UDC 513.88+517.51

MATHEMATICS

A. Kh. Gudiev

EMBEDDING THEOREMS FOR SPACES WITH MIXED NORM

(Presented by Academician S. L. Sobolev on 21 V 1965)

Recently, in connection with the introduction of new functional spaces \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega)\) \((^{4-6})\), works have appeared \((^{7-10})\) in which the differential properties of traces of functions of many variables on hyperplanes of arbitrary dimensions are studied, depending on the membership of the function itself in one or another space. The importance of studying this question was pointed out by S. L. Sobolev and S. M. Nikol’skii in a joint report at the Fourth All-Union Mathematical Congress \((^1)\).

All results obtained on this question up to now have had a one-sided character, i.e., in all works embedding theorems of one and the same type \(M \to N\) are proved; by \(M\) we mean an ordinary function space \(\bigl(W_p^{(l)}, H_p^{(r)}, B_p^{(r)}, W_{p_1,p_2,\ldots,p_k}^{(l_1,l_2,\ldots,l_k)}\bigr)\), and by \(N\) a space with mixed norm; the sign \(\to\) denotes the embedding of the space \(M\) into the space \(N\) with preservation of the corresponding inequality, and in all such theorems restrictions \(\rho_1 \leq \rho_2 \leq \cdots \leq \rho_h\) are imposed on the parameters \(\rho_1,\rho_2,\ldots,\rho_h\) of the spaces with mixed norm.

The results of the present work make it possible to get rid of these restrictions. As it turned out, these restrictions were not essential, but arose from the method of proof. Moreover, the main results of the present paper are Theorems 1–3. Theorem 1 characterizes the action of an operator of potential type in spaces with mixed norm and is the most general theorem of this type. Particular cases of this theorem were considered at different times in works \((^{2-16})\). In the case of spaces with mixed norm, closest to our result (Theorem 1) is the result of A. Benedek and R. Panzone \((^5,\) p. 321). It should be noted that their result is the simplest particular case \((R^m = R^s\) and \(R^{m_i} = R^{s_i},\ i=1,2,\ldots,k)\) of our theorem and can be proved in a few lines. However, if even \(R^m = R^s\), but \(R^{m_i} \ne R^{s_i}\) for at least one value of \(i\), then the theorem is not so simple to establish.

Theorem 1 makes it possible to obtain a general embedding theorem of the type \(\widetilde N \to N\), where \(\widetilde N\) and \(N\) are both spaces with mixed norm (Theorems 2 and 3).

In order to formulate the main results, let us introduce notation. Let \(R^n\) be \(n\)-dimensional Euclidean space and let \(n_i\) be positive integers such that \(\sum_1^h n_i = n\); let \(\Omega_i\) be an \(n_i\)-dimensional domain in \(n_i\)-dimensional Euclidean space \(R^{n_i}\); \(\Omega_i\) may coincide with \(R^{n_i}\); let \(\rho_1,\rho_2,\ldots,\rho_h\) be positive real numbers; let \(F(t)\) be a function defined in the domain \(\Omega_1 \times \Omega_2 \times \cdots \times \Omega_h\). Denote

\[ A_{(\Omega_h,\ldots,\Omega_1)}^{(\rho_h,\ldots,\rho_1)}[F(t)] = \left( \int_{\Omega_h} \left( \int_{\Omega_{h-1}} \cdots \int_{\Omega_2} \left( \int_{\Omega_1} F(t)\,d\omega_1 \right)^{\rho_2/\rho_1} d\omega_2 \right)^{\rho_3/\rho_2} \cdots d\omega_{h-1} \right)^{\rho_h/\rho_{h-1}} d\omega_h \right)^{1/\rho_h}, \tag{1} \]

where \(d\omega_i\) is the volume element of the domain \(\Omega_i\), \(1 \leq i \leq h\).

The space with mixed norm \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)\) is defined as the set of functions \(F(t)\), defined in \(\Omega_1 \times \Omega_2 \times \cdots \times \Omega_h\), for which the norm introduced by the equality

\[ \|F\|_{L_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1\times\Omega_2\times\cdots\times\Omega_h)} = A^{(\rho_h,\ldots,\rho_1)}_{(\Omega_h,\ldots,\Omega_1)}[|F|^{\rho_1}], \]

is finite.

Let us note that, generally speaking,

\[ \|F\|_{L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^{n_1},R^{n_2},\ldots,R^{n_h})} \ne \|F\|_{L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^{n'_1},R^{n'_2},\ldots,R^{n'_h})}, \]

unless \(R^{n_i}\) coincides with \(R^{n'_i}\) for all \(i=1,2,\ldots,h\). Therefore it is expedient to use the notation \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^{n_1},\ldots,R^{n_h})\) instead of the accepted and more compact notation \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^n)\), for \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^n)\), even with a specified representation \(R^n=R^{n_1}\times\cdots\times R^{n_h}\) (if the sequence in which the \(R^{n_i}\) are taken in (1) is not indicated), will be a notation, generally speaking, for \(h!\) different spaces.

It is interesting to note that, although, generally speaking, these spaces cannot be ordered with respect to the inclusion operation \(\subseteq\), the following still holds.

Lemma 1. Let \(\rho_1,\rho_2,\ldots,\rho_h\) be positive real numbers, \(1\le \rho_i<\infty\), \(i=1,2,\ldots,h\); let \(\rho_{11},\rho_{12},\ldots,\rho_{1h}\) and \(\rho_{21},\rho_{22},\ldots,\rho_{2h}\) be the same numbers as \(\rho_1,\rho_2,\ldots,\rho_h\), but arranged in nondecreasing and, respectively, nonincreasing order. Then

\[ L_{(\rho_{11},\rho_{12},\ldots,\rho_{1h})}(R^{n_{11}},R^{n_{12}},\ldots,R^{n_{1h}}) \subseteq L_{(\rho_1,\rho_2,\ldots,\rho_h)}(R^{n_1},R^{n_2},\ldots,R^{n_h}) \subseteq \]

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

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

This lemma allows one to remove the restrictions \(p_1\le p_2\le\cdots\le p_n\) in the embedding theorems \(H_{p_1}^r\to H^{r_1}_{(p_1,p_2,\ldots,p_n)}\), obtained by S. M. Nikol’skii in paper (7).

Let \(R^m\) and \(R^s\) be, respectively, \(m\)- and \(s\)-dimensional Euclidean spaces, with \(R^m\cap R^s\ne 0\), and let \(\sum_{i=1}^{\tau} m_i=m\), \(\sum_1^k s_j=s\); \(m_i\) and \(s_j\) are integers or rational numbers.

Consider the spaces \(L_{(p_1,p_2,\ldots,p_k)}(R^{s_1},R^{s_2},\ldots,R^{s_k})\), \(L_{(q_1,q_2,\ldots,q_\tau)}(R^{m_1},R^{m_2},\ldots,R^{m_\tau})\). Let \(M\) and \(N\) be 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 \(R^{m_i}\cap R^s\ne 0\), and by \(M_2\) the set \(M\setminus M_1\). Similarly, by \(N_1\) denote the set of those natural numbers \(j\in N_1\) for which \(R^{s_j}\cap R^m\ne 0\), and define \(N_2\) by the equality \(N_2=N\setminus N_1\). Further, 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. \(\bar p_1\le \bar p_2\le\cdots\le \bar p_k\).

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 \(R^{s_i}\) is denoted 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 a different order. The quantities \(\bar q_1,\bar q_2,\ldots,\bar q_\tau\) and \(R^{\bar m_1},R^{\bar m_2},\ldots,R^{\bar m_\tau}\) are defined analogously.

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

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

where \(q_{1i},p_{1j}\) are positive real numbers greater than one and satisfying the conditions

a) \(p_{1k}\leq p_{1\,k-1}\leq \cdots \leq p_{11}<q_{1i},\ i\in M\), or b) \(p_{1j}<q_{11}\leq q_{12}\leq \cdots \leq q_{1\tau},\ j\in N\), then the inequality holds

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

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

Lemma 3. If the real positive numbers \(p_{3j}, q_{3i}\) \((i\in M;\ j\in N)\) satisfy the conditions
\(p_{31}\leq p_{32}\leq \cdots \leq p_{3k}<q_{3\tau}\leq q_{3\,\tau-1}\leq \cdots \leq q_{31}\), then one can choose real positive numbers \(p_{1j}, p_{2j}, q_{1i}, q_{2i}\) \((i\in M;\ j\in N)\) such that:

a) \(p_{1k}\leq p_{1\,k-1}\leq \cdots \leq p_{11}<p_{1i},\ i\in M;\) \hfill (2)

b) \(p_{2j}<q_{21}\leq q_{22}\leq \cdots \leq q_{2\tau},\ j\in N,\) \hfill (3)

and, moreover, satisfying the conditions

\[ \frac{t}{p_{2j}}+\frac{1-t}{p_{1j}}=\frac{1}{p_{3j}}, \qquad \frac{t}{q_{2i}}+\frac{1-t}{q_{1i}}=\frac{1}{q_{3i}}, \qquad i\in M,\ j\in N, \tag{4} \]

where \(0<t<1\).

Lemma 4. If \(f(\bar y)\in L_{(p_{31},p_{32},\ldots,p_{3k})}(R^{s_1},R^{s_2},\ldots,R^{s_k})\) and

\[ \lambda=\sum_{i=1}^{\tau}\frac{\bar m_i}{q_{3i}}+\sum_{j=1}^{k}\frac{\bar s_j}{p_{3j}}, \]

where \(p_{3j}, q_{3i}\) are real positive numbers satisfying the conditions
\(1\leq p_{31}\leq p_{32}\leq \cdots \leq p_{3k}<q_{3\tau}\leq q_{3\,\tau-1}\leq \cdots \leq q_{31}\), then

\[ A_{(R^{m_\tau},\,R^{m_{\tau-1}},\,\ldots,\,R^{m_1})}^{(q_{3\tau},\,q_{3\,\tau-1},\,\ldots,\,q_{31})} \left[ \left(\int_{R^s} f(\bar y)r^{-\lambda}\,d\bar y\right)^{q_{31}} \right] \leq cA_{(R^{s_k},\,R^{s_{k-1}},\,\ldots,\,R^{s_1})}^{(p_{3k},\,p_{3\,k-1},\,\ldots,\,p_{31})} \left[\,|f|^{p_{31}}\right], \]

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

The validity of this lemma is established on the basis of Lemmas 2, 3 and the Riesz–Thorin theorem (5).

Theorem 1. If
\(f(\bar y)\in L_{(p_1,p_2,\ldots,p_k)}(R^{s_1},R^{s_2},\ldots,R^{s_k})\),

\[ \lambda=\sum_{i=1}^{\tau}\frac{m_i}{q_i}+\sum_{j=1}^{k}\frac{s_j}{p_j}; \]

\(q_i, p_j\) \((i\in M,\ j\in N)\) are real positive numbers greater than one and satisfying the conditions

\[ q_i>p_j,\quad \text{if } i\in M_1,\ j\in N_1; \tag{5} \]

\[ q_i\geq \min_{j\in M_1} q_j,\quad \text{if } i\in M_2; \tag{6} \]

\[ q_i\leq \max_{j\in N_1} p_j,\quad \text{if } i\in N_2, \tag{7} \]

then

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

and, moreover, the inequality holds

\[ \|U\|_{L_{(q_1,\ldots,q_\tau)}(R^{m_1},\ldots,R^{m_\tau})} \leq c\|f\|_{L_{(p_1,p_2,\ldots,p_k)}(R^{s_1},R^{s_2},\ldots,R^{s_k})}, \]

where \(r=|\bar x-\bar y|\), \(c\) is a constant independent of \(f\).

Remark. If \(R^m=R^s\), then conditions (5), (6), (7) reduce to the single condition (5); if \(R^m\supset R^s\), then to two conditions (5) and (6); if \(R^m\subset R^s\), then to two conditions (5) and (7).

Let \(l\) be a positive integer, \(1\leq h\leq n\), \(1\leq \rho_i<\infty\), \(i=1,\ldots,h\).

Definition of the classes \(L_{(\rho_1,\rho_2,\ldots,\rho_h)}^{(l)}(\Omega_1,\Omega_2,\ldots,\Omega_h)\). We say that the function \(f(\bar x)\) belongs to the class
\(L_{(\rho_1,\rho_2,\ldots,\rho_h)}^{(l)}(\Omega_1,\Omega_2,\ldots,\Omega_h)\), if:

a) \(f(\bar{x})\) has all generalized derivatives, in the sense of Sobolev, of order \(l\) in the domain \(\Omega_1\times\Omega_2\times\cdots\times\Omega_h\), and

b)
\[ \|f\|_{L^{(l)}_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)} = A^{(\rho_h,\ldots,\rho_1)}_{(\Omega_h,\ldots,\Omega_1)} \left[ \sum_{|\bar{\alpha}|=l} |D^{\bar{\alpha}} f|^{\rho_1} \right]<\infty . \]

Definition of the classes \(W^{(l)}_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1,\ldots,\Omega_h)\). We shall say that a function \(f(\bar{x})\), defined in \(\Omega_1\times\Omega_2\times\cdots\times\Omega_h\), belongs to the Sobolev space with mixed norm \(W^{(l)}_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_h,\ldots,\Omega_1)\) if it has all generalized derivatives, in the sense of Sobolev, of order \(l\) in the domain \(\Omega_1\times\Omega_2\times\cdots\times\Omega_h\) and satisfies the conditions:

a)
\[ \|f\|_{L_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)}<\infty, \]

b)
\[ \|f\|_{L^{(l)}_{(\rho_1,\rho_2,\ldots,\rho_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)}<\infty . \]

Let \(\Omega_i\) be star-shaped with respect to some ball \(C_i\), \(1\le i\le h\); then the domain \(\Omega_1\times\Omega_2\times\cdots\times\Omega_h\) will also be star-shaped with respect to some ball. Let, further, \(\Omega_i'\) be an \(n_i'\)-dimensional domain in \(R^{n_i'}\), \(1\le i\le h\).

Theorem 2. If
\[ f\in W^{(l)}_{(p_1,p_2,\ldots,p_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h); \qquad \sum_{i=1}^{h}\frac{n_j}{p_j}-l=\sum_{i=1}^{\tau}\frac{n_i'}{q_i}; \]
\(q_i, p_j\) are positive real numbers satisfying the conditions
\[ 1<p_j<q_i,\quad i=1,2,\ldots,\tau;\quad j=1,2,\ldots,h, \]
then
\[ f\in L_{(q_1,q_2,\ldots,q_\tau)}(\Omega_1',\Omega_2',\ldots,\Omega_\tau') \]
and, moreover, the inequality
\[ \|f\|_{L_{(q_1,q_2,\ldots,q_\tau)}(\Omega_1',\Omega_2',\ldots,\Omega_\tau')} \le c\, \|f\|_{W^{(l)}_{(p_1,p_2,\ldots,p_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)} \]
holds, where \(c\) is a constant independent of \(f\).

The spaces \(\widetilde{\Phi}_{(p_1,p_2,\ldots,p_h)}(X;\Omega_1\times\cdots\times\Omega_h)\) and \(\Psi^{(l)}_{(p_1,p_2,\ldots,p_h)}(X;\Omega_1\times\cdots\times\Omega_h)\) are defined in the same way as \(\Phi_p(X;\Omega)\) and \(\Psi^{(l)}_p(X;\Omega)\), except that in these spaces, instead of the norm \(\|\cdot\|_{L_p}\), one must take the norm
\[ \|\cdot\|_{L_{(p_1,p_2,\ldots,p_h)}(\Omega_1,\Omega_2,\ldots,\Omega_h)} . \]

Theorem 3. If
\[ \varphi(E)\in\Psi^{(l)}_{(p_1,p_2,\ldots,p_h)}(X;\Omega_1\times\cdots\times\Omega_h) \]
and
\[ \sum_{i=1}^{h}\frac{n_j}{p_j}-l=\sum_{i=1}^{\tau}\frac{n_i'}{q_i}; \]
\(p_j, q_i\) are positive real numbers satisfying the conditions
\[ 1<p_j<q_i,\quad i=1,2,\ldots,h;\quad j=1,2,\ldots,\tau, \]
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
\[ \|\varphi\|_{\widetilde{\Phi}_{(q_1,q_2,\ldots,q_\tau)}(X;\Omega_1'\times\Omega_2'\times\cdots\times\Omega_\tau')} \le c\, \|\varphi\|_{\Psi^{(l)}_{(p_1,p_2,\ldots,p_h)}(X;\Omega_1\times\Omega_2\times\cdots\times\Omega_h)} \]
holds, where \(c\) is a constant independent of \(\varphi\).

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

Received
14 V 1965

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