MATHEMATICS

V. P. IL’IN

ON THE APPROXIMATION OF FUNCTIONS FROM THE SPACES \(\widetilde W_p^{(l)}(D)\) AND \(W_p^{(l)}(D)\) BY CONTINUOUSLY DIFFERENTIABLE FUNCTIONS

(Presented by Academician V. I. Smirnov on 28 XI 1960)

1. Let \(D\) be a finite or infinite domain of \(n\)-dimensional Euclidean space \(E_n\), and let \(l\) be a positive (not necessarily integer) number.

By \(\widetilde W_p^{(l)}(D)\) \((p \ge 1)\) we shall denote the set of all functions \(f(X)\) \((X=(x_1,\ldots,x_n))\), defined in \(D\), possessing all generalized derivatives in the sense of S. L. Sobolev \((^1)\) of order \(\bar l=[l]\) (\([l]\) is the integer part of \(l\)), satisfying the conditions:

\[ 1)\quad \|f\|_{L_p(D)}=\left[\int_D |f(X)|^p\,dX\right]^{1/p}<\infty; \]

\[ 2)\quad \|f\|_{L_p^{(l)}(D)} = \sum_{i_1,\ldots,i_l=1}^{n} \left[ \int_D \left| \frac{\partial^l f(X)}{\partial x_{i_1}\cdots \partial x_{i_l}} \right|^p dX \right]^{1/p} <\infty, \]

if \(l\) is an integer, or

\[ 2')\quad \|f\|_{L_p^{(l)}(D)} = \sum_{i_1,\ldots,i_{\bar l}=1}^{n} \left[ \int_D \left( \int_D \frac{ \left| \frac{\partial^{\bar l} f(X)}{\partial x_{i_1}\cdots \partial x_{i_{\bar l}}} - \frac{\partial^{\bar l} f(Y)}{\partial x_{i_1}\cdots \partial x_{i_{\bar l}}} \right|^p }{ |X-Y|^{\,n+(l-\bar l)p} } \,dY \right) dX \right]^{1/p} <\infty, \]

if \(l\) is not an integer.

Put

\[ \|f\|_{\widetilde W_p^{(l)}(D)} = \|f\|_{L_p(D)}+\|f\|_{L_p^{(l)}(D)}. \]

From the class of functions belonging to \(\widetilde W_p^{(l)}(D)\), we single out the subclass of functions having all possible generalized derivatives of order up to \(\bar l\) inclusive, belonging to \(L_p(D)\). We shall denote this class of functions by \(W_p^{(l)}(D)\).

Put

\[ \|f\|_{W_p^{(l)}(D)} = \sum_{k=0}^{l}\|f\|_{L_p^{(k)}(D)}, \]

if \(l\) is an integer, and

\[ \|f\|_{W_p^{(l)}(D)} = \sum_{k=0}^{\bar l}\|f\|_{L_p^{(k)}(D)} + \|f\|_{L_p^{(l)}(D)} \]

if \(l\) is not an integer.

In the study of the spaces \(\widetilde W_p^{(l)}(D)\) and \(W_p^{(l)}(D)\), an important role is played by the problem of approximating a function \(f \in \widetilde W_p^{(l)}(D)\) or \(f \in W_p^{(l)}(D)\), in the norm \(\widetilde W_p^{(l)}(D)\) or, respectively, \(W_p^{(l)}(D)\), by means of a sequence of functions \(\varphi_\nu(X)\) \((\nu = 1, 2, \ldots)\) belonging to \(C^{(l)}(\overline D)\), where \(\overline D = D + \Gamma\), \(\Gamma\) is the boundary of \(D\).

If \(D = E_n\), then such an approximation with any degree of accuracy is carried out by means of averaging functions \((^1)\). If, however, \(D\) does not coincide with the whole space \(E_n\), then, as is known, it is not always possible. For the case when the boundary of the domain \(D\) belongs to the class \(C^{(l)}\), the possibility of such an approximation for functions \(f \in W_p^{(l)}(D)\) was proved by V. M. Babich \((^2)\) (for integer \(l\)) and by L. N. Slobodetskii \((^3)\) (for noninteger \(l\)). For integer \(l\), for the same class of functions, E. Gagliardo \((^4)\) obtained an analogous result under the assumption that \(D\) is a bounded domain whose boundary belongs to the class Lip 1. For functions belonging to \(\widetilde W_p^{(l)}(D)\), the possibility of the above approximation was proved for the case when \(D\) is a bounded domain star-shaped with respect to some interior point \(D\) \((^5)\).

The results given below concern the same circle of questions.

II. Let \(D\) be a finite or infinite domain of the Euclidean space \(E_n\). We introduce the following definitions:

1) We shall say that \(D \in C(H,\sigma)\) if for each point \(X \in D\) there exists an \(n\)-dimensional spherical sector with vertex at \(X\), radius \(H\), and solid angle \(\sigma\), wholly contained in \(D\).

2) We shall say that the domain \(D\) belongs to the class \(C(H,\sigma,K,\lambda)\) and write \(D \in C(H,\sigma,K,\lambda)\) if it satisfies the following condition: for any two points \(X\) and \(Y\) in \(D\) for which \(|X-Y| \le H\), there exist \(n\)-dimensional spherical sectors with solid angle \(\sigma\) and radius \(\le K|X-Y|\), with vertices at \(X\) and \(Y\), contained in \(D\), and such that, if by \(G\) we denote the intersection of these sectors, then the inequality holds:
\[ mG \ge \lambda |X-Y|^n, \]
where \(H,\sigma,K,\lambda\) are positive numbers fixed for the given domain.

It is evident that if \(D \in C(H,\sigma,K,\lambda)\), then \(D \in C(H/2,\sigma)\).

3) By \(D_\delta\) we shall denote the domain consisting of the points of \(D\) whose distance from the boundary of \(D\) is greater than \(\delta\).

4) We shall say that the domain \(D\) has property \(A(N,\varkappa)\), and write \(D \in A(N,\varkappa)\), if there exist two finite systems of \(n\)-dimensional domains \(S_1,\ldots,S_N\) and \(S'_1,\ldots,S'_N\), each of which forms a covering of \(D\), satisfying the following conditions: a) \(\overline S_i \subset S''_i\) \((i=1,\ldots,N)\), and if \(X \in S_i\), \(Y \in \overline{S''_i}\), then \(|X-Y| \ge \varkappa > 0\); b) the sets \(D_i = D \cdot S_i\), \(D'_i = D \cdot (S'_i)_{\varkappa/2} = D \cdot S'_i\), \(D''_i = D \cdot S''_i\) \((i=1,\ldots,N)\), as well as the sets \(D'_i \cdot D'_k\) \((i,k = 1,\ldots,N)\), if they are nonempty, are finitely connected; c) for each set \(D'_i = D \cdot S'_i\) \((i=1,\ldots,N)\) there exists a vector \(Q_i\) such that the translation of \(D'_i\) by the vector \(tQ_i\), for arbitrary \(t\), \(0 < t \le 1\), carries \(D'_i\) into a domain \(\Omega''_{it}\) interior with respect to \(D\), i.e., such that
\[ \rho(\Omega''_{it}, E_n - D) > 0 \quad (i=1,\ldots,N). \]

We note that if some domain \(S'_i\) is a strictly interior subdomain of the domain \(D\), then the corresponding vector \(Q_i\) may be taken to be zero.

We also note that if \(D\) is a bounded domain whose boundary belongs to the class Lip 1, then there exist positive numbers \(H,\sigma,K,\lambda,N,\varkappa\) such that \(D \in C(H,\sigma,k,\lambda)\) and \(D \in A(N,\varkappa)\).

III. In all the theorems given below, \(\varphi_\nu(X)\) will denote functions having continuous derivatives of every order in the whole space \(E_n\).

Theorem 1*. If \(f\in W_p^{(l)}(D)\), \(p\geqslant 1\), \(l\) is an integer, \(D\in A(N,\varkappa)\), then there exists a sequence of functions \(\varphi_\nu(X)\) \((\nu=1,\ldots)\) such that

\[ \lim_{\nu\to\infty}\|f-\varphi_\nu\|_{W_p^{(l)}(D)}=0. \]

Lemma. Let \(f(X)\in \widetilde W_p^{(l)}(D)\) and have continuous derivatives up to order \(\bar l=[l]\) in \(D\), \(D\in C(H,\sigma)\). Then for any integer \(s\), \(0\leqslant s\leqslant \bar l\), the inequality

\[ \|f\|_{L_p^{(s)}(D)} \leqslant C_1\|f\|_{L_p(D)}^{1-s/l} \|f\|_{\widetilde W_p^{(l)}(D)}^{s/l}. \]

holds.

If \(l\) is not an integer, \(D\in C(H,\sigma,K,\lambda)\), then the inequality

\[ \|f\|_{L_p^{(s+l-\bar l)}(D)} \leqslant C_2\|f\|_{L_p(D)}^{\frac{\bar l-s}{l}} \|f\|_{\widetilde W_p^{(l)}(D)}^{1-\frac{\bar l-s}{l}} . \]

also holds.

The constants \(C_1\) and \(C_2\) do not depend on \(f\).

With the help of this lemma the following theorems are proved.

Theorem 2. If \(f\in \widetilde W_p^{(l)}(D)\), \(p\geqslant 1\), \(l\) is an integer, \(D\in A(N,\varkappa)\) and \(D\in C(H,\sigma)\), then:

1) \(f\in W_p^{(l)}(D)\);

2) there exists a sequence of functions \(\varphi_\nu(X)\) \((\nu=1,\ldots)\) such that

\[ \lim_{\nu\to\infty}\|f-\varphi_\nu\|_{W_p^{(l)}(D)}=0. \]

Theorem 3. If \(f\in \widetilde W_p^{(l)}(D)\), \(p\geqslant 1\), \(l\) is not an integer, \(D\in A(N,\varkappa)\), \(D\in C(H,\sigma,K,\lambda)\), then:

1) \(f\in W_p^{(l)}(D)\);

2) there exists a sequence of functions \(\varphi_\nu(X)\) \((\nu=1,\ldots)\) such that

\[ \lim_{\nu\to\infty}\|f-\varphi_\nu\|_{W_p^{(l)}(D)}=0. \]

Remark. Let \(f(X)\) and the domain \(D\) satisfy the conditions of Theorem 2 if \(l\) is an integer, or the conditions of Theorem 3 if \(l\) is not an integer. Suppose, in addition, that for any integer \(m\), \(0\leqslant m\leqslant n\), and all derivatives of order \(\bar l=[l]\) the inequalities

\[ \sup_{D_m} \left[ \int_{(|D_m|_{n-m}^{d})}\cdots\int |D^l f(X)|^p\,dX \right]^{1/p} \leqslant M d^{\alpha_m}, \tag{1} \]

hold if \(l\) is an integer, or

\[ \sup_{D_m} \left[ \int_{(|D_m|_{n-m}^{d})}\cdots\int \left( \int_{(D)}\cdots\int \frac{|D^{\bar l}f(X)-D^{\bar l}f(Y)|^p}{|X-Y|^{n+(l-\bar l)p}}\,dY \right)dX \right]^{1/p} \leqslant M d^{\alpha_m}, \tag{2} \]

if \(l\) is not an integer, where \(M>0\), \(\alpha_m\) \((m=0,1,\ldots,n)\) are constants, with

\[ \alpha_0\geqslant \alpha_1\geqslant \cdots \geqslant \alpha_n=0,\qquad \alpha_m\leqslant \frac{n-m}{p}, \]

and \(|D_m|_{n-m}^{d}\) is the set of points of the domain \(D\) at distance no greater than \(d\) from some section \(D_m\) of the domain \(D\) by the hyperplane \(x_{m+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\).

* If \(D\) is a bounded domain, then Theorem 1 is implicitly contained in the results of E. Gagliardo (4).

It can be shown that inequalities (1) and (2) will also hold for the functions \(\varphi_\nu(X)\) constructed respectively in Theorems 2 and 3.

This remark makes it possible to extend the results of works \({}^{6,7}\) also to functions \(f(X) \in \widetilde{W}_p^{(l)}(D)\), if \(D\) satisfies the conditions of Theorem 2 or 3.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
17 XII 1960

References

\({}^{1}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
\({}^{2}\) V. M. Babich, Uspekhi Mat. Nauk, 8, issue 2 (54) (1953).
\({}^{3}\) L. N. Slobodetskii, Scientific Notes of the Leningrad State Pedagogical Institute named after A. I. Herzen, 197, 54 (1958).
\({}^{4}\) E. Gagliardo, Ricerche di Matematica, 8, Fasc. 1 (1958).
\({}^{5}\) V. I. Smirnov, A Course of Higher Mathematics, 5, 1959.
\({}^{6}\) V. P. Il’in, Proceedings of the V. I. Steklov Mathematical Institute, Academy of Sciences of the USSR, 53, 64 (1959).
\({}^{7}\) V. P. Il’in, Dokl. Akad. Nauk SSSR, 135, no. 4 (1960).