MATHEMATICS

S. V. USPENSKII

AN EMBEDDING THEOREM FOR THE S. L. SOBOLEV CLASSES \(W_p^r\) OF FRACTIONAL ORDER

(Presented by Academician I. M. Vinogradov on 17 X 1959)

Let \(R\) be \(n\)-dimensional space. We shall say that a function \(f\) belongs to the class \(W_p^r(R_n)\), where \(r=\bar r+\alpha\), \(\bar r\) is an integer, \(0<\alpha<1\), if:

1) \(f\) belongs to the Sobolev class \(W_p^{\bar r}(R_n)\) (for \(r=0\), \(f\in L_p(R_n)\));

2)
\[ \left\|D_i^{\bar r}f\right\|_{W_p^\alpha} = \sum_{j=1}^{n} \left( \int_{0}^{1}\int_{R_n} \frac{ \left|D_i^{\bar r}f(x_1,\ldots,x_j+h,\ldots,x_n) - D_i^{\bar r}f(x_1,\ldots,x_j,\ldots,x_n)\right|^p }{ h^{1+p\alpha} }\,dR_n\,dh \right)^{1/p} <\infty, \]
\[ i=1,2,\ldots,n; \tag{1} \]
here \(D_i^{\bar r}f\) is the \(\bar r\)-th Sobolev derivative with respect to \(x_i\).

We introduce a norm in the space \(W_p^r(R_n)\):
\[ \|f\|_{W_p^r} = \|f\|_{W_p^{\bar r}} + \sum_{i=1}^{n}\left\|D_i^{\bar r}f\right\|_{W_p^\alpha}. \tag{2} \]

Such classes of functions were considered for \(p=2\) in the works \((^{1-3,\,8})\); for \(p\geqslant 1\) in \((^{5,\,7,\,9})\), where the embedding theorems of S. L. Sobolev and V. I. Kondrashov \((^{10})\) were strengthened. In \((^8)\), L. N. Slobodetskii, with a somewhat different normalization, proved embedding theorems for classes with \(p=2\). It can be shown, however, that the norm introduced there is equivalent to (2).

The following theorem holds, analogous to the corresponding theorem on embeddings of the classes \(H_p^r\) of S. M. Nikol’skii \((^4)\).

Main theorem. Let \(1<p\leqslant q<\infty\), \(r>0\), \(1\leqslant m\leqslant n\),
\[ \chi = 1-\frac{1}{r}\left(\frac{n}{p}-\frac{m}{q}\right)>0 \quad (\geqslant 0 \text{ when } p<q\leqslant 2). \tag{3} \]

Then, if a function \(f\) belongs to the class \(W_p^r(R_n)\) (\(r\) non-integer when \(m=n,\ q<2\)), it also belongs, in the variables \(x_1,\ldots,x_m\), for any fixed \(x_{m+1},\ldots,x_n\), to the class \(W_q^\rho(R_m)\) (\(\rho\) non-integer when \(q>2\)), where
\[ \rho=\chi r, \tag{4} \]
\[ \|f\|_{W_q^\rho}\leqslant c\|f\|_{W_p^r}; \tag{5} \]
\(c\) depends only on \(m,n,p,q\).

The proof of this theorem is based on a number of propositions formulated below, which are also of independent interest.

Let \(g\) be a bounded \(n\)-dimensional domain with sufficiently smooth boundary \(\Lambda\). Define on \(g\) a function \(\sigma=\sigma(P)\). We shall assume that \(\sigma(P)\) is bounded on \(g\), and near the boundary has the order of the distance from the point \(P\) to \(\Lambda\).

The following theorem strengthens the corresponding result of L. D. Kudryavtsev \((^6)\).

Theorem 1. Let the function \(f\) have all generalized partial derivatives up to order \(l\) inclusive, and suppose that

\[ \|f\|_{W^{\,l}_{p,\alpha}}^{p} = \sum_{j=0}^{l}\int_{G}\sigma^{\alpha} \sum_{\alpha_1+\cdots+\alpha_n=j} \left| \frac{\partial^{j} f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^{p}\,dq<\infty . \tag{6} \]

Then, if \(\alpha-kp>-1\), we have

\[ \int_{g}\sigma^{\alpha-kp} \sum_{\alpha_1+\cdots+\alpha_n=l-k} \left| \frac{\partial^{\,l-k} f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^{p}\,dg \le c\|f\|_{W^{\,l}_{p,\alpha}}^{p}, \tag{7} \]

and for sufficiently small \(\rho\),

\[ \int_{\Lambda_{\rho}} \sum_{\alpha_1+\cdots+\alpha_n=l-k} \left| \frac{\partial^{\,l-k} f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^{p}\,d\Lambda_{\rho} = O(\rho^{-\alpha+kp-1}). \tag{8} \]

Here \(\Lambda_{\rho}\) is the surface at distance \(\rho\) from \(\Lambda\); \(c\) does not depend on \(f\).

The theorem is also valid in the case where \(g\) is an \(n\)-dimensional half-space.

Definition. Let \(\Lambda\) be a sufficiently smooth surface. We shall say that \(F|_{\Lambda}=f\), if along almost all normals to \(\Lambda\), \(F\) has a limit equal to \(f\).

Theorem 2. Let \(f\in W_p^r(R_n)\), \(p>1\) when \(r\) is nonintegral and \(p\ge 2\) when \(r\) is integral.

Then there exists a function \(F\), defined in the domain \(R_{n+1}\{0\le x_{n+1}\le \infty,\ (x_1,\ldots,x_n)\in R_n\}\), such that:

1) \(F\) is harmonic in \(\overline{R}_{n+1}\) and \(F|_{R_n}=f\);

\[ \text{2)}\qquad \sum_{\alpha_1+\cdots+\alpha_{n+1}=l} \int_{\overline{R}_{n+1}} x_{n+1}^{\,p(l-r)-1} \left| \frac{\partial^{l}F}{\partial x_1^{\alpha_1}\cdots \partial x_{n+1}^{\alpha_{n+1}}} \right|^{p} \,d\overline{R}_{n+1} \le c\|f\|_{W_p^r}^{p}, \tag{9} \]

(for arbitrary \(r\) when \(p\ge 2\), and for nonintegral \(r\) when \(p<2\)), where \(l\) is an integer, \(l-r>0\), and \(c\) does not depend on \(f\).

Theorem 3. If \(f\in W_p^r(R_n)\), then for any mixed derivative of the function \(f\) of order \(\bar r\),

\[ D^{\bar r}f= \frac{\partial^{\bar r}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}, \qquad \alpha_1+\cdots+\alpha_n=\bar r, \]

\[ \|D^{\bar r}f\|_{W_p^{\alpha}}^{p} = \int_{0}^{1}\int_{R_n} \frac{ \left| D^{\bar r}f(x_1,\ldots,x_i+h,\ldots,x_n) - D^{\bar r}f(x_1,\ldots,x_n) \right|^{p} }{h^{1+p\alpha}} \,dR\,dh <\infty \tag{10} \]

(for all \(i=1,2,\ldots,n\)).

Received
12 X 1959

CITED LITERATURE

1. N. Aronszajn, Boundary Values of Functions with Finite Dirichlet Integral, No. 14, Kansas, 1955.
2. L. V. Slobodetskii, V. M. Babich, DAN, 106, No. 4 (1955).
3. A. A. Vasharin, DAN, 117, No. 5 (1957).
4. S. M. Nikol’skii, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
5. E. Gogliardo, Rend. Semin. Mat. Padova, V, XXVII, 284 (1957).
6. L. O. Kudryavtsev, Continuation of Functions and Embedding of Classes of Functions, Doctoral dissertation, Mathematical Institute named after V. A. Steklov, Academy of Sciences of the USSR, 1956.
7. P. I. Lizorkin, DAN, 126, No. 4 (1959).
8. L. N. Slobodetskii, DAN, 118, No. 2 (1958).
9. L. N. Slobodetskii, DAN, 120, No. 3 (1958).
10. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950, pp. 64–94.