Abstract Generated abstract
This paper studies Sobolev type classes with fractional smoothness on Euclidean spaces, bounded domains, and differentiable manifolds. It formulates trace and converse extension theorems for the classes \(W_p^r\), extending known results for Nikol’skii classes and for the Hilbert case \(p=2\), and proves an extension theorem from bounded domains to the whole space under smoothness assumptions on the boundary. The paper also introduces weighted classes \(W_{p,\alpha}^r(G,\Lambda_m)\) defined by powers of the distance to a boundary manifold, and states extension, trace, and embedding results connecting these weighted spaces with fractional Sobolev classes \(W_p^r\). These results generalize earlier theorems of Slobodetskii, Nikol’skii, Kudryavtsev, Vasharin, and Lizorkin in several ranges of parameters.
Full Text
MATHEMATICS
S. V. USPENSKII
PROPERTIES OF THE CLASSES \(W_p^r\) WITH FRACTIONAL DERIVATIVE ON DIFFERENTIABLE MANIFOLDS
(Presented by Academician I. M. Vinogradov on 23 XII 1959)
This work is a further development of the results published in our paper \((^1)\).
Let \(g\) be a domain in the space \(R_n\); \(R_m\) is the \(m\)-dimensional subspace of points \((x_1,\ldots,x_m,x_{m+1},\ldots,x_n)\), where \(x_{m+1},\ldots,x_n\) are fixed.
A function \(f\) belongs to the class \(W_p^r(g)\), \(1<p<\infty\), for \(r=\bar r+\alpha\), where \(\bar r\) is an integer, \(0<\alpha<1\), if:
1) \(f\) belongs to the Sobolev class \(W_p^{\bar r}(g)\);
2)
\[
\|f^{\bar r}\|_{W_p^\alpha(g)}
=
\iint\limits_{g\,g}
\frac{|f^{\bar r}(x_1,\ldots,x_n)-f^{\bar r}(y_1,\ldots,y_n)|^p}
{|x-y|^{\,n+p\alpha}}
\,dg_x\,dg_y<\infty,
\]
where
\[
|x-y|=\left[\sum_{i=1}^{n}(x_i-y_i)^2\right]^{1/2},\quad i=1,2,\ldots,n.
\]
Put
\[
\|f\|_{W_p^r(g)}
=
\|f\|_{W_p^{\bar r}(g)}
+
\sum \|f^{\bar r}\|_{W_p^\alpha(g)}.
\]
The following theorems for the classes \(W_p^r(R_n)\) are analogous to the corresponding theorems for the classes \(H_p^r\) of S. M. Nikol’skii \((^2)\).
Theorem 1 (direct). Let \(f\in W_p^r(R_n)\), \(1\le m<n\), \(\lambda_j\ge0\),
\[
\rho=r-\frac{(n-m)}{p}-\sum_{j=m+1}^{n}\lambda_j>0.
\]
Then the partial derivative
\[
\psi(x_1,\ldots,x_m)
=
\frac{\partial^{\lambda_{m+1}+\cdots+\lambda_n} f(x_1,\ldots,x_n)}
{\partial x_{m+1}^{\lambda_{m+1}}\cdots \partial x_n^{\lambda_n}}
\]
as a function of \(x_1,\ldots,x_m\), for any fixed \(x_{m+1},\ldots,x_n\), belongs to \(W_p^\rho(R_m)\) (\(\rho\) not an integer when \(p>2\)),
\[
\|f\|_{W_p^\rho(R_m)}\le c\|f\|_{W_p^r(R_n)},
\]
where \(c\) does not depend on \(f\).
Theorem 2 (converse). Given a number \(r>0\) and all possible systems \(\{\lambda\}\) of nonnegative numbers \(\lambda_{m+1},\ldots,\lambda_n\), for which
\[ \rho^\lambda=r-\frac{(n-m)}{p}-\sum_{j=m+1}^{n}\lambda_j\geq 0. \]
Let to each system \(\{\lambda\}\) there correspond a function \(\varphi_\lambda(x_1,\ldots,x_m)\) belonging to \(W_p^{\rho}(R_m)\) \((\rho\) nonintegral for \(1<p<2)\). Then one can construct a function \(f(x_1,\ldots,x_n)\in W_p^r(R_n)\) such that
\[ \left. \frac{\partial^{\lambda_{m+1}+\cdots+\lambda_n} f(x_1,\ldots,x_n)} {\partial x_{m+1}^{\lambda_{m+1}}\cdots \partial x_n^{\lambda_n}} \right|_{R_m} = \varphi_\lambda(x_1,\ldots,x_m), \]
\[ \|f\|_{W_p^r(R_n)} \leq c\sum_{\{\lambda\}}\|\varphi_\lambda\|_{W_p^\rho(R_m)}. \]
Theorems 1 and 2 for \(p=2\) are due to L. N. Slobodetskii \((^7)\). S. M. Nikolskii in \((^3)\) and V. M. Babich obtained a theorem on the extension of classes \(W_p^r\) (\(r\) integral) from a bounded domain \(g\) to the whole space \(R_n\). The following theorem generalizes this result to the case where \(r\) is fractional (for \(p=2\) it is due to L. N. Slobodetskii).
Theorem 3. If \(f\in W_p^r(G)\), \(G\) is bounded, then, whatever \(\eta>0\), there exists a function \(F\in W_p^r(R_n)\) such that
\[ F|_{G_\eta}=f,\qquad \|F\|_{W_p^r(R_n)}\leq c\|f\|_{W_p^r(G)}, \]
where \(c\) depends only on \(\eta\).
If the boundary \(\Lambda\) of the domain \(G\) is sufficiently smooth \((\Lambda\in C^{(r+1)})\), then \(\eta\) may be set equal to \(0\).
With the aid of this theorem, by known methods proposed by S. M. Nikolskii \((^2)\), Theorems 1 and 2, as well as all the theorems of our work \((^1)\), are carried over to bounded domains \(G\) with sufficiently smooth boundary.
Definition. Let \(G\) be an \(n\)-dimensional bounded domain, and let \(\Lambda_m\) be its sufficiently smooth \(m\)-dimensional boundary. We shall say that a function \(\sigma=\sigma(x_1,\ldots,x_n)\), defined on \(G\), satisfies the inequality
\[ c_1\rho(x,\Lambda_m)\leq \sigma(x)\leq c_2\rho(x,\Lambda_m), \]
where \(\rho(x,\Lambda_m)\) is the distance from the point \(x\) to \(\Lambda_m\), and \(c_1,c_2\) are constants depending only on the domain \(G\). We shall say that a function \(f=f(x_1,\ldots,x_n)\) belongs to the class \(W_{p,\alpha}^r(G,\Lambda_m)\), if it is defined on \(G\), has on \(G\) all generalized partial derivatives up to order \(r\) (\(r\) integral) inclusive,
\[ \|f\|_{W_{p,\alpha}^r(G,\Lambda_m)} = \sum_{l=0}^{r} \int_G \sigma^\alpha \sum_{\alpha_1+\cdots+\alpha_n=l} \left| \frac{\partial^l f} {\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^p \,dg <\infty, \qquad -k<\alpha<pr. \]
Such classes were considered in works \((^{4-6})\), etc.
L. D. Kudryavtsev obtained extension and embedding theorems for the classes \(W_{p,\alpha}^r(G,\Lambda_m)\) into \(H_p^r(G)\). Below are formulated theorems on the extension and embedding of the classes \(W_{p,\alpha}^r(G,\Lambda_m)\) into \(W_p^r(G)\), which generalize the results of A. A. Vasharin \((^5)\) and P. I. Lizorkin \((^6)\).
Theorem 4. Let \(f\in W_p^r(\Lambda_m)\) (\(r\) nonintegral for \(1<p<2\)), \(\Lambda_m\in C^{l+1}\), \(l>r\). Then there exists a function \(F\), defined on \(G\), such that
\[ F\big|_{\Lambda_m}=f,\qquad F\in W^{\,l}_{p,\,p(l-r)-k}(G,\Lambda_m), \]
\[ \|F\|_{W^{\,l}_{p,\,p(l-r)-k}(G,\Lambda_m)} \le c\|f\|_{W_p^r(\Lambda_m)}. \]
Theorem 5. Let \(f\in W^r_{p,\alpha}(G,\Lambda_m)\), and let \(\Lambda_k\) be a \(k\)-dimensional, sufficiently smooth closed manifold belonging to \(\overline G\). Then, if
\[ \rho=r-\frac{(n-k)}{p}-\sum_{i=1}^{n}\lambda_i-\frac{\alpha}{p}>0,\qquad 1\le k<n, \]
then
\[ \left. \frac{\partial^{\lambda_1+\cdots+\lambda_n} f(x_1,\ldots,x_n)} {\partial x_1^{\lambda_1}\cdots \partial x_n^{\lambda_n}} \right|_{\Lambda_k} =\psi(\Lambda_k)\in W_p^\rho(\Lambda_k) \]
\[ (\rho\text{ nonintegral for }p>2), \]
\[ \|\psi\|_{W_p^\rho(\Lambda_k)} \le c\|f\|_{W^r_{p,\alpha}(G,\Lambda_m)}. \]
Theorems 4 and 5 were obtained by A. A. Vasharin for \(p=2\) and \(m=k=n-1\), and by P. I. Lizorkin for \(0\le r<1\), \(m=k=n-1\), \(1<p<\infty\).
Theorem 6. Let \(f\in W^r_{p,\alpha}(G,\Lambda_m)\), \(\Lambda_m\in C^{r+1}\). Then \(f\in W_p^{r-\alpha/p}(G)\) (\(\alpha\ge0\)) and \(f\in W_p^r(G)\) (\(\alpha<0\)),
\[ \|f\|_{W_p^{r-\alpha/p}(G)} \le c\|f\|_{W^r_{p,\alpha}(G,\Lambda_m)}. \]
Theorem 7. Let \(f\in W^r_{p,\alpha}(G,\Lambda_m)\), \(\Lambda_m\in C^{r+1}\). Then, if
\[ \alpha-vp>-k,\qquad v=1,2,\ldots,r, \]
then
\[ f\in W^{r-v}_{p,\alpha-vp}(G,\Lambda_m), \]
\[ \|f\|_{W^{r-v}_{p,\alpha-vp}(G,\Lambda_m)} \le c\|f\|_{W^r_{p,\alpha}(G,\Lambda_m)}, \]
\[ \int_{\Lambda_m} \sum_{\alpha_1+\cdots+\alpha_n=r-v} \left| \frac{\partial^{r-v}f} {\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^p \,d\Lambda_m =o\!\left(\rho^{-(\alpha-vp+k)}\right). \]
This theorem strengthens the corresponding theorem of L. D. Kudryavtsev in work \((4)\)*.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
10 XII 1959
REFERENCES
- S. V. Uspenskii, DAN, 130, No. 5 (1950).
- S. M. Nikol’skii, Matem. sborn., 33 (75), 261 (1953).
- S. M. Nikol’skii, DAN, 88, 409 (1953).
- L. D. 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.
- A. A. Vasharin, DAN, 117, No. 5 (1957).
- P. I. Lizorkin, DAN, 126, No. 4 (1959).
- L. N. Slobodetskii, DAN, 118, No. 2 (1958).
* This work arose as a result of participation in the seminar of V. I. Kondrashov, L. D. Kudryavtsev, and S. M. Nikol’skii. In preparing the work for publication it became known that, independently and simultaneously, Theorem 6, Theorem 5 for \(\rho\) nonintegral and \(m=k\), and also Theorem 2 for \(\rho\) nonintegral and Theorem 4 for \(r\) nonintegral had been obtained by another participant of the seminar, P. I. Lizorkin.