UDC 517.51

MATHEMATICS

A. P. UNINSKII

EMBEDDING THEOREMS FOR A CLASS OF FUNCTIONS WITH MIXED NORM

(Presented by Academician S. L. Sobolev, May 24, 1965)

In this note we consider a class of differentiable functions defined on the \(n\)-dimensional space \(R_n\), whose derivatives of a certain order have finite mixed norm \((^1)\), while the higher derivatives satisfy a multiple Hölder condition in this same norm. For functions of this class, which we denote by \(S_{\mathbf p}^{(\mathbf r)}H(R_n)\), where \(\mathbf r=(r_1,\ldots,r_n)\), \(\mathbf p=(p_1,\ldots,p_n)\) (for the definition of the class \(S_{\mathbf p}^{(\mathbf r)}H(R_n)\), see \((^2)\)), Ya. S. Bugrov \((^2)\) proved a theorem on representation in the form of a series of entire functions satisfying certain conditions.

Theorem (main). In order that a function \(f(\mathbf x)\) belong to the class

\[ S_{\mathbf p}^{(\mathbf r)}H(R_n),\qquad \mathbf r=(r_1,\ldots,r_n),\quad r_i>0\quad (i=1,\ldots,n), \]
\[ \mathbf p=(p_1,\ldots,p_n),\qquad 1\le p_i\le \infty\quad (i=1,\ldots,n), \]

it is necessary and sufficient that it be representable in the form

\[ f(\mathbf x)=\sum_{e\subseteq e_n}\sum_{\mathbf k^e\ge 0} Q_{\mathbf k^e}^{e}, \]

where the outer sum, of a finite number of terms (series), is extended over all possible subsets \(e\subseteq e_n\), including the empty set. The inner sum is extended over all integer nonnegative vectors \(\mathbf k^e=(k_1^e,\ldots,k_i^e)\), \(k_j^e\ge 0\). The functions \(Q_{\mathbf k^e}(\mathbf x)\) are entire of degree \(2^{k_j^e}\) in \(x_j,\ j\in e\) (thus, of degree \(1\) in \(x_j,\ j\in e_n-e\)), and satisfy the inequalities

\[ \|Q_{\mathbf k^e}\|_{\mathbf p}\le M\cdot 2^{-(\mathbf k,\mathbf r^e)}, \]

where \(M\) is a constant.

Using the conditions of the main theorem, we have proved several embedding theorems for the classes \(S_{\mathbf p}^{(\mathbf r)}H(R_n)\), generalizing results of S. M. Nikol’skii \((^3)\).

Theorem 1. Let the function \(f(\mathbf x)\) belong to the class \(S_{\mathbf p}^{(\mathbf r)}H(R_n)\), and let \(0<\boldsymbol\rho\le \mathbf r\), where \(\mathbf r=(r_1,\ldots,r_n)\), \(r_j>0\), \(\boldsymbol\rho=(\rho_1,\ldots,\rho_n)\), \(\rho_j>0\), \(\mathbf p=(p_1,\ldots,p_n)\). Then

\[ f(\mathbf x)\in S_{\mathbf p}^{(\boldsymbol\rho)}H(R_n), \]
\[ \|f\|_{S_{\mathbf p}^{(\boldsymbol\rho)}H(R_n)} \le c\|f\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)}. \]

Theorem 2. Let the function \(f(\mathbf x)\) belong simultaneously to the classes

\[ S_{\mathbf p}^{(\mathbf r^1)}H(R_n),\ldots,S_{\mathbf p}^{(\mathbf r^N)}H(R_n). \]

and

\[ \sum_{1}^{N}\lambda_\nu \leqslant 1,\qquad \lambda_\nu \geqslant 0 \quad (\nu=1,\ldots,N),\qquad \mathbf r=\sum_{1}^{N}\lambda_\nu \mathbf r^\nu . \]

Then

\[ f(x)\in S_{\mathbf p}^{(\mathbf r)}H(R_n); \]

\[ \|f\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)} \leqslant c\prod_{\nu=1}^{N}\|f\|_{S_{\mathbf p}^{(\mathbf r^\nu)}H(R_n)}^{\lambda_\nu}, \qquad \text{for }\sum_{1}^{N}\lambda_\nu=1; \]

\[ \|f\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)} \leqslant c\prod_{\nu=1}^{N}\|f\|_{S_{\mathbf p}^{(\mathbf r^\nu)}H(R_n)}^{\lambda_\nu/\alpha}, \qquad \text{for }\sum_{1}^{N}\lambda_\nu=\alpha<1; \]

\[ \|f\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)} \leqslant c\sum_{\nu=1}^{N}\|f\|_{S_{\mathbf p}^{(\mathbf r^\nu)}H(R_n)}, \qquad \text{for }\sum_{1}^{N}\lambda_\nu\leqslant 1. \]

Theorem 3. Let the function \(f(x)\) belong to the class

\[ S_{\mathbf p}^{(\mathbf r)}H(R_n),\qquad \mathbf r=(r_1,\ldots,r_n)>0\quad (e_{\mathbf r}=e_n), \]

\[ \mathbf p=(p_1,\ldots,p_n),\qquad p_1\geqslant p_2\geqslant\cdots\geqslant p_n\geqslant 1,\qquad \mathbf q=(q_1,\ldots,q_n), \]

\[ p_k<q_k\leqslant\infty\qquad (k=1,n). \]

Then \(f(x)\in S_{\mathbf q}^{(\vec\rho)}H(R_n)\), where \(\vec\rho=(\rho_1,\ldots,\rho_n)\),

\[ \rho_j=r_j-\left(\frac1{p_j}-\frac1{q_j}\right)>0\qquad (j=1,\ldots,n), \]

\[ \|f\|_{S_{\mathbf q}^{(\vec\rho)}H(R_n)} \leqslant c\|f\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)}. \]

Theorem 4. Let the function \(f(x)\) belong to the class

\[ S_{\mathbf p}^{(\mathbf r)}H(R_n),\qquad \mathbf r=(r_1,\ldots,r_n)>0\quad (e_{\mathbf r}=e_n); \]

\(x_{m+1}^0,\ldots,x_n^0\) are fixed coordinates of the point
\(\mathbf x=(x_1,\ldots,x_m,x_{m+1},\ldots,x_n)\), \(1\leqslant m<n\).

Suppose, moreover, that for some nonnegative numbers \(\lambda_{m+1},\ldots,\lambda_n\), forming the system \((\vec\lambda)\), the inequalities

\[ r_j-\lambda_j-\frac1{p_j}>0\qquad (j=m+1,\ldots,n) \]

are satisfied. Then

\[ f^{(\lambda)}= \frac{\partial^{\lambda_{m+1}+\cdots+\lambda_n} f(x_1,\ldots,x_m,x_{m+1}^0,\ldots,x_n^0)} {\partial^{\lambda_{m+1}}x_{m+1}\cdots \partial^{\lambda_n}x_n} = \psi(\mathbf y,\mathbf z_0) \]

as a function of \((x_1,\ldots,x_m)\) belongs to the class
\(S_{\mathbf p}^{(\vec\rho)}H(R_m)\), where

\[ \vec\rho=(r_1,\ldots,r_m)=\mathbf r^{e_m},\qquad \mathbf y=(x_1,\ldots,x_m),\qquad \mathbf z_0=(x_{m+1}^0,\ldots,x_n^0), \]

and

\[ \|\psi\|_{S_{\mathbf p}^{(\vec\rho)}H(R_m)} \leqslant c\|\psi\|_{S_{\mathbf p}^{(\mathbf r)}H(R_n)}, \]

\[ \|\psi(\mathbf y,\mathbf z)-\psi(\mathbf y,\mathbf z_0)\|_{(p_1,\ldots,p_m)}\to 0 \quad \text{as }|\mathbf z-\mathbf z_0|\to 0, \]

where \(\mathbf z=(x_{m+1},\ldots,x_n)\), \(\psi(\mathbf y,\mathbf z)=f^{(\vec\lambda)}(\mathbf x)\).

If

\[ \sum_{m+1}^{n}\lambda_j=0, \]

then \(f^{(\vec\lambda)}\) is the function \(f\) itself.

Theorem 5. Let a system of functions be given

\[ \varphi_{\vec{(\lambda)}}(x_1,\ldots,x_m), \qquad \vec{(\lambda)}=(\lambda_{m+1},\ldots,\lambda_n), \]

where \(\lambda_i\) are nonnegative integers \((i=m+1,\ldots,n)\), belonging to the class \(S_{\vec{\rho}}^{(p)}H(R_m)\), \(\vec{\rho}=(r_1,\ldots,r_m)\), \(r_i>0\) \((i=1,\ldots,m)\).

Then one can construct a function \(f(x_1,\ldots,x_n)\) of \(n\) variables having the following properties:

a) \(f \in S_p^{(\mathbf r)}H(R_n)\), \(\mathbf r=(r_1,\ldots,r_n)\), where for \(r_j\) \((j \ge m+1)\) one may take arbitrarily large natural numbers, and, moreover,

\[ \|f\|_{S_p^{(\mathbf r)}H(R_n)} \le c \sum_{\vec{(\lambda)}} \|\varphi_{\vec{(\lambda)}}\|_{S_p^{(\vec{\rho})}H(R_m)}; \]

b)

\[ \frac{ \partial^{\lambda_{m+1}+\cdots+\lambda_n} f(x_1,\ldots,x_m,0,\ldots,0) }{ \partial^{\lambda_{m+1}}x_{m+1}\cdots \partial^{\lambda_n}x_n } = \varphi_{\vec{(\lambda)}}(x_1,\ldots,x_m). \]

The last equality is understood in the sense of convergence in norm (1).

If

\[ \sum_{m+1}^{n}\lambda_j=0, \]

then the left-hand side in condition b) is the function \(f\) itself.

Remark. In the proof of the theorems, an essential role was played by inequalities (11), (16), obtained by I. I. Ibragimov\({}^{4}\).

Blagoveshchensk Agricultural Institute

Received
13 V 1965

REFERENCES

\({}^{1}\) Ya. S. Bugrov, Embedding theorems for a class of functions with mixed norm, Reports of the III Siberian Conference on Mathematics and Mechanics, Tomsk, 1964, p. 54.
\({}^{2}\) Ya. S. Bugrov, DAN, 163, No. 4 (1965).
\({}^{3}\) S. M. Nikol’skii, Siberian Mathematical Journal, 4, No. 6, 1342 (1963).
\({}^{4}\) I. I. Ibragimov, Dokl. Azerb. SSR, 20, No. 4, 13 (1964).