, 9–15, 17, 18].

In the case of spaces with mixed norm, the result closest to our result (Theorem 1) is the result obtained in [9] (p. 321). The case considered by A. Benedek and Panzone (\(R^m = R^s;\ R^{m_i} = R^{s_i},\ i = 1, 2, \ldots, k\)) requires only a successive application of the theorem of S. L. Sobolev ([11], p. 474). However, if even \(R^m = R^s\), but \(R^{m_i} \ne R^{s_i}\), although for at least one value of \(i\), then the proof, unfortunately, becomes considerably more complicated.

Theorem 1 makes it possible to obtain general embedding theorems of type III for the spaces

\[ W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k); \]

\[ \Psi^{(l)}_{(p_1,p_2,\ldots,p_k)}(X;\ \Omega_1 \times \Omega_2 \times \cdots \times \Omega_k). \]

§ 1. PRELIMINARY REMARKS

Let \(R^n\) be \(n\)-dimensional Euclidean space; let \(n_i\) be positive integers such that

\[ \sum_{1}^{k} n_i = n; \]

let \(\Omega_i\) be an \(n_i\)-dimensional domain in the \(n_i\)-dimensional Euclidean space \(R^{n_i}\); in particular, \(\Omega_i\) may coincide with \(R^{n_i}\). Let \(F\) be a function defined in the domain \(\Omega_1 \times \Omega_2 \times \cdots \times \Omega_k\); and let \(p_1, p_2, \ldots, p_k\) be positive real numbers.

Introduce the notation

\[ A^{(p_j,\ldots,p_i)}_{(\Omega_j,\ldots,\Omega_i)}[F] = \left( \int_{\Omega_j} \left( \int_{\Omega_{j-1}} \cdots \left( \int_{\Omega_{i+1}} \left( \int_{\Omega_i} F\,d\omega_i \right)^{\frac{p_{i+1}}{p_i}} d\omega_{i+1} \right)^{\frac{p_{i+2}}{p_{i+1}}} \cdots d\omega_{j-1} \right)^{\frac{p_j}{p_{j-1}}} \times d\omega_j \right)^{\frac{1}{p_j}}, \tag{1} \]

where \(1 \leq i \leq j \leq k\); \(d\omega_\sigma\) is the volume element of the domain \(\Omega_\sigma\); \(1 \leq \sigma \leq k\).

Spaces with mixed norm \(L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\), introduced in [8, 9, 15], are defined as the set of functions \(F\), given on \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\), for which the norm defined by the equality

\[ \|F\|_{L_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)} = A_{(\Omega_k,\ldots,\Omega_1)}^{(p_k,\ldots,p_1)}\bigl(|F|^{p_1}\bigr) \]

is finite.

Let us note that, generally speaking,

\[ \|F\|_{L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k})} \ne \|F\|_{L_{(p_1,p_2,\ldots,p_k)}(R^{n'_1},R^{n'_2},\ldots,R^{n'_k})}, \]

provided only that \(R^{n_i}\) does not coincide with \(R^{n'_i}\) for all \(1\le i\le k\)
\((R^{n_1}\times R^{n_2}\times\cdots\times R^{n_k} = R^{n'_1}\times R^{n'_2}\times\cdots\times R^{n'_k}=R^n)\).
Therefore it is expedient to use the notation
\(L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k})\)
instead of the accepted and more compact notation
\(L_{(p_1,p_2,\ldots,p_k)}(R^n)\), for even with a fixed representation
\(R^n=R^{n_1}\times R^{n_2}\times\cdots\times R^{n_k}\) (if the sequence in which the \(R^{n_i}\) are taken in (1) is not specified) it will denote, generally speaking, \(k!\) different spaces. In general, using the given numbers \(p_1,p_2,\ldots,p_k\) and subspaces \(R^{n_1},\ldots,R^{n_k}\) (if the order in which they are taken in (1) is not fixed), one can define \((k!)^2\) different spaces. It is interesting to note that, although, generally speaking, these spaces cannot be ordered with respect to the operation of “inclusion” \(\subseteq\), nevertheless the following holds.

Lemma 1. Let \(p_1,p_2,\ldots,p_k\) be positive real numbers:
\(1\le p_i<\infty\), \(i=1,2,\ldots,k\), and let
\(p_{11},p_{12},\ldots,p_{1k};\ p_{21},p_{22},\ldots,p_{2k}\) be the same numbers as \(p_1,p_2,\ldots,p_k\), but arranged, respectively, in nondecreasing and nonincreasing order. Then

\[ L_{(p_{11},p_{12},\ldots,p_{1k})}(R^{n_{11}},R^{n_{12}},\ldots,R^{n_{1k}}) \subseteq L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k}) \subseteq \]

\[ \subseteq L_{(p_{21},p_{22},\ldots,p_{2k})}(R^{n_{21}},R^{n_{22}},\ldots,R^{n_{2k}}), \]

where \(R^{n_{1i}}=R^{n_j}\) \((R^{n_{2i}}=R^{n_j})\) if the number \(p_j\) in the system of numbers
\(p_{11},p_{12},\ldots,p_{1k}\) \((p_{21},p_{22},\ldots,p_{2k})\) is denoted by
\(p_{1i}\) \((p_{2i})\).

To prove this lemma it is enough, in a certain order depending on
\(L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k})\), to apply the required number of times the generalized integral Minkowski inequality to the expression

\[ \|F\|_{L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k})}, \]

if the assertion

\[ L_{(p_{11},p_{12},\ldots,p_{1k})}(R^{n_{11}},R^{n_{12}},\ldots,R^{n_{1k}}) \subseteq L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k}), \]

is being proved, and to the expression

\[ \|F\|_{L_{(p_{21},p_{22},\ldots,p_{2k})}(R^{n_{21}},R^{n_{22}},\ldots,R^{n_{2k}})}, \]

if the assertion

\[ L_{(p_1,p_2,\ldots,p_k)}(R^{n_1},R^{n_2},\ldots,R^{n_k}) \subseteq L_{(p_{21},p_{22},\ldots,p_{2k})}(R^{n_{21}},R^{n_{22}},\ldots,R^{n_{2k}}) \]

is being proved. Lemma 1 allows one to remove the restriction

\[ p_1\le p_2\le \cdots \le p_n \]

in the embedding theorems

\[ H_{p_1}^{r}\to H_{(p_1,p_2,\ldots,p_n)}^{(\rho)}, \]

obtained by S. M. Nikol’skii in [5].

§ 2. ON INTEGRALS OF POTENTIAL TYPE

We now consider \(s\)- and \(m\)-dimensional subspaces \(R^s, R^m\) of the space \(R^n\), which, in particular, may coincide with \(R^n\) and satisfy the condition \(R^s\cap R^m\ne 0\).

Introduce for consideration the spaces
\(L_{(p_1,p_2,\ldots,p_k)}(R^{s_1}, R^{s_2}, \ldots, R^{s_k})\) and
\(L_{(q_1,q_2,\ldots,q_\tau)}(R^{m_1}, R^{m_2}, \ldots, R^{m_\tau})\).
Here \(\sum_1^k s_i=s;\ \sum_1^\tau m_i=m;\ 1\le s,\ m\le n;\)
\(R^{s_1}\times R^{s_2}\times\cdots\times R^{s_k}=R^s;\)
\(R^{m_1}\times R^{m_2}\times\cdots\times R^{m_\tau}=R^m\).
It is important to note that the representations
\(R^s=R^{s_1}\times R^{s_2}\times\cdots\times R^{s_k}\) and
\(R^m=R^{m_1}\times R^{m_2}\times\cdots\times R^{m_\tau}\) are in no way dependent on each other. We represent each point \(\bar x\in R^m\) in the form
\(\bar x=(\bar x^{(1)}, \bar x^{(2)}, \ldots, \bar x^{(\tau)})\), where
\(\bar x^{(i)}=(x_1^{(i)}, x_2^{(i)}, \ldots, x_{m_i}^{(i)})\) is a point of the space \(R^{m_i}\). Similarly, we represent each point \(\bar y\) of the space \(R^s\) in the form
\(\bar y=(\bar y^{(1)}, \bar y^{(2)}, \ldots, \bar y^{(k)})\), where
\(\bar y^{(j)}=(y_1^{(j)}, y_2^{(j)}, \ldots, y_{s_j}^{(j)})\) is a point of the space \(R^{s_j}\).

Let \(M\) and \(N\) be the sets of natural numbers from 1 to \(\tau\) and from 1 to \(k\), respectively.

Denote by \(M_1\) the set of those natural numbers \(i\in M\) for which \(R^{m_i}\cap R^s\ne0\), and by \(M_2\) denote the set \(M\setminus M_1\). Similarly, by \(N_1\) denote the set of those natural numbers \(j\in N\) for which \(R^{s_j}\cap R^m\ne0\), and define \(N_2\) by the equality \(N_2=N\setminus N_1\).

Let \(N_{ij}\) denote the set of those numbers \(j\) for which \(R^{m_i}\cap R^{s_j}\ne0\), and \(M_{ij}\) the set of those numbers \(i\) for which \(R^{m_i}\cap R^{s_j}\ne0\).

Theorem 1. If \(\bar f(\bar y)\in L_{(p_1,p_2,\ldots,p_k)}(R^{s_1}, R^{s_2}, \ldots, R^{s_k})\);
\(\lambda=\sum_1^\tau \dfrac{m_i}{q_i}+\sum_1^k \dfrac{s_j}{p_j'};\)
\(p_j, q_i\ (i\in M;\ j\in N)\) are real positive numbers greater than one and satisfying the conditions:

a) \(q_i>p_j\), if \(i\in M_1;\ j\in N_{ij}\),

b) \(q_i\ge \min_{j\in M_1} q_j\), if \(i\in M_2\),

c) \(p_i\le \max_{i\in N_1} p_j\), if \(i\in N_2\),

then

\[ u(\bar x)=\int_{R^s} f(\bar y)\, r^{-\lambda}\,d\bar y \in L_{(q_1,q_2,\ldots,q_\tau)}(R^{m_1}, R^{m_2}, \ldots, R^{m_\tau}) \tag{2} \]

and, moreover, the inequality

\[ \|u\|_{L_{(q_1,q_2,\ldots,q_\tau)}(R^{m_1},R^{m_2},\ldots,R^{m_\tau})} \le c\|f\|_{L_{(p_1,p_2,\ldots,p_k)}(R^{s_1},R^{s_2},\ldots,R^{s_k})}, \tag{3} \]

holds, where \(r=|\bar x-\bar y|;\ \bar x\in R^m,\ \bar y\in R^s,\ c\) is a constant independent of \(f\).

Remark. If \(R^m=R^s\), then conditions a), b), c) reduce to one condition a). If \(R^m \supset R^s\), then to two— a) and b). If \(R^m \subset R^s\), then to two— a) and c).

The proof of this theorem rests on a number of auxiliary lemmas. In order to formulate them, we introduce notation.

Let \(\bar p_1,\bar p_2,\ldots,\bar p_k\) be the same numbers as \(p_1,p_2,\ldots,p_k\), but arranged in nondecreasing order, i.e. \(\bar p_1\leq \bar p_2\leq\cdots\leq \bar p_k\). If the number \(p_i\) in the system of numbers \(\bar p_1,\bar p_2,\ldots,\bar p_k\) is denoted by \(\bar p_j\), then, for convenience in the subsequent transformations, we shall denote \(R^{s_i}\) by \(R^{\bar s_j}\). Consequently, \(R^{\bar s_1}, R^{\bar s_2},\ldots,R^{\bar s_k}\) are the same as \(R^{s_1}, R^{s_2},\ldots,R^{s_k}\), but arranged, possibly, in another order. The \(\bar q_1,\bar q_2,\ldots,\bar q_\tau\) and \(R^{\bar m_1}, R^{\bar m_2},\ldots,R^{\bar m_\tau}\) are defined analogously.

With this notation the sets \(M_1,M_2,N_1,N_2,N_{ij},M_{ij}\) pass into new sets, which we shall denote respectively by \(\bar M_1,\bar M_2,\bar N_1,\bar N_2,\bar N_{ij},\bar M_{ij}\).

Lemma 2. If \(f(\bar y)\in L_{(p_{11},p_{12},\ldots,p_{1k})}(R^{\bar s_1},R^{\bar s_2},\ldots,R^{\bar s_k})\),

\[ \lambda=\sum_1^\tau \frac{\bar m_i}{q_{1i}}+\sum_1^k \frac{\bar s_j}{p_{1j}}, \]

and \(q_{1i},p_{1j}\ (i\in M;\ j\in N)\) are real positive numbers greater than one and satisfying the conditions:

\[ \text{I) }\quad p_{1k}\leq p_{1k-1}\leq\cdots\leq p_{11}<q_{1i},\quad i\in M \]

or

\[ \text{II) }\quad p_{1j}<q_{11}\leq q_{12}\leq\cdots\leq q_{1\tau},\quad j\in N, \]

then the inequality holds

\[ A_{(R^{\bar m_\tau},\ldots,R^{\bar m_1})}^{(q_{1\tau},\ldots,q_{11})} \left[ \left(\int_{R^s} f(\bar y)r^{-\lambda}\,d\bar y\right)^{q_{11}} \right] \leq cA_{(R^{\bar s_k},\ldots,R^{\bar s_1})}^{(p_{1k},\ldots,p_{11})} \left[|f|^{p_{11}}\right], \tag{4} \]

where \(c\) is a constant independent of \(f\).

Before passing to the proof of this lemma, we introduce notation. Let \(R^{\bar m_i}\cap R^{s_j}=R^{m_{ij}}\) for \(i\in M;\ j\in N\); \(R^{\bar m_i}\cap R^s=R^{s^{(i)}}\) for \(i\in M\). A point of the space \(R^{m_{ij}}\) will be denoted by \(\bar x^{(i;j)}\) or \(\bar y^{(i;j)}\), and a point of \(R^{s^{(i)}}\) by \(\bar x_s^{(i)}\) or \(\bar y_s^{(i)}\). We note that

\[ \bar x^{(i;j)}=(x_{\gamma_1}^{(i)},x_{\gamma_2}^{(i)},\ldots,x_{\gamma_{m_{ij}}}^{(i)});\quad \bar y^{(i;j)}=(y_{\delta_1}^{(j)},y_{\delta_2}^{(j)},\ldots,y_{\delta_{m_{ij}}}^{(j)}), \]

where \(\gamma_1,\gamma_2,\ldots,\gamma_{m_{ij}}\) \((\delta_1,\delta_2,\ldots,\delta_{m_{ij}})\) are part of the set of natural numbers \(1,2,\ldots,m_i\) \((1,2,\ldots,s_j)\),

\[ \bar x_s^{(i)}=(\bar x^{(i;1)},\bar x^{(i;2)},\ldots,\bar x^{(i;k)});\quad \bar y_s^{(i)}=(\bar y^{(i;1)},\bar y^{(i;2)},\ldots,\bar y^{(i;k)}). \]

It may happen that \(R^{\bar m_i}\setminus(R^{\bar m_i}\cap R^s)\ne0\); therefore we introduce the notation

\[ R^{\bar m_i}\setminus(R^{\bar m_i}\cap R^s)=R^{m_{i;k+1}}. \]

A point of the space \(R^{m_{i;k+1}}\) will be denoted by \(\bar x^{(i;k+1)}\). Analogously we denote

\[ R^{s_j}\setminus(R^{s_j}\cap R^m)=R^{m_{\tau+1;j}}, \]

and its point by \(\bar y^{(\tau+1;j)}\). We now pass to the proof of Lemma 2.

1. The case \(R^m\supset R^s\) and \(p_{1k}\leq p_{1k-1}\leq\cdots\leq p_{11}<q_{1i}\). In this case we have

\[ r^2=|\bar x-\bar y|^2=\sum_{i=1}^{\tau}\sum_{j=1}^{k+1} r_{ij}^{\,2}. \]

where

\[ r_{ij}= \begin{cases} \left|\bar x^{(i,j)}-\bar y^{(i,j)}\right|, & \text{if } i\in \overline M,\ j\in N,\\ \left|\bar x^{(i,j)}\right|, & \text{if } i\in M,\ j=k+1, \end{cases} \]

and

\[ \lambda=\sum_{i=1}^{\tau}\frac{\bar m_i}{q_{1i}}+\sum_{j=1}^{k}\frac{\bar s_j}{p'_{1j}} =\sum_{i=1}^{\tau}\sum_{j=1}^{k+1}\lambda_{ij}, \]

where

\[ \lambda_{ij}= \begin{cases} m_{ij}\left(\dfrac{1}{q_{1i}}+\dfrac{1}{p'_{1j}}\right), & \text{if } i\in \overline M_1,\ j\in N,\\[6pt] m_{ij}\dfrac{1}{q_{1i}}, & \text{if } i\in M,\ j=k+1. \end{cases} \]

For convenience in the subsequent computations we denote

\[ \bar r_\delta=\sum_{i=\delta}^{\tau}\sum_{j=1}^{k+1} r_{ij}, \qquad \bar\lambda_\delta=\sum_{i=\delta}^{\tau}\sum_{j=1}^{k+1}\lambda_{ij}. \]

In this notation \(\bar r_1=r\).

Using the general form of a linear functional in spaces with mixed norm [9], one can show that, in order to prove inequality (4) of Lemma 2 under the hypotheses of the lemma, it is enough to prove the validity of the following inequality:

\[ \int_{R^m}\int_{R^s} f(\bar y)\varphi(\bar x)r^{-\lambda}\,d\bar x\,d\bar y \le c A_{\left(R^{\bar s_k},\ldots,R^{\bar s_1}\right)}^{(p_{1k},\ldots,p_{11})}\bigl[\lvert f\rvert^{p_{11}}\bigr]\, A_{\left(R^{\bar m_\tau},\ldots,R^{\bar m_1}\right)}^{(q'_{1\tau},\ldots,q'_{11})} \bigl[\lvert\varphi\rvert^{q'_{11}}\bigr]. \tag{5} \]

for every
\(\varphi(\bar x)\in L_{(q'_{11},\ldots,q'_{1\tau})}(\bar R^{m_1},\ldots,\bar R^{m_\tau})\);
\(q'_{1i}=\dfrac{q_{1i}}{q_{1i}-1}\), \(i\in M\).
From the condition of Theorem 1 and Lemma 2 it follows that
\(\bar R^{m_\tau}\cap R^s\ne 0\). Consequently,
\(\bar R^{m_\tau}\cap R^{s_j}\ne 0\) at least for one value \(j\in N\).
We denote this value of the number \(j\) by \(e\); then
\(\bar R^{m_\tau}\cap \bar R^{s_e}\ne 0\). The following estimate holds:

\[ \int_{R^m}\int_{R^s} f(\bar y)f(\bar x)r^{-\lambda}\,d\bar x\,d\bar y \le \sum_{i=1}^{3} \int_{R^m\setminus R^{m_\tau e}} \int_{R^s\setminus R^{s_e}} \int\!\!\int_{D_i} f\varphi r^{-\lambda}\,d\bar x^{(\tau,e)} \times \]

\[ \times d\bar y^{(\tau,e)}\,d\bar x_{m-m_\tau e}\,d\bar y_{s-m_\tau e}, \]

where \(\overline D_i\) is a \(2m_{\tau e}\)-dimensional domain in
\(R^{m_\tau e}\times R^{m_\tau e}\), defined by the inequalities

\[ \begin{cases} r_k\le r_i,\\ r_j\le r_i, \end{cases} \qquad k\ne j\ne i\ne k;\quad k,j,i=1,2,3. \]

Here

\[ r_1=\left|\bar y^{(\tau,e)}\right|,\qquad r_2=\left|\bar x^{(\tau,e)}\right|,\qquad r_3=\left|\bar x^{(\tau,e)}-\bar y^{(\tau,e)}\right|, \]

the remaining notation is clear. On the basis of the lemma of S. L. Sobolev ([1], p. 474) and its generalization [6], when estimating the integral standing on the left-hand side of the last inequality, we may assume that
\(f=f^*(r_1,\bar y_{s-m_\tau e})\) is a nonincreasing function of \(r_1\) for all \(\bar y_{s-m_\tau e}\);
\(\varphi=\varphi^*(r_2,\bar x_{m-m_\tau e})\) is a nonincreasing function of \(r_2\) for all \(\bar x_{m-m_\tau e}\).

Consider and estimate one of these three summands, for example the second. Taking into account that the following inequality holds:
\[ \iint_{D_2} f\varphi r^{-\lambda}\,d\bar{x}^{(\tau;e)}\,d\bar{y}^{(\tau;e)} \le c\int_0^\infty \varphi r_2^{\frac{m_{\tau e}-1}{p'_{1e}}} F(r_2,\bar{y}_{s-m_{\tau e}}) \left(\int_0^{r_2}\frac{r_3^{m_{\tau e}-1}\,dr_3}{r^\lambda}\right)dr_2, \]
where
\[ F(r_2,y_{s-m_{\tau e}}) = r_2^{-\frac{m_{\tau e}}{p'_{1e}}-\frac{1}{p_{1e}}} \int_0^{r_2} f r_1^{m_{\tau e}-1}\,dr_1, \]
to obtain which it is sufficient to use the lemma of S. L. Sobolev ([1], p. 476), we obtain
\[ J_2\equiv \int_{R^m\times R^{m_{\tau e}}} \int_{R^s\times R^{m_{\tau e}}} \iint_{D_2} f\varphi r^{-\lambda}\,d\bar{x}^{(\tau;e)}\,d\bar{y}^{(\tau;e)} \,d\bar{x}_{m-m_{\tau e}}\,d\bar{y}_{s-m_{\tau e}} \le \]
\[ \le c\int_0^\infty r_2^{\frac{m_{\tau e}-1}{p'_{1e}}} \left[ \int_{R^m\setminus R^{m_{\tau e}}} \int_{R^s\setminus R^{m_{\tau e}}} \varphi F \left(\int_0^{r_2}\frac{r_3^{m_{\tau e}-1}\,dr_3}{r^\lambda}\right) d\bar{x}_{m-m_{\tau e}}\,d\bar{y}_{s-m_{\tau e}} \right]dr_2 = \]
\[ = c\int_0^\infty r_2^{\frac{m_{\tau e}-1}{p'_{1e}}} \left[ \int_{\Omega_\tau}\int_{\Omega_{\tau-1}}\cdots\int_{\Omega_1} \varphi F \left(\int_0^{r_2} r_3^{m_{\tau e}-1}r^{-\lambda}\,dr_3\right) d\omega_1\,d\omega_2\cdots d\omega_\tau \right]dr_2, \tag{6} \]
where
\[ \Omega_i= \begin{cases} R^{\bar m_i}\times R^{s^{(i)}}, & \text{if } i\in \overline{M}_1 \text{ and } i\ne \tau,\\[4pt] \bigl(R^{\bar m_i}\setminus R^{m_{\tau e}}\bigr)\times \bigl(R^{s^{(i)}}\setminus R^{m_{\tau e}}\bigr), & \text{if } i=\tau,\\[4pt] R^{\bar m_i}, & \text{if } i\in \overline{M}_2, \end{cases} \]
\(d\omega_i\) is the volume element of the domain \(\Omega_i\), \(i\in M\). Let
\(\Omega_{ij}=R^{m_{ij}}\times R^{m_{ij}}\), and let \(D_i\) be an \(m_i^{\,k+1}\)-dimensional domain in \(R^{m_i;k+1}\), which can take two values: one equal to \(C_{r_2}^{m_i;k+1}\), the ball in \(R^{m_i;k+1}\) of radius \(r_2\) and with center at the origin, and the second \(R^{m_i;k+1}\setminus C_{r_2}^{m_i;k+1}\); \(\delta_i\) are such positive real numbers that \(\delta=\sum_1^\tau \delta_i\) satisfies the condition
\[ m_{\tau e}\left(\frac{1}{q_{1\tau}}+\frac{1}{p'_{1e}}\right)+\delta<m_{\tau e}. \]

We now estimate the expression
\[ B_1\equiv \int_{\Omega_1} \varphi F \left(\int_0^{r_2} r^{-\lambda}r_3^{m_{\tau e}-1}\,dr_3\right)d\omega_1 = \int_{R^{m_1}}\int_{R^{s(1)}} \varphi F \left(\int_0^{r_2} r^{-\lambda}r_3^{m_{\tau e}-1}\,dr_3\right) d\bar{y}_s^{(1)}\,d\bar{x}^{(1)} = \]
\[ = \sum \int_{\Omega_{1k}}\int_{\Omega_{1k-1}}\cdots\int_{\Omega_{11}}\int_{D_1} \varphi F \left(\int_0^{r_2} r^{-\lambda}r_3^{m_{\tau e}-1}\,dr_3\right) d\omega_{1;k+1}\,d\omega_{11}\,d\omega_{12}\cdots d\omega_{1k}. \tag{7} \]

where the sum is extended over the possible values of \(D_1\), i.e., in this sum there are two terms differing from one another by the domain \(D_1\): in one term \(D_1=C_{r_2}^{m_1;k+1}\), and in the other \(D_1=R^{m_1;k+1}\setminus C_{r_2}^{m_1;k+1}\):

\[ \int_{\Omega_{1k}}\int_{\Omega_{1k-1}}\cdots \int_{\Omega_{11}}\int_{D_1} \varphi F\left(\int_0^{r_2} r^{-\lambda} r_3^{m_\tau-1}\,dr_3\right) \,d\omega_{1;k+1}\,d\omega_{11}\,d\omega_{12}\cdots d\omega_{1k}\le \]

\[ \le \left(\int_0^{r_2}\bar r_2^{-\bar\lambda_2-\sigma_1}r_3^{m_\tau-1}\,dr_3\right) \int_{\Omega_{1k}}\frac{d\omega_{1k}}{r_{1k}^{\lambda_{1k}}} \int_{\Omega_{1k-1}}\frac{d\omega_{1k-1}}{r_{1k-1}^{\lambda_{1k-1}}} \cdots \int_{\Omega_{11}}\frac{F\,d\omega_{11}}{r_{11}^{\lambda_{11}}} \int_{D_1}\frac{\varphi\,d\omega_{1k+1}}{r_{1;k+1}^{\lambda_{1;k+1}+\sigma_1}}, \]

where

\[ \sigma_1= \begin{cases} \delta_1, & \text{if } D_1=C_{r_2}^{m_1;k+1},\\ -\delta_1, & \text{if } D_1=R^{m_1;k+1}\setminus C_{r_2}^{m_1;k+1}. \end{cases} \]

Applying Hölder’s inequality to the integral over \(D_1\), and then successively applying the theorem of S. L. Sobolev ([1], p. 477) to the remaining integrals, we shall have

\[ B_1\le c\sum r_2^{\sigma_1} \left(\int_0^{r_2}\bar r_2^{-\bar\lambda_2-\sigma_1}r_3^{m_\tau-1}\,dr_3\right) A_{(R^{m_{1k}},\ldots,R^{m_{11}})}^{(p_{1k},\ldots,p_{11})}\,[|F|^{p_{11}}]\, A_{(R^{m_1})}^{(q'_{11})}\,[|\varphi|^{q'_{11}}], \]

where the sum is extended over the possible values of the number \(\sigma_1\). Taking this estimate into account, we obtain

\[ J_2\le c\sum \int_0^\infty r_2^{\frac{m_\tau-1}{p'_{1e}}+\sigma_1} \left[ \int_{\Omega_\tau}\cdots\int_{\Omega_2} \varphi_1 F_1 \left(\int_0^{r_2}\bar r_2^{-\bar\lambda_2-\sigma_1}r_3^{m_\tau-1}\,dr_3\right) \times \right. \]

\[ \left. \times\,d\omega_2\cdots d\omega_\tau \right]\,dr_2, \]

where

\[ F_1= A_{(R^{m_{1k}},\ldots,R^{m_{11}})}^{(p_{1k},\ldots,p_{11})}\,[|F|^{p_{11}}]; \qquad \varphi_1= A_{(R^{m_1})}^{(q'_{11})}\,[|\varphi|^{q'_{11}}]. \]

By analogous estimates we obtain that

\[ \int_{\Omega_2}\varphi_1 F_1 \left(\int_0^{r_2}\bar r_2^{-\bar\lambda_2-\sigma_1}r_3^{m_\tau-1}\,dr_3\right) \,d\omega_2\le \]

\[ \le \sum r_2^{\sigma_2} \left(\int_0^{r_2}\bar r_3^{-\bar\lambda_3-\sigma_1-\sigma_2}r_3^{m_\tau-1}\,dr_3\right) A_{(R^{m_{2k}},\ldots,R^{m_{21}})}^{(p_{1k},\ldots,p_{11})}\,[|F_1|^{p_{11}}]\, A_{(R^{\bar m_2})}^{(q'_{12})}\,[|\varphi_1|^{q'_{12}}], \]

therefore

\[ J_2\le c\sum_{\sigma_2}\sum_{\sigma_1}\int_0^\infty r_2^{\frac{m_\tau-1}{p'_{1e}}+\sigma_1+\sigma_2} \left[ \int_{\Omega_\tau}\cdots\int_{\Omega_3} \left(\int_0^{r_2}\bar r_3^{-\bar\lambda_3-\sigma_1-\sigma_2}r_3^{m_\tau-1}\,dr_3\right) \times \right. \]

\[ \left. \times A_{(R^{m_{1k}+m_{2k}},\ldots,R^{m_{11}+m_{21}})}^{(p_{1k},\ldots,p_{11})}\,[|F|^{p_{11}}]\, A_{(R^{\bar m_2},R^{\bar m_1})}^{(q'_{12},q'_{11})}\,[|\varphi|^{q'_{11}}]\, d\omega_3\cdots d\omega_\tau \right]\,dr_2. \]

Continuing these estimates, at the \(\tau\)-th step we obtain

\[ \begin{aligned} J_2 \le c \sum_{\sigma_\tau}\sum_{\sigma_{\tau-1}}\cdots\sum_{\sigma_1} \int_0^\infty &r_2^{\frac{m_{\tau e}-1}{p'_{1e}}+\sum_1^\tau \sigma_i} \Bigg[ \left( \int_0^{r_2} r_3^{-\lambda_{\tau e}-\sum_1^\tau \sigma_i} r_3^{m_{\tau e}-1}\,dr_3 \right) \\ &\times A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|F|^{p_{11}}\right] \\ &\cdot A_{\left(R^{\bar m_\tau}\times R^{m_{\tau e}},R^{\bar m_{\tau-1}},\ldots,R^{\bar m_1}\right)}^{(q'_{1\tau},q'_{1\tau-1},\ldots,q'_{11})} \left[|\varphi|^{q'_{11}}\right] \Bigg]\,dr_2 \\ \le c\int_0^\infty &r_2^{\frac{m_{\tau e}-1}{q'_{1\tau}}-\frac{1}{p'_{1e}}} A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|F|^{p_{11}}\right] \\ &\cdot A_{\left(R^{\bar m_\tau}\times R^{m_{\tau e}},\ldots,R^{\bar m_1}\right)}^{(q'_{1\tau},\ldots,q'_{11})} \left[|\varphi|^{q'_{11}}\right]\,dr_2 \\ \le c\int_0^\infty &\left(r_2^{\frac{m_{\tau e}-1}{q'_{1\tau}}}\varphi_\tau\right) \left(F_\tau\right)^{\frac{p_{1e}}{q_{1\tau}}} \left(r_2^{-\frac{1}{p_{1e}}}F_\tau\right)^{1-\frac{p_{1e}}{q_{1\tau}}}\,dr_2, \end{aligned} \]

where

\[ \varphi_\tau = A_{\left(R^{\bar m_\tau}\times R^{m_{\tau e}},R^{\bar m_{\tau-1}},\ldots,R^{\bar m_1}\right)}^{(q'_{1\tau},q'_{1\tau-1},\ldots,q'_{11})} \left[|\varphi|^{q'_{11}}\right], \]

\[ F_\tau = A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|F|^{p_{11}}\right]. \]

The following estimate holds:

\[ \begin{aligned} r_2^{\frac{1}{p_{1e}}}F_\tau &= r_2^{\frac{1}{p_{1e}}} A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|F|^{p_{11}}\right] \\ &\le r_2^{-\frac{m_{\tau e}}{p'_{1e}}} \int_0^{r_2} A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|f|^{p_{11}}\right] r_1^{m_{\tau e}-1}\,dr_1 \\ &\le c\, A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \left[|f|^{p_{11}}\right]. \end{aligned} \]

Thus,

\[ \begin{aligned} J_2 &\le c\left( A_{\left(R^{\bar S_k},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{11})} \left[|f|^{p_{11}}\right] \right)^{1-\frac{p_{1e}}{q_{1\tau}}} \int_0^\infty \left(r_2^{\frac{m_{\tau e}-1}{q'_{1\tau}}}\varphi_\tau\right) \left(F_\tau\right)^{\frac{p_{1e}}{q_{1\tau}}}\,dr_2 \\ &\le c\left( A_{\left(R^{\bar S_k},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{11})} \left[|f|^{p_{11}}\right] \right)^{1-\frac{p_{1e}}{q_{1\tau}}} \left( \int_0^\infty r_2^{m_{\tau e}-1}\varphi_\tau^{q'_{1\tau}}\,dr_2 \right)^{\frac{1}{q'_{1\tau}}} \left( \int_0^\infty F_\tau^{p_{1e}}\,dr_2 \right)^{\frac{1}{q_{1\tau}}}. \tag{8} \end{aligned} \]

It is not difficult to establish the following estimates:

\[ \left( \int_0^\infty r_2^{m_{\tau e}-1}\varphi_\tau^{q'_{1\tau}}\,dr_2 \right)^{\frac{1}{q'_{1\tau}}} \le c\, A_{\left(R^{\bar m_\tau},\ldots,R^{\bar m_1}\right)}^{(q'_{1k},\ldots,q'_{11})} \left[|\varphi|^{q'_{11}}\right], \tag{9} \]

\[ \begin{aligned} \left( \int_0^\infty F_\tau^{p_{1e}}\,dr_2 \right)^{\frac{1}{q_{1\tau}}} \le \Bigg[ \int_0^\infty &r_2^{-\left(\frac{m_{\tau e}}{p'_{1e}}+\frac{1}{p_{1e}}\right)p_{1e}} \Bigg( \int_0^{r_2} A_{\left(R^{\bar S_k},\ldots,R^{\bar S_e}\times R^{m_{\tau e}},\ldots,R^{\bar S_1}\right)}^{(p_{1k},\ldots,p_{1e},\ldots,p_{11})} \\ &\times \end{aligned} \]

\[ \times \left[|f|^{p_{11}}\right)^{p_{1e}} r_1^{m_\tau e-1}\,dr_1\right]^{\frac1{q_{1\tau}}} \leq \]

\[ \leq \left( A_{\left(R^k,\ldots,R^e,\ldots,R^1\right)}^{\left(\frac{p_{1k}}{s_k},\ldots,\frac{p_{1e}}{s_e},\ldots,\frac{p_{11}}{s_1}\right)} \left[|f|^{p_{11}}\right] \right)^{\frac{p_{1e}}{q_{1\tau}}}. \tag{10} \]

Taking into account the estimates (9), (10), we obtain from (8) the required inequality (5). For \(J_1\) and \(J_3\) we obtain analogous estimates. Consequently, in the case under consideration the lemma is proved.

1. The case \(R^m=R^s\) and \(p_{1k}\leq p_{1k-1}\leq \cdots \leq p_{11}<q_{1i}\) \((i=1,2,\ldots,\tau)\) is proved similarly.

2. The case \(R^m\subset R^s\) and \(p_{1k}\leq p_{1k-1}\leq \cdots \leq p_{11}<q_{1i}\), \(i\in M\). In this case it may turn out that
   \[ R^{s_j}\setminus\left(R^{s_j}\cap R^m\right)=R^{s_{\tau+1;j}}\ne0, \]
   therefore we denote by \(\bar y^{(\tau+1;j)}\) the point of \(R^{s_{\tau+1;j}}\). Then

\[ r_{ij}= \begin{cases} |x^{(i;j)}-\bar y^{(i;j)}|, & \text{if } i\in M,\ j\in \bar N_1,\\ |\bar y^{(i;j)}|, & \text{if } i=\tau+1,\ j\in N; \end{cases} \]

\[ \lambda_{ij}= \begin{cases} m_{ij}\left(\dfrac1{q_{1i}}+\dfrac1{p'_{1j}}\right), & \text{if } i\in M,\ j\in \bar N_1,\\[6pt] s_{ij}\dfrac1{p'_{1j}}, & \text{if } i=\tau+1,\ j\in N. \end{cases} \]

Arguing in the same way as in case 1), we obtain for \(J_2\), instead of (6), the following estimate:

\[ J_2\leq c\int_0^\infty r_2^{\frac{m_\tau e-1}{p_{1e}}} \left[ \int_{\Omega_\tau}\cdots\int_{\Omega_1} \int_{\Omega'_k}\cdots\int_{\Omega'_1} \varphi F\left(\int_0^{r_2} r^{-\lambda} r_3^{m_\tau e-1}\,dr_3\right) \times \right. \]

\[ \left. \times d\omega'_1\cdots d\omega'_k\,d\omega_1\cdots d\omega_\tau \right]\,dr_2, \tag{11} \]

where
\[ \Omega'_j=R^{\bar s_j}\setminus\left(R^{\bar s_j}\cap R^m\right)=R^{s_{\tau+1;j}},\qquad j\in N; \]
\(d\omega'_j\) is the volume element of the domain \(\Omega'_j\). Inequality (11) can be written in the following form:

\[ J_2\leq c\int_0^\infty r_2^{\frac{m_\tau e-1}{p_{1e}}} \left[ \int_{\Omega_\tau}\cdots\int_{\Omega_1} \varphi\left\{ \sum \int_{D'_k}\cdots\int_{D'_1} F\left(\int_0^{r_2} r^{-\lambda} r_3^{m_\tau e-1}\,dr_3\right) \times \right. \]

\[ \left. \times d\omega'_1\cdots d\omega'_k \right\} d\omega_1\cdots d\omega_\tau \right]\,dr_2, \]

where each \(D'_j\) is an \(s_{\tau+1;j}\)-dimensional domain in \(R^{s_{\tau+1;j}}\), which may take two values: one equal to \(C_{r_2}^{s_{\tau+1;j}}\), the \(s_{\tau+1;j}\)-dimensional ball in \(R^{s_{\tau+1;j}}\) of radius \(r_2\) centered at the origin, and the second, \(R^{s_{\tau+1;j}}\setminus C_{r_2}^{s_{\tau+1;j}}\). The sum is taken over the possible values of \(D'_j\), \(j\in N\).

Let \(\varepsilon_j\) \((j\in N)\) be positive real numbers such that
\[ \varepsilon=\sum_{1}^{k}\varepsilon_i \]
satisfies the condition

\[ m_{\tau e}\left(\frac{1}{q_{1\tau}}+\frac{1}{p'_{1e}}\right)+\varepsilon<m_{\tau e}, \]

then the following estimate holds

\[ \begin{aligned} J_2 \leq C \int_0^\infty r_2^{\frac{m_{\tau e}-1}{p'_{1e}}} \Biggl[ \int_{\Omega_\tau}\int_{\Omega_{\tau-1}}\cdots\int_{\Omega_1} \varphi\left(\int_0^{r_2} r_*^{-\lambda_*-\sigma} r_3^{m_{\tau e}-1}\,dr_3\right) \times {}\\ \times \left\{ \int_{D'_k} \frac{d\omega'_k}{r_{\tau+1;k}^{\lambda_{\tau+1;k}-\sigma_k}} \int_{D'_{k-1}} \frac{d\omega'_{k-1}}{r_{\tau+1;k-1}^{\lambda_{\tau+1;k-1}-\sigma_{k-1}}} \cdots \int_{D'_1} \frac{F\,d\omega'_1}{r_{\tau+1;1}^{\lambda_{\tau+1;1}-\sigma_1}} \right\} d\omega_1\cdots d\omega_\tau \Biggr]\,dr_2 , \end{aligned} \]

where \(\sigma=\sum_1^k \sigma_j\), and

\[ \sigma_j = \begin{cases} \varepsilon_j, & \text{if } D'_j=C_{r_2}^{s_{\tau+1;j}},\\ -\varepsilon_j, & \text{if } D'_j=R^{s_{\tau+1;j}}\setminus C_{r_2}^{s_{\tau+1;j}}; \end{cases} \]

\[ r_*^2=\sum_{i\in M_1}\sum_{j\in N} r_{ij}^2;\qquad \lambda_*=\sum_{i\in M_1}\sum_{j\in N}\lambda_{ij}. \]

Applying Hölder’s inequality successively to the expression enclosed in braces on the right-hand side of the last inequality, we obtain

\[ \begin{aligned} &\int_{D'_k} \frac{d\omega'_k}{r_{\tau+1;k}^{\lambda_{\tau+1;k}-\sigma_k}} \int_{D'_{k-1}} \frac{d\omega'_{k-1}}{r_{\tau+1;k-1}^{\lambda_{\tau+1;k-1}-\sigma_{k-1}}} \cdots \int_{D'_1} \frac{F\,d\omega'_1}{r_{\tau+1;1}^{\lambda_{\tau+1;1}-\sigma_1}} \leq {}\\ &\qquad \leq C r_2^\sigma A_{\left(R^{s_{\tau+1;k}},\,R^{s_{\tau+1;k-1}},\,\ldots,\,R^{s_{\tau+1;1}}\right)}^{(p_{1k},\,p_{1k-1},\,\ldots,\,p_{11})} \left[|F|^{p_{11}}\right] \equiv C r_2^\sigma \psi_1 . \end{aligned} \tag{12} \]

Therefore, for \(J_2\) we obtain the estimate

\[ \begin{aligned} J_2 \leq C \sum_{\sigma_1}\sum_{\sigma_2}\cdots\sum_{\sigma_k} \int_0^\infty r_2^{\frac{m_{\tau e}-1}{p'_{1e}}+\sigma} \Biggl[ \int_{\Omega_\tau}\int_{\Omega_{\tau-1}}\cdots\int_{\Omega_1} \varphi\psi_1 \times {}\\ \times \left(\int_0^{r_1} r_*^{-\lambda_*-\sigma}r_3^{m_{\tau e}-1}\,dr_3\right) d\omega_1\cdots d\omega_\tau \Biggr]\,dr_2 . \end{aligned} \tag{13} \]

Proceeding in the same way as in case 1, one can establish the validity of the estimate

\[ \begin{aligned} &\int_{\Omega_\tau}\int_{\Omega_{\tau-1}}\cdots\int_{\Omega_1} \varphi\psi_1 \left(\int_0^{r_2} r_*^{-\lambda_*-\sigma}r_3^{m_{\tau e}-1}\,dr_3\right) d\omega_1\cdots d\omega_{\tau-1}\,d\omega_\tau \leq {}\\ &\qquad \leq C r_2^{\,m_{\tau e}-m_{\tau e}\left(\frac{1}{q_{1\tau}}+\frac{1}{p'_{1e}}\right)-\sigma} A_{\left(R^{m_\tau}\times R^{m_{\tau e}},\,R^{m_{\tau-1}},\,\ldots,\,R^{m_1}\right)}^{(q'_{1\tau},\,q'_{1\tau-1},\,\ldots,\,q'_{11})} \left[|\varphi|^{q'_{11}}\right]\times \end{aligned} \]

\[ \times A_{\left(R^{\bar s_k}\times R^{s_{\tau+1};k},\,\ldots,\,(R^{\bar s_e}\times R^{m_{\tau e}})\times R^{s_{\tau+1};e},\,\ldots,\,R^{\bar s_1}\times R^{s_{\tau+1};1}\right)} ^{(p_{1k},\,\ldots,\,p_{1e},\,\ldots,\,p_{11})} \left[|\psi_1|^{p_{11}}\right]. \]

Since

\[ A_{\left(R^{\bar s_k}\times R^{s_{\tau+1};k},\,\ldots,\,(R^{\bar s_e}\times R^{m_{\tau e}})\times R^{s_{\tau+1};e},\,\ldots,\,R^{\bar s_1}\times R^{s_{\tau+1};1}\right)} ^{(p_{1k},\,\ldots,\,p_{1e},\,\ldots,\,p_{11})} \left[|\psi_1|^{p_{11}}\right]\leq \]

\[ \leq A_{\left(R^{\bar s_k},\,\ldots,\,R^{\bar s_e}\times R^{m_{\tau e}},\,\ldots,\,R^{\bar s_1}\right)} ^{(p_{1k},\,\ldots,\,p_{1e},\,\ldots,\,p_{11})} \left[|F|^{p_{11}}\right], \]

we shall have

\[ J_2\leq C\int_0^\infty r_2^{\frac{m_{\tau e}-1}{q'_{1\tau}}-\frac{1}{p'_{1e}}} A_{\left(R^{\bar s_k},\,\ldots,\,R^{\bar s_e}\times R^{m_{\tau e}},\,\ldots,\,R^{\bar s_1}\right)} ^{(p_{1k},\,\ldots,\,p_{1e},\,\ldots,\,p_{11})} \left[|F|^{p_{11}}\right]\times \]

\[ \times A_{\left(R^{\bar m_\tau}\times R^{m'_{\tau e}},\,\ldots,\,R^{m_1}\right)} ^{(q'_{1\tau},\,\ldots,\,q'_{11})} \left[|\varphi|^{q'_{11}}\right]\,dr_2 . \]

The subsequent arguments are analogous to case 1.

1. The case \(R^m\supseteq R^s\); \(p_{1j}<q_{11}\leq q_{12}\leq\cdots\leq q_{1\tau}\). By the condition of the lemma, \(\bar R^{m_i}\cap R^{s_k}\ne 0\) for at least one value \(i\in M\). Denote one such value of the number \(i\) by \(h\). Put \(r_1=|\bar y^{(h;k)}|\), \(r_2=|\bar x^{(h;k)}|\), \(r_3=|\bar x^{(h;k)}-\bar y^{(h;k)}|\). In this notation, for \(J_2\) we obtain the estimate

\[ J_2\leq C\int_0^\infty r_2^{\frac{m_{h;k}-1}{p'_{1k}}} \left[ \int_{\Omega_k}\cdots\int_{\Omega_1} \int_{\Omega'_\tau}\cdots\int_{\Omega'_1} \varphi F\left(\int_0^{r_2} r^{-\lambda}r_3^{m_{hk}-1}\,dr_3\right)\times \right. \]

\[ \left. \times d\omega'_1\cdots d\omega'_\tau\,d\omega_1\cdots d\omega_k \right]\,dr_2, \]

where

\[ \Omega_j= \begin{cases} R^{\bar s_j}\times\left(R^{\bar s_j}\cap R^m\right), & \text{if } i\ne h,\\ \left(R^{\bar s_j}\setminus R^{m_{hk}}\right)\times\left(R^{\bar s_j}\cap R^m\right)\setminus R^{m_{hk}}, & \text{if } i=h, \end{cases} \]

\[ \Omega'_i=\bar R^{m_i}\setminus R^{s^{(i)}},\quad d\omega_j\ (d\omega'_i)\text{ is the volume element of the domain }\Omega_j\ (\Omega'_i). \]

The subsequent estimates are analogous to case III.

1. Let \(R^m\subset R^s\) and \(p_{1j}<q_{11}\leq q_{12}\leq\cdots\leq q_{1\tau}\). In this case we shall have

\[ J_2\leq C\int_0^\infty r_2^{\frac{m_{hk}-1}{p'_{1k}}} \left[ \int_{\Omega_k}\cdots\int_{\Omega_1} \varphi F\left(\int_0^{r_2} r^{-\lambda}r_3^{m_{hk}-1}\,dr_3\right) \,d\omega_1\cdots d\omega_k \right]\,dr_2, \]

where

\[ \Omega_j= \begin{cases} R^{\bar s_j}\times\left(R^{\bar s_j}\cap R^m\right), & \text{if } j\in \bar N_1 \text{ and } j\ne h,\\ \left(R^{\bar s_j}\setminus R^{m_{hk}}\right)\times\left(R^{\bar s_j}\cap R^m\right)\setminus R^{m_{hk}}, & \text{if } j=h,\\ R^{\bar s_j}, & \text{if } j\in \bar N_2. \end{cases} \]

The subsequent estimates are analogous to case 1.

1. The case \(R^m\setminus (R^m\cap R^s)\ne 0,\ R^s\setminus (R^s\cap R^m)\ne 0\) and \(p_{1k}\leq p_{1k-1}\leq\cdots\leq p_{11}<q_{1i};\ i\in M\). By the condition of the lemma, \(\overline{R^{m_\tau}}\cap \overline{R^{s_k}}\ne 0\); therefore, denoting
   \(r_1=|\overline{y^{(\tau;k)}}|,\ r_2=|\overline{x^{(\tau;k)}}|,\ r_3=|\overline{x^{(\tau;k)}}-\overline{y^{(\tau;k)}}|\), we obtain for \(J_2\) the estimate

\[ J_2 \leq c \int_0^\infty r_2^{\frac{m_\tau t_k-1}{p_{1k}}} \left[ \int_{\Omega_{\alpha_e}}\cdots \int_{\Omega_{\alpha_1}} \int_{\Omega'_\tau} \int_{\Omega''_k}\cdots \int_{\Omega'_1} \varphi F \left( \int_0^{r_2} r^{-\lambda} r_3^{m_\tau t_k-1}\,dr_3 \right) \times \right. \]

\[ \left. \times d\omega_1\cdots d\omega''_k\,d\omega'_1\cdots d\omega_\tau\,d\omega_1\cdots d\omega_e \right]\,dr_2, \]

where \(\alpha_1,\alpha_2,\ldots,\alpha_e\) are elements of the set \(\overline{M}_1\);

\[ \Omega_{\alpha_i}= \begin{cases} \left(R^{\overline{m}_{\alpha_i}}\times R^{s^{(\alpha_i)}}\right), & \text{if } \alpha_i\in \overline{M}_1 \text{ and } \alpha_i\ne \tau,\\[4pt] \left(R^{\overline{m}_{\alpha_i}}\setminus R^{m_{\tau k}}\right)\times \left(R^{s^{(i)}}\setminus R^{m_{ik}}\right), & \text{if } \alpha_i=\tau, \end{cases} \]

\[ \Omega'_i=R^{\overline{m}_i}\setminus R^{s^{(i)}},\qquad \Omega''_j=R^{\overline{s}_j}\setminus (R^{\overline{s}_j}\cap R^m). \]

The right-hand side of the last inequality is estimated analogously to the preceding case, if one takes into account that in this case

\[ r_{ij}= \begin{cases} |\overline{x^{(i;j)}}-\overline{y^{(i;j)}}|, & \text{if } i\in \overline{M}_1,\ j\in \overline{N}_1,\\ |\overline{x^{(i;j)}}|, & \text{if } i\in \overline{M}_2,\ j=k+1,\\ |\overline{y^{(i;j)}}|, & \text{if } j\in \overline{N}_2,\ i=\tau+1; \end{cases} \]

\[ \lambda_{ij}= \begin{cases} m_{ij}\left(\dfrac{1}{q_{1i}}+\dfrac{1}{p_{1j}}\right), & \text{if } i\in \overline{M}_1,\ j\in \overline{N}_1,\\[6pt] m_{ij}\dfrac{1}{q_{1i}}, & \text{if } i\in \overline{M}_2,\ j=k+1,\\[6pt] s_{ij}\dfrac{1}{p_{1j}}, & \text{if } j\in \overline{N}_2,\ i=\tau+1. \end{cases} \]

1. The case \(R^m\setminus (R^m\cap R^s)\ne 0,\ R^s\setminus (R^s\cap R^m)\ne 0;\ p_{1j}\leq q_{11}\leq q_{12}\leq \cdots \leq q_{1\tau}\) is proved in the same way as case 6.

Lemma 3. If the positive real numbers \(p_{3j},q_{3i}\ (i\in M;\ j\in N)\) satisfy the conditions

\[ p_{31}\leq p_{32}\leq \cdots \leq p_{3k}<q_{3\tau}\leq q_{3\tau-1}\leq \cdots \leq q_{31}, \tag{1'} \]

then one can choose positive real numbers \(p_{1j},p_{2j},q_{1i},q_{2i}\) \((i\in M;\ j\in N)\) such that

\[ \text{a) }\quad p_{1k}\leq p_{1k-1}\leq \cdots \leq p_{11}<q_{1i};\quad i\in M; \tag{2'} \]

\[ \text{b) }\quad p_{2j}<q_{21}\leq q_{22}\leq \cdots \leq q_{2\tau};\quad j\in N; \tag{3'} \]

and, moreover, satisfying the conditions

\[ \frac{t}{p_{2j}}+\frac{1-t}{p_{1j}}=\frac{1}{p_{3j}},\qquad j\in N \tag{4'} \]

and

\[ \frac{t}{q_{2i}}+\frac{1-t}{q_{1i}}=\frac{1}{q_{3i}},\qquad i\in M, \tag{5'} \]

where \(0<t<1\).

Without giving a detailed proof of the lemma, let us note that each of the equalities \((4')\), \((5')\) with respect to \(p_{2j}\) and \(p_{1j}\) \((q_{2i}\) and \(q_{1i})\) represents a hyperbola \(XY=(1-t)tp_{3j}^2\) \([XY=(1-t)tq_{3i}^2]\) with center at the point \((tp_{3j},(1-t)p_{3j})\) \([(tq_{3i},(1-t)q_{3i})]\) and with asymptotes parallel to the original coordinate axes. By simple arguments one can show how, from \(p_{3j}\) \((q_{3i})\), to choose the numbers \(p_{2j}\) and \(p_{1j}\) \((q_{2i}\) and \(q_{1i})\) satisfying conditions \((2')\), \((3')\), \((4')\), \((5')\). Namely, we choose the numbers \(q_{1i}\) so that the inequalities

\[ q_{11}>(1-t)q_{31},\quad q_{1i-1}\geq q_{1i}>(1-t)q_{3i},\quad i\in M \]

hold. Then, from \(q_{1i}\), by the equalities \((5')\), we find \(q_{2i}\). We choose the numbers \(p_{1j}\) so that the inequalities

\[ q_{1\tau}>p_{11}>(1-t)p_{31},\quad p_{1j-1}\geq p_{1j}>(1-t)p_{3j},\quad j\in N \]

hold. Then, from \(p_{1j}\), by \((4')\), we find \(p_{2j}\).

Lemma 4. If the conditions of Theorem 1 are satisfied, then the inequality

\[ A_{(R^{\bar m_\tau},\ldots,R^{\bar m_1})}^{(\bar q_\tau,\ldots,\bar q_1)} \left[ \left( \int_{R^s} f(\bar y)\,r^{-\lambda}\,d\bar y \right)^{\bar q_1} \right] \leq cA_{(R^{s_k},\ldots,R^{s_1})}^{(\bar p_k,\ldots,\bar p_1)} \left[\,|f|^{\bar p_1}\right], \tag{14} \]

holds, where \(c\) is a constant independent of \(f\).

Indeed, by the condition we have

\[ \bar p_k\leq \bar p_{k-1}\leq \cdots \leq \bar p_1<\bar q_\tau\leq q_{\tau-1}\leq \cdots \leq \bar q_1. \]

On the basis of Lemma 3 there exist numbers \(p_{2j},p_{1j},q_{2i}\) and \(q_{1i}\) \((i\in M;\ j\in N)\), satisfying conditions \((2')\), \((3')\), \((4')\), \((5')\) of Lemma 3.

On the basis of Lemma 2 we have

\[ A_{(R^{\bar m_\tau},\ldots,R^{\bar m_1})}^{(q_{1\tau},\ldots,q_{11})} \left[ \left( \int_{R^s} f(\bar y)\,r^{-\lambda}\,d\bar y \right)^{q_{11}} \right] \leq cA_{(R^{s_k},\ldots,R^{s_1})}^{(p_{1k},\ldots,p_{11})} \left[\,|f|^{p_{11}}\right], \tag{15} \]

\[ A_{(R^{\bar m_\tau},\ldots,R^{\bar m_1})}^{(q_{2\tau},\ldots,q_{21})} \left[ \left( \int_{R^s} f(\bar y)\,r^{-\lambda}\,d\bar y \right)^{q_{21}} \right] \leq A_{(R^{s_k},\ldots,R^{s_1})}^{(p_{2k},\ldots,p_{21})} \left[\,|f|^{p_{21}}\right]. \tag{16} \]

If one takes into account \((4)\) and \((5)\), then from \((15)\) and \((16)\), on the basis of the Riesz–Thorin theorem [9], we obtain \((14)\). From Lemmas 1 and 4 follows the validity of Theorem 1.

§ 3. Embedding Theorems

1. Let \(l\) be a positive integer; \(1\leq k\leq n\); \(1\leq p_i<\infty\).

Definition of the classes \(L_{(p_1,p_2,\ldots,p_k)}^{(l)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\). We shall say that a function \(f\) belongs to the class

\[ L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k), \]

if:

a) \(f\) has all generalized derivatives, in the sense of Sobolev, of order \(l\) in the domain \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\), and

b)

\[ \|f\|_{L_{(p_1,p_2,\ldots,p_k)}^{(l)}(\Omega_1,\Omega_2,\ldots,\Omega_k)} = \sum_{|\alpha|=l} A_{(\Omega_k,\ldots,\Omega_1)}^{(p_k,\ldots,p_1)} \left[\,|D^\alpha f|^{p_1}\right] <\infty. \]

Definition of the classes \(W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\). We shall say that a function \(f(\bar x)\), defined in \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\), belongs to the Sobolev space \(W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\) with mixed norm if \(f\) has all generalized derivatives, in the sense of Sobolev, of order \(l\) in the domain \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\) and satisfies the conditions:

\[ a)\quad \|f\|_{L(p_1,p_2,\ldots,p_k)(\Omega_1,\Omega_2,\ldots,\Omega_k)}<\infty; \]

\[ b)\quad \|f\|_{L^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)}<\infty. \]

We introduce the norm in \(W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\) by the equality

\[ \|f\|_{W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)} = \|f\|_{L(p_1,\ldots,p_k)(\Omega_1,\ldots,\Omega_k)} + \|f\|_{L^{(l)}_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)}. \]

Let \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\) be the union of a finite number of domains, each of which is star-shaped with respect to some ball, and let \(\Omega'_i\) be an \(m_i\)-dimensional domain in \(R^{m_i}\), \(1\le i\le \tau\).

Theorem 2. If \(f\in W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\),

\[ \sum_{j=1}^{k}\frac{n_j}{p_j}-l = \sum_{i=1}^{\tau}\frac{m_i}{q_i}; \]

\(p_j,q_i\) are positive real numbers satisfying the conditions

\[ 1<p_j<q_i\le\infty;\quad i\in M,\ j\in N_{ij}, \]

then

\[ f\in L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau) \]

and, moreover, the inequality holds

\[ \|f\|_{L(q_1,q_2,\ldots,q_\tau)(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)} \le c\|f\|_{W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)}, \]

where \(c\) is a constant independent of \(f\).

Let \(f\in W^{(l)}_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)\). Then, on the basis of the integral identity of S. L. Sobolev ([2], p. 62),

\[ f(\bar x) = \sum_{|\alpha|\le l-1} x_1^{\alpha_1}\cdots x_n^{\alpha_n} \int_c \xi_\alpha(\bar y)f(\bar y)\,d\bar y + \int_{\Omega}\frac{1}{r^{n-l}} \sum_{|\alpha|=l} \omega_\alpha(\bar x,\bar y)D^{\bar\alpha}f(\bar y)\,d\bar y. \]

For convenience, denote

\[ f_1(\bar x) = \sum_{|\alpha|\le l-1} x_1^{\alpha_1}\cdots x_n^{\alpha_n} \int_c \xi_\alpha(\bar y)f(\bar y)\,d\bar y, \]

\[ f_{1\bar\alpha}(\bar x) = \int_{\Omega}\frac{1}{r^{n-l}} \omega_\alpha(\bar x,\bar y)D^{\bar\alpha}f(\bar y)\,d\bar y, \]

then

\[ f(\bar x)=f_1(\bar x)+\sum_{|\alpha|=l}f_{1\bar\alpha}(\bar x). \]

Let us consider and estimate the norm of \(f(\bar x)\) in \(L_{(q_1,\ldots,q_\tau)}(\Omega'_1,\ldots,\Omega'_\tau)\):

\[ \|f\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le \|f_1\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)}+ \]

\[ + \sum_{|\alpha|=l} \|f_{1\alpha}\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} . \]

Let us estimate each term on the right-hand side of the inequality separately:

\[ \|f_{11}\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} = \]

\[ = \left\| \sum_{|\bar\alpha|<l-1} x_1^{\alpha_1}\cdots x_n^{\alpha_n} \int_C \zeta_{\bar\alpha}(\bar y) f(\bar y)\,d\bar y \right\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le \]

\[ \le c\|f\|_{L(p_1,\ldots,p_k)(\Omega_1,\ldots,\Omega_k)} . \]

\[ \|f_{\bar\alpha}\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} = \left\| \int_\Omega \frac{\omega_{\bar\alpha}(\bar x,\bar y)}{r^{\,n-l}} D^{\bar\alpha} f(\bar y)\,d\bar y \right\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le \]

\[ \le c\left\| \int_\Omega \frac{1}{r^{\,n-l}} D^{\bar\alpha} f(\bar y)\,d\bar y \right\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} . \]

By the hypothesis of the theorem,

\[ \sum_{j=1}^{k}\frac{n_j}{p_j}-l=\sum_{i=1}^{\tau}\frac{m_i}{q_i}, \]

whence

\[ n-l=\sum_{i=1}^{\tau}\frac{m_i}{q_i}+\sum_{j=1}^{k}\frac{n_j}{p_j}. \]

Moreover, by the hypothesis of Theorem 2,

\[ D^{\bar\alpha} f(\bar y)\in L(p_1,\ldots,p_k)(\Omega_1,\ldots,\Omega_k), \qquad |\bar\alpha|=l, \]

therefore, on the basis of Theorem 1 (the case \(R^s=R^m=R^n\), if \(p_i<\infty\), and the case \(R^m\subset R^s=R^n\), if \(p_i\le\infty\)), we obtain

\[ \left\| \int_\Omega \frac{1}{r^{\,n-l}} D^{\bar\alpha} f(\bar y)\,d\bar y \right\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le c\|D^{\bar\alpha}f\|_{L(p_1,\ldots,p_k)(\Omega_1,\ldots,\Omega_k)} . \]

Consequently, we finally obtain

\[ \|f\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le c\|f\|_{W^{(l)}_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)}, \]

as was required to prove.

1. Let \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k=\Omega\), \(r_1=|\bar x+\Delta\bar x-\bar y|\), and \(f\in L(p_1,p_2,\ldots,p_k)\times(\Omega_1,\Omega_2,\ldots,\Omega_k)\). We shall regard \(f(\bar y)\) as equal to zero outside \(\Omega\) and extended to the whole space.

Consider the expression

\[ U(\bar x,\Delta\bar x) = \int_{r<K} \frac{(r+r_1)^{\lambda-1}}{r^\lambda r_1^\lambda} f(\bar y)\,d\bar y, \]

where \(K\) is any number greater than the diameter of the domain \(\Omega\).

Lemma 5. If \(f\in L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)\),

\[ \lambda<\sum_{1}^{\tau}\frac{m_i}{q_i}+\sum_{1}^{k}\frac{n_j}{p_j'}, \]

\[ \sum_{1}^{\tau} m_i=\sum_{1}^{k} n_j=n;\quad q_i,\ p_j \text{ are real numbers satisfying the conditions} \]

\[ 1<p_j<q_i,\qquad i=1,2,\ldots,\tau;\quad j=1,2,\ldots,k, \]

then

\[ |\Delta \bar{x}|\,|U|_{L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)} \le c\,|f|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)}\,|\Delta \bar{x}|^{-\beta}, \tag{*} \]

where \(\Omega'_i\) is an \(m_i\)-dimensional domain in \(R^{m_i}\); \(c\) is a constant independent of \(f\) and \(|\Delta \bar{x}|\);

\[ \beta= \begin{cases} \displaystyle \sum_{1}^{k}\frac{n_j}{p'_j} +\frac{p_1}{p_1+\delta}\sum_{1}^{\tau}\frac{m_i}{q_i} -\lambda, & \displaystyle \text{if }\sum_{1}^{k}\frac{n_j}{p'_j} +\sum_{1}^{\tau}\frac{m_i}{q_i}-\lambda<1, \\[2ex] \displaystyle 1-\sigma-\sum_{1}^{\tau}\frac{m_i}{q_i} \left(1-\frac{p_1}{p_1+\sigma}\right), & \displaystyle \text{if }\sum_{1}^{k}\frac{n_j}{p'_j} +\sum_{1}^{\tau}\frac{m_i}{q_i}-\lambda\ge 1; \end{cases} \]

\(\sigma\) is an arbitrarily small positive number.

Denote

\[ \sum_{1}^{\tau}\frac{m_i}{q_i}+\sum_{1}^{k}\frac{n_j}{p'_j}-\lambda=\varepsilon, \]

and let \(\varepsilon=k\varepsilon_1+\tau\varepsilon_2\), where

\[ \varepsilon_1= \begin{cases} \displaystyle \frac{\varepsilon}{k+\tau}, & \text{if } \varepsilon<1,\\[1.5ex] \displaystyle \frac{1}{k+\tau}+\frac{\sigma}{k}, & \text{if } \varepsilon=1,\\[1.5ex] \displaystyle \frac{1}{k+\tau}+\frac{\sigma+\eta}{k}, & \text{if } 1<\varepsilon=1+\eta; \end{cases} \]

\[ \varepsilon_2= \begin{cases} \displaystyle \frac{\varepsilon}{k+\tau}, & \text{if } \varepsilon<1,\\[1.5ex] \displaystyle \frac{1}{k+\tau}-\frac{\sigma}{\tau}, & \text{if } \varepsilon=1,\\[1.5ex] \displaystyle \frac{1}{k+\tau}-\frac{\sigma}{\tau}, & \text{if } \varepsilon=1+\eta,\ \eta>0; \end{cases} \]

\(\sigma\) is an arbitrarily small positive number. Further, let \(q\) be a number satisfying the condition

\[ p_1<q<p_1\cdot \min_{1\le i\le \tau}\frac{m_i}{m_i-\varepsilon_2 q_i}. \]

We shall denote points of the space \(R^{n_i}\) by \(\bar{x}^{(i)},\bar{y}^{(i)}\), and points of the space \(R^{m_i}\) by \(\tilde{x}^{(i)},\tilde{y}^{(i)}\). The estimate holds ...

\[ |u(\bar{x},\Delta \bar{x})|\le \int_{\Omega_k}\int_{\Omega_{k-1}}\cdots\int_{\Omega_1} |f|^{\left(1-\frac{p_1}{q}\right)} \left[ \frac{(r+r_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)}} {(rr_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1\right)}} \right]\times \]

\[ \times \left[ |f|^{\frac{p_1}{q}} \frac{(r+r_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)}} {(rr_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2\right)}} \right] \,d\bar y^{(1)}\,d\bar y^{(2)}\ldots d\bar y^{(k)}. \]

Consider the inner integral and apply to it Hölder’s inequality with exponents \(q,\ p'_1,\ \dfrac{p_1q}{q-p_1}\). Then

\[ J_1= \int_{\Omega_1} |f|^{1-\frac{p_1}{q}} \left[ \frac{(r+r_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)}} {(rr_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1\right)}} \right]\times \]

\[ \times \left[ |f|^{\frac{p_1}{q}} \frac{(r+r_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)}} {(rr_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2\right)}} \right] \,d\bar y^{(1)}\le \]

\[ \le \|f\|_{L_{p_1}(\Omega_1)}^{\,1-\frac{p_1}{q}} \left[ \int_{\Omega_1} \frac{(r+r_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)p'_1}} {(rr_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1\right)p'_1}} \,d\bar y^{(1)} \right]^{\frac1{p'_1}} \times \]

\[ \times \left[ \int_{\Omega_1} |f|^{p_1} \frac{(r+r_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q}} {(rr_1)^{\sum_1^\tau\left(\frac{m_i}{q_i}-\varepsilon_2\right)q}} \,d\bar y^{(1)} \right]^{\frac1q}. \]

Since

\[ \int_{\Omega_1} \frac{(r+r_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)p'_1}} {(rr_1)^{\sum_1^k\left(\frac{n_j}{p'_j}-\varepsilon_1\right)p'_1}} \,d\bar y^{(1)} \le \]

\[ \le \prod_{j=2}^{k} \frac{(r_{s_j}+r'_{s_j})^{\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)p'_1}} {(r_{s_j}r'_{s_j})^{\left(\frac{n_j}{p'_j}-\varepsilon_1\right)p'_1}} \int_{\Omega_1} \frac{(r_{s_1}+r'_{s_1})^{\,n_1-\left(\varepsilon_1+\frac{1}{k+\tau}\right)p'_1}} {(r_{s_1}r'_{s_1})^{\,n_1-\varepsilon_1p'_1}} \,d\bar y^{(1)} \le \]

\[ \leqslant \left( B_1+B'_1|\Delta \bar{x}|^{\left(\varepsilon_1-\frac{1}{k+\tau}\right)p'_1} \right) \prod_{j=2}^{k} \frac{ \left(r_{s_j}+r'_{s_j}\right)^{\left(\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}\right)p'_1} }{ \left(r_{s_j}r'_{s_j}\right)^{\left(\frac{n_j}{p'_j}-\varepsilon_1\right)p'_1} }, \]

where \(B_1, B'_1\) are constants and \(r_{s_j}=|\bar{x}^{(j)}-\bar{y}^{(j)}|\), \(r'_{s_j}=|\bar{x}^{(j)}+\Delta\bar{x}-\bar{y}^{(j)}|\); then

\[ J_2\leqslant \left( B_1+B'_1|\Delta \bar{x}|^{\left(\varepsilon_1-\frac{1}{k+\tau}\right)p'_1} \right)^{\frac{1}{p'_1}} \|f\|_{L_{p_1}(\Omega_1)}^{\,1-\frac{p_1}{q}} \prod_{j=2}^{k} \frac{ \left(r_{s_j}+r'_{s_j}\right)^{\frac{n_j}{p'_j}-\varepsilon_1-\frac{1}{k+\tau}} }{ \left(r_{s_j}r'_{s_j}\right)^{\frac{n_j}{p'_j}-\varepsilon_1} } \times \]

\[ \times \left[ \int_{\Omega_1} \|f\|^{p_1} \frac{ (r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q} }{ (rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)q} } \,d\bar{y}^{(1)} \right]^{\frac{1}{q}} . \]

Making analogous estimates successively for the integrals over \(\Omega_2,\Omega_3,\ldots,\Omega_k\), we obtain at the \(k\)-th step the following estimate:

\[ |u(\bar{x},\Delta x)| \leqslant \prod_{j=1}^{k} \left( B_j+B'_j|\Delta\bar{x}|^{\left(\varepsilon_1-\frac{1}{k+\tau}\right)p'_j} \right)^{\frac{1}{p'_j}} \|f\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)}^{\,1-\frac{p_1}{q}} \times \]

\[ \times \left( A_{(\Omega'_k,\ldots,\Omega'_1)}^{(p_k,\ldots,p_1)} \left[ |f|^{p_1} \frac{ (r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q} }{ (rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)q} } \right] \right)^{\frac{p_1}{q}} . \]

Taking the norm in \(L_{(q_1)}(\Omega'_1)\) of both sides of the last inequality, we obtain

\[ \|u\|_{L_{q_1}(\Omega'_1)} \leqslant \prod_{j=1}^{k} \left( B_j+B'_j|\Delta\bar{x}|^{\left(\varepsilon_1-\frac{1}{k+\tau}\right)p'_j} \right)^{\frac{1}{p'_j}} \|f\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)}^{\,1-\frac{p_1}{q}} \times \]

\[ \times \left\{ \int_{\Omega'_1} \left( A_{(\Omega'_k,\ldots,\Omega'_1)}^{(p_k,\ldots,p_1)} \left[ |f|^{p_1} \frac{ (r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q} }{ (rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)q} } \right] \right)^{\frac{p_1q_1}{q}} \,d\bar{x}^{(1)} \right\}^{\frac{1}{q_1}} \leqslant \]

\[ \leqslant c\cdot c_1 \left\{ \int_{\Omega'_1} \left( A_{(\Omega'_k,\ldots,\Omega'_1)}^{(p_k,\ldots,p_1)} \left[ |f|^{p_1} \frac{ (r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q} }{ (rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)q} } \right] \right)^{q_1} \,d\bar{x}^{(1)} \right\}^{\frac{p_1}{q_1q}}, \]

since \(q>p_1\). Here \(c\) is a constant independent of \(f\), and

\[ c_1=\prod_{j=1}^{k}\left(B_j+B'_j|\Delta x|^{-\left(\varepsilon_1-\frac{1}{\tau+k}\right)p'_j}\right)^{\frac{1}{p'_j}} \|f\|_{L^{1-\frac{p_1}{q}}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)} . \]

Taking into account that \(q_1>p_j\) by assumption, we shall have

\[ J_2 \le \left\{ \int_{\Omega'_1} \left( A^{(p_k,\ldots,p_1)}_{(\Omega_k,\ldots,\Omega_1)} \left[ |f|^{p_1} \frac{(r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q}} {(rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)q}} \right] \right)^{q_1} \,d\tilde{x}^{(1)} \right\}^{\frac{p_1}{q_1q}} \le \]

\[ \le \left\{ A^{(p_k,\ldots,p_1)}_{(\Omega_k,\ldots,\Omega_1)} \left[ \left( \int_{\Omega'_1} |f|^{q_1} \frac{(r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)\frac{qq_1}{p_1}}} {(rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)\frac{qq_1}{p_1}}} \,d\tilde{x}^{(1)} \right)^{\frac{p_1}{q_1}} \right] \right\}^{\frac{p_1}{q}} . \]

If we take into account that

\[ \frac{(r+r_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)\frac{qq_1}{p_1}}} {(rr_1)^{\sum_{1}^{\tau}\left(\frac{m_i}{q_i}-\varepsilon_2\right)\frac{qq_1}{p_1}}} \le \prod_{i=1}^{\tau} \frac{(\tilde r_{s_i}+\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)\frac{qq_1}{p_1}}} {(\tilde r_{s_i}\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2\right)\frac{qq_1}{p_1}}}, \]

where \(\tilde r_{s_i}=|x^{(i)}-y^{(i)}|\), \(\tilde r'_{s_i}=|x^{(i)}+\Delta x-y^{(i)}|\), and that

\[ \left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)\frac{qq_i}{p_1} = m_i-\bar\varepsilon_i\frac{qq_i}{p_1} -\frac{1}{k+\tau}\frac{qq_i}{p_1}, \]

where

\[ 0<\bar\varepsilon_i=\varepsilon_2-\frac{m_i}{q_i}\left(1-\frac{p_1}{q}\right), \]

then we obtain

\[ J_2\le \left\{ A^{(p_k,\ldots,p_1)}_{(\Omega_k,\ldots,\Omega_1)} \left[ |f|^{p_1} \prod_{i=2}^{\tau} \frac{(\tilde r_{s_i}+\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q}} {(\tilde r_{s_i}\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2\right)q}} \times \right. \right. \]

\[ \left. \left. \times \left( \int_{\Omega'_1} \frac{(\tilde r_{s_1}+\tilde r'_{s_1})^{m_1-\left(\bar\varepsilon_1-\frac{1}{k+\tau}\right)\frac{qq_1}{p_1}}} {(\tilde r_{s_1}\tilde r'_{s_1})^{m_1-\bar\varepsilon_1\frac{qq_1}{p_1}}} \,d\tilde{x}^{(1)} \right)^{\frac{p_1}{q_1}} \right] \right\}^{\frac{p_1}{q}} \le \]

\[ \le \left(A_1+A'_1|\Delta x|^{-\left(\bar\varepsilon_1-\frac{1}{k+\tau}\right)\frac{qq_1}{p_1}}\right)^{\frac{p_1}{qq_1}} \times \]

\[ \times\left\{ A_{(\Omega'_k,\ldots,\Omega'_1)}^{(p_k,\ldots,p_1)} \left[ |f|^{p_1} \prod_{i=2}^{\tau} \frac{ (\tilde r_{s_i}+\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2-\frac{1}{k+\tau}\right)q} }{ (\tilde r_{s_i}\tilde r'_{s_i})^{\left(\frac{m_i}{q_i}-\varepsilon_2\right)q} } \right] \right\}^{\frac{p_1}{q}} . \]

Continuing these estimates, at the \(\tau\)-th step we obtain
\[ \|u\|_{L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)} \leq \]
\[ \leq c c_1 \prod_{i=1}^{\tau} \left( A_i+A'_i|\Delta \bar x|^{\left(\varepsilon_i-\frac{1}{k+\tau}\right)\frac{q q_i}{p_1}} \right)^{\frac{p_1}{q q_i}} \left( A_{(\Omega'_k,\ldots,\Omega'_1)}^{(p_k,\ldots,p_1)}(|f|^{p_1}) \right)^{\frac{p_1}{q}} = \]
\[ = c c_1 \prod_{i=1}^{\tau} \left( A_i+A'_i|\Delta \bar x|^{\left(\varepsilon_i-\frac{1}{k+\tau}\right)\frac{q q_i}{p_1}} \right)^{\frac{p_1}{q q_i}} \|f\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)}^{\frac{p_1}{q}} . \]

Substituting the value of \(c_1\) into the last inequality, we obtain
\[ \|u\|_{L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)} \leq c\|f\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k)} \times \]
\[ \times \prod_{j=1}^{k} \left( B_j+B'_j|\Delta \bar x|^{\left(\varepsilon_1-\frac{1}{k+\tau}\right)p'_j} \right)^{\frac{1}{p'_j}} \prod_{i=1}^{\tau} \left( A_i+A'_i|\Delta \bar x|^{\left(\varepsilon_i-\frac{1}{k+\tau}\right)\frac{q q_i}{p_1}} \right)^{\frac{p_1}{q q_i}} . \]

From the last inequality, taking into account the values of \(\varepsilon_1\) and \(\varepsilon_2\), we obtain
\[ \|u\|_{L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)} \leq \]
\[ \leq c\|f\|_{L_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)} \left\{ \begin{array}{ll} |\Delta \bar x|^{\displaystyle \varepsilon-1-\sum_{1}^{\tau}\frac{m_i}{q_i}\left(1-\frac{p_1}{q}\right)}, & \text{if } \varepsilon<1, \\[1.2em] |\Delta \bar x|^{\displaystyle -\sigma-\sum_{1}^{\tau}\frac{m_i}{q_i}\left(1-\frac{p_1}{q}\right)}, & \text{if } \varepsilon\geq 1 . \end{array} \right. \tag{**} \]

The last inequality implies \((*)\).

Remark. The estimate \((**)\) can be sharpened, but for this it is necessary to carry out more delicate arguments; we shall not do this, since our main purpose is to establish the complete continuity of the embedding operator from
\[ W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k) \]
into
\[ L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau). \]
For our purpose the estimates obtained here are sufficient.

Theorem 3. If
\[ \sum_{1}^{k}\frac{n_j}{p_j}-l < \sum_{1}^{\tau}\frac{m_i}{q_i}, \qquad \sum_{1}^{\tau} m_i=\sum_{1}^{k} n_j=n, \qquad 1<p_j<q_i, \]
\(i\in M,\ j\in N\), then the embedding operator from
\[ W^{(l)}_{(p_1,p_2,\ldots,p_k)}(\Omega_1,\Omega_2,\ldots,\Omega_k) \]
into
\[ L_{(q_1,q_2,\ldots,q_\tau)}(\Omega'_1,\ldots,\Omega'_\tau) \]
is completely continuous.

Indeed, let
\[ X=\{\varphi\}\subset W^{(l)}_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k) \]
and
\[ \|\varphi\|_{W^{(l)}_{(p_1,\ldots,p_k)}(\Omega_1,\ldots,\Omega_k)}<A. \]
We shall show that \(X\) is compact in
\[ L_{(q_1,q_2,\ldots,q_\tau)}\times(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau). \]
It is not difficult to observe, on the basis of the theorem ([9], p. 320)

and by simple arguments, that for the proof of this theorem it is sufficient to establish, for the function

\[ \tilde\varphi(\bar x)=\int\limits_{\Omega}\frac{1}{r^{\,n-l}}\sum_{|\alpha|=l}\omega_\alpha D^{\bar\alpha}\varphi(\bar y)\,d\bar y,\qquad \varphi\in X, \]

the validity of the inequality

\[ \|\tilde\varphi(\bar x+\Delta\bar x)-\tilde\varphi(\bar x)\|_{L(q_1,q_2,\ldots,q_\tau)(\Omega'_1,\Omega'_2,\ldots,\Omega'_\tau)}<\varepsilon \tag{1_3} \]

for \(|\Delta\bar x|<\delta(\varepsilon)\), where \(\delta(\varepsilon)\to0\) as \(\varepsilon\to0\).

The estimate holds ([1], p. 93)

\[ |\tilde\varphi(\bar x+\Delta\bar x)-\tilde\varphi(\bar x)| \le c|\Delta\bar x|\int\limits_{\Omega} \frac{(r+r_1)^{n-l-1}}{r^{n-l}r_1^{n-l}} \sum_{|\bar\alpha|}\left|D^{\bar\alpha}\varphi(\bar y)\right|\,d\bar y . \]

Hence, on the basis of Lemma 5, we shall have

\[ \|\tilde\varphi(\bar x+\Delta\bar x)-\tilde\varphi(\bar x)\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le c|\Delta\bar x|\times \]

\[ {}\times \sum \left\| \int\limits_{\Omega} \frac{(r+r_1)^{n-l-1}}{r^{n-l}r_1^{n-l}} \left|D^{\bar\alpha}\varphi(\bar y)\right|\,d\bar y \right\|_{L(q_1,\ldots,q_\tau)(\Omega'_1,\ldots,\Omega'_\tau)} \le \]

\[ \le c\sum \|D^{\bar\alpha}\varphi\|_{L(p_1,\ldots,p_k)(\Omega_1,\ldots,\Omega_k)} |\Delta\bar x|^\beta, \tag{2_3} \]

where

\[ \beta= \begin{cases} \displaystyle \sum_1^k\frac{n_j}{p'_j} + \frac{p}{p_1+\sigma}\sum_1^\tau\frac{m_i}{q_i} -\lambda, & \displaystyle \text{if }\sum_1^k\frac{n_j}{p'_j}+\sum_1^\tau\frac{m_i}{q_i}-\lambda<1,\\[1.2em] \displaystyle 1-\sigma-\sum_1^\tau\frac{m_i}{q_i} \left(1-\frac{p_1}{p_1+\sigma}\right), & \displaystyle \text{if }\sum_1^k\frac{n_j}{p'_j}+\sum_1^\tau\frac{m_i}{q_i}-\lambda\ge1. \end{cases} \tag{3_3} \]

From \((2_3)\) follows \((1_3)\). Consequently, the theorem is proved.

3. Definition of the classes \(\tilde\Phi_{(p_1,p_2,\ldots,p_k)}(X;\Omega_1\times\Omega_2\times\cdots\times\Omega_k)\). We shall say that an abstract additive set function \(\varphi(E)\) \((E\in\mathcal E\), where \(\mathcal E\) is the set of all Lebesgue-measurable subsets of \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k)\) with values in the \(B\)-space \(X\) belongs to the space

\[ \tilde\Phi_{(p_1,p_2,\ldots,p_k)}(X;\Omega_1\times\Omega_2\times\cdots\times\Omega_k), \]

if the norm defined by the equality

\[ \|\varphi\|_{\tilde\Phi_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)} = \sup_{\tau} \frac{\left\|\int \tau(\bar x)\,d_x\varphi(E)\right\|_X} {\|\tau\|_{L(p'_1,\ldots,p'_k)(\Omega_1,\ldots,\Omega_k)}}, \]

is finite, \(\tau(\bar x)\) is a real measurable function taking only a finite number of nonzero values; \(\dfrac1{p_i}+\dfrac1{p'_i}=1,\;1<p_i<\infty\).

Remark. The sign \(\sim\) over \(\Phi\) is placed in order to distinguish the spaces
\(\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\) from the spaces
\(\Phi_{(p_1,\ldots,p_k)}(X;S\cap\Omega)\), introduced in [16], which differ not only in the domain of definition.

Definition of the classes \(\psi_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\). By
\(\psi_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\) we shall mean the collection of all abstract functions from
\(\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\) that are continuous with respect to translation in the metric
\(\|\cdot\|_{\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)}\).

Definition of the classes \(\psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\). We shall say that an abstract additive set function \(\varphi(E)\), defined for all \(L\)-measurable sets \(E\) from \(\Omega_1\times\Omega_2\times\cdots\times\Omega_k\) with values in the \(B\)-space \(X\), belongs to the space
\(\psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\), if:

a) \(\varphi(E)\in\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\),

b) all derivatives of order \(l\):

\[ \frac{\partial^l p(E)}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \equiv D^\alpha\varphi(E)\in \psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k). \]

The norm in this space is defined by the formula

\[ \|\varphi\|_{\psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)} = \|\varphi\|_{\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1,\ldots,\Omega_k)} + \]

\[ + \sum_{|\alpha|=l} \|D^\alpha\varphi\|_{\widetilde{\Phi}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)}. \]

Theorem 4. If
\(\varphi(E)\in\psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)\) and

\[ \sum_1^k \frac{n_i}{p_i}-l = \sum_1^\tau \frac{m}{q_j}, \]

where \(p_i,q_j\) are real positive numbers satisfying the conditions
\(1<p_i<q_j,\quad i=1,2,\ldots,k;\quad j=1,2,\ldots,\tau,\) then

\[ \varphi(E)\in \widetilde{\Phi}_{(q_1,q_2,\ldots,q_\tau)} (X;\Omega'_1\times\Omega'_2\times\cdots\times\Omega'_\tau) \]

and the inequality

\[ \|\varphi\|_{\widetilde{\Phi}_{(q_1,\ldots,q_\tau)} (X;\Omega'_1\times\cdots\times\Omega'_\tau)} \leq c\|\varphi\|_{\psi^{(l)}_{(p_1,\ldots,p_k)}(X;\Omega_1\times\cdots\times\Omega_k)}, \]

holds, where \(c\) is a constant independent of \(\varphi(E)\).

This theorem is proved in the same way as the corresponding embedding theorem of type 1 of S. L. Sobolev ([3], p. 321). Therefore, not wishing to repeat ourselves, we shall omit the proof. We note only that in the proof of Theorem 3 one must use Sobolev’s integral identity for continuously differentiable abstract functions of a point ([3], p. 321) and Theorem 1.

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Received by the editors
September 7, 1965.

Institute of Mathematics, Siberian Branch of the Academy of Sciences of the USSR