Abstract Generated abstract
The paper develops a framework for additive set functions with values in Banach spaces, introducing norms, absolute continuity, translation continuity, generalized derivatives, and related spaces of Sobolev type. It establishes embedding and compactness criteria for these spaces and identifies several classes of dual-valued set functions isometrically or topologically with spaces of linear operators into Lebesgue, Sobolev, and Hölder-type function spaces. These identifications are used to transfer classical embedding and extension theorems to abstract set functions, including a result that functions in a sufficiently smooth class admit continuous Banach-valued densities when the smoothness exponent satisfies the Sobolev condition.
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V. B. KOROTKOV
ABSTRACT SET FUNCTIONS AND EMBEDDING THEOREMS
(Presented by Academician S. L. Sobolev on 31 III 1962)
Let \(\Omega\) be a simply connected domain of \(n\)-dimensional Euclidean space \(R_n\). Denote by \(\Sigma(\Omega)\) the collection of all subsets of \(\Omega\) having finite Lebesgue measure. The collection of all abstract additive set functions defined on \(\Sigma(\Omega)\) and taking values in a \((B)\)-space \(X\) forms a vector manifold \(\Phi(X,\Omega)\).
We shall say that \(\Phi(E)\) from \(\Phi(X,\Omega)\) belongs to \(\Phi_1(X,\Omega)\) \({}^{(1)}\) (respectively, to \(\Phi_p(X,\Omega)\), \(1 < p < \infty\) \({}^{(1)}\)), if the inequality (1) (respectively, the inequality (2)) holds:
\[ \|\Phi(E)\|_{\Phi_1(X,\Omega)} = \sum_{\substack{E_1\in\Sigma(\Omega),\,E_2\in\Sigma(\Omega)\\ E_1\cap E_2=\varnothing}} \|\Phi(E_1)-\Phi(E_2)\|_X <\infty; \tag{1} \]
\[ \|\Phi(E)\|_{\Phi_p(X,\Omega)} = \sup_{\omega} \frac{ \left\|\int_{\Omega}\omega(x)\,d_x\Phi(E)\right\|_X }{ \|\omega(x)\|_{L_{p'}(\Omega)} } <\infty, \tag{2} \]
where \(\varnothing\) is the empty set, \(1 < p < \infty\), \(1/p+1/p'=1\), and the least upper bound is taken over all finite linear combinations of characteristic functions of sets from \(\Sigma(\Omega)\). We shall call such functions step functions. Note that
\[ \sup_{E\in\Sigma(\Omega)}\|\Phi(E)\|_X \le \|\Phi(E)\|_{\Phi_1(X,\Omega)} \le 2\sup_{E\in\Sigma(\Omega)}\|\Phi(E)\|_X . \tag{3} \]
We shall call a function \(\Phi(E)\in\Phi(X,\Omega)\): 1) countably additive, if
\[ \lim_{N\to\infty} \left\| \Phi\left(\bigcup_{i=1}^{\infty}E_i\right) - \sum_{i=1}^{N}\Phi(E_i) \right\|_X =0 \]
for every sequence \(\{E_i\}\) of pairwise disjoint sets from \(\Sigma(\Omega)\) such that \(\bigcup_{i=1}^{\infty}E_i\subset \Sigma(\Omega)\); 2) absolutely continuous in the norm \(\|\ \|_{\Phi_p}\), \(1\le p<\infty\), if for every \(\varepsilon>0\) there is a \(\delta>0\) such that \(\|\Phi(E)\|_{\Phi_p(X,\Lambda)}<\varepsilon\), if \(m\Lambda<\delta\); 3) continuous under translation in the norm \(\|\ \|_{\Phi_p}\), \(1\le p<\infty\), if
\[ \lim_{|\vec h|\to 0} \|\Phi(E+\vec h)-\Phi(E)\|_{\Phi_p(X,\Omega)} =0, \]
where \(\Phi(E+\vec h)=\Phi((E+\vec h)\cap\Omega)\). If the function \(\Phi\) is countably additive, then \(\|\Phi\|_{\Phi_1(X,\Omega)}<\infty\) (\({}^{(2)}\), p. 319).
Consider the following spaces: \(\widetilde{\Phi}_1(X,\Omega)\) \({}^{(3)}\) is the space of all countably additive functions with norm (1); \(\widetilde{\Phi}_p(X,\Omega)\), \(1<p<\infty\) \({}^{(1)}\), is the space of all functions absolutely continuous in the norm \(\|\ \|_{\Phi_p}\), with norm \(\|\ \|_{\Phi_p}\); \(\Psi_1(X,\Omega)\) \({}^{(1)}\) is the space of all functions from \(\widetilde{\Phi}_1(X,\Omega)\) continuous under translation in the norm (1); \(\Psi_p(X,\Omega)\), \(1<p<\infty\) \({}^{(1)}\), is the space of all functions from \(\Phi_p(X,\Omega)\) continuous under translation in the norm (2).
All these spaces are Banach spaces, and, in the case of a bounded domain \(\Omega\), the following embeddings hold:
\[ \Phi_1(X,\Omega)\leftarrow \widetilde{\Phi}_1(X,\Omega)\leftarrow \widetilde{\widetilde{\Phi}}_1(X,\Omega)\leftarrow \Psi_1(X,\Omega); \tag{4} \]
\[ \widetilde{\Phi}_1(X,\Omega)\leftarrow \Phi_p(X,\Omega)\leftarrow \widetilde{\Phi}_p(X,\Omega)\leftarrow \Psi_p(X,\Omega). \tag{5} \]
The embeddings (4) are easily verified with the aid of Pettis’ theorem (see, for example, \((^2)\), p. 318). The embedding \(\Phi_p(X,\Omega)\to \widetilde{\widetilde{\Phi}}_1(X,\Omega)\) follows from (3) and the following estimate (1):
\[ \|\Phi(I)\|_X = (mI)^{1/p'}\frac{\|\Phi(I)\|_X}{(mI)^{1/p'}} = (mI)^{1/p'} \frac{ \left\|\int_\Omega \chi_I(x)\,d_x\Phi(E)\right\|_X }{ \|\chi_I(x)\|_{L_{p'}(\Omega)} } \leq (mI)^{1/p'}\|\Phi\|_{\Phi_p(X,\Omega)}. \tag{6} \]
The embedding \(\Psi_p(X,\Omega)\to \widetilde{\Phi}_p(X,\Omega)\) was proved in \((^1)\).
Let \(\Phi(E)\in \Phi(Y^*,\Omega)\), where \(Y^*\) is the space conjugate to the \(B\)-space \(Y\). We shall call \(\Phi\in \Phi(Y^*,\Omega)\) \((*)\)-weakly absolutely continuous (and denote the totality of all such functions by \(\widetilde{\Phi}_{1^*}(Y^*,\Omega)\)) if, for every \(g\in Y\), the set function \(\Phi(E)g\) is absolutely continuous. It can be shown that if \(\Phi(E)\in \widetilde{\Phi}_{1^*}(Y^*,\Omega)\), then \(\|\Phi\|_{\Phi_1(Y^*,\Omega)}<\infty\), and \(\widetilde{\Phi}_{1^*}(Y^*,\Omega)\) is a \(B\)-space with norm (1).
Let \(\Phi(E)\in \widetilde{\Phi}_{1^*}(Y^*,\Omega)\). Then
\[ \Phi(E)g=\int_E \varphi_g(x)\,dx,\qquad \varphi_g(x)=\frac{d}{dx}[\Phi(E)g],\qquad g\in Y, \tag{7} \]
where \(\dfrac{d}{dx}[\Phi(E)g]\) is the Radon–Nikodym derivative \((^4)\).
Lemma 1. 1) Let \(\Phi(E)\in \widetilde{\Phi}_{1^*}(Y^*,\Omega)\); then
\[ \|\Phi\|_{\Phi_1(Y^*,\Omega)} = \sup_{\|g\|_Y\leq 1}\|\varphi_g\|_{L_1(\Omega)}. \]
2) Let \(1<p<\infty\), \(\Phi(E)\in \Phi_p(Y^*,\Omega)\); then
\[ \|\Phi\|_{\Phi_p(Y^*,\Omega)} = \sup_{\|g\|_Y\leq 1}\|\varphi_g\|_{L_p(\Omega)}. \]
Let \(B_1\) and \(B_2\) be \((B)\)-spaces. By \((B_1\to B_2)\) we denote the space of all linear continuous operators acting from \(B_1\) into \(B_2\). The set of linear operators from \((B_1\to B_2)\) possessing some property \(N\) will be denoted by \((B_1\to B_2,N)\). An operator \(T\in(B_1\to B_2)\) will be called an isomorphism if \(TB_1=B_2\) and \(T\) maps \(B_1\) onto \(B_2\) one-to-one.
Theorem 1. 1) \(\widetilde{\Phi}_{1^*}(Y^*,\Omega)\leftrightarrow (Y\to L_1(\Omega))\); 2) if \(m\Omega<\infty\), then \(\widetilde{\Phi}_1(Y^*,\Omega)\leftrightarrow (Y\to L_1(\Omega),\) weakly completely continuous\()\); 3) if \(1<p<\infty\), then \(\Phi_p(Y^*,\Omega)\leftrightarrow (Y\to L_p(\Omega))\); 4) if \(\Omega\) is a bounded domain and \(1\leq p<\infty\), then \(\Psi_p(Y^*,\Omega)\leftrightarrow (Y\to L_p(\Omega),\) completely continuous\()\). Here the symbol \(\leftrightarrow\), connecting two spaces, means that these spaces are isometrically isomorphic, and the isometric isomorphism is defined by the equations
\[ T_\Phi(g):=\frac{d}{dx}[\Phi(E)g],\qquad g\in Y; \tag{8} \]
\[ \Phi_T(E)g=\int_E T(g)(x)\,dx,\qquad g\in Y. \tag{9} \]
Indeed, in \((^5)\) (see also \((^2)\), p. 498) it is shown that (8), (9) define an isomorphism between the corresponding spaces of items 1–3. The isometry follows from Lemma 1. We note that earlier only the following estimate was known:
\[ \sup_{E\in\Sigma(\Omega)}\|\Phi(E)\|_{Y^*} \leq \|T_\Phi\| \leq 2\sup_{E\in\Sigma(\Omega)}\|\Phi(E)\|_{Y^*}. \]
4) immediately
follows from the isometry, if one observes that
\[ \left\|\Phi(E+\vec h)-\Phi(E)\right\|_{\Phi_p(Y^*,\Omega)} = \sup_{\|g\|_Y\le 1} \left\|T_\Phi(g)(x+\vec h)-T_\Phi(g)\right\|_{L_p(\Omega)} . \]
Theorem 2. The following four assertions are equivalent, if \(\Omega\) is a bounded domain, \(1\le p<\infty\): 1) \(\Phi(E)\in\Psi_p(X,\Omega)\); 2) \(\Phi(E)\in \widetilde{\Phi}_p(X,\Omega)\cap\Psi_1(X,\Omega)\); 3) \(\Phi(E)\in\widetilde{\Phi}_p(X,\Omega)\) and the set of values of the function \(\Phi\) is compact in \(L_p(\Omega)\); 4) the set
\[ M_{p,\Phi}\left\{\varphi_f(x)=\frac{d}{dx}[f\Phi(E)],\ \|f\|_X\le 1\right\} \]
is compact in \(L_p(\Omega)\).
Theorem 3. The set of values of a function \(\Phi\) from \(\widetilde{\Phi}_1(X,\Omega)\) is separable.
Definition 1. We shall say that \(\Phi(E)\in\Phi_p^{(l)}(X,\Omega)\) if \(\Phi(E)\in\widetilde{\Phi}_1(X,\Omega)\) and all generalized derivatives* \(\Phi^{(\alpha)}(E)\) of order \(l\) of the function \(\Phi(E)\) belong to \(\Phi_p(X,\Omega)\). The norm in \(\Phi_p^{(l)}(X,\Omega)\) is defined by the equality
\[ \|\Phi(E)\|_{\Phi_p^{(l)}(X,\Omega)} = \Phi(E)\|_{\Phi_1(X,\Omega)} + \sum_{|\alpha|=l}\|\Phi^{(\alpha)}(E)\|_{\Phi_p(X,\Omega)} . \tag{10} \]
By definition we put \(\Phi_p^{(0)}(X,\Omega)=\Phi_p(X,\Omega)\). From the fact that \(\Phi(E)\in\Phi_p^{(l)}(Y^*,\Omega)\) it follows that
\[ \Phi(E)g=\int_E \varphi_g(x)\,dx,\qquad \Phi^{(\alpha)}(E)g=\int_E D^{(\alpha)}\varphi_g(x)\,dx,\qquad g\in Y . \tag{11} \]
Definition 2. Let \(\lambda\ge 0,\ \lambda=\bar\lambda+\alpha,\ \bar\lambda\) an integer, \(0<\alpha<1\). By definition \(\Phi(E)\in\Phi_p^{(\lambda)}(Y^*,\Omega)\) if \(\Phi(E)\in\Phi_p^{(\bar\lambda)}(Y^*,\Omega)\) and if the norm is finite (see (11))
\[ \|\Phi(E)\|_{\Phi_p^{(\lambda)}(Y^*,\Omega)} = \Phi(E)\|_{\Phi_p^{(\bar\lambda)}(Y^*,\Omega)} + \]
\[ + \sum_{|\gamma|=\bar\lambda} \sup_{\|g\|_Y\le 1} \left( \int_{\Omega\Omega} \frac{\left|D^{(\gamma)}\varphi_g(x)-D^{(\gamma)}\varphi_g(y)\right|^p} {|x-y|^{n+p\alpha}} \,dx\,dy \right)^{1/p}. \tag{12} \]
If \(X\) is an arbitrary \((B)\)-space, then we shall say that \(\Phi(E)\in\Phi_p^{(\lambda)}(X,\Omega)\) if \(A\Phi(E)\in\Phi_p^{(\lambda)}(X^{**},\Omega)\), where the operator \(A\) is defined by the equation
\[ Ax=E,\qquad x\in X,\qquad E\in X^{**}\ \text{and for every } f\in X^*\quad E(f)=f(x). \tag{13} \]
Definition 3. We shall say that \(\Phi(E)\in H_p^{(r)}(X,R_n)\); \(r=\bar r+\alpha,\ \bar r\) an integer, \(0<\alpha\le 1\), if \(\Phi(E)\in\Phi_p(X,\Omega)\) and has all unmixed generalized derivatives up to order \(\bar r\), belonging to \(\Phi_p(X,\Omega)\), and moreover
\[ \left\|\Phi_{x_i}^{(\bar r)}(E+\vec h_i)-\Phi_{x_i}^{(\bar r)}(E)\right\|_{\Phi_p(X,R_n)} < M|\vec h_i|^\alpha, \]
\[ \text{if }0<\alpha<1;\quad i=1,2,\ldots,n; \tag{14} \]
\[ \left\|\Phi_{x_i}^{(\bar r)}(E+\vec h_i)-2\Phi_{x_i}^{(\bar r)}(E)+\Phi_{x_i}^{(\bar r)}(E-\vec h_i)\right\|_{\Phi_p(X,R_n)} < M|\vec h_i|, \]
\[ \text{if }\alpha=1;\quad i=1,2,\ldots,n. \tag{15} \]
\[ \text{* The generalized derivative }\Phi^{(\alpha)}(E),\ \alpha=(\alpha_1,\ldots,\alpha_n),\ |\alpha|=\alpha_1+\cdots+\alpha_n,\ \text{is defined} \]
by the integral identity
\[ \int_\Omega \frac{\partial^\alpha \omega}{\partial x^\alpha}\,d_x\Phi(E) = (-1)^{|\alpha|} \int_\Omega \omega\,d_x\Phi^{(\alpha)}(E), \qquad \omega\in C_1^0(\Omega)\ (1). \]
\(H_p^{(r)}(X,R_n)\) becomes a \((B)\)-space if one introduces the norm
\[ \|\Phi\|_{H_p^{(r)}(X,R_n)}=\|\Phi\|_{\Phi_p(X,R_n)}+M_\Phi, \tag{16} \]
where \(M_\Phi\) is the smallest constant for which inequalities (14), (15), \(i=1,2,\ldots,n\), hold.
Theorem 4. 1) \(\Phi_p^{(l)}(Y^*,\Omega)\sim (Y\to W_p^{(l)}(\Omega))\); 2) \(\Phi_p^{(\lambda)}(Y^*,\Omega)\sim (Y\to W_p^{(\lambda)}\Omega)\); 3) \(H_p^{(r)}(Y,\Omega)\sim (Y\to H_p^{(r)}(R_n))\); here the symbol \(\sim\), connecting two spaces, means that these spaces are isomorphic and the isomorphism is defined by equations (8), (9); moreover
\[ \|T_\Phi\|\leq \|\Phi\|_{\Phi_p^{(l)}(Y^*,\Omega)}\leq (1+N_l)\|T_\Phi\|, \]
\[ \|T_\Phi\|=\sup_{\|g\|Y\leq 1}\left\|\frac{d}{dx}[\Phi(E)g]\right\|_{W_p^{(l)}(\Omega)}; \]
\[ \|T_\Phi\|\leq \|\Phi\|_{\Phi_p^{(\lambda)}(Y^*,\Omega)}\leq (1+2N_\lambda)\|T_\Phi\|,\qquad \|T_\Phi\|=\sup_{\|g\|Y\leq 1}\left\|\frac{d}{dx}[\Phi(E)g]\right\|_{W_p^{(\lambda)}(\Omega)}; \]
\[ \|T_\Phi\|\leq \|\Phi\|_{H_p^{(r)}(Y^*,R_n)}\leq 2\|T_\Phi\|, \]
\[ \|T_\Phi\|=\sup_{\|g\|Y\leq 1}\left\|\frac{d}{dx}[\Phi(E)g]\right\|_{H_p^{(r)}(R_n)}, \]
where \(N_l\) is the number of all distinct generalized derivatives of order \(l\).
The theorem establishes the general form of a linear continuous operator acting from a \((B)\)-space into \(W_p^{(l)}\), \(W_p^{(\lambda)}\), \(\lambda\geq 0\), \(H_p^{(r)}(R_n)\).
Theorem 5. Let \(1<p_1,p_2<\infty\). Then: 1) if \(W_{p_1}^{(\lambda_1)}(\Omega_{n_1})\to W_{p_2}^{(\lambda_2)}(\Omega_{n_2})\), then \(\Phi_{p_1}^{(\lambda_1)}(X,\Omega_{n_1})\to \Phi_{p_2}^{(\lambda_2)}(X,\Omega_{n_2})\); 2) if \(H_{p_1}^{(r_1)}(R_{n_1})\to H_{p_2}^{(r_2)}(R_{n_2})\), then \(H_{p_1}^{(r_1)}(X,R_{n_1})\to H_{p_2}^{(r_2)}(X,R_{n_2})\). Here \(\to\) denotes, as usual, an embedding or an extension (if \(\dim\Omega_{n_1}<\dim\Omega_{n_2}\)), accompanied by the corresponding inequality for the norms ((6), p. 66). It is assumed that the extension (in the case when \(\dim\Omega_{n_1}<\dim\Omega_{n_2}\)) is carried out by means of a linear continuous operator \(V\).
Theorem 6. Let \(\Phi(E)\in \Phi_p^{(\lambda)}(X,\Omega)\) and \(\lambda p>n\). Then
\[ \Phi(E)=\int_E \varphi(x)\,dx,\qquad E\in \Sigma(\Omega), \]
where \(\varphi(x)\) is a continuous abstract function of the point \(x\in\Omega\) with values in \(X\), and
\[ \sup_{x\in\Omega}\|\varphi(x)\|_X\leq c\|\Phi\|_{\Phi_p^{(\lambda)}(X,\Omega)}, \]
where the constant \(c\) does not depend on the function \(\Phi\).
Theorem 6 is a strengthening of one theorem of S. L. Sobolev ((1), Theorem 27).
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
7 III 1962
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