Abstract Generated abstract
This paper extends Sobolev-type embedding theorems for abstract additive set functions with values in a Banach space. It introduces a broader class of normal completely additive set functions, defines corresponding spaces with shift-continuous generalized derivatives, and proves that such functions are absolutely continuous and shift-continuous in the appropriate norm. These results allow the author to derive analogues of classical Sobolev embeddings, including continuity when lp exceeds the dimension and trace-type statements on lower-dimensional manifolds when lp is at most the dimension. The paper also notes that compactness of the embedding holds in finite-dimensional target spaces but generally fails for infinite-dimensional Banach spaces, and it shows by example that the normality assumption is essential.
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MATHEMATICS
Yu. I. GILDERMAN
ON EMBEDDING THEOREMS FOR ABSTRACT FUNCTIONS
(Presented by Academician S. L. Sobolev, 18 V 1961)
In the paper (¹) S. L. Sobolev considered the \(B\)-space \(\Phi_1\) of abstract additive functions \(\varphi\), defined on the totality of all measurable sets \(E\) from a certain domain \(\Omega \subset R_n\) with values in the \(B\)-space \(X\),
\[ \|\varphi\|_{\Phi_1} = \sup_{\substack{E_1,E_2 \in \Omega\\ E_1 \cap E_2 = 0}} \|\varphi(E_1)-\varphi(E_2)\|_X . \]
From this space one singles out the subspace \(\Psi_p^{(l)}\) of functions \(\varphi(E)\) such that \(\varphi_i(E)\in \Phi_{1+\varepsilon}\), \(\varepsilon>0\), and for all \(\alpha=(a_1,a_2,\ldots,a_n)\), \(|\alpha|=l\), there exists a generalized derivative \(\Psi^\alpha(E)\), continuous with respect to translation in the norm \(\Phi_p\), \(p>1\).
\[ \|\varphi\|_{\Phi_p} = \sup_{\omega\in L_{p'}} \frac{\left\|\int_\Omega \omega(x)\,d\varphi(E)\right\|_X} {\|\omega(x)\|_{L_{p'}}}, \qquad \frac1p+\frac1{p'}=1 . \]
For the spaces \(\Psi_p^{(l)}\) embedding theorems are proved, analogous to the corresponding theorems for the spaces of numerical functions \(W_p^{(l)}\).
Theorem 1. If \(lp>n\), the function \(\varphi(E)\in \Psi_p^{(l)}\) is the integral of a certain continuous point function \(\varphi(x)\). The modulus of continuity of \(\varphi(x)\) is equal to \(A|\Delta x|^\beta\), where \(\beta>1\),
\[ \|\varphi(x)\|_X \le A\|\varphi(E)\|_{\Psi_p^{(l)}} , \]
where \(A\) is a constant independent of \(\varphi\).
Theorem 2. If \(lp\le n\), the function \(\varphi(E)\in \Psi_p^{(l)}\) is defined on all smooth manifolds of dimension \(s\), where \(s>n-pl\), and represents a function of a set \(\varphi(I)\) belonging to \(\Phi_q\), where \(s/q=n/p-l\).
Moreover, for any \(q^*<q\) the function \(\varphi(I)\) is continuous with respect to translation in the norm \(\Phi_{q^*}\), and
\[ \|\varphi(I)\|_{\Phi_q} \le A\|\varphi(E)\|_{\Psi_p^{(l)}} . \]
By slightly modifying the proof given in (¹), one can obtain the indicated theorems starting from somewhat broader assumptions. Namely, for a function \(\varphi(E)\in\Phi_1\), in addition to the existence of a generalized derivative \(\Psi^\alpha(E)\in\Phi_p\) continuous with respect to translation, absolute continuity is required:
\[ \|\varphi(E)\|_X < \varepsilon \quad \text{when } mE<\delta(\varepsilon). \]
However, embedding theorems 1 and 2 are also valid for broader spaces of abstract functions of sets.
Denote by $\Phi_1$ the totality of additive and normal abstract set functions, i.e. such functions whose norm in the space $X$ tends to $0$ on a vanishing sequence of measurable sets.
Theorem 3. In order that an additive abstract function $\varphi(E)$ be normal, it is necessary and sufficient that it be completely additive.
Theorem 4. The $B$-space $\overline{\Phi}_1$ with norm $\Phi_1$ is a regular part of the space $\Phi_1$.
Obviously, $\overline{\Phi}_1$ is broader than the subspace of functions $\varphi(E) \in \Phi_1$ that are absolutely continuous.
Let $\varphi(E) \in \Phi_1$ and let $\omega(x)$ be a numerical stepwise bounded function taking only a finite number of values: $\omega(x)=a_k$, $x \in E_k$, $\sum_{k=1}^{N} E_k=\Omega$, the $E_k$ do not intersect.
Then, by definition, set
\[ \int_{\Omega} \omega(x)\,d\varphi(E)=\sum_{k=1}^{N} a_k\varphi(E_k). \]
Using the completeness of the space $X$, this operator can be extended to arbitrary bounded $\omega(x)$.
In this case
\[ \left\|\int_{\Omega}\omega(x)\,d\varphi(E)\right\|_{X} \leq \max_{x\in\Omega}|\omega(x)|\,\|\varphi(E)\|_{\Phi_1}. \tag{1} \]
Lemma. Let $\varphi(E)\in\overline{\Phi}_1$, $E_{k+1}\subset E_k$, $E_0=\lim_{k\to\infty}E_k$; let $\omega(x)$ be a bounded numerical function and $\omega(x)=a_0$ for $x\in E_0$.
Then
\[ \lim_{k\to\infty}\int_{E_k}\omega(x)\,d\varphi(E)=a_0\varphi(E_0). \tag{2} \]
(In particular, $E_0$ may be of zero measure.)
Define $\overline{\Psi}^{(l)}_p$ as the totality of normal set functions $\varphi(E)$ having shift-continuous derivatives $\Psi^\alpha \in \Phi_p$.
From (1) and (2) it follows:
Theorem 5. If $\varphi(E)\in\overline{\Psi}^{(l)}_p$, then it is absolutely continuous.
Hence it follows:
Theorem 6. If $\varphi(E)\in\overline{\Psi}^{(l)}_p$, then it is shift-continuous in the norm $\Phi_1$ and, consequently, is the limit of integrals of abstract functions continuous at a point.
This circumstance makes it possible to pass to the limit in the integral identity
\[ \varphi_h(x)=\int_{\Omega} K(x,y)\,d\varphi_h(E)+\int_{\Omega}\sum_{\alpha}\frac{K_\alpha(x,y)}{r^{\,n-l}}\,d\Psi^\alpha_h(E), \]
where $\varphi_h(x)$ is the mean function for $\varphi(E)$. From the equality obtained in the limit as $h\to 0$, embedding theorems 1 and 2 follow.
If $X$ is finite-dimensional, then the embedding operator is completely continuous. (The proof does not differ from the corresponding proof for numerical functions.)
In the case of an infinite-dimensional space $X$, the embedding operator, generally speaking, is not completely continuous (V. I. Kondrashov’s theorem does not hold).
Indeed, the set \(\{\varphi_k(x)=i_k\}\) for \(x\in\Omega\), where \(i_k=(0,\ldots,0,1,0,\ldots)\), \(k=1,2,\ldots\), while bounded in the norm \(\overline{\Psi}^{(l)}_p\), is at the same time not compact in \(X=l_2\).
Let us note that the normality condition on \(\varphi(E)\) in the definition of \(\Psi^{(l)}_p\) is essential. Indeed, let \(X=R_1\), \(\Omega=(0,1)\), \(\delta>0\), and
\[ \varphi(E)= \begin{cases} mE, & \text{if } (0,\delta)\notin E \text{ for any value of } \delta,\\ mE-1, & \text{if } (0,\delta)\in E \text{ for at least one } \delta . \end{cases} \]
It is easy to see that the function \(\varphi(E)\) thus constructed is additive, is not normal, belongs to \(\Phi_1(\Omega)\), and has a generalized derivative of any order \(l\ge 1\) equal to zero. However, for it the embedding theorems 1 and 2 do not hold, since by either of these theorems the function \(\varphi(E)\) must be not only normal but also absolutely continuous.
In conclusion I express my deep gratitude to Academician S. L. Sobolev, whose attention I unfailingly enjoyed in carrying out this work.
Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR
Received
13 V 1961
REFERENCES
- S. L. Sobolev, Fund. math., 47, 277 (1959).
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.