Ya. S. Bugrov

A Representation Theorem for a Class of Functions

(Presented by Academician S. L. Sobolev on 19 I 1965)

In the present paper a class of differentiable functions is introduced, defined on the \(n\)-dimensional space, whose generalized derivatives of a certain order have a finite mixed norm, while the higher derivatives satisfy a multiple Hölder condition in the same norm. For this class a representation theorem is proved in the form of a series in entire functions satisfying certain conditions.

Let \(R_n\) be the \(n\)-dimensional space of points \(x=(x_1,\ldots,x_n)\), \(-\infty < x_i < \infty\) \((i=1,\ldots,n)\); \(e_n=\{1,\ldots,n\}\). By \(e\) we shall denote any subset of the set \(e_n\), including the empty set; \(r=(r_1,\ldots,r_n)\), where \(r_j \ge 0\) \((j=1,\ldots,n)\); \(r^e=(r_1^e,\ldots,r_n^e)\), where \(r_j^e=r_j\) if \(j \in e\), and \(r_j^e=0\) if \(j \in e_n-e\); if \(\alpha=(\alpha_1,\ldots,\alpha_n)\), then \(r^\alpha=r_1^{\alpha_1}\cdots r_n^{\alpha_n}\). For an integral vector \(k=(k_1,\ldots,k_n)\) put \(D^k=\partial^{|k|}/\partial x_1^{k_1}\cdots\partial x_n^{k_n}\), where \(k_i \ge 0\) are integers, \(|k|=\sum_{i=1}^n k_i\);

\[ \Delta_{h_i}^2 f(x)=f(x_1,\ldots,x_i+2h_i,\ldots,x_n) -2f(x_1,\ldots,x_i+h_i,\ldots,x_n)+f(x_1,\ldots,x_i,\ldots,x_n), \]

\(\Delta_h^{2\omega^e} f(x)\), where \(h=(h_1,\ldots,h_n)\), \(h_i>0\) \((i=1,\ldots,n)\), \(\omega=(1,\ldots,1)\), denote the second \(h\)-differences applied successively \(n\) times to the function \(f(x)\) with respect to all variables \(x_i\) for which \(i \in e\); \((k,r)=\sum_{i=1}^n k_i r_i\).

Let now \(p=(p_1,\ldots,p_n)\), where \(1 \le p_i \le \infty\) \((i=1,\ldots,n)\). Denote

\[ \|f\|_p=\|f\|_{p_1,\ldots,p_n} =\{\|\cdots\|(\|f\|_{p_1})\|_{p_2}\cdots\|_{p_{n-1}}\}\|_{p_n}, \tag{1} \]

where

\[ \|f\|_{p_i} =\left(\int_{-\infty}^{\infty}|f(x)|^{p_i}\,dx_i\right)^{1/p_i} \qquad (i=1,\ldots,n). \]

If \(\|f\|_p<\infty\), then we shall write \(f(x)\in L_p(R_n)\equiv L_{(p_1,\ldots,p_n)}(R_n)\). If some \(p_i=\infty\), then with respect to the variables \(x_i\), instead of the integral (in the Lebesgue sense), we take the essential maximum.

Definition. Let \(r=(r_1,\ldots,r_n)\), \(r_i=\bar r_i+\alpha_i\), where \(\bar r_i\) are nonnegative integers, \(0<\alpha_i\le 1\) \((i=1,\ldots,n)\). We shall say that a function \(f(x)\) belongs to the class \(S_p^{(r)}H(R_n)\), \(p=(p_1,\ldots,p_n)\), if: 1) the function \(f\) and its generalized derivatives (in the sense of Sobolev) \(D^k f\), where \(0\le k_i\le \bar r_i\) \((i=1,\ldots,n)\), are bounded in the sense of the norm (1); 2) the derivatives \(D^{\bar r^e}f\), \(\bar r=(\bar r_1,\ldots,\bar r_n)\), for any \(e\subseteq e_n\) satisfy the condition

\[ \sup_h \left\|\Delta_h^{2\omega^e}D^{\bar r^e}f/h^{\alpha^e}\right\|_p =M_p^{(r^e)}(f)<\infty, \]

where \(h=(h_1,\ldots,h_n)\), \(h_i>0\) \((i=1,\ldots,n)\); \(\alpha=(\alpha_1,\ldots,\alpha_n)\).

In the class \(S_{\mathbf p}'{}^{(r)}H(R_n)\) the norm is introduced by

\[ \|f\|_{S_{\mathbf p}^{(r)}H(R_n)} = \sum_{e\subseteq e_n} M_{\mathbf p}^{(r^e)}(f), \]

where \(M_{\mathbf p}^{(r^0)}(f)=\|f\|_{\mathbf p}\).

We note that this class of functions was considered by us in paper (1). For \(p_1=\cdots=p_n\) the class \(S_{\mathbf p}^{(r)}H(R_n)\) becomes the class \(S_{p_1}^{(r)}H(R_n)\), which was introduced and studied by S. M. Nikol’skii (2).

Theorem. In order that a function \(f\in S_{\mathbf p}^{(r)}H(R_n)\), where \(r=(r_1,\ldots,r_n)\), \(r_i>0\) \((i=1,\ldots,n)\); \(\mathbf p=(p_1,\ldots,p_n)\), \(1\le p_i\le\infty\) \((i=1,\ldots,n)\), it is necessary and sufficient that it be representable in the form

\[ f(x)=\sum_{e\subseteq e_n}\sum_{k^e>0} Q_{k^e}(x). \]

Here the outer sum of a finite number of terms (series) extends over all possible subsets \(e\subseteq e_n\), including the empty set. The inner sum extends over all possible integer nonnegative vectors \(k^e=(k_1^e,\ldots,k_n^e)\), \(k_i^e\ge0\). The functions \(Q_{k^e}(x)\)—integral powers \(2^{k_j^e}\) in \(x_j\), \(j\in e\) (thus, powers \(1\) in \(x_i\), \(j\in e_n-e\))—satisfy the inequalities

\[ \|Q_{k^e}\|_{\mathbf p}\le M2^{-(k,r^e)}, \]

where \(M\) is a constant.

In proving necessity we obtain \(M\le c\|f\|_{S_{\mathbf p}^{(r)}H(R^n)}\), and in proving sufficiency

\[ \|f\|_{S_{\mathbf p}^{(r)}H(R_n)}\le cM. \]

This theorem is a generalization of results of S. M. Nikol’skii (2). On the basis of this theorem A. P. Uninskii proved a number of embedding theorems for the class \(S_{\mathbf p}^{(r)}H(R_n)\).

Blagoveshchensk State
Pedagogical Institute
named after M. I. Kalinin

Received
18 XII 1964

References Cited

1. Ya. S. Bugrov, Reports of the III Siberian Conference on Mathematics and Mechanics, Tomsk Univ. Press, 1964, p. 54.
2. S. M. Nikol’skii, Siberian Mathematical Journal, 4, No. 6, 1342 (1963).