A. Kh. Gudiev

Classes \(L_{(p_1,p_2,\ldots,p_k)}(\Omega_m)\) and an Embedding Theorem for Abstract Set Functions

(Presented by Academician S. L. Sobolev on 12 VII 1962)

Let \(\Omega_m\) be a bounded domain of the \(m\)-dimensional Euclidean space \(R^m\) \((m \leq n)\). We denote each point \(\bar x \in R^m\) by
\(\bar x = (x_{s_1}, x_{s_2}, \ldots, x_{s_k})\), where \(s_i\) are positive integers satisfying the condition
\(s_1 + s_2 + \cdots + s_k = m\);
\(\bar x_{s_1} = (x_1, x_2, \ldots, x_{s_1})\),
\(\bar x_{s_2} = (x_{s_1+1}, x_{s_1+2}, \ldots, x_{s_1+s_2})\), \(\ldots\),
\(\bar x_{s_j} = (x_{s_1+s_2+\cdots+s_{j-1}+1}, x_{s_1+s_2+\cdots+s_{j-1}+2}, \ldots, x_{s_1+s_2+\cdots+s_{j-1}+s_j})\), \(\ldots\).

Next, denote by \(R^{s_j}\) the space of vectors \(\bar x_{s_j}\),

\[ \Omega_{s_j}=R^{s_j}\cap \Omega_m,\qquad |x-y|=r=r_n=\sqrt{r_{s_1}^2+r_{s_2}^2+\cdots+r_{s_k}^2}, \]

\[ r_{s_j}= \left[ \sum_{i=s_1+s_2+\cdots+s_{j-1}+1}^{s_1+s_2+\cdots+s_{j-1}+s_j} (x_i-y_i)^2 \right]^{1/2}, \]

\[ r_{\,n-(s_1+s_2+\cdots+s_{j-1})} = \left(r_{s_j}^2+r_{s_{j+1}}^2+\cdots+r_{s_k}^2\right)^{1/2}. \]

We shall consider the set of functions \(f(\bar x)\) defined in \(\Omega_m\). Each function \(f(\bar x)\) may be regarded as a function of the variable vectors
\(\bar x_{s_1}, \bar x_{s_2}, \ldots, \bar x_{s_k}\). Under such a consideration, to almost every vector \(\bar x_{s_i}\) there corresponds an element of an abstract space—the function
\(f(\bar x_{s_1}, \bar x_{s_2}, \ldots, \bar x_{s_{i-1}}, \bar x_{s_i}^{\,0}, \bar x_{s_{i+1}}, \ldots, \bar x_{s_k})\) of the variable vectors
\(\bar x_{s_1}, \bar x_{s_2}, \ldots, \bar x_{s_{i-1}}, \bar x_{s_{i+1}}, \ldots, \bar x_{s_k}\). Thus one may introduce for consideration the set of abstract functions
\(f(\bar x_{s_1}, \bar x_{s_2}, \ldots, \bar x_{s_k})\) for which the expression

\[ \left( \int_{\Omega_{s_k}} \left( \int_{\Omega_{s_{k-1}}} \cdots \left( \int_{\Omega_{s_2}} \left( \int_{\Omega_{s_1}} |f|^{p_1}\,d\bar x_{s_1} \right)^{p_2/p_1} d\bar x_{s_2} \right)^{p_3/p_2} \cdots d\bar x_{s_{k-1}} \right)^{p_k/p_{k-1}} d\bar x_{s_k} \right)^{1/p_k} \]

is bounded.

We denote the set of such functions by \(L_{(p_1,p_2,\ldots,p_k)}(\Omega_m)\). We introduce a norm in it by the equality

\[ \|f\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_m)}= \]

\[ = \left( \int_{\Omega_{s_k}} \left( \int_{\Omega_{s_{k-1}}} \cdots \left( \int_{\Omega_{s_2}} \left( \int_{\Omega_{s_1}} |f|^{p_1}\,d\bar x_{s_1} \right)^{p_2/p_1} d\bar x_{s_2} \right)^{p_3/p_2} \cdots d\bar x_{s_{k-1}} \right)^{p_k/p_{k-1}} d\bar x_{s_k} \right)^{1/p_k}. \]

Next, let \(\Omega_s\) denote either a domain of an \(s\)-dimensional Euclidean space, or a smooth manifold of \(s\) dimensions in a Euclidean space of a larger number of dimensions. We assume the \(s\)-dimensional measure of \(\Omega_s\) to be finite and different from zero, and set

\[ U(\bar x)=\int_{\Omega_s}\tau(\bar y)\,r^{-\lambda}\,d\bar y. \]

For \(s=n\), \(U(\bar x)\) was studied in the works of S. L. Sobolev \(({}^{1,2})\), V. I. Kondrashov \(({}^{4})\), V. P. Il’in \(({}^{5})\), and L. V. Kantorovich \(({}^{6})\); for \(s<n\) and \(m=n\), in the works of L. V. Kantorovich \(({}^{6})\) and Kh. L. Smolitskii \(({}^{7})\).

In this note we present results that generalize (in the case of a finite exponent) some results for \(U(\bar x)\) obtained in the works cited above. These results made it possible to generalize the embedding theorem of S. L. Sobolev \(({}^{3})\) for abstract set functions, i.e., made it possible to formulate and prove an embedding theorem for the trace on arbitrary hyperplanes of dimension \(s\le n\).

Theorem 1. If \(\tau(\bar y)\in L_p(\Omega_s)\); \(\lambda<\dfrac{s}{p'}+\sum_{i=1}^{k}\dfrac{s_i}{p_i}\), where \(s_i\) are positive rational numbers satisfying the condition \(\sum_{i=1}^{k}s_i=n\), \(s\le n\), and \(1<p\le p_1\le p_2,p_3,\ldots,p_k<\infty\), then \(U(\bar x)\in L_{(p_1,p_2,\ldots,p_k)}(\Omega_n)\) and, moreover, the inequality
\[ \|U\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_n)} \le c\|\tau\|_{L_p(\Omega_s)} \]
holds.

Let the number \(\varepsilon\) be such that
\[ \lambda=\frac{s}{p'}+\sum_{i=1}^{k}\frac{s_i}{p_i}-\varepsilon. \]
Put \(\varepsilon=\varepsilon_1+\varepsilon_2\); then
\[ |U(\bar x)|\le \int_{\Omega_s} r^{-s/p'+\varepsilon_1}\,|\tau(\bar y)|^{p\left(\frac1p-\frac1{p_1}\right)} |\tau(\bar y)|^{p/p_1} r^{-\left(\sum_{i=1}^{k}\frac{s_i}{p_i}-\varepsilon_2\right)} \,d\bar y . \tag{1} \]

Applying Hölder’s inequality to three factors, after obvious transformations we obtain
\[ |U(\bar x)|^{p_1}\le c_1\|\tau\|_{L_p(\Omega_s)}^{p_1(1-p/p_1)} \int_{\Omega_s} |\tau(\bar y)|^{p} r^{-\left(\sum_{i=1}^{k}\frac{s_i}{p_i}-\varepsilon_2\right)p_1} \,d\bar y . \tag{2} \]

Integrating (2) over \(\Omega_{s_1}\), changing the order of integration and simplifying, we shall have
\[ \int_{\Omega_{s_1}} |U(\bar x)|^{p_1}\,d\bar x_{s_1} \le c_1\|\tau\|_{L_p(\Omega_s)}^{p_1(1-p/p_1)} \int_{\Omega_s} |\tau(\bar y)|^p r_{\,n-s_1}^{-\left(\sum_{i=2}^{k}\frac{s_i}{p_i}-\varepsilon_2\right)p_1} \,d\bar y . \tag{3} \]

Raising both sides of inequality (3) to the power \(p_2/p_1\), integrating over \(\Omega_{s_2}\), applying the generalized Minkowski inequality \((({}^{9}),\ \text{p. }179)\), and simplifying, we obtain
\[ \int_{\Omega_{s_2}} \left( \int_{\Omega_{s_1}} |U(\bar x)|^{p_1}\,d\bar x_{s_1} \right)^{p_2/p_1} d\bar x_{s_2} \le c_2\|\tau\|_{L_p(\Omega_s)}^{p_2(1-p/p_1)} \left( \int_{\Omega_s} |\tau(\bar y)|^p r_{\,n-s_1-s_2}^{-\left(\sum_{3}^{k}\frac{s_i}{p_i}-\varepsilon_2\right)p_1} \,d\bar y \right)^{p_2/p_1}. \]

Continuing this process, at the \(k\)-th step we obtain
\[ \|U\|_{L_{(p_1,p_2,\ldots,p_k)}(\Omega_n)}^{p_k} \le c_k\|\tau\|_{L_p(\Omega_s)}^{p_k}. \]

Remark 1. For: 1) \(s_1=n,\ p_1=\infty,\ s_2=s_3=\cdots=s_k=0\); 2) \(s_1=n,\ p_1<\infty\), we obtain assertions a) and b) of Theorem 2 of Kh. L. Smolitskii (7) and, in a somewhat different formulation, Theorem 1 of L. V. Kantorovich (6). For \(s=n,\ s_1<n,\ s_2=s_3=\cdots=s_k=0,\ p=2,\ p_1<\infty,\ p_2=\infty\), we obtain S. L. Sobolev’s result for an indefinite exponent (2); for \(s=n,\ s_1<n,\ s_3=s_4=\cdots=s_k=0,\ p_1<\infty,\ p_2=\infty\), V. I. Kondrashov’s result.

Theorem 2. If \(0<p_1\leq p_2,p_3,\ldots,p_k<\infty,\ k\leq s\), then

\[ \mathbf L_{(p_1,p_2,\ldots,p_k)}(\Omega_s)\to \mathbf L_{p_1}(\Omega_s), \]

where \(\to\) denotes embedding (see (8)).

Theorem 3. If \(\tau(\bar y)\in \mathbf L_{(q',p_2,\ldots,p_k)}(\Omega_s)\);

\[ \lambda < \frac{s}{q}+\frac{s_1}{p'}\sum_{i=2}^{k}\frac{s_i}{p_i}, \]

where \(s_i\) are positive integral rational numbers satisfying the condition

\[ \sum_{i=1}^{k}s_i=n,\qquad k\leq s \]

and

\[ p\leq q\leq \max(q,p')\leq p_2,p_3,\ldots,p_k<\infty, \]

then \(U(\bar x)\in \mathbf L_{p'}(\Omega_n)\), and, moreover, the inequality

\[ \|U\|_{\mathbf L_{p'}(\Omega_n)} \leq c\|\tau\|_{\mathbf L_{(q',p_2,\ldots,p_k)}(\Omega_s)} . \tag{4} \]

holds.

Remark 2. For \(s_2=s_3=\cdots=s_k=0\), from Theorem 3 we obtain the assertion of S. L. Sobolev ((3), p. 307).

Theorem 3 makes it possible to generalize S. L. Sobolev’s embedding theorem to abstract functions of sets. In order to formulate and prove the result obtained, let us consider the class \(\Phi_{(p_1,p_2,\ldots,p_k)}(X,\Omega\cap S_s)\) of abstract additive functions of sets \(\varphi(I)\) \((I\in \mathcal E_s\), where \(\mathcal E_s\) is the set of all Lebesgue-measurable subsets of \(\Omega\cap S_s\)) with values in a Banach space \(X\), for which the norm introduced by the equality

\[ \|\varphi\|_{\Phi_{(p_1,p_2,\ldots,p_k)}(X,\Omega\cap S_s)} = \sup_{\tau} \frac{\left\|\int_S \tau(x)\,d_x\varphi(I)\right\|_{X}} {\|\tau\|_{\mathbf L_{(p_1',p_2,\ldots,p_k)}(\Omega\cap S_s)}} \tag{5} \]

is bounded. As is not difficult to see, from \(\Phi_{(p_1,p_2,\ldots,p_k)}\), as a special case one obtains the known space \(\Phi_{p_1}\) of S. L. Sobolev.

Theorem 4. If \(\varphi(E)\in \psi_p^{(l)}(X,\Omega)\),

\[ n-l<\frac{s}{q}+\frac{s_1}{p'}+\sum_{i=2}^{k}\frac{s_i}{p_i}, \]

where \(s_i\) are positive integral rational numbers satisfying the condition

\[ \sum_{i=1}^{k}s_i=n,\qquad k\leq s\leq n, \]

and

\[ p\leq q\leq \max(q,p')\leq p_2,p_3,\ldots,p_k<\infty, \]

then \(\varphi\) is defined on all smooth manifolds \(S_s\cap\Omega\) of \(s\) dimensions and represents a function of sets \(\tilde\varphi(I)\) belonging to the class
\(\Phi_{(q,p_2,\ldots,p_k)}(X,\Omega\cap S_s)\). Moreover, the inequality

\[ \|\tilde\varphi\|_{\Phi_{(q,p_2,\ldots,p_k)}(X,\Omega\cap S_s)} \leq c\|\varphi\|_{\psi_p^{(l)}(X,\Omega)}; \tag{6} \]

holds; \(c=\mathrm{const}\) does not depend on \(\varphi\).

Consider the integral identity

\[ \tilde\varphi_h(I) = \int_{\Omega} K(I,\bar y)\,d_y\varphi_h(E) + \sum_{|\bar\alpha|=l}\int_{\Omega} K_{\bar\alpha}(I,\bar y)\,d_y D^{\bar\alpha}\varphi_h(E), \]

where

\[ \widetilde{\varphi}_h(I)=\int_I \overline{\varphi}(\overline{y})\,d\overline{y}; \qquad \varphi_h(E)=\int_E \overline{\varphi}(\overline{x})\,d\overline{x}; \qquad D^{\overline{\alpha}}\varphi_h(E)=\int_E D^{\overline{\alpha}}\varphi(\overline{y})\,d\overline{y}; \]

\[ K(I,\overline{y})=\int_I \sum_{|\overline{\alpha}|\leq l-1} x_1^{\alpha_1}\cdots x_n^{\alpha_n}\xi_{\overline{\alpha}}(\overline{y})\,d\overline{x}; \qquad K_{\overline{\alpha}}(I,\overline{y})=\int_I \frac{\omega_{\overline{\alpha}}(\overline{x},\overline{y})}{r^{\,n-l}}\,d\overline{x}. \]

Let us estimate the expression

\[ \|\widetilde{\varphi}\|_{\Phi_{(q,p_2,\ldots,p_k)}(X,\Omega\cap S_s)} \leq \sup_{\tau} \frac{ \left\|\int \tau(\overline{x})\,d_x \left[\int_{\Omega} K(I,\overline{y})\,d_y\varphi(E)\right]\right\|_X }{ \|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)} } + \]

\[ + \sum_{|\overline{\alpha}|=l}\sup_{\tau} \frac{ \left\|\int \tau(\overline{x})\,d_x \left[\int_{\Omega} K_{\overline{\alpha}}(I,\overline{y})\,d_yD^{\alpha}\varphi_h(E)\right]\right\|_X }{ \|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)} }. \]

Using the definition of the integral (3) and making the necessary transformations, we obtain

\[ \|\widetilde{\varphi}\|_{\Phi_{(q,p_2,\ldots,p_k)}(X,\Omega\cap S_s)} \leq \]

\[ \leq \sup_{\tau} \frac{ \left\|\int_{\Omega}\left[\int_I \tau(\overline{x})\,d_x K(I,\overline{y})\right]\,d_y\varphi_h(E)\right\|_X }{ \left\|\int \tau(\overline{x})\,d_xK(I,\overline{y})\right\|_{L_{p'}(\Omega)} } \, \frac{ \left\|\int \tau(\overline{x})\,d_xK(I,\overline{y})\right\|_{L_{p'}(\Omega)} }{ \|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)} } + \]

\[ + \sum_{|\overline{\alpha}|=l}\sup_{\tau} \frac{ \left\|\int_{\Omega}\left[\int_I \tau(\overline{x})\,d_xK_{\overline{\alpha}}(I,\overline{y})\right]\, d_yD^{\alpha}\varphi_h(E)\right\|_X }{ \left\|\int \tau(\overline{x})\,d_xK_{\overline{\alpha}}(I,\overline{y})\right\|_{L_{p'}(\Omega)} } \, \frac{ \left\|\int \tau(\overline{x})\,d_xK_{\overline{\alpha}}(I,\overline{y})\right\|_{L_{p'}(\Omega)} }{ \|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)} }. \tag{7} \]

On the basis of Theorem 3 and simple estimates we have

\[ \left\|\int_I \tau(\overline{x})\,d_xK_{\overline{\alpha}}(I,\overline{y})\right\|_{L_{p'}(\Omega)} \leq c_1\|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)}; \tag{8} \]

\[ \left\|\int_I \tau(\overline{x})\,d_xK(I,\overline{y})\right\|_{L_{p'}(\Omega)} \leq c_2\|\tau\|_{L_{(q',p_2,\ldots,p_k)}(\Omega\cap S_s)}. \tag{9} \]

From (7), (8), and (9) we obtain

\[ \|\widetilde{\varphi}_h\|_{\Phi_{(q,p_2,\ldots,p_k)}(X,\Omega\cap S_s)} \leq c\|\varphi_h\|_{\psi_p^{(s)}(X,\Omega)}. \tag{10} \]

Passing to the limit in equality (10) as \(h\to 0\), we obtain (6).

Remark 3. For \(s_1=n,\ s_2=s_3=\cdots=s_k=0\), we obtain the theorem of S. L. Sobolev (\(^{3}\), p. 321).

Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
10 VII 1962

References

\(^{1}\) S. L. Sobolev, Matem. sborn., 4 (46), No. 3, 471 (1938).
\(^{2}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^{3}\) S. L. Sobolev, Fund. Math., 17 (1959).
\(^{4}\) V. I. Kondrashov, DAN, 48, 363 (1945).
\(^{5}\) V. P. Il’in, DAN, 96, 908 (1954).
\(^{6}\) L. V. Kantorovich, UMN, 11, 2 (68), 3 (1956).
\(^{7}\) Kh. L. Smolitskii, UMN, 12, 4 (76), 349 (1957).
\(^{8}\) S. M. Nikol’skii, UMN, 16, 5 (101) (1961).
\(^{9}\) G. G. Hardy, J. E. Littlewood, G. Pólya, Inequalities, 1948.