MATHEMATICS

Academician S. L. SOBOLEV

EMBEDDING THEOREMS FOR ABSTRACT FUNCTIONS OF SETS

In a previous note \((^{1})\) we constructed a space \(\Psi_p\) of abstract functions of sets, which is the closure, in the sense of the norm \(\|\ \|_{\Phi_p}\), of the set of step measurable functions. Generally speaking, \(\Psi_p\) is a proper subspace of \(\Phi_p\) and consists of functions that are absolutely continuous and continuous under shifts.

Some embedding theorems, which have found applications in the theory of partial differential equations \((^{2,3})\), can be extended to the elements of \(\Psi_p\). We shall show how this can be done.

Define the integrals

\[ \int \omega(P)\,d\varphi(E), \tag{1} \]

where the function \(\varphi(E)\in\Psi_p\), and \(\omega(P)\) is an element of \(L_{p'}\). (For \(p=1\), \(\omega(P)\) must be regarded as a bounded measurable function.) The integral (1) always exists and is an element of \(X\), as is proved by means of a passage to the limit from step functions.

We shall study integrals

\[ U(Q)=\int \omega(Q,P)\,d\varphi(E), \tag{2} \]

which take values in \(X\), where \(\omega(P,Q)\) is an abstract function of the point \(Q\), whose values are elements of \(L_{p'}\) as functions of the point \(P\).

Theorem 1. Let \(\varphi(E)\in\Psi_p\), and let \(\omega(Q,P)\) be continuous as a function of the point \(Q\in\Omega\). Then \(U(Q)\) will be a continuous abstract function of the point \(Q\).

The proof of this theorem is carried out by direct computation of

\[ \|U(Q+\Delta Q)=U(Q)\|_X. \tag{3} \]

As an application of this theorem we give Theorem 2.

Theorem 2 (on integrals of potential type). The integral

\[ U(Q)=\int \frac{K(P,Q)}{r^\lambda}\,d\varphi(E), \tag{4} \]

where \(K(P,Q)\) is a bounded function of the pair of points \(P,Q\) from \(R_n\), continuous in each of them when \(P\ne Q\); \(r\) is the distance from \(P\) to \(Q\); \(\lambda<n/p'\), and \(\varphi(E)\in\Phi_p\), is a continuous abstract function of the point \(Q\), taking values in \(X\).

Indeed, the kernel \(\omega(Q,P)=K(P,Q)/r^\lambda\) satisfies the conditions of the preceding theorem, whence our assertion follows.

Consider in Euclidean \(m\)-dimensional space \(R_m(y_1,y_2,\ldots,y_m)\) some \(s\)-dimensional linear manifold \(S_s\), which we shall specify by the equations \(y_{s+1}=y_{s+2}=\cdots=y_m=0\), together with a system of parallel manifolds \(S_s(y_{s+1},\ldots,y_m)\) corresponding to constant values \(y_{s+1},\ldots,y_m\). Let, in some domain \(\Omega\) of the variables \(y_1,y_2,\ldots,y_s\), there be given an abstract function \(\varphi(E_s,y_{s+1},\ldots,y_m\mid P)\) of sets \(E_s\subset\Omega\), depending also on the variables \(y_{s+1},y_{s+2},\ldots,y_m\), and taking values in \(L_{p'}\) as a function of \(P\in R_n\). Suppose that this function is continuous with respect to \(E_s\) and continuous with respect to translation in \(m\)-dimensional space (this notion needs no explanation).

Theorem 3. The abstract function

\[ U(E_s,y_{s+1},\ldots,y_m)=\int \omega(E_s,y_{s+1},\ldots,y_m\mid P)\,d\varphi(E), \tag{5} \]

where \(\varphi(E)\in\Psi_p\), is a function with values in \(X\), absolutely continuous with respect to \(E_s\) and continuous with respect to translation in \(R_m(y_1,\ldots,y_m)\).

The proof of Theorem 3 is also obvious.

As an application of Theorem 3 we indicate Theorem 4.

Theorem 4. The integral

\[ U(E_s,y_{s+1},\ldots,y_n)= \left[\iint_{E_s}\frac{K(Q_s,y_{s+1},\ldots,y_n,P)}{r^\lambda}\,dQ_s\right]\,d\varphi(E), \tag{6} \]

where \(\varphi(E)\in\Psi_p\), \(n/p'<\lambda<s/q\), and \(K(Q,P)\) is a bounded function of \(P\) and \(Q\), continuous for \(P\ne Q\), is an abstract function of \(E_s,y_{s+1},\ldots,y_n\), absolutely continuous and continuous with respect to translation, with values in \(X\).

Theorem 4 is reduced to Theorem 3 if one observes that the kernel

\[ \omega(E_s,y_{s+1},\ldots,y_n\mid P) = \int_{E_s}\frac{K(Q_s,y_{s+1},\ldots,y_n\mid P)}{r^\lambda}\,dQ_s \tag{7} \]

satisfies the conditions of the preceding theorem. This last assertion has essentially been proved, for example, in the author’s book \((^2)\).

Let us introduce the notion of a derivative of an abstract function of sets. Let \(\psi_{\alpha_1\ldots\alpha_n}(E)\) be such that for any continuously differentiable \(k\)-times finite function \(\omega(P)\) the identity

\[ \int \omega\,d\psi_{\alpha_1\ldots\alpha_k}(E) = (-1)^{\alpha_1+\alpha_2\cdots+\alpha_k} \int \frac{\partial^k\omega}{\partial x_1^{\alpha_1}\cdots\partial x_k^{\alpha_k}}\,d\varphi(E). \tag{8} \]

Then we shall say that

\[ \psi_{\alpha_1\ldots\alpha_n} = \frac{\partial^k\varphi}{\partial x_1^\alpha\cdots\partial x_{n_1}^{\alpha_n}}. \tag{9} \]

The functions \(\varphi(E)\) for which all derivatives of order \(l\) belong to \(\Phi_p\) form the space \(\Phi_p^{(l)}\). As the norm in this space \(\|\varphi\|_{\Phi_p^{(l)}}\) it is convenient to take, for example, the sum of the norms in \(\Phi_p\) of all its derivatives of order \(l\) and the norm of \(\varphi(E)\) itself in \(\Phi_1\):

\[ \|\varphi(E)\|_{\Phi_p^{(l)}}= \|\varphi(E)\|_{\Phi_1} + \sum \left\| \frac{\partial^l\varphi}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \right\|_{\Phi_p}. \]

Theorem 5. If \(lp>n\), every element \(\varphi(E)\) of \(\Psi_p^{(l)}\) is a continuous function of the point \(\mathbf Q\). The modulus of continuity of this function is equal to \(\delta(\varepsilon)=\varepsilon^\beta\), where \(\beta=\min\left(1-0,\; l-\dfrac{n}{p}\right)\).

The inequality

\[ \|\varphi(E)\|_C=\max_E \|\varphi(E)\|_X \leq A\|\varphi(E)\|_{\Phi_p^{(l)}}. \]

is valid.

Theorem 6. If \(lp<n\), every element \(\varphi(E)\) of \(\Psi_p^{(l)}\) on any manifold of dimension \(s>n-lp\) is an element of \(\Psi_q\), where \(\dfrac{s}{q}\leq \dfrac{n}{p}-l\). The function \(\varphi(E_s)\) on such manifolds is continuous with respect to translation. Its modulus of continuity is equal to \(\delta(\varepsilon)=\varepsilon^\beta\), where \(\beta=\min\left(1-0,\; l-\dfrac{n}{p}-\dfrac{s}{q}-0\right)\) when \(\dfrac{s}{q}<\dfrac{n}{p}-l\).

The inequality

\[ \|\varphi(E_s)\|_{\Phi_q}\leq A\|\varphi(E)\|_{\Phi_p^{(l)}}. \]

is valid.

The proof of the theorems stated is based on passing to mean functions. As was noted in the preceding note \((^1)\), these functions are continuous functions of a point. For the mean functions \(\varphi_h(E)\), in the abstract case the identity remains valid

\[ \varphi_h(\mathbf Q) = \int K_0(\mathbf Q,\mathbf P)\varphi_h(\mathbf P)\,dP + \sum \int K_{\alpha_1\ldots\alpha_n}(\mathbf Q,\mathbf P) \frac{\partial^l\varphi_h(\mathbf P)} {\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}\,dP, \tag{10} \]

which expresses the value \(\varphi_h(\mathbf P)\) in terms of derivatives and is usually used to prove embedding theorems for numerical functions \((^2)\).

From identity (10) our theorems are obtained by applying the theory of integrals of potential type. The limiting passage then yields the corresponding theorem for functions of sets.

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

Received
22 II 1957

CITED LITERATURE

\(^1\) S. L. Sobolev, DAN, 114, No. 6 (1957).
\(^2\) S. L. Sobolev, Mat. sborn., 4 (46), 3 (1938).
\(^3\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^4\) S. L. Sobolev, Mat. sborn., 5 (47), 1 (1939).