MATHEMATICS

V. B. Korotkov

On Direct and Inverse Embedding Theorems for Some Spaces of Abstract Set Functions

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

Let \(\Phi(E)\) be an abstract additive set function, defined on all Lebesgue-measurable subsets \(E\) of a bounded domain \(\Omega\) of Euclidean space \(R_n\), with values in a Banach space \(X\); denote by \(\bar X\) the space conjugate to \(X\). We shall say that the function \(\Phi(E)\) belongs to the space \(\Phi_p(X,\Omega)\), introduced by S. L. Sobolev \((^1)\), if

\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)} = \sup_{\tilde\omega} \frac{ \left\|\int_{\Omega}\tilde\omega(x)\,d_x\Phi(E)\right\|_{X} }{ \|\tilde\omega\|_{L_{p'}(\Omega)} } <\infty, \]

where \(\tilde\omega(x)\) is a real measurable function taking only a finite number of nonzero values, \(\frac1p+\frac1{p'}=1,\ p>1\). Denote by \(\tilde\Psi_p(X,\Omega)\) \((^1)\) the totality of all abstract set functions from \(\Phi_p\) that are absolutely continuous in the metric \(\|\ \|_{\Phi_p}\); the totality of all abstract functions from \(\Phi_p\) continuous with respect to shifts in the metric \(\|\ \|_{\Phi_p}\) will be denoted by \(\Psi_p(X,\Omega)\) \((^1)\).

It is known that every function \(\Phi(E)\) from \(\Phi_p(X,\Omega)\), \(p>1\), is absolutely continuous in the metric of the space \(X\) \((^{1,2})\). Indeed, let \(\tilde\omega=\chi_e\), where \(\chi_e(x)\) is the characteristic function of the set \(e\); then

\[ \|\Phi(e)\|_{X} = (\operatorname{mes} e)^{1/p'} \frac{\|\Phi(e)\|_{X}}{(\operatorname{mes} e)^{1/p'}} = \]

\[ = \frac{ \left\|\int_{\Omega}\chi_e(x)\,d_x\Phi(E)\right\|_{X} }{ (\operatorname{mes} e)^{1/p'} } (\operatorname{mes} e)^{1/p'} \le (\operatorname{mes} e)^{1/p'} \|\Phi(E)\|_{\Phi_p}. \]

Consequently, for any \(f\in\bar X\) the real-valued set function \(f\Phi(E)\) is absolutely continuous and therefore can be represented in the form

\[ f\Phi(E)=\int_E \varphi_f(x)\,dx. \]

Lemma 1. Let \(\Phi(E)\in\Phi_p(X,\Omega)\), \(p>1\). Then

\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)} = \sup_{\tilde\omega} \frac{ \left\|\int_{\Omega}\tilde\omega\,d_x\Phi(E)\right\|_{X} }{ \|\tilde\omega\|_{L_{p'}} } = \sup_{\|f\|\le 1}\|\varphi_f\|_{L_p(\Omega)}. \]

For \(p=1\) an analogous assertion holds for functions \(\Phi(E)\) absolutely continuous in the metric \(X\), forming the space \(\bar\Phi_1\), defined in \((^2)\). The lemma remains valid if \(p>1\) and \(\operatorname{mes}\Omega=\infty\), with the corresponding definition of the space \(\Phi_p(X,\Omega)\).

Corollary. Let \(\Phi(E)\in \Phi_p(X,\Omega)\). Then the correspondence \(f\to \varphi_f(x)\) is a linear continuous operator \(\Phi\) acting from \(\overline X\) into \(L_p(\Omega)\), and
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)}=\|\Phi\|. \]
In this case we shall say that the operator \(\Phi\) is generated by the function \(\Phi(E)\). We denote by \(\mathfrak M_\Phi\) the set \(\{\varphi_f(x), \|f\|\leqslant 1\}\); \(\mathfrak M_\Phi\subset L_p(\Omega)\).

Definition 1. By \((B_1\to B_2;\mathfrak M)\) we denote the space of all linear operators acting from \(B_1\) into \(B_2\) and possessing a certain property \(\mathfrak M\).

Definition 2. We shall call an operator
\[ \Phi\in(\overline X\to L_p(\Omega);\ \text{cont.}) \]
absolutely continuous if its norm is arbitrarily small when \(\operatorname{mes}\Omega\to 0\), i.e., if \(\mathfrak M_\Phi\) is a family of functions with uniformly absolutely continuous norms \(\bigl((^3),\ \text{p. }117\bigr)\).

Theorem 1. A function \(\Phi(E)\) from \(\Phi_p(X,\Omega)\) generates an operator
\[ \Phi\in(\overline X\to L_p(\Omega);\ \text{cont.}), \]
a function \(\Phi(E)\) from \(\widetilde\Psi_p(X,\Omega)\) generates an operator
\[ \Phi\in(\overline X\to L_p(\Omega);\ \text{abs. cont.}), \]
and a function \(\Phi(E)\) from \(\Psi_p(X,\Omega)\) generates an operator
\[ \Phi\in(\overline X\to L_p(\Omega);\ \text{completely cont.}). \]
Moreover
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)}=\|\Phi\|. \]

Thus, \(\Phi(E)\in\Phi_p\) (respectively \(\Phi(E)\in\widetilde\Psi_p\), respectively \(\Phi(E)\in\Psi_p\)) if and only if \(\mathfrak M_\Phi\) is a family of functions bounded (respectively with uniformly absolutely continuous norms, respectively compact) in \(L_p(\Omega)\).

Hence, and from the compactness criterion of M. A. Krasnosel’skii \((^3)\), there follows immediately

Theorem 2.
\[ \widetilde\Psi_p=\Psi_1\cap\widetilde\Psi_p,\qquad p>1. \]

A natural generalization of Krasnosel’skii’s criterion is

Theorem 3. Let \(\mathfrak M\subset\widetilde\Psi_p(X,\Omega)\). Then \(\mathfrak M\) is compact in \(\widetilde\Psi_p(X,\Omega)\) if and only if \(\mathfrak M\) is a family of functions uniformly absolutely continuous in the metric \(\|\ \|_{\Phi_p}\), whose set is compact in \(\Phi_1(X,\Omega)\).

Now let \(X\) be the space conjugate to some Banach space \(B\), i.e. \(X=\overline B\). As above, one can show that if \(\Phi(E)\in\Phi_p(X,\Omega)\), then for any \(g\in B\) the scalar function \(\Phi(E)g\) is absolutely continuous and therefore representable in the form
\[ \Phi(E)g=\int_E \varphi_g(x)\,dx. \]

Lemma 2.
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)} = \sup_{\|g\|_B\leqslant 1}\|\varphi_g\|_{L_p(\Omega)}. \]

Thus, a function \(\Phi(E)\in\Phi_p(X,\Omega)\) generates an operator
\[ \Phi\in(B\to L_p(\Omega);\ \text{cont.}), \]
and moreover
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)}=\|\Phi\|. \]
Conversely, let
\[ \Phi\in(B\to L_p(\Omega);\ \text{cont.}). \]
Then
\[ \left|\int_E \Phi(g)\,dx\right| \leqslant \|\Phi\|(\operatorname{mes}\Omega)^{1/p'}\|g\|, \]
i.e.
\[ \int_E \Phi(g)\,dx \]
is a continuous linear functional on \(B\). Taking
\[ \int_E \Phi(g)\,dx=\Phi(E)g, \]
we are convinced that \(\Phi(E)\) is an additive abstract function of sets, and
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)}=\|\Phi\|. \]

Theorem 4. If \(X=\overline B\), then
\[ \Phi_p(X,\Omega)\leftrightarrow(B\to L_p(\Omega);\ \text{cont.}), \]
\[ \widetilde\Psi_p(X,\Omega)\leftrightarrow(B\to L_p(\Omega);\ \text{abs. cont.}), \]
\[ \Psi_p(X,\Omega)\leftrightarrow(B\to L_p(\Omega);\ \text{completely cont.}), \]
where the sign \(\leftrightarrow\) denotes an isometric and isomorphic correspondence defined by the formula
\[ \int_E \Phi(g)\,dx=\Phi(E)g. \tag{*} \]

Example 1. Let \(X=R_m\); then
\[ \Phi_p(R_m,\Omega)=\widetilde\Psi_p(R_m,\Omega)=\Psi_p(R_m,\Omega). \]

Example 2. Let \(X=L_2(\Omega)\), \(p=2\), \(\Phi=I\), where \(I\) is the identity operator. From (*) it is seen that \(I\) is generated by the function \(\Phi(E)=\chi_E(x)\), \(\chi_E(x)\) being the characteristic function of the set \(E\). It is easy to see that
\(\Phi(E)\in \Phi_2(L_2(\Omega),\Omega)\setminus \Psi_2(L_2(\Omega);\Omega)\).

Example 3. Let \(X=L_2(\Omega)\), \(1<p<2\), \(\Phi=V\), where \(V\) is the operator of embedding \(L_2(\Omega)\) into \(L_p(\Omega)\). Then \(\Phi(E)=\chi_E\) and
\(\Phi(E)\in \Psi_p(L_2(\Omega),\Omega)\setminus \Psi_p(L_2(\Omega),\Omega)\)

Let
\[ \Psi(E)=\frac{\partial^l\Phi(E)}{\partial x_1^{\,l_1}\ldots \partial x_n^{\,l_n}} \]
be the generalized, in the sense of S. L. Sobolev, derivative of the function \(\Phi(E)\) (1). Denote by \(\Phi_p^{(l)}(X,\Omega)\) the collection of those and only those functions \(\Phi(E)\) from \(\overline{\Phi}_1(X,\Omega)\) all of whose generalized derivatives of order \(l\) belong to \(\Phi_p(X,\Omega)\). In \(\Phi_p^{(l)}(X,\Omega)\) introduce the norm
\[ \|\Phi(E)\|_{\Phi_p^{(l)}(X,\Omega)} = \|\Phi(E)\|_{\Phi_1(X,\Omega)} + \sum_{l_1+\ldots+l_n=l} \left\| \frac{\partial^l\Phi(E)}{\partial x_1^{\,l_1}\ldots \partial x_n^{\,l_n}} \right\|_{\Phi_p(X,\Omega)} . \]

From the fact that \(\Phi(E)\in \Phi_p^{(l)}\) it follows that
\[ \Phi(E)g=\int_E \varphi_g(x)\,dx,\qquad \frac{\partial^l\Phi(E)}{\partial x_1^{\,l_1}\ldots \partial x_n^{\,l_n}}g = \int_E \frac{\partial^l}{\partial x_1^{\,l_1}\ldots \partial x_n^{\,l_n}} \varphi_g(x)\,dx . \]

From Lemma 2 there follows the equivalence of the norm \(\|\Phi(E)\|_{\Phi_p^{(l)}}\) to the norm
\[ {}^{(1)}\|\Phi(E)\|_{\Phi_p^{(l)}(X,\Omega)} = \sup_{\|g\|_B\le 1}\|\varphi_g\|_{W_p^{(l)}}, \]
which we shall call the canonical norm. The space \(\Psi_p^{(l)}(X,\Omega)\) (1,2) of all functions \(\Phi(E)\) from \(\overline{\Phi}_1(X,\Omega)\) having all generalized derivatives of order \(l\), continuous with respect to shifts in the metric \(\|\ \|_{\Phi_p}\), forms a subspace in \(\Phi_p^{(l)}(X,\Omega)\).

Now let \(\lambda>0\), \(\lambda\) noninteger, and \(\lambda=\bar\lambda+\alpha\), \(\bar\lambda\) integer, \(0<\alpha<1\), \(1<p<\infty\). We shall say that \(\Phi(E)\in \Phi_p^{(\lambda)}(X,\Omega)\) if \(\Phi(E)\in \Phi_p^{(\bar\lambda)}(X,\Omega)\) and, for any generalized derivative \(D^{(\bar\lambda)}\) of order \(\bar\lambda\) of the function \(\Phi(E)\), the inequality
\[ \sup_{\|g\|_B\le 1} \int_\Omega\int_\Omega \frac{\left|D^{(\bar\lambda)}\varphi_g(x)-D^{(\bar\lambda)}\varphi_g(y)\right|^p} {|x-y|^{\,n+p\alpha}} \,dx\,dy <\infty \]
holds.

The norm in \(\Phi_p^{(\lambda)}\), defined by the equality
\[ \|\Phi(E)\|_{\Phi_p^{(\lambda)}} = \|\Phi(E)\|_{\Phi_p^{(\bar\lambda)}}+ \]
\[ + \sum_{\lambda_1+\ldots+\lambda_n=\bar\lambda} \sup_{\|g\|_B\le 1} \left( \int_\Omega\int_\Omega \frac{\left|D^{(\bar\lambda)}\varphi_g(x)-D^{(\bar\lambda)}\varphi_g(y)\right|^p} {|x-y|^{\,n+p\alpha}} \,dx\,dy \right)^{1/p}, \]
is equivalent to the canonical norm
\[ {}^{(1)}\|\Phi(E)\|_{\Phi_p^{(\lambda)}}= \sup_{\|g\|_B\le 1}\|\varphi_g\|_{W_p^{(\lambda)}}, \]
where \(W_p^{(\lambda)}(\Omega)\) is the space considered by L. N. Slobodetskii (4), O. V. Besov (5), and others. The formally narrower class
\[ \overline{\Phi}_p^{(\lambda)}(X,\Omega) = \Psi_p^{(\bar\lambda)}\cap \Phi_p^{(\lambda)} \]
in fact coincides with \(\Phi_p^{(\lambda)}\). Let us also define the space \(\Psi_p^{(\lambda)}(X,\Omega)\). \(\Phi(E)\in \Psi_p^{(\lambda)}\) if \(\Phi(E)\in \Phi_p^{(\lambda)}\) and
\[ \lim_{h\to 0}\|\Phi_h(E)-\Phi(E)\|_{\Phi_p^{(\lambda)}}=0, \]
where \(\Phi_h(E)\) is the mean function (1).

Theorem 5. Let \(X=\overline B\), \(1<p<\infty\), \(r>0\), and let the norms in the spaces \(\Phi_p^{(r)}\) and \(\Psi_p^{(r)}\) be canonical. Then
\[ \Phi_p^{(r)}(X,\Omega)\leftrightarrow (B\to W_p^{(r)}(\Omega);\ \text{cont.}),\qquad \Psi_p^{(r)}(X,\Omega)\leftrightarrow (B\to W_p^{(r)}(\Omega);\ \text{completely cont.}), \]
where the symbol \(\leftrightarrow\) denotes an isometric and isomorphic correspondence defined by formula \((*)\).

The correspondences obtained in Theorem 5 make it possible to prove, for the spaces \(\Phi_p^{(r)}\) and \(\Psi_p^{(r)}\), theorems analogous to the known embedding and extension theorems for the spaces \(W_p^{(r)}\). Let us prove, for example, the following theorem (see, for example, \({}^{6}\), p. 111):

Let \(X=\overline B\); \(0\leq k=l-\dfrac np+\dfrac m{p'}\); \(1<p<p'<\infty\); \(\Omega\) be a domain all of whose points are attainable by means of a fixed cone; then \(\Phi_p^{(l)}(X,\Omega_n)\) is embedded in \(\Phi_{p'}^{(k)}(X,\Omega_m)\).

Indeed, \(\Phi(E)\in\Phi_p^{(l)}(X,\Omega)\) generates an operator \(\Phi\in(B\to W_p^{(l)}(\Omega_n),\ \text{cont.})\). In turn, the operator \(W=V\Phi\), where \(V\) is the embedding operator of \(W_p^{(l)}(\Omega_n)\) into \(W_{p'}^{(k)}(\Omega_m)\), generates a function \(\widetilde\Phi(I)\in\Phi_{p'}^{(k)}(X,\Omega_m)\), which is the trace of the function \(\Phi(E)\), and moreover
\[ \|\widetilde\Phi(I)\|_{\Phi_{p'}^{(k)}(X,\Omega_m)} =\|W\|\leq \leq \|V\|\cdot\|\Phi\|=\|V\|\cdot \|\Phi(E)\|_{\Phi_q^{(l)}(X,\Omega_n)}. \]

The corresponding theorems on extension of abstract functions of sets are proved analogously.

Let us note, however, that in the case when \(X=\overline B\) is an infinite-dimensional Banach space, the following two features occur:

1) The embedding operator is not completely continuous, although estimates of the type of the estimates of V. I. Kondrashev \({}^{7}\) do hold. This is connected with the fact that the compactness criterion in \(L_p(\Omega)\), \(p>1\), due to M. Riesz \({}^{8}\), cannot be generalized to the space \(\Psi_p(X,\Omega)\).

2) For \(p>1\), \(lp>n\), \(l\) natural, every function \(\Phi(E)\in\Phi_p^{(l)}(X,\Omega)\) is representable in the form of an indefinite integral of I. M. Gelfand \({}^{9}\) of a weakly continuous abstract function of points \(x\in\Omega\), i.e.
\[ \Phi(E)=\int_E \varphi(x)\,dx, \]
and
\[ \sup_{\|g\|B\leq 1}\max_{x\in\Omega}|\varphi(x)g| \leq c\|\Phi(E)\|_{\Phi_p^{(l)}}. \]

Let \(X\) be an arbitrary Banach space. The application of Lemma 1 and of the apparatus of mean functions makes it possible to obtain in a simple way, for the spaces \(\Psi_p^{(r)}\), analogues of some known embedding and extension theorems \({}^{10}\).

Finally, we note that a number of embedding theorems in which the dimension of the domain \(\Omega\) remains unchanged (of the type of the Gagliardo–Nirenberg theorems \({}^{11,12}\), etc.) also hold for the spaces \(\Phi_p^{(l)}(X,\Omega)\). In this case \(X\) may also be an arbitrary Banach space.

In conclusion I express my deep gratitude to Prof. L. D. Kudryavtsev for posing the problem and for his attention to this work, and also to Acad. S. L. Sobolev and all the participants of his seminar, especially Yu. I. Gilderman, for valuable advice and comments expressed during the discussion of this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
20 XII 1961

CITED LITERATURE

\({}^{1}\) S. L. Sobolev, Fund. Math., 47, 3 (1959).
\({}^{2}\) Yu. I. Gilderman, DAN, 140, No. 4 (1961).
\({}^{3}\) M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, M., 1958.
\({}^{4}\) L. N. Slobodetskii, DAN, 118, No. 2 (1958).
\({}^{5}\) O. V. Besov, DAN, 126, No. 6 (1959).
\({}^{6}\) S. M. Nikol’skii, UMN, 16, issue 5 (1961).
\({}^{7}\) V. I. Kondrashev, DAN, 48, No. 8, 563 (1945).
\({}^{8}\) M. Riesz, Acta Sci. Math. Szeged, 6, 184 (1933).
\({}^{9}\) I. M. Gelfand, Zap. N.-i. inst. matem. i mekh. pri Kharkovsk. gos. univ., 13, ser. 4 (1936).
\({}^{10}\) V. B. Korotkov, DAN, 141, No. 2, 308 (1961).
\({}^{11}\) E. Gagliardo, Ricerche di Mat., 8, No. 1 (1959).
\({}^{12}\) L. Nirenberg, Ann. Scuola Norm. Sup. di Pisa, 13 (1959).