Abstract
Full Text
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}
|\Phi(E_1)-\Phi(E_2)|_X
<\infty;
\tag{1}
]
[
|\Phi(E)|{\Phi_p(X,\Omega)}
=
\sup
\frac{
\left|\int_{\Omega}\omega(x)\,d_x\Phi(E)\right|X
}{
|\omega(x)|}(\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)|
\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)|), (1\le p<\infty), if}<\varepsilon), if (m\Lambda<\delta); 3) continuous under translation in the norm (|\ |_{\Phi_p
[
\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 (|\ |), is the space of all functions from (\Phi_p(X,\Omega)) continuous under translation in the norm (2).}), 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)
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)|}(\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|.
]
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|.
]
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)|
+
\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
\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|+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|,}(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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