A. Kh. Gudiev

On S. L. Sobolev’s embedding theorems for abstract functions

(Presented by Academician S. L. Sobolev on 16 VI 1962)

In the well-known works of S. L. Sobolev \((^{1-3})\), attention is drawn to the importance of studying abstract functions of sets \(\varphi(E)\) in connection with their possible application in the theory of nonlinear differential equations. In addition, in the indicated works the classes \(\Phi_p(X,\Omega)\), \(\Psi_p^{(l)}(X,\Omega)\) are introduced, analogous to the well-known classes \(L_p(\Omega)\), \(W_p^{(l)}(\Omega)\) \((^4)\), for which embedding theorems are proved. The works of V. B. Korotkov \((^5)\) and Yu. I. Gil'derman \((^6)\) are also devoted to this question.

In the present work the classes \(\operatorname{Lip}\beta(X,S_s\cap\Omega)\), \(\operatorname{Lip}^{(\beta)}\beta(X,\Omega)\) are introduced, and embedding theorems for the classes \(\Psi_p^{(l)}(\Omega)\) into these classes are proved.

For abstract functions of a point \(\overline{\varphi}(x)\), continuously differentiable up to order \(l\) inclusive, the identity \((^3)\) holds

\[ \overline{\varphi}(x)= \sum_{\Sigma a_i\le l-1} x_1^{a_1}\cdots x_n^{a_n}\int_{\Omega}\xi_{\overline{a}}(y)\overline{\varphi}(y)\,dy + \sum_{\Sigma a_i=l}\int_{\Omega} \frac{\omega_{\overline{a}}(x,y)}{r^{\,n-l}}D^{\overline{a}}\overline{\varphi}(y)\,dy, \tag{1} \]

where \(\xi_{\overline{a}}(y)\) are continuous and vanish outside the star-shaped ball of the domain \(\Omega\); \(\omega_{\overline{a}}(x,y)\) are functions bounded in any bounded domain of variation of the variables \(x,y\).

Integrating both sides of equality (1) over a measurable set \(I\in\varepsilon_s\), where \(\varepsilon_s\) is the set of all measurable subsets of \(S_s\cap\Omega\), \(0<s\le n\), and changing the order of integration, we shall have:

\[ \widetilde{\varphi}(I)= \int_{\Omega}K(I,y)\,d_y\varphi(E) + \sum_{\Sigma a_i=l}\int_{\Omega}K_{\overline{a}}(I,y)\,d_yD^{\overline{a}}\varphi(E), \tag{2} \]

where

\[ \widetilde{\varphi}(I)=\int_I\overline{\varphi}(x)\,dx;\qquad \varphi(E)=\int_E\overline{\varphi}(y)\,dy;\qquad D^{\overline{a}}\varphi(E)=\int_E D^{\overline{a}}\overline{\varphi}(x)\,dy; \]

\[ K(I,y)=\int_I \sum_{\Sigma a_i\le l-1} x_1^{a_1}\cdots x_n^{a_n}\xi_{\overline{a}}(y)\,dx;\qquad K_{\overline{a}}(I,y)=\int_I \frac{\omega_{\overline{a}}(x,y)}{r^{\,n-1}}\,dx. \tag{3} \]

Lemma 1. If \(\varphi(E)=\displaystyle\int_E\overline{\varphi}(x)\,dx\) and \(\overline{\varphi}(x)\) has continuous derivatives up to order \(l\) inclusive, then

\[ \frac{\partial^l\varphi(E)} {\partial x_1^{a_1}\cdots\partial x_n^{a_n}} = \int_E \frac{\partial^l\overline{\varphi}(x)} {\partial x_1^{a_1}\cdots\partial x_n^{a_n}}\,dx, \qquad |\overline{a}|=l. \tag{4} \]

From (2), with the aid of Lemma 1, it is not difficult to establish that

\[ \frac{\partial^{\overline{\beta}}\widetilde{\varphi}(I)} {\partial x_1^{\beta_1}\cdots\partial x_s^{\beta_s}} = \int_{\Omega}D^{\overline{\beta}}K(I,y)\,d_y\varphi(E) + \sum_{\Sigma a_i=l}\int_{\Omega}D^{\overline{\beta}}K_{\overline{a}}(I,y)\,d_yD^{\overline{a}}\varphi(E), \tag{5} \]

where

\[ D^{\bar\beta}K(I,y)=\int_I \sum_{\Sigma\alpha_i<l-1-|\bar\beta|} x_1^{\alpha_1}\cdots x_n^{\alpha_n}\xi_{\bar\alpha}^{\bar\beta}(y)\,dx; \qquad D^{\bar\beta}K_{\bar\alpha}^{-}(I,y)=\int_I \frac{\omega_{\bar\alpha}^{\bar\beta}(x,y)}{r^{\,n-l+|\bar\beta|}}\,dx. \]

Lemma 2. For fixed \(I\), the function \(K(I,y)\), as a numerical function of the variable \(y\), is continuous and bounded; moreover, the inequality

\[ |K(I,y)|\leq c_1\,\operatorname{mes} I \tag{6} \]

holds.

Lemma 3. For fixed \(I\) and \(n-l<s/q\), the function \(K_{\bar\alpha}(I,y)\), as a numerical function of the variable \(y\), is continuous and bounded; moreover, the inequality

\[ |K_{\bar\alpha}^{-}(I,y)|\leq c_2(\operatorname{mes} I)^{1-l/q} \tag{7} \]

holds.

Lemma 4. If \(s/q\leq \lambda<s/q+n/q'\), then for fixed \(I\) the function \(K_{\bar\alpha}^{-}(I,y)\), as a numerical function of the variable \(y\), belongs to \(L_{p'}(\Omega)\), where

\[ p'\leq \frac{n}{\lambda-s/q}, \]

and, moreover, the estimate

\[ \|K_{\bar\alpha}^{-}(I,y)\|_{L_{p'}(\Omega)} \leq c_3(\operatorname{mes} I)^{1-l/q}. \tag{8} \]

holds.

Lemma 5. If \(n\leq lp\), \(|\bar\beta|=l-[n/p]-1\), then for fixed \(I\) the function \(D^{\bar\beta}K_{\bar\alpha}^{-}(I,y)\), as a numerical function of the variable \(y\), belongs to \(L_{p'}(\Omega)\), and, moreover, the inequality

\[ \|D^{\bar\beta}K_{\bar\alpha}^{-}(I,y)\|_{L_{p'}(\Omega)} \leq c_4\,\operatorname{mes} I \tag{9} \]

holds.

Definition. We shall say that an abstract additive set function \(\varphi(E)\) with values in a Banach space \(X\) satisfies the Lipschitz condition with exponent \(\bar\beta\), if

\[ \|\varphi(E_1)-\varphi(E_2)\|_X \leq M\rho^{\bar\beta}(E_1,E_2), \]

where \(M=\mathrm{const}\), \(0<\bar\beta\leq 1\), \(\rho(E_1,E_2)=\operatorname{mes}(E_1\cup E_2\setminus E_1\cap E_2)\).

Consider the set of abstract additive set functions \(\varphi(E)\) with values in the Banach space \(X\), satisfying, together with their \(\rho\)-th generalized derivatives, the Lipschitz condition with exponent \(\beta\). For this set we introduce the norm by the equality

\[ \|\varphi\|= \sup_{\substack{E_1,E_2\in\Omega\\ |\bar\alpha|=\rho}} \frac{\|D^{\bar\alpha}\varphi(E_1)-D^{\bar\alpha}\varphi(E_2)\|_X} {\rho^\beta(E_1,E_2)} +\sum_{|\bar\alpha|=\rho}\max \|D^{\bar\alpha}\varphi(E)\|_X. \tag{10} \]

This set with norm (10) will be denoted by \(\operatorname{Lip}^{(\rho)}\beta(X,\Omega)\). We introduce also the set of abstract additive functions \(\widetilde{\varphi}(I)\), defined on \(\varepsilon_s\), with values in the Banach space \(X\), and satisfying the Lipschitz condition with exponent \(\bar\beta\). We denote this set by \(\operatorname{Lip}\beta(X,S_s\cap\Omega)\) and define the norm in it by the equality:

\[ \|\widetilde{\varphi}\|_{\operatorname{Lip}\beta(X,S_s\cap\Omega)} = \sup_{I_1,I_2\in\overline{E}_s} \frac{\|\widetilde{\varphi}(I_1)-\widetilde{\varphi}(I_2)\|_X} {\rho^\beta(I_1,I_2)} + \max_{I\in\varepsilon_s}\|\widetilde{\varphi}(I)\|_X. \tag{11} \]

Theorem 1. If \(\varphi(E)\in\Psi_p^{(l)}(X,\Omega)\), \(n<lp\) and \(\rho=l-[n/p]-1\), then

\[ \varphi(E)\in \operatorname{Lip}^{(\rho)}1(X,\Omega). \]

Moreover, the inequalities

\[ \|\varphi\|_{\operatorname{Lip}^{(\rho)}1(X,\Omega)} \leq c_5\|\varphi\|_{\Psi_p^{(l)}(X,\Omega)} \tag{12} \]

hold.

Theorem 2. If \(\varphi(E)\in\Psi_p^{(l)}(X,\Omega)\), \(n\geq lp\), \(s>n-lp\), \(q\leq \dfrac{sp}{\,n-lp\,}\), then \(\varphi\) is defined on all smooth manifolds \(S_s\cap\Omega\) of dimension \(s\) and represents a set function \(\tilde{\varphi}(I)\) belonging to the set

\[ \operatorname{Lip}\left(1-\frac1q\right)(X,S_s\cap\Omega), \]

where \(I\) belongs to \(\varepsilon_s\), and the inequality

\[ \|\tilde{\varphi}\|_{\operatorname{Lip}\left(1-\frac1q\right)(X,S_s\cap\Omega)} \leq c_6\|\varphi\|_{\Psi_p^{(l)}(X,\Omega)} \tag{13} \]

holds.

We indicate the path of proof of one of these theorems, for example theorem 2.
For the functions \(\tilde{\varphi}_h(I)\) the equality holds

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

therefore

\[ \begin{aligned} \|\tilde{\varphi}_h\|_{\operatorname{Lip}\left(1-\frac1q\right)(X,S_s\cap\Omega)} \leq \Bigg\{ &\sup_{I_1,I_2\in\varepsilon_s} \frac{\|\tilde{\varphi}'_h(I_1)-\tilde{\varphi}'_h(I_2)\|_X} {[\rho(I_1,I_2)]^{1-1/q}} + \max_{I\in\varepsilon_s}\|\tilde{\varphi}'_h(I)\|_X \\ &+ \sum_{|\bar{\alpha}|=l} \left[ \sup_{I_1,I_2\in\varepsilon_s} \frac{\|\tilde{\varphi}^{\bar{\alpha}}_h(I_1)-\tilde{\varphi}^{\bar{\alpha}}_h(I_2)\|_X} {[\rho(I_1,I_2)]^{1-1/q}} + \max_{I\in\varepsilon_s}\|\varphi^{\bar{\alpha}}_h(I)\|_X \right] \Bigg\}, \tag{14} \end{aligned} \]

where

\[ \tilde{\varphi}'_h(I)=\int_{\Omega}K(I,y)\,d_y\varphi_h(E); \qquad \tilde{\varphi}^{\bar{\alpha}}_h(I)= \int_{\Omega}K_{\bar{\alpha}}^{-}(I,y)\,d_yD^{\bar{\alpha}}\varphi_h(E). \]

Let us estimate separately each of the norms \(\|\ \|_X\) entering into each term of the right-hand side of inequality (14):

\[ \begin{aligned} \|\tilde{\varphi}^{\bar{\alpha}}_h(I_1)-\tilde{\varphi}^{\bar{\alpha}}_h(I_2)\|_X &= \left\| \int_{\Omega} \{K_{\bar{\alpha}}^{-}(I_1,y)-K_{\bar{\alpha}}^{-}(I_2,y)\} \,d_yD^{\bar{\alpha}}\varphi_h(E) \right\|_X \\ &= \left\| \int_{\Omega} \left( \int_{I_1}\frac{\omega_{\bar{\alpha}}^{\bar{\beta}}(x,y)}{r^{\,n-l}}\,dx - \int_{I_2}\frac{\omega_{\bar{\alpha}}^{\bar{\beta}}(x,y)}{r^{\,n-l}}\,dx \right) d_yD^{\bar{\alpha}}\varphi_h(E) \right\|_X \\ &= \left\| \int_{\Omega} \left( \int_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\bar{\alpha}}^{\bar{\beta}}(x,y)}{r^{\,n-l}}\,dx \right) d_yD^{\bar{\alpha}}\varphi_h(E) \right\|_X . \end{aligned} \]

Since \(p<q\), it follows from the conditions of the theorem that

\[ n-l<\frac{s}{q}+\frac{n}{p'}. \tag{15} \]

With respect to \(n-l\) one can make two assumptions: \(n-l<s/q\) and \(n-l\geq s/q\).

If \(n-l<s/q\), then, by lemma 3,

\[ \left| \int_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\bar{\alpha}}^{\bar{\beta}}(x,y)}{r^{\,n-l}}\,dx \right| \leq c_2\,[\operatorname{mes}(I_1\cup I_2\setminus I_1\cap I_2)]^{1-1/q}, \]

therefore

\[ \left\| \int\limits_{\Omega}\left( \int\limits_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\alpha}^{\bar\beta}(x,y)}{r^{\,n-l}}\,dx \right)d_yD^{\bar\alpha}\varphi_h(E) \right\|_X \le \]

\[ \le c_3\,[\operatorname{mes}(I_1\cup I_2\setminus I_1\cap I_2)]^{1-1/q} \left\|D^{\bar\alpha}\varphi_h\right\|_{\Phi_p(X,\Omega)} . \tag{16} \]

If, however, \(n-l\ge s/q\), then, taking into account condition (15) and Lemma 4, we obtain that

\[ \int\limits_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\alpha}^{\bar\beta}(x,y)}{r^{\,n-l}}\,dx\in L_{p'}(\Omega), \]

therefore

\[ \left\| \int\limits_{\Omega}\left( \int\limits_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\alpha}^{\bar\beta}(x,y)}{r^{\,n-l}}\,dx \right)d_yD^{\bar\alpha}\varphi_h(E) \right\|_X \le \]

\[ \le \left\| \int\limits_{I_1\cup I_2\setminus I_1\cap I_2} \frac{\omega_{\alpha}^{\bar\beta}(x,y)}{r^{\,n-l}}\,dx \right\|_{L_{p'}(\Omega)} \left\|D^{\bar\alpha}\varphi_h\right\|_{\Phi_p(X,\Omega)} \le \]

\[ \le c_4\,[\operatorname{mes}(I_1\cup I_2\setminus I_1\cap I_2)]^{1-1/q} \left\|D^{\bar\alpha}\varphi_h\right\|_{\Phi_p(X,\Omega)} . \tag{17} \]

Denoting \(c_5=\max(c_3,c_4)\), we shall have

\[ \left\|\bar{\varphi}_h^{\bar\alpha}(I_1)-\bar{\varphi}_h^{\bar\alpha}(I_2)\right\|_X \le c_5\,[\operatorname{mes}(I_1\cup I_2\setminus I_1\cap I_2)]^{1-1/q} \left\|D^{\bar\alpha}\varphi_h\right\|_{\Phi_p(X,\Omega)} . \tag{18} \]

By analogous arguments we obtain:

\[ \left\|\widetilde{\varphi}_h(I_1)-\widetilde{\varphi}_h(I_2)\right\|_X \le c_7\,[\operatorname{mes}(I_1\cup I_2\setminus I_1\cap I_2)]^{1-1/q} \left\|\varphi_h\right\|_{\Phi_1(X,\Omega)}; \tag{19} \]

\[ \left\|\widetilde{\varphi}_h(I)\right\|_X \le c_8\,(\operatorname{mes} I)\left\|\varphi_h\right\|_{\Phi_1(X,\Omega)}; \tag{20} \]

\[ \left\|\widetilde{\varphi}_h^{\bar\alpha}(I)\right\|_X \le c_9\,(\operatorname{mes} I)^{1-1/q} \left\|D^{\bar\alpha}\varphi_h\right\|_{\Phi_p(X,\Omega)} . \tag{21} \]

Taking into account inequalities (18), (19), (20), (21), (14), we obtain

\[ \left\|\widetilde{\varphi}_h\right\|_{\operatorname{Lip}\left(1-\frac1q\right)(X,S_s\cap\Omega)} \le c_{10}\left\|\varphi_h\right\|_{\Psi_p^{(l)}(X,\Omega)} . \tag{22} \]

Passing to the limit in (22) as \(h\to0\), we obtain (13).

I express my sincere gratitude to Academician S. L. Sobolev for valuable advice and attention to this work.

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

Received
11 VI 1962

REFERENCES

\({}^{1}\) S. L. Sobolev, DAN, 114, No. 6 (1957).
\({}^{2}\) S. L. Sobolev, DAN, 115, No. 1 (1957).
\({}^{3}\) S. L. Sobolev, Fund. Math., 47, No. 3 (1959).
\({}^{4}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\({}^{5}\) V. B. Korotkov, DAN, 141, No. 2 (1961).
\({}^{6}\) Yu. I. Gil’derman, DAN, 142, No. 4 (1962).