MATHEMATICS

A. S. DZHAFAROV

EMBEDDING THEOREMS WITH A WEIGHT

(Presented by Academician A. N. Kolmogorov on 28 VIII 1961)

The embedding theorems of S. L. Sobolev \((^{1,2})\), whose fundamental importance in the theory of boundary-value problems of mathematical physics is well known, have very many diverse generalizations, additions, and applications. Among them one should especially note the investigations of S. M. Nikol’skii \((^{3-6})\) on the embedding of the classes
\(H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}[M]\), who was the first to apply, for these purposes, methods of the theory of best approximation of functions of several variables. We also note that embedding theorems with a weight were first proved by L. D. Kudryavtsev \((^{7,8})\). Works \((^{9,10})\) are also devoted to these questions.

In the present paper we give results of the type of S. M. Nikol’skii’s embedding theorems for integral norms containing a general weight, whose character may differ from that of the weight functions used in the above-mentioned works. These results were obtained on the basis of the theorems of \((^{12})\), which relate the behavior of best approximations of functions of several variables in norms containing a weight, by means of entire functions of finite degrees, to the differential properties of the functions under consideration.

Below, in addition to the notation and definitions of \((^{12})\), we shall also use the following definitions. We shall say that a function \(f(x_1,\ldots,x_n)\) belongs to the class \(H_{p,\varphi,x_k}[\psi]\) if \(\|f\|_{p,\varphi}^{(n)}<\infty\) and the inequality
\(\|f(x_1,\ldots,x_{k-1},x_k+h,x_{k+1},\ldots,x_n)-2f(x_1,\ldots,x_n)+f(x_1,\ldots,x_{k-1},x_k-h,x_{k+1},\ldots,x_n)\|_{p,\varphi}^{(n)}\le \psi(|h|)\) holds for all \(h\) for which \(\psi(|h|)\) has meaning, where \(\psi(|h|)\) is a function tending to zero as \(h\to0\). In what follows we shall assume that \(f(x_1,\ldots,x_n)\) belongs to the class \(H_{p,\varphi,n}[\psi]\) if it simultaneously belongs to all the classes \(H_{p,\varphi,x_k}[\psi]\).

Let \(\ln_1 x=\ln x\), \(\ln_j x=\ln(\ln_{j-1}x)\) \((j=2,3,\ldots)\). We shall say that a function \(f(x_1,\ldots,x_n)\) belongs to the class
\(H_{p,\varphi,x_k}^{(r)}\left[M\prod_{j=1}^{N}\ln_j^{s_j}\right]\), where \(r=\bar r+\alpha>0\) (\(\bar r\) is a nonnegative integer, \(0<\alpha\le1\)), and \(s_j\) are arbitrary real numbers, if \(f\in L_{p,\varphi}^{(n)}\) has a generalized partial derivative \(\partial^{\bar r}f/\partial x_k^{\bar r}\) belonging to the class \(H_{p,\varphi,x_k}[\psi]\), where

\[ \psi(|h|)=M|h|^\alpha\prod_{j=1}^{N}\left(\ln_j\frac1{|h|}\right)^{s_j}, \]

and \(M\) does not depend on \(h\) (when \(s_j=0\), \(j=1,\ldots,N\), we shall simply say that \(f\in H_{p,\varphi,x_k}^{(r)}[M]\)).

Further we shall assume that \(f(x_1,\ldots,x_n)\) belongs to the class

\[ H_{p,\varphi}^{(r_1,\ldots,r_n)} \left[ M_1\prod_{j=1}^{N}\ln_j^{s_j^{(1)}},\ldots, M_n\prod_{j=1}^{N}\ln_j^{s_j^{(n)}} \right], \tag{1} \]

if it belongs simultaneously to the classes
\(H_{p,\varphi,x_k}^{(r_k)}\left[M_k\prod_{j=1}^{N}\ln_j^{s_j^{(k)}}\right]\)
\((k=1,2,\ldots,n)\). For \(\varphi\equiv1\), \(s_j^{(i)}=0\), \((i=1,2,\ldots,n;\ j=1,2,\ldots,N)\)

the class (1) coincides with the class \(H_p^{(r_1,\ldots,r_n)}[M_1,\ldots,M_n]\) of S. M. Nikol’skii. Let \(r_i>0,\ 1\le p_i\le \infty\) \((i=1,2,\ldots,n)\); we shall say that \(f(x_1,\ldots,x_n)\) belongs to the class

\[ H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}[M], \tag{2} \]

if, in each of the variables \(x_i\), it belongs to the class \(H_{p_i,\varphi,x_i}^{(r_i)}[M]\) \((i=1,\ldots,n)\). For \(\varphi\equiv1\) the class (2) was first introduced by S. M. Nikol’skii \({}^{(6)}\).

Theorem 1. Let \(1\le p\le p'\le \infty,\ r_i>0\) \((i=1,2,\ldots,n)\), \(1\le m\le n\),

\[ \varkappa_m = 1-\left(\frac1p-\frac1{p'}\right)\sum_{i=1}^{m}\frac1{r_i} -\frac1p\sum_{i=m+1}^{n}\frac1{r_i}, \]

\[ \theta_m^{(j)} = \left(\frac1p-\frac1{p'}\right)\sum_{i=1}^{m}\frac{s_i^{(j)}}{r_i} -\frac1p\sum_{i=m+1}^{n}\frac{s_i^{(j)}}{r_i}, \]

and let \(f\) belong to the class (1).

Then:

1) If \(\varkappa_m>0\), then, for fixed \(x_{m+1},\ldots,x_n\), the function \(f\), in the variables \(x_1,\ldots,x_m\), belongs to the class

\[ H_{p',\varphi}^{(\rho_1,\ldots,\rho_m)} \left[ M^*\prod_{j=1}^{N}\ln_j^{\,\varkappa_m s_j^{(1)}+\theta_m^{(j)}}, \ldots, M^*\prod_{j=1}^{N}\ln_j^{\,\varkappa_m s_j^{(m)}+\theta_m^{(j)}} \right], \]

where \(\rho_i=\varkappa_m r_i,\ M^*\le c_1\left(\|f\|_{p,\varphi}^{(n)}+\sum_{1}^{n}M_k\right)\), and \(C_1\) does not depend on \(f,\ M_k,\ x_{m+1},\ldots,x_n,\ h\).

2) If \(\varkappa_m=0\), but there exists a natural number \(l\) such that for \(j=1,2,\ldots,l-1\) \(\theta_m^{(j)}=-1\) and \(\theta_m^{(l)}<-1\), then, for fixed \(x_{m+1},\ldots,x_n\), the function \(f\), with respect to \(x_1,\ldots,x_m\), belongs to the class

\[ H_{p',\varphi,m} \left[ M^*\ln_l^{\,\theta_m^{(l)}+1} \prod_{j=l+1}^{N}\ln_j^{\,\theta_m^{(j)}} \right], \]

where \(M^*\) is defined above.

In both cases

\[ \|f\|_{p',\varphi}^{(m)} \le C_2\left(\|f\|_{p,\varphi}^{(n)}+\sum_{i=1}^{n}M_i\right), \]

where \(C_2\) does not depend on \(f,\ x_{m+1},\ldots,x_n\).

Theorem 2. Let the function \(f\) belong to the class (1), and let \(\lambda_1,\ldots,\lambda_n\) be nonnegative integers,

\[ \sigma=1-\sum_{k=1}^{n}\frac{\lambda_k}{r_k}, \qquad \Delta_j=\sum_{i=1}^{n}\frac{s_j^{(i)}\lambda_i}{r_i}, \]

\[ r_i'=\sigma r_i,\qquad \alpha_j^{(i)}=\sigma s_j^{(i)}+\Delta_j. \]

Then, if: 1) \(\sigma>0\), or else 2) \(\sigma=0\), but there exists a natural number \(l\) such that for \(j=1,2,\ldots,l-1\) \(\Delta_j=-1\) and \(\Delta_l<-1\), then on \(E_n\) there exists the partial derivative \(\partial^{\lambda_1+\cdots+\lambda_n}f/\partial x_1^{\lambda_1}\cdots\partial x_n^{\lambda_n}\), belonging in case 1) to the class

\[ H_{p,\varphi}^{(r_1',\ldots,r_n')} \left[ \overline M\prod_{j=1}^{N}\ln_j^{\alpha_j^{(1)}}, \ldots, \overline M\prod_{j=1}^{N}\ln_j^{\alpha_j^{(n)}} \right]. \]

and in case 2) to the class

\[ H_{p,\varphi,n}\left[\overline M \ln_l^{\Delta_l+1}\prod_{j=l+1}^N \ln_j^{\Delta_j}\right]. \]

In both cases

\[ M+\left\|\frac{\partial^{\lambda_1+\cdots+\lambda_n} f}{\partial x_1^{\lambda_1}\cdots \partial x_n^{\lambda_n}}\right\|_{p,\varphi}^{(n)} \leqslant C_3\left(\|f\|_{p,\varphi}^{(n)}+\sum_{k=1}^n M_k\right), \]

where \(C_3\) does not depend on \(f, M_k, h\).

Theorem 3. Let \(r_i>0;\ 1\leqslant p_i\leqslant q\leqslant\infty;\ n, m\) be natural numbers for which \(1\leqslant m\leqslant n\),

\[ \rho^{(i)}=\frac{r_i\varkappa}{\varkappa^{(i)}}>0\qquad (i=1,\ldots,n), \]

where

\[ \varkappa= \left| \begin{matrix} 1-\displaystyle\sum_{j=1}^n \frac{\frac1{p_j}-\frac1q}{r_j} & -\displaystyle\frac1q\sum_{j=1}^n \frac1{r_j} \\[1.2em] -\displaystyle\sum_{j=m+1}^n \frac{\frac1{p_j}-\frac1q}{r_j} & 1-\displaystyle\frac1q\sum_{j=m+1}^n \frac1{r_j} \end{matrix} \right|, \]

\[ \varkappa^{(i)}=1-\sum_{j=1}^n \frac{\frac1{p_j}-\frac1{p_i}}{r_j}\qquad (i=1,\ldots,n). \]

If the function \(f(x_1,\ldots,x_n)\) belongs to the class \(H_{p_1,\ldots,p_n;\varphi}^{(r_1,\ldots,r_n)}[M]\), then, for any fixed \(x_{m+1},\ldots,x_n\), the function \(f\), as a function of \(x_1,\ldots,x_m\), belongs to the class \(H_{q,\varphi}^{(\rho^{(1)},\ldots,\rho^{(m)})}[\overline M]\), and the inequality

\[ \|f\|_{q,\varphi}^{(m)}+\overline M < C_4\left(\min_{1\leqslant i\leqslant n}\|f\|_{p_i,\varphi}^{(n)}+M\right) \]

holds, where the constant \(C_4\) does not depend on \(f, M, x_{m+1},\ldots,x_n\).

Remark. The theorems stated above, for \(\varphi\equiv 1,\ s_j^{(i)}=0\ (i=1,\ldots,n;\ j=1,\ldots,N)\), coincide completely with the corresponding theorems of S. M. Nikol’skii \((^{3\text{–}6})\). For \(\varphi\equiv 1\), Theorems 1 and 2 were obtained earlier in \((^{11})\).

Let us note that if, by \(B_{p,\theta,\varphi}^{(r_1,\ldots,r_n)}\), we denote the space of functions \(f\in L_{p,\varphi}^{(n)}\) having on \(E_n\) partial generalized unmixed derivatives in the sense of S. L. Sobolev
\[ \partial^k f/\partial x_i^k\in L_{p,\varphi}^{(n)}\quad (k=0,1,\ldots,r_i;\ i=1,2,\ldots,n) \]
with norm

\[ \|f\|_{B_{p,\theta,\varphi}^{(r_1,\ldots,r_n)}}= \|f\|_{p,\varphi}^{(n)} + \sum_{i=1}^n \left\{ \int_0^1 \frac{ \omega_{1+[\alpha_i],x_i}^{\theta} \left(t;\frac{\partial^{r_i}f}{\partial x_i^{r_i}}\right)_{p,\varphi}^{(n)} }{ t^{\theta\alpha_i+1} }\,dt \right\}^{1/\theta} <\infty, \]

where \(r_i=\bar r_i+\alpha_i>0\), \(\bar r_i\) are nonnegative integers, \(0<\alpha_i\leqslant 1\), then

Theorems 1–3 of O. V. Besov\(^{13}\) remain valid if, in their formulations, \(B_{p,\theta}^{(r_1,\ldots,r_n)}\) is replaced in the corresponding way by \(B_{p,\theta,\varphi}^{(r_1,\ldots,r_n)}\).

For \(\varphi \equiv 1\), the space \(B_{p,\theta,\varphi}^{(r_1,\ldots,r_n)}\) coincides with O. V. Besov’s space \(B_{p,\theta}^{(r_1,\ldots,r_n)}\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
21 VII 1961

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