MATHEMATICS

P. PILIKA

A SUPPLEMENT TO S. M. NIKOLSKII’S EMBEDDING THEOREM FOR THE CLASS \(H^{*(r_1,\ldots,r_n)}_{p_1,\ldots,p_n}\) AND THE IMPOSSIBILITY OF IMPROVING ONE ESTIMATE

(Presented by Academician I. M. Vinogradov, 7 VII 1960)

In this note an embedding theorem is proved for the classes \(H^{*(r_1,\ldots,r_n)}_{p_1,\ldots,p_n}\) by S. M. Nikolskii’s method. In addition, it is proved that one of his estimates for the classes \(H^{(r_1,\ldots,r_n)}_p\) cannot be improved.

S. M. Nikolskii proved in \((^6)\) an embedding theorem for the class \(H^{(r_1,\ldots,r_n)}_{p_1,\ldots,p_n}\) in the case when \(G=R_n\) is the whole \(n\)-dimensional space and when \(p_i\leq p\) \((i=1,2,\ldots,n)\). We consider, by the same methods, the more general case when

\[ 1\leq p_1\leq p_2\leq\cdots\leq p_l\leq p\leq p_{l+1}\leq\cdots\leq p_n\leq\infty \qquad (l=1,2,\ldots,n) \]

for the class \(H^{*(r_1,\ldots,r_n)}_{p_1,\ldots,p_n}\) (periodic case).

The proof in this case is again carried out on the basis of approximations of a function \(f\) by entire functions of finite degree. In doing so we use the generalized Jackson theorem, proved by S. M. Nikolskii \((^6)\) as applied to the classes \(H^{*(r_1,\ldots,r_n)}_{p_1,\ldots,p_n}\). In addition, two inequalities for trigonometric polynomials \(T=T_{\nu_1,\ldots,\nu_n}\) of orders \(\nu_1,\ldots,\nu_n\), respectively in the variables \(x_1,\ldots,x_n\), are an essential tool in the proof:

\[ \|T\|_{L_{p'}^{(n)}}\leq 2^n\left(\prod_1^n \nu_i\right)^{1/p-1/p'}\|T\|_{L_p^{(n)}} \qquad (1\leq p<p'\leq\infty), \tag{1} \]

\[ \|T\|_{L_p^{(m)}}\leq 2^n\left(\prod_{m+1}^n \nu_i\right)^{1/p}\|T\|_{L_p^{(n)}} \qquad (1\leq p\leq\infty;\;1\leq m\leq n), \tag{2} \]

where

\[ \|f\|_{L_p^{(m)}}= \left( \int_0^{2\pi}\cdots\int_0^{2\pi} |f(x_1,\ldots,x_m,x_{m+1},\ldots,x_n)|^p \,dx_1\cdots dx_m \right)^{1/p}. \]

These inequalities were obtained in the works of S. M. Nikolskii \((^3,^4)\). Inequality (1), for \(n=1\) and \(p'=\infty\), becomes an inequality obtained by Jackson \((^1)\).

In addition, the following lemma is used, generalizing the inverse approximation theorem of Bernstein:

Lemma 1. Let \(r>0\), \(f\in L_p^{(n)}\), and suppose that for some sequence \(T_{\nu,\infty,\ldots,\infty}\), for all \(\nu\) running through the geometric progression \(\nu=a^s\) \((s=0,1,2,\ldots;\;a>1)\), the inequality

\[ \|f-T_{\nu,\infty,\ldots,\infty}\|_{L_p^{(n)}}<\frac{K}{\nu^r} \tag{3} \]

holds.

Then

\[ f \in H_{p x_1}^{*(r)}(M), \]

where

\[ M < C\left(\|f\|_{L_p^{(n)}}+K\right), \tag{4} \]

where the constant \(C\) does not depend on the multiplier standing next to it.

For the proof see paper \((^4)\), Theorem 8 (the formulation and the remark on p. 264 at the end of the proof of the theorem).

We have proved the following theorem:

Theorem 1. Let, for the numbers considered below, the inequalities hold:

\[ r_i>0; \]

\[ 1 \leq p_1 \leq \cdots \leq p_l \leq p \leq p_{l+1} \leq \cdots \leq p_n \leq \infty; \]

\(n, m, l\) are natural numbers, and

\[ 1 \leq m \leq n,\qquad 1 \leq l \leq n. \]

Moreover:

a) when \(l \leq m \leq n\)

\[ \rho_i= \frac{ r_i\left[1-\sum_{1}^{l}\left(\frac{1}{p_d}-\frac{1}{p}\right)\frac{1}{r_d}\right] \left[1-\frac{1}{p}\sum_{m+1}^{n}\frac{1}{r_d}\right] }{ 1+\left(\frac{1}{p_i}-\frac{1}{p}\right)\sum_{1}^{n}\frac{1}{r_d} -\sum_{1}^{l}\left(\frac{1}{p_d}-\frac{1}{p}\right)\frac{1}{r_d} } >0,\quad i=1,2,\ldots,l; \tag{5} \]

\[ \rho_i=r_i\left(1-\frac{1}{p}\sum_{m+1}^{n}\frac{1}{r_d}\right)>0, \qquad i=l+1,\ldots,n; \]

b) when \(m<l\leq n\)

\[ \rho_i=r_i \frac{ \left(1-\frac{1}{p}\sum_{l+1}^{n}\frac{1}{r_d}\right) \left[1-\sum_{1}^{m}\left(\frac{1}{p_d}-\frac{1}{p}\right)\frac{1}{r_d}\right] }{ 1-\sum_{1}^{l}\frac{1}{r_d}\left(\frac{1}{p_d}-\frac{1}{p}\right) +\left(\frac{1}{p_i}-\frac{1}{p}\right)\sum_{l+1}^{n}\frac{1}{r_d} } - \]

\[ -r_i \frac{ \left(1+\frac{1}{p}\sum_{1}^{m}\frac{1}{r_d}\right) \sum_{m+1}^{l}\frac{1}{p_d r_d} -\frac{1}{p}\sum_{1}^{m}\frac{1}{p_d r_d}\sum_{m+1}^{l}\frac{1}{r_d} }{ 1-\sum_{1}^{l}\frac{1}{r_d}\left(\frac{1}{p_d}-\frac{1}{p}\right) +\left(\frac{1}{p_i}-\frac{1}{p}\right)\sum_{l+1}^{n}\frac{1}{r_d} } >0,\quad i=1,2,\ldots,l; \tag{6} \]

\[ \rho_i=r_i \frac{ \left(1-\frac{1}{p}\sum_{l+1}^{n}\frac{1}{r_d}\right) \left[1-\sum_{1}^{m}\left(\frac{1}{r_d}-\frac{1}{p}\right)\frac{1}{r_d}\right] }{ 1-\sum_{1}^{l}\frac{1}{r_d}\left(\frac{1}{p_d}-\frac{1}{p}\right) } - \]

\[ -r_i \frac{ \left(1+\frac{1}{p}\sum_{1}^{m}\frac{1}{r_d}\right) \sum_{m+1}^{l}\frac{1}{p_d r_d} -\frac{1}{p}\sum_{1}^{m}\frac{1}{p_d r_d}\sum_{m+1}^{l}\frac{1}{r_d} }{ 1-\sum_{1}^{l}\frac{1}{r_d}\left(\frac{1}{p_d}-\frac{1}{p}\right) } >0,\quad i=l+1,\ldots,n. \]

Let, further, a function \(f(x_1,\ldots,x_n)\), defined in the \(n\)-dimensional space \(R_n\), belong to the class

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

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 \(H_p^{*_{(\rho_1,\ldots,\rho_m)}}(\overline M)\). Moreover, the inequality

\[ \|f\|_{L_p^{(m)}}+\overline M < C\left(\min_{1\le i\le n}\|f\|_{L_{p_i}^{(n)}}+M\right), \tag{7} \]

holds, where the constant \(C\) does not depend on the set standing next to it.

Remark. For \(l=n\) one obtains the theorem of S. M. Nikol’skii \({}^{(6)}\) for the class \(H_{p_1,\ldots,p_n}^{*(r_1,\ldots,r_n)}(M)\).

A function \(f\) belonging to \(L_p^{(n)}\) is defined in \(R_n\) up to a set of measure zero; therefore, in order to be able to speak of its values on hyperplanes of smaller dimension, the following definition is introduced \({}^{(5,8)}\): a function \(\varphi(x_1,\ldots,x_m)\), where \(0<m<n\), is called the value of the function \(f\) in the sense of convergence in the mean (for fixed \(p\), \(1\le p\le \infty\)) on the \(m\)-dimensional hyperplane \(x_{m+1}=x_{m+1}^{(0)},\ldots,x_n=x_n^{(0)}\), if the function \(f\) can be modified on a set of measure zero so that the limit

\[ \lim_{\sum_{i=m+1}^{n} h_i^2\to 0} \left\{ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left|f(x_1,\ldots,x_m,x_{m+1}^{(0)}+h_{m+1},\ldots,x_n^{(0)}+h_n) -\varphi(x_1,\ldots,x_m)\right|^p \,dx_1\cdots dx_m \right\}^{1/p}=0 \tag{8} \]

exists.

The boundary value of the function \(f\) defined in this way is unique up to the class of equivalent functions with respect to \(m\)-dimensional measure; we shall denote it by

\[ f(x_1,\ldots,x_m,x_{m+1}^{(0)},\ldots,x_n^{(0)}). \]

S. M. Nikol’skii found the order with which the convergence to zero in (8) takes place (see \({}^{(2)}\)). We formulate this result here in the case when \(m=n-1\).

Let \(f\in H_p^{(r_1,\ldots,r_n)}(M)\), \(\beta_1=r_1-1/p\), where \(0<\beta_1\le 1\), and let some values of \(x_1,h\) be fixed. Then there exist constants \(C_1>0\) and \(C_2>0\), independent of \(M,f,x_1,h\), such that:

for \(\beta_1<1\),

\[ \|\Delta_{x_1}^{(1)}(f;h)\|_{L_p^{(m)}}= \left\{ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left|f(x_1+h,x_2,\ldots,x_n)-f(x_1,\ldots,x_n)\right|^p \,dx_2\cdots dx_n \right\}^{1/p} \le (C_1M+C_2\|f\|_{L_p^{(n)}})|h|^{\beta_1}; \tag{9} \]

for \(\beta_1=1\),

\[ \|\Delta_{x_1}^{(2)}(f;h)\|_{L_p^{(m)}}= \]

\[ = \left\{ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \left|f(x_1+h,x_2,\ldots,x_n)-2f(x_1,\ldots,x_n)+f(x_1-h,x_2,\ldots,x_n)\right|^p \,dx_2\cdots dx_n \right\}^{1/p} \le (C_1M+C_2\|f\|_{L_p^{(n)}})|h|. \tag{10} \]

We have proved that this estimate cannot be improved. In doing so we used a lemma proved by us in (7):

Lemma 2. The function

\[ F_N(x)=\left(\frac{\sin \tfrac12 Nx}{x}\right)^2 \]

has the following properties:

\[ |F_N^{(k)}(x)|>CN^{k+3}x,\qquad k=0,1,2,\ldots, \tag{11} \]

for \(0<Nx\le \pi/5\) \((N\ge 1)\) and

\[ \left\|F_N^{(k)}(x)\right\|_{L_p^{(1)}}=K_p N^{2-1/p},\qquad p\ge 1. \]

Our theorem is formulated as follows:

Theorem 2. The function

\[ f(x_1,\ldots,x_n)=\sum_{\nu=1}^{\infty} \frac{ \displaystyle \prod_{j=1}^{n} F_{\frac{a^{r_1\ldots r_n}}{r_j}}\,(x_j) }{ \displaystyle r_1\ldots r_n\sqrt[\;a\;]{\,1+\left(1+\frac1q\right)\sum_{i=1}^{n}\frac1{r_i}\,} }, \tag{12} \]

where convergence of the series is understood in the sense of \(L_p^{(n)}\), and where

\[ F_N(x)=\left(\frac{\sin \tfrac12 x}{x}\right)^2,\qquad 1\le p\le \infty,\qquad \frac1p+\frac1q=1, \]

for any \(a>1\) belongs to the class \(H_p^{(r_1,\ldots,r_n)}\) (9).

Moreover, for sufficiently large \(a>1\) one can find for it a sequence \(h\to 0\) \((h>0)\) such that, for some number \(K>0\) independent of \(h\), and for \(x_1=0\), the inequalities hold:

\[ \left\|\Delta_{x_1=0}^{(1)}(f;h)\right\|_{L_p^{(n-1)}}>Kh^{\beta_1} \quad\text{for }0<\beta_1<1; \tag{13} \]

\[ \left\|\Delta_{x_1=0}^{2}(f;h)\right\|_{L_p^{(n-1)}}>Kh \quad\text{for }\beta_1=1. \tag{14} \]

This theorem proves that the above-formulated theorem of S. M. Nikol’skii cannot be improved.

Tirana State University
Tirana, Albania

Received
29 VI 1960

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