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Reports of the Academy of Sciences of the USSR
- Volume 160, No. 3
MATHEMATICS
V. B. KOROTKOV
ON CRITERIA OF COMPACTNESS IN SPACES OF ABSTRACT FUNCTIONS AND ON COMPLETE CONTINUITY OF THE EMBEDDING OPERATOR
(Presented by Academician S. L. Sobolev on 16 VII 1964)
Definition 1 \((^1)\). 1) Denote by \(\sum (R_n)\) the collection of all \(L\)-measurable subsets of finite measure of Euclidean space \(R_n\). Denote by \(\Lambda\) the collection of all finite linear combinations of characteristic functions of sets \(E \in \sum (R_n)\). By \(\Phi_p(X, R_n)\), \(1 < p < \infty\), we denote the collection of all additive set functions \(\varphi(E)\), defined on \(\sum (R_n)\), taking values in the \(B\)-space \(X\) and having finite norm
\[ \|\varphi\|_{\Phi_p(X,R_n)}=\sup_{\omega\in\Lambda}\left\|\int_{R_n}\omega\,d\varphi\right\|_X \big/ \|\omega\|_{L_{p'}},\qquad \frac{1}{p}+\frac{1}{p'}=1. \tag{1} \]
2) A function \(\varphi\) from \(\Phi_p\) is said to belong to \(\Psi_p(X,R_n)\) if
\[ \lim_{|h|\to 0}\|\varphi(E+h)-\varphi(E)\|_{\Phi_p}=0. \tag{2} \]
3) In what follows, by \(\Omega\) we shall everywhere mean a bounded simply connected domain in \(R_n\). One says that \(\varphi \in \Phi_p(X,\Omega)\) (respectively \(\varphi \in \Psi_p(X,\Omega)\)) if \(\varphi \in \Phi_p(X,R_n)\) (respectively \(\varphi \in \Psi_p(X,R_n)\)) and \(\varphi(E)=\varphi(E\cap\Omega)\), \(E\in \sum(R_n)\).
For every function \(\omega=\sum \alpha_i\chi_{E_i}\) from \(\Lambda\), by definition we put
\[ \int \omega\,d\varphi=\sum \alpha_i\varphi(E_i). \]
In view of (1),
\[ \left\|\int \omega\,d\varphi\right\|\le \|\omega\|_{L_{p'}}\|\varphi\|_{\Phi_p}. \tag{3} \]
Using (3) and the density of \(\Lambda\) in \(L_{p'}\), one may, by a limiting passage, define for any function \(g\in L_{p'}\) the integral \(\int g\,d\varphi\), which makes it possible, by means of a known integral identity, to introduce for some functions from \(\Phi_p\) the notion of generalized derivative \((^1)\) and thereby to construct analogues of the spaces \(W_p^l\) \((^2)\) and \(H_p^r\) \((^3)\)—the spaces \(\Phi_p^l(X,\Omega)^*\) and \(H_p^r(X,\Omega)\) \((^4)\), for which \((^1,^4,^5)\) the same embedding theorems are valid as for \(W_p^l\), \(H_p^r\).
The specific feature of these embedding theorems is that, in the case of infinite-dimensional \(X\), the embedding operator will not be completely continuous.
Example 1. Let \(A\) be a bounded noncompact set in \(X\). Consider the functions \(\varphi_x(E)=xmE\), \(x\in A\), \(E\in \sum(\Omega)\). It is clear that \(D^\alpha\varphi_x=0\) for \(|\alpha|\ne 0\). Consequently,
\[ \|\varphi_x\|_{\Phi_p^{\,l}}=m^{1/p}\Omega\,\|x\|. \]
The trace of the function \(\varphi_x\) under the embedding \(\Phi_q^{l}(X,\Omega)\) into \(\Phi_q^{l_1}(\Omega_s)\) has the form \(\widetilde{\varphi}_x(e)=xme\), \(e\in \sum(\Omega_s)\), and
\[ \|\widetilde{\varphi}_x\|_{\Phi_q^{\,l_1}}=m^{1/q}\Omega_s\,\|x\|; \]
hence it follows that the set \(\widetilde{\Phi}=\{\widetilde{\varphi}_x:\ x\in A\}\) is noncompact in \(\Phi_q^{\,l_1}(X,\Omega_s)\).
\[ \text{* The space } \Phi_p^l \text{ for integer } l \text{ was defined in } (^{1}), \text{ and for noninteger } l \text{ in } (^{4}). \]
The main result of the article is contained in Theorem 2 and is based on the lemmas given below.
Lemma 1. In order that a set \(F\{f\}\) from the space \(C(X,\Omega)\) of continuous abstract functions be compact, it is necessary and sufficient that:
1) the set
\[
\Xi=\{f(y);\ y\in\Omega,\ f\in F\}
\]
be compact in \(X\);
2) the functions \(f\in F\) be equicontinuous.
The necessity of conditions 1) and 2) is obvious. The proof of their sufficiency can be carried out by analogy with the proof of Arzelà’s theorem given in \((^{6})\), p. 519. The example of the set \(f_x(t)=x,\ x\in A\), where \(A\) is the set of Example 1, shows that condition 1) cannot, in the case of infinite-dimensional \(X\), be replaced by the condition that the set \(F\) be bounded in \(C(X,\Omega)\).
Lemma 2 \((^{5})\), p. 31. The set of values of a function \(\varphi\) from \(\Psi_p(X,\Omega)\) is compact in \(X\).
Corollary. If \(\varphi\in\Psi_p(X,\Omega)\) and \(h(y)\) is a continuous function, then the set of values of the function
\[
\widetilde{\varphi}(E)=\int_E h(y)\,d\varphi
\]
is compact in \(X\).
Lemma 3. For compactness of a set \(\Phi\{\varphi\}\) from \(\Psi_p(X,\Omega)\) it is necessary and sufficient that:
1) the set
\[
\Xi=\{\varphi(E);\ E\in\sum(\Omega),\ \varphi\in\Phi\}
\]
be compact in \(X\);
2) \(\|\varphi\|_{\Phi_p}\le K\) for all \(\varphi\in\Phi\);
3) condition 2) be satisfied uniformly with respect to \(\varphi\) from \(\Phi\).
Proof. The necessity of conditions 2), 3) is obvious. The necessity of condition 1) follows from the existence of a finite \(\varepsilon/2\)-net in \(\Phi\) and from Lemma 2.
Sufficiency. Denote by \(K_h(y)\), \(h>0\), the \(n\)-dimensional cube with center at the point \(y\) and edge \(2h\), and by \(\Omega_h\) the set of all points \(y\in R_n\) whose distance from \(\Omega\) is not more than \(h\). Consider the functions
\[
\varphi_h(y)=(2h)^{-n}\varphi(K_h(y)),\quad \varphi\in\Phi.
\]
Condition 1) of Lemma 3 ensures the fulfillment of condition 1) of Lemma 1. The equicontinuity of the functions \(\varphi_h\) \((\varphi\in\Phi)\) follows from the estimate, easily verified with the aid of (3),
\[
\|\varphi_h(y_1)-\varphi_h(y_2)\|_X\le K(2h)^{-n}m^{1/p'}(K_h(y_1)\setminus K_h(y_2)).
\]
Thus, by Lemma 1 the family of functions \(\varphi_h\) \((\varphi\in\Phi)\) is compact in \(C(X,\Omega_h)\). Consequently, the family of functions of sets
\[
\varphi_h(E)=\int_E \varphi_h(y)\,dy,\quad E\in\sum(\Omega),
\]
is compact in \(\Psi_p(X,\Omega)\). Using (1), pp. 314–316, one can obtain the following inequality*
\[
\|\varphi_h(E)-\varphi(E)\|_{\Phi_p}
\le
(2h)^{-n}\int_{K_h(0)}
\|\varphi(E+y)-\varphi(E)\|_{\Phi_p}\,dy.
\tag{4}
\]
From (4), condition 2), and the compactness of the family \(\varphi_h(E)\), it follows that the set \(\Phi\) is compact. The set of functions of Example 1 shows that condition 1) cannot be discarded in the case of infinite-dimensional \(X\).
Lemma 4. Let \(\varphi_{x_i}'\in\Psi_p(X,\Omega)\) be the generalized derivative of a function \(\varphi\in\Phi_p(X,\Omega)\). Then
\[
\left\|
\frac{\varphi(E+h_i)-\varphi(E)}{|h_i|}
-\varphi_{x_i}'(E)
\right\|_{\Phi_p}
\le
\frac{1}{|h_i|}
\int_0^{|h_i|}
\|\varphi_{x_i}'(E+\xi_i)-\varphi_{x_i}'(E)\|_{\Phi_p}\,d\xi_i.
\tag{5}
\]
* This inequality can also be obtained with the aid of the isometry of \(\Phi_p\), proved in \((^{4,5})\), to a certain subspace of the space of linear bounded operators acting from \(X^*\) into \(L_p\).
Definition 2. Let \(r=\bar r+\alpha\), \(\bar r>0\) an integer, \(0<\alpha<1\). We say that \(\varphi\in H_p^r(X,R_n)\) if \(\varphi\in \Phi_p(X,R_n)\), has all unmixed generalized derivatives up to order \(\bar r\) from \(\Phi_p(X,R_n)\), and the norm is finite
\[ \|\varphi\|_{H_p^r(X,R_n)} = \|\varphi\|_{\Phi_p} + \max_{i=1,2,\ldots,n}\sup_{h_i} \frac{\|\varphi_{x_i}^{\bar r}(E+h_i)-\varphi_{x_i}^{\bar r}(E)\|_{\Phi_p}}{|h_i|^\alpha}. \]
Theorem 1. If a set \(\Phi\) in \(H_p^r(X,R_n)\) satisfies the conditions:
1) the set \(\Xi=\{\varphi(E);\ E\in \sum(R_n),\ \varphi\in\Phi\}\) is compact in \(X\);
2) \(\|\varphi\|_{H_p^r(X,R_n)}\le K\) for all \(\varphi\in\Phi\);
3) \(\varphi(E)=\varphi(E\cap\Omega)\) for all \(\varphi\in\Phi\);
4) uniformly with respect to \(\varphi\in\Phi\),
\[ \|\varphi_{x_i}^{\bar r}(E+h_i)-\varphi_{x_i}^{\bar r}(E)\|_{\Phi_p} = o(|h_i|^\alpha), \qquad i=1,2,\ldots,n, \tag{6} \]
then \(\Phi\) is compact in \(H_p^r(X,R_n)\).
Proof. Using conditions 2) and 3), one can show that for all \(\varphi\in\Phi\)
\[ \|\varphi_{x_i}^{k}\|_{\Phi_p}\le K_1, \qquad i=1,2,\ldots,n;\quad k=0,1,2,\ldots,\bar r. \tag{7} \]
Hence, and from Lemma 4, we obtain
\[ \|\varphi_{x_i}^{k}(E+h_i)-\varphi_{x_i}^{k}(E)\|_{\Phi_p} \le K_1|h_i|; \]
\[ i=1,2,\ldots,n;\quad k=0,1,\ldots,\bar r-1; \tag{8} \]
\[ \lim_{|h_i|\to0} \left\| \frac{1}{|h_i|} \bigl(\varphi_{x_i}^{k}(E+h_i)-\varphi_{x_i}^{k}(E)\bigr) - \varphi_{x_i}^{k+1}(E) \right\|_{\Phi_p} =0, \]
\[ k=0,1,2,\ldots,\bar r-1, \tag{9} \]
uniformly with respect to \(\varphi\in\Phi\). Using condition 1) and (9) ((6) for \(k=\bar r\)), it is not difficult to show successively that the functions of the families \(\Phi\) and \(\Phi_{ik}=\{\varphi_{x_i}^{k};\ \varphi\in\Phi\}\), \(i=1,2,\ldots,n\), \(k=1,2,\ldots,\bar r\), satisfy condition 1) of Lemma 3, whence, together with (7), (8) ((6) for \(k=\bar r\)), by virtue of Lemma 3, it follows that the families \(\Phi\) and \(\Phi_{ik}\) are compact in \(\Psi_p(X,\Omega)\). Thus, from any infinite subset of \(\Phi\) one can select a subsequence \(\varphi_k\), converging in the norm \(\Phi_p\) together with the derivatives of order \(\bar r\). Consider the expression
\[ |h_i|^{-\alpha} \left\| \bigl(D_{x_i}^{\bar r}\varphi_k(E+h_i)-D_{x_i}^{\bar r}\varphi_l(E+h_i)\bigr) - \bigl(D_{x_i}^{\bar r}\varphi_k(E)-D_{x_i}^{\bar r}\varphi_l(E)\bigr) \right\|_{\Phi_p}. \tag{10} \]
By virtue of (6), for any \(\varepsilon>0\) one can choose \(\delta>0\) so that (10) is less than \(\varepsilon/2\) for \(|h_i|<\delta\). Using the invariance of the norm \(\Phi_p\) with respect to shifts and the convergence of \(D_{x_i}^{\bar r}\varphi_k\) in the norm \(\Phi_p\), one can further, for all \(|h_i|\ge\delta\), choose a number \(N\) such that for \(k>N\), \(l>N\), (10) also becomes less than \(\varepsilon/2\).
Corollary*. The set \(\Phi\) of Theorem 1 is compact in \(H_p^{r'}(X,R_n)\), \(r'<2\).
Denote by \(B_1\overset{V}{\to}B_2\) the embedding of a \(B\)-space \(B_1\) into a \(B\)-space \(B_2\) with bounded embedding operator \(V\), and by \(VM\) the image of a set \(M\subset B_1\) under this embedding.
Theorem 2. Let \(\Phi\) be a bounded set in \(\Phi_p^r(X,\Omega_n)\), satisfying condition 1) of Theorem 1, and let \(\Phi_p^r(X,\Omega_n)\overset{V}{\to}\)
* Cf. for \(X=R_1\) with (3), p. 81.
\[
\to \Phi_{p_1}^{r_1}(X,\Omega_{n_1'}),\quad \text{where } r_1<r^*=r-\frac{n}{p}-\frac{n_1}{p_1}\ \text{ and } \Omega_{n_1'} \text{ is a strictly interior subdomain of the } n_1\text{-dimensional section } \Omega_n.
\]
Then \(\Phi\) is compact in \(\Phi_{p_1}^{r_1}(X,\Omega_{n_1'})\).
An analogous theorem is valid for \(H\)-classes.
Proof. Let \(\Omega_n''\) be a strictly interior subdomain of \(\Omega_n\) containing \(\Omega_n'\). A function \(\varphi\in\Phi_p^r(X,\Omega_n)\) can be extended \({}^{(5)}\) from \(\Omega_n'\) to \(R_n\) with preservation of the class, and the extension operator is linear, bounded, and has the form
\[ P\varphi=\widetilde{\varphi}(I)=\int_I H(y)\,d\varphi,\qquad I\in\sum(R_n), \tag{11} \]
where \(H(y)\) is a certain continuous function equal to 1 in \(\Omega_n\) and finite in \(\Omega_n''\), \(\Omega_n'\subset\Omega_n''\subset\Omega_n\). From the corollary to Lemma 2 it follows that the set of values of all functions \(\widetilde{\varphi}\), \(\varphi\in\Phi\), is compact in \(X\). In \({}^{(5)}\) it was shown that
\[
\Phi_p^r(X,R_n)\to H_p^{r-\varepsilon}(X,R_n)\to H_p^{r-2\varepsilon}(X,R_n)\to \Phi_p^{r-3\varepsilon}(X,R_n).
\]
Hence, by virtue of the corollary to Theorem 1, the set \(\widetilde{\Phi}=\{\widetilde{\varphi};\ \varphi\in\Phi\}\) is compact in \(H_p^{r-2\varepsilon}\), and consequently also in \(\Phi_p^{r-3\varepsilon}\). Since, by hypothesis, \(r<r^*\), for sufficiently small \(\varepsilon>0\)
\[
\Phi_p^{r-3\varepsilon}(X,\Omega_n)\xrightarrow{V_\varepsilon}\Phi_{p_1}^{r_1}(X,\Omega_{n_1'})
\]
and, moreover, \(V_\varepsilon\widetilde{\Phi}\) is compact in \(\Phi_{p_1}^{r_1}(X,\Omega_{n_1'})\). But, by the properties of the function \(H(x)\), \(V\Phi=V_\varepsilon\widetilde{\Phi}\). The theorem is proved.
Remark 1. If there exists an extension (11) from the domain \(\Omega_n\) itself, then in Theorem 2 one may put \(\Omega_{n_1'}=\Omega_{n_1}\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
4 VII 1964
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