MATHEMATICS

V. B. KOROTKOV

ON S. L. SOBOLEV’S EMBEDDING THEOREMS FOR ABSTRACT FUNCTIONS

(Presented by Academician S. L. Sobolev on 21 VI 1961)

S. L. Sobolev \((^1)\) introduced the classes \(\Psi_p^{(l)}\) of abstract functions and obtained embedding theorems for these classes.

In the present paper we prove, for the classes \(\Psi_p^{(l)}\), an embedding theorem in the case of the limiting exponent \((^3)\) and a theorem on composite functions. These theorems answer questions posed by S. L. Sobolev \((^1)\). In the present paper we also study properties of boundary values of functions from \(\Psi_p^{(l)}\). For this purpose the classes \(\Psi_p^{(\lambda)}\) with arbitrary nonintegral \(\lambda \ge 0\) are defined. For boundary values of functions from \(\Psi_p^{(1)}\) a theorem is proved that generalizes Gagliardo’s theorem \((^4)\).

Let \(D\) be a bounded domain in the \(n\)-dimensional Euclidean space \(E^n\), and let \(\Phi(E)\) be an abstract additive function of Lebesgue-measurable subsets of the set \(D\), with values in the Banach space \(X\). The space conjugate to \(X\) will be denoted by \(\overline X\).

Lemma 1. If \(\Phi(E) \in \Psi_p(D)\), \(\Phi(E)=\int_E \varphi(x)\,dx\), where \(\varphi(x)\) is a continuous abstract function with values in \(X\), then

\[ \|\Phi(E)\|_{\Psi_p(D)} = \sup_{\|f\|=1} \left[ \int_D |f\varphi(x)|^p dx \right]^{1/p}, \tag{1} \]

where \(f \in \overline X^*\).

Lemma 2. If \(\varphi(x)\) is a continuous abstract function of the point \(x \in D\) with values in \(X\), then

\[ \max_{x\in D}\|\varphi(x)\|_X = \sup_{\|f\|=1}\max_{x\in D}|f\varphi(x)|, \tag{2} \]

where \(f \in \overline X\).

Lemma 1 is used essentially in the proof of the following theorem.

Theorem 1. Let \(lp<n,\ s>n-lp\). Then the function \(\Phi(E)\) from \(\Psi_p^{(l)}(D)\), on every flat section of the domain \(D\), belongs to \(\Psi_q(D_s)\), where

\[ q=\frac{sp}{n-lp}. \]

The embedding operator is bounded.**

Let \(\Phi(E)=\int_E \varphi(x)\,dx\), where \(\varphi(x)\) is an abstract function continuous together with derivatives up to order \(l\). By the trace of the function \(\Phi(E)\) on \(D_s\) we shall mean the function \(\overline\Phi(I)=\int_{I\subset D_s}\overline\varphi(x)\,dx\), where \(\overline\varphi(x)=\varphi(x)|_{D_s}\). From Lemma 1 it follows that, for any fixed functional \(f,\ f\in X,\ \|f\|=1\), the real function \(f\varphi(x)\in W_p^{(l)}(D)\).

* In the proof of the lemma, properties of the Gelfand integral \((^7)\) are used.

** For \(q<\dfrac{sp}{n-lp}\) the theorem was proved by S. L. Sobolev \((^1)\).

Applying to the function \(f\varphi(x)\) the embedding theorem for the limiting exponent \((^{2,3})\), we obtain

\[ \left[\int_{D_s} |f\varphi(x)|^q\,dx\right]^{1/q} \leq C_1\left\{ \left[\int_D |f\varphi(x)|^p\,dx\right]^{1/p} + \sum_{l_1+\cdots+l_n=l} \left[\int_D |fD^l\varphi(x)|^p\,dx\right]^{1/p} \right\}. \tag{3} \]

\(C_1\) does not depend on the function \(f\varphi(x)\).

On the basis of Lemma 1 we have

\[ \|\widetilde{\Phi}(I)\|_{\Psi_q(D_s)} \leq C_1\|\Phi(E)\|_{\Psi_p^{(l)}(D)}. \tag{4} \]

Inequality (4) makes it possible to define the trace on \(D_s\) of any function from \(\Psi_p^{(l)}(D)\), using the density in \(\Psi_p^{(l)}(D)\) of the set of mean functions and the completeness of \(\Psi_q(D_s)\).

We pass to the theorem on composite functions. Define the domains \(D, D_y, \Omega, \Omega(t), D_1\) exactly as was done in \((^2)\), p. 228. Consider a bounded abstract function \(\varphi(t,x_1,\ldots,x_n;y_1,\ldots,y_m)\), defined in the domain \(D\) with values in \(X\). We shall assume the function \(\varphi\) to be continuous together with its derivatives with respect to \(y_1,\ldots,y_m\) up to order \(l\) inclusive. For the functions

\[ \varphi_{\beta_1,\ldots,\beta_m} = \frac{\partial^\beta\varphi} {\partial y_1^{\beta_1}\cdots \partial y_m^{\beta_m}} \qquad (0\leq \beta_1+\cdots+\beta_m=\beta\leq l) \]

we shall assume the existence of generalized derivatives up to order \(l\) inclusive with respect to \(t,x_1,\ldots,x_n\) for each fixed system of values \(y_1,\ldots,y_m\).

Definition. The function \(\varphi(t,x_1,\ldots,x_n;y_1,\ldots,y_m)\) has property \(T\) (see \((^2)\), p. 228) if there is a \(p>1\), \(p>n/l\), such that for any system of functions \(y_i=\eta_i(t,x_1,\ldots,x_n)\) the composite functions

\[ \left[ \varphi_{\beta_1,\ldots,\beta_m}^{\alpha_0,\alpha_1,\ldots,\alpha_n} (t,x_1,\ldots,x_n) \right]_{y_i=\eta_i} = \frac{ \partial^\alpha \varphi(t,x_1,\ldots,x_n,\eta_1(t,x_1,\ldots,x_n),\eta_m(t,x_1,\ldots,x_n)) } {\partial t^{\alpha_0}\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \tag{5} \]

for each fixed \(t\) belong to:

\[ \Psi_{\frac{1}{\frac{1}{p}-\frac{l-\alpha}{n}}}(\Omega(t)), \qquad \text{if } l\geq \alpha > l-\frac{n}{p}, \]

\[ K, \qquad \text{if } \alpha < l-\frac{n}{p}, \]

where \(K\) is the space of abstract continuous functions with the uniform metric; moreover it is assumed that the functions (5) have in the indicated spaces a bound \(A_\Omega\) (see \((^2)\), p. 231) independent of \(\alpha,\beta\) and of the system of functions from \(D_y\).

If the functions \(y_i\) also depend on \(z_1,\ldots,z_k\):

\[ y_i=\eta_i(t,x_1,\ldots,x_n,z_1,\ldots,z_k), \tag{6} \]

and, for \(z_1,\ldots,z_k\) from \(D_z\), the values \(y_i\) lie in \(D_y\), then the result of substituting the functions (6) into the function \(\varphi(t,x_1,\ldots,x_n;y_1,\ldots,y_m)\) is a composite function of \(t,x_1,\ldots,x_n,z_1,\ldots,z_k\):

\[ \psi(t,x_1,\ldots,x_n,z_1,\ldots,z_k) = \varphi(t,x_1,\ldots,x_n,\eta_1,\ldots,\eta_m). \]

Theorem 2. If the functions \(\varphi,\eta_1,\ldots,\eta_m\) have property \(T\), then the function \(\psi\) also has property \(T\).

We shall prove the theorem under the assumption that the functions \(\varphi_{\beta_1,\ldots,\beta_m}\) have strong derivatives with respect to \(t, x_1,\ldots,x_n\). For any fixed functional \(f\in \bar X\), \(\|f\|=1\), the function \(f\varphi(t,x_1,\ldots,x_n,y_1,\ldots,y_m)\) has property T with a boundary \(A_2\) independent of \(f\). Application of the lemmas completes the proof of this case. The general case is reduced to the one just considered.

Let \(S^n\) be the unit cube of \(n\)-dimensional Euclidean space:
\[ S^n\equiv \{0\le x_i\le 1,\ i=1,2,\ldots,n\}. \]
Let \(\lambda=l+\alpha>0\), where \(l\) is an integer, \(0<\alpha<1\). By \(\Psi_p^{(\lambda)}(S^n)\) we denote the totality of all abstract additive functions of sets from \(\Psi_p^{(l)}(S^n)\) with values in \(X\) satisfying the inequality
\[ \begin{aligned} \|\Phi(E)\|_{\Psi_p^{(\lambda)}(S^n)} &=\|\Phi(E)\|_{\Psi_p^{(l)}(S^n)} \\ &\quad+\sum_{i=1}^n \sup_{\|f_i\|=1} \left\{ \int_0^1 t^{-1-p\alpha} \left[ \sup_{\tilde\omega(x)} \frac{\left|\displaystyle\int_{S_i^t}\tilde\omega(x)\,d_x\bigl(f\Phi(E+\bar x_i^0 t)-f\Phi(E)\bigr)\right|} {\|\tilde\omega(x)\|_{L_{p'}(S_i^t)}} \right]^p dt \right\}^{1/p}<+\infty, \end{aligned} \tag{7} \]
where
\[ S_i^t\equiv \begin{cases} 0\le x_j\le 1, & j=1,2,\ldots,n,\ j\ne i,\\ 0\le x_j\le 1-t, & j=i \end{cases}; \]
\(\tilde\omega(x)\) denotes a real step function taking only a finite number of nonzero values; \(E+\bar x_i^0 t\) denotes the set obtained by shifting the set \(E\) by \(t\) in the direction of the \(x_i\)-axis; \(1/p+1/p'=1\).

It is easy to see that when \(X=E^1\), \(\Psi_p^{(\lambda)}\) is the space of absolutely continuous real functions of sets, isomorphic and isometric to the space \(W_p^{(\lambda)}(S^n)\) of functions of \(n\) variables. Such spaces were considered by L. N. Slobodetskii \((^5)\) and by other authors.

The space \(\Psi_p^{(\lambda)}(S^n)\) is a Banach space with norm (7). The mean functions \(\Phi_h(E)\) form a dense set in \(\Psi_p^{(\lambda)}(S^n)\). Moreover, the inequalities
\[ \|\Phi_h(E)\|_{\Psi_p^{(l)}(D)} \le C_2\|\Phi(E)\|_{\Psi_p^{(l)}(D)} \qquad (l=0,1,2,\ldots), \]
\[ \|\Phi_h(E)\|_{\Psi_p^{(\lambda)}(D)} \le C_3\|\Phi(E)\|_{\Psi_p^{(\lambda)}(D)}, \qquad \lambda>0\ \text{nonintegral}, \]
hold; \(C_2\) and \(C_3\) do not depend on \(\Phi(E)\) and \(h\).

Theorem 3. Let \(\Phi(E)\in\Psi_p^{(l)}(S^n)\), where \(p>1\), \(l\) is a natural number. Then the traces on
\[ S^{n-1}\equiv \{0\le x_i\le 1,\ i=1,2,\ldots,n-1;\ x_n=0\} \]
of the generalized \(n\)-derivatives
\[ \partial^k\Phi(E)/\partial x_n^k,\qquad k=0,1,\ldots,l-1, \]
exist in the sense of the definition given above (see the proof of Theorem 1), and belong to the spaces
\[ \Psi_p^{(l-k-1/p)}(S^{n-1}). \]
Moreover,
\[ \left\| \left. \frac{\partial^k\Phi(E)}{\partial x_n^k} \right|_{x_n=0} \right\|_{\Psi_p^{(l-k-1/p)}(S^{n-1})} \le C_4\|\Phi(E)\|_{\Psi_p^{(l)}(S^n)}, \]
where \(C_4\) does not depend on \(\Phi(E)\).

It suffices to prove the theorem for \(l=1\). But in this case application of Lemma 1 reduces the proof to Gagliardo’s theorem \((^4)\).

Theorem 4. Let \(\widetilde\Phi(I)\in\Psi_p^{(1-1/p)}(S^{n-1})\). Then there exists a function \(\Phi(E)\in\Psi_p^{(1)}(S^n)\) such that its trace on \(S^{n-1}\) is the function \(\Phi(I)\).

and the inequality holds

\[ \|\Phi(E)\|_{\Psi_p^{(1)}(S^n)} \leq C_5\|\widetilde{\Phi}(I)\|_{\Psi_p^{(1-1/p)}(S^{n-1})}; \]

\(C_5\) does not depend on \(\widetilde{\Phi}(I)\).

For the mean functions

\[ \Phi_h(I)=\int_{I\subset S^{n-1}}\widetilde{\varphi}_h(x)\,dx \]

the desired extensions are provided by the functions

\[ \Phi_h(E)=\int_E \frac{1}{x_n^{\,n-1}} \int_{x_1}^{x_1+x_n} d\xi_1 \ldots \int_{x_{n-1}}^{x_{n-1}+x_n} \varphi_h(\xi_1,\ldots,\xi_{n-1})\,d\xi_{n-1}\,dx_1,\ldots,dx_n . \]

For \(p=1\) the following holds.

Theorem 5. The trace of a function \(\Phi(E)\) from \(\Psi_1^{(1)}(S^n)\) on \(S^{n-1}\) is a function \(\widetilde{\Phi}(I)\) from \(\Psi_1(S^{n-1})\), and

\[ \|\widetilde{\Phi}(I)\|_{\Psi_1(S^{n-1})} \leq C_6\|\Phi(E)\|_{\Psi_1^{(1)}(S^n)} . \tag{8} \]

Conversely, let \(\widetilde{\Phi}^{\,1}(I)\in\Psi_1(S^{n-1})\). One can construct a function \(\Phi(E)\in\Psi_1^{(1)}(S^n)\) such that its trace on \(S^{n-1}\) is the function \(\widetilde{\Phi}(I)\), and moreover the inequality

\[ \|\Phi(E)\|_{\Psi_1^{(1)}(S^n)} \leq C_7\|\widetilde{\Phi}(I)\|_{\Psi_1(S^{n-1})} \tag{9} \]

is satisfied;

\(C_6, C_7\) do not depend on \(\widetilde{\Phi}(I)\).

Here too the extension begins with mean functions. In this case the extension operator is nonlinear. In obtaining inequalities (8) and (9), the equivalence of the norm \(\|\ \|_{\Psi_1}\) with the uniform norm (1) is essentially used.

Remark 1. The assertions of Theorems 3 and 4 are also valid for the cube \(S^m\), where \(m\), in the case of Theorem 3, satisfies the inequality \(l-k-\dfrac{n-m}{p}>0\), \(k=0,1,2,\ldots,l-1\), and, in the case of Theorem 4, the inequality \(1-\dfrac{n-m}{p}>0\).

Remark 2. Using Bochner’s norm \((^6)\), one can introduce the spaces \(B_p^{(l)}(S^n)\), with \(l\) a natural number, and \(B_p^{(\lambda)}(S^n)\) with nonintegral \(\lambda\geq 0\). For such spaces, Theorems 1–5 are formulated and proved analogously.

Remark 3. Lemmas 1 and 2 make it possible to give a new proof of the embedding theorems obtained by S. L. Sobolev with the aid of theorems on integrals of potential type \((^1)\).

In conclusion I express my deep gratitude to Prof. L. D. Kudryavtsev for valuable advice and attention to this work.

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

Received
5 VI 1961

REFERENCES

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