Characterization of mixed modulus of smoothness in weighted $L^{p}$ spaces$^{\ast ,a}$

Ramazan Akgün$^{1,2,3}$

August 22nd, 2016.

$^1$This work has been supported by Balikesir University Scientific Research Project 2016/58.

$^2$ Results of this work were presented in the International Conference on Analysis and Its Applications (ICAA-2016), which held in 12-15 of July at the Ahi Evran University, Kirsehir, Turkey.

$^3$Department of Mathematics, Faculty of Arts and Sciences, University of Balikesir, Cagis Yerleskesi, 10145, Balikesir, Turkey, e-mail: rakgun@balikesir.edu.tr.

The paper is concerned with estimates for the mixed modulus of smoothness in Lebesgue spaces with Muckenhoupt weights, Steklov type averages.

MSC. 42A10, 41A17.

Keywords. Modulus of smoothness, Lebesgue spaces, Muckenhoupt weights, partial de la Vallée Poussin means, Fourier series.

1 Introduction

The main aim of this research is to investigate the approximation properties of some means of two dimensional Fourier series in Lebesgue spaces $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ with weights $\omega $ in the Muckenhoupt’s class $A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, where $\mathbb {J}$ is the set of rectangles in $\mathbb {T}^{2}:=\mathbb {T\times T}$, $\mathbb {T}:=[0,2\pi ]$ with sides parallel to coordinate axes. Trigonometric approximation by "angle" and mixed K-functional will be the main tools. We obtain the main properties of the weighted mixed modulus of smoothness $\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }$ in $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $1{\lt}p{\lt}\infty $. Note that, in general, in weighted spaces, such as $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) ,$ the classical translation operators are not bounded. Instead of classical translation operators we use Steklov type operators to define the weighted mixed modulus of smoothness $\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }$ in $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ (see [ 2 ] ). Starting from 70s, in the classical nonweighted Lebesgue spaces $L^{p}\left( \mathbb {T}^{2}\right) $ (defined on $\mathbb {T}^{2}$ or $\mathbb {T}^{d}$, $d\geq 1$), some problems related to the classical nonweighted mixed modulus of smoothness $\omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p}$ have been actively studied by mathematicians: M. K. Potapov [ 14 , 17 ] , [ 18 ] , [ 15 , 16 ] ; Potapov, Simonov, Lakovich [ 20 ] ; Potapov, Simonov, Tikhonov [ 22 ] , [ 19 , 21 ] ; A. F. Timan [ 27 ] ; M. F. Timan [ 28 , Chapter 2 ] . Among these problems we mention direct and inverse theorems of angular approximation [ 14 , 17 ] , [ 18 ] ; Hardy-Littlewood, Marcinkiewicz-Littlewood-Paley and embedding results [ 15 , 16 ] ; transformed Fourier series; embedding results of the Besov-Nikolski and Weyl-Nikolskii classes [ 19 , 21 ] , Ulyanov type inequalities [ 23 ] ; mixed $K$-functionals [ 6 ] , [ 25 ] ; fractional order classical mixed modulus of smoothness [ 24 ] .

In what follows, $A\lesssim B$ will mean that, there exists a positive constant $C_{u,v,\ldots }$, depending only on the parameters $u,v,\ldots $ and can be different in different places, such that the inequality $A\leq CB$ holds. If $A\lesssim B$ and $B\lesssim A$ we will write $A\approx B$.

It is well known that the main property of modulus of smoothness
$\Omega _{r}\left( \cdot ,\delta _{1},\delta _{2}\right) _{p,w}$ is that it decreases to zero as $\max \left\{ \delta _{1},\delta _{2}\right\} \rightarrow 0$. This rate can be characterized by some class $\Phi _{a_{1},a_{2}}$ defined below: the class $\Phi _{a_{1},a_{2}}$ $\left( a_{1},a_{2}\in \mathbb {R}\times \mathbb {R}\right) $ consists of functions $\psi \left( \cdot ,\cdot \right) $ satisfying conditions

$\psi \left( t_{1},t_{2}\right) \geq 0$ bounded on $(0,\infty )\times (0,\infty ),\qquad $
$\psi \left( t_{1},t_{2}\right) \rightarrow 0$ as $\max \left\{ t_{1},t_{2}\right\} \rightarrow 0,$
$\psi \left( t_{1},t_{2}\right) $ is non-decreasing in $t_{1}$ and $t_{2} $, (d) $t_{i}^{-a_{i}}\psi \left( t_{i}\right) $ is non-increasing in $t_{i}$ $\left( i=1,2\right) $.

We suppose that $\mathbb {J}$ is the set of rectangles in $\mathbb {T}^{2}$ with the sides parallel to coordinate axes. A function $\omega :\mathbb {T}^{2}\rightarrow \mathbb {R}^{\geq }:=[0,\infty )$ is called a weight on $\mathbb {T}^{2}$ if $\omega \left( x_{1},x_{2}\right) $ is measurable and positive almost everywhere on $\mathbb {T}^{2}$. We denote by $A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, ($1{\lt}p{\lt}\infty $) the collection of locally integrable weights $\omega :\mathbb {T}^{2}\rightarrow \mathbb {R}^{\geq }$ such that $\omega \left( x_{1},x_{2}\right) $ is $2\pi $-periodic with respect to each variable $x,y$ and

\begin{equation} C\text{:=}\underset {G\in \mathbb {J}}{\sup }\bigg( \tfrac {1}{\left\vert G\right\vert }\int \limits _{G}\omega \left( x_{1},x_{2}\right) dx_{1}dx_{2}\bigg) \bigg( \tfrac {1}{\left\vert G\right\vert }\int \limits _{G}\left[ \omega \left( x_{1},x_{2}\right) \right] ^{\frac{1}{1-p}}dx_{1}dx_{2}\bigg) ^{p-1}<\infty . \label{wei} \end{equation}

The least constant $C$ in (1) will be called the Muckenhoupt’s constant of $\omega $ and denoted by $\left[ \omega \right] _{A_{p}}$.

The main result of this work is the characterization of the modulus of smoothness, given in the following theorem.

Theorem 1

Let $r\in \mathbb {N},$ $p\in \left( 1,\infty \right) $ and $w\in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) .$

If $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, then there exists $\psi \in \Phi _{2r,2r}$ such that

\begin{equation} \Omega _{r}\left( f,t_{1},t_{2}\right) _{p,w}\approx \psi \left( t_{1},t_{2}\right) \label{ma1} \end{equation}
2

holds for all $t_{1},t_{2}\in \left( 0,\infty \right) \times \left( 0,\infty \right) $ with equivalence constants depending only on $r$ and $\left[ w\right] _{A_{p}}$.
If $\psi \in \Phi _{2r,2r}$ then there exist $f_{0}\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ and the positive real numbers $t_{0},t_{3}$ such that

\begin{equation} \Omega _{r}\left( f_{0},\delta _{1},\delta _{2}\right) _{p,w}\approx \psi \left( \delta _{1},\delta _{2}\right) \label{ma2} \end{equation}
3

holds for all $\delta _{1},\delta _{2}\in \left( 0,t_{0}\right) \times \left( 0,t_{3}\right) $ with equivalence constants depending only on $r$ and $\left[ w\right] _{A_{p}}.$

For functions in $L_{\omega }^{p}\left( \mathbb {T}\right) ,$ $p\in (1,\infty ),$ $\omega \in A_{p}\left( \mathbb {T}\right) $ Theorem 1 was obtained by the author in [ 1 ] . In this work we simplify the (long) proof given in [ 1 ] .

This type characterization theorem was proved in [ 26 ] (one dimensional case) for the spaces $L^{p}\left( \mathbb {T}\right) ,$ $p\in \lbrack 1,\infty )$, with classical moduli of smoothness of fractional order. The class $\Phi _{\varrho }$ describes completely the class of all majorants for the moduli of smoothness $\omega _{r}\left( \cdot ,\delta \right) _{p}$ in the space $L^{p}\left( \mathbb {T}\right) ,$ $p\in \lbrack 1,\infty )$. For $\omega _{r}\left( \cdot ,\delta \right) _{p}$, $r\in \mathbb {N}$ the characterization problem was investigated by O. V. Besov, S. B. Stechkin [ 4 ] , V. I. Kolyada [ 12 ] ; for $\omega _{r}\left( \cdot ,\delta \right) _{p}$, $r{\gt}0$ the characterization theorem was obtained by S. Tikhonov [ 26 ] .

2 Preliminaries

Let $L^{1}\left( \mathbb {T}^{2}\right) $ be the collection of Lebesgue integrable functions $f\left( x_{1},x_{2}\right) :\mathbb {T}^{2}\rightarrow \mathbb {R}$ such that $f\left( x_{1},x_{2}\right) $ is $2\pi $-periodic with respect to each variable $x_{1},x_{2}.$ Let $1{\lt}p{\lt}\infty ,$ $\omega \left( x_{1},x_{2}\right) \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, and let $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ be the collection of Lebesgue integrable functions $f\left( x_{1},x_{2}\right) :\mathbb {T}^{2}\rightarrow \mathbb {R}$ such that $f\left( x_{1},x_{2}\right) $ is $2\pi $-periodic with respect to each variable $x_{1},x_{2}$ and

\[ \left\Vert f\right\Vert _{p,\omega }:=\left( \iint \limits _{\mathbb {T}^{2}}\left\vert f\left( x_{1},x_{2}\right) \right\vert ^{p}\omega \left( x_{1},x_{2}\right) dx_{1}dx_{2}\right) ^{1/p}{\lt}\infty . \]

When $\omega \left( x_{1},x_{2}\right) \equiv 1$ we denote $\left\Vert f\right\Vert _{p,1}=:\left\Vert f\right\Vert _{p}$ and $L_{1}^{p}\left( \mathbb {T}^{2}\right) =:L^{p}\left( \mathbb {T}^{2}\right) $ for $1\leq p{\lt}\infty $; $L_{1}^{\infty }\left( \mathbb {T}^{2}\right) =:L^{\infty }\left( \mathbb {T}^{2}\right) $.

We define Steklov type averages by

\begin{align*} \sigma _{h_{1},h_{2}}f\left( x_{1},x_{2}\right) =& \tfrac {1}{4h_{1}h_{2}}\int _{x_{1}-h_{1}}^{x_{1}+h_{1}}\int _{x_{2}-h_{2}}^{x_{2}+h_{2}}f\left( t,\tau \right) dtd\tau . \\ \sigma _{h_{1},\circ }f\left( x_{1},x_{2}\right) =& \tfrac {1}{2h}\int _{x_{1}-h_{1}}^{x_{1}+h_{1}}f\left( t,\tau \right) dt,\\ \sigma _{\circ ,h_{2}}f\left( x_{1},x_{2}\right) =& \tfrac {1}{2k}\int _{x_{2}-h_{2}}^{x_{2}+h_{2}}f\left( t,\tau \right) d\tau . \end{align*}

Lemma 2

[ 8 , Theorem 3.3 ] , [ 2 ] If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, then

\begin{equation} \left\{ \left\Vert \sigma _{h_{1},h_{2}}f\right\Vert _{p,\omega },\left\Vert \sigma _{h_{1},\circ }f\right\Vert _{p,\omega },\left\Vert \sigma _{\circ ,h_{2}}f\right\Vert _{p,\omega }\right\} \lesssim \left\Vert f\right\Vert _{p,\omega }, \label{bo sigmaH} \end{equation}

uniformly in $h_{1},h_{2}$, where the constants depend only on $\left[ \omega \right] _{A_{p}}$ and $p$.

For $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, $h_{1},h_{2},$ $r\in \mathbb {N}$, we define the mixed differences by

\begin{eqnarray*} \bigtriangledown _{h_{1},\circ }^{r,\circ }f\left( x_{1},x_{2}\right) & =& \left( \mathbb {I}-\sigma _{h_{1},\circ }\right) ^{r}f\left( x_{1},x_{2}\right) , \\ \bigtriangledown _{\circ ,h_{2}}^{\circ ,r}f\left( x_{1},x_{2}\right) & =& \left( \mathbb {I}-\sigma _{\circ ,h_{2}}\right) ^{r}f\left( x_{1},x_{2}\right) , \\ \bigtriangledown _{h_{1},h_{2}}^{r,r}f\left( x_{1},x_{2}\right) & =& \bigtriangledown _{h_{1},\circ }^{r,\circ }\left( \bigtriangledown _{\circ ,h_{2}}^{\circ ,r}f\right) \left( x_{1},x_{2}\right) , \end{eqnarray*}

where $\mathbb {I}$ is identity operator on $\mathbb {T}^{2}$. Using the inequalities (4) we get

\begin{equation} \left\{ \left\Vert \bigtriangledown _{h,\circ }^{r,\circ }f\right\Vert _{p,\omega },\left\Vert \bigtriangledown _{\circ ,k}^{\circ ,r}f\right\Vert _{p,\omega },\left\Vert \bigtriangledown _{h,k}^{r,r}f\right\Vert _{p,\omega }\right\} \lesssim \left\Vert f\right\Vert _{p,\omega }, \label{del1} \end{equation}

for $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) ,$ $r\in \mathbb {N}$, with constants depending only on $\left[ \omega \right] _{A_{p}}$ and $p,r$.

The mixed modulus of smoothness of $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) ,$ $1{\lt}p{\lt}\infty ,$ $\omega \left( x,y\right) \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $r\in \left\{ 0\right\} \cup \mathbb {N}$, can be defined as

\begin{equation} \Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }=\left\{ \begin{tabular}{cc} $\underset {0\leq h_{2}\leq \delta _{2}}{\underset {0\leq h_{1}\leq \delta _{1}}{\sup }}\left\Vert \bigtriangledown _{h_{1},h_{2}}^{r,r}f\right\Vert _{p,\omega }$ & , $r\in \mathbb {N}$, \\ $\left\Vert f\right\Vert _{p,\omega }$ & , $r=0$. \end{tabular}\right. \label{c0} \end{equation}

If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, $r\in \mathbb {N}$, then from (6) and (5) $\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }\lesssim \left\Vert f\right\Vert _{p,\omega }$ with constant depending only on $\left[ \omega \right] _{A_{p}} $ and $p,r$.

Note that from the definition of $\Omega _{r}\left( f,\cdot ,\cdot \right) _{p,\omega }$, it has the following properties when $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, $r\in \mathbb {N}$:

$\Omega _{r}\left( f,0,0\right) _{p,\omega }=0.$
$\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }$ is subadditive with respect to $f$.
$\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }\leq \Omega _{r}\left( f,t_{1},t_{2}\right) _{p,\omega }$ for $0\leq \delta _{i}\leq t_{i};\quad i=1,2.$

When $\omega \left( x_{1},x_{2}\right) \equiv 1$ we denote $\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,1}=:\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p}$ for $1\leq p{\lt}\infty $; $\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{\infty ,1}=:\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{\infty }$.

Let $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, and $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) ,$ then there is $\lambda \in \left( 1,\infty \right) $ such that $f\in L^{\lambda }\left( \mathbb {T}^{2}\right) $, namely, we have $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) \subset L^{\lambda }\left( \mathbb {T}^{2}\right) $ and this gives possibility to define the corresponding Fourier series of $f$.

Lemma 3

[ 2 ] If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, and $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, then we have

\begin{equation} L^{\infty }\left( \mathbb {T}^{2}\right) \subset L_{\omega }^{p}\left( \mathbb {T}^{2}\right) \subset L^{\lambda }\left( \mathbb {T}^{2}\right) \label{Ap} \end{equation}

for some $\lambda {\gt}1$.

We define $\mathcal{T}_{m,n}$ as the set of all trigonometric polynomials of degree at most $m$ with respect to variable $x_{1}$ and of degree at most $n$ with respect to variable $x_{2}.$ Then

\begin{equation*} Y_{m_{1},m_{2}}(f)_{p,\omega }=\inf \left\{ \Big\Vert f-\textstyle \sum \limits _{i=1}^{2}T_{i}\Big\Vert _{p,\omega }:T_{i}\in \mathcal{T}_{m_{i}}\right\} , \end{equation*}

where $\mathcal{T}_{m_{i}}$ is the set of all two dimensional trigonometric polynomials of degree at most $m_{i}$ with respect to variable $x_{i}$ ($i=1,2$)

Let $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $ and $\sum \limits _{n_{1}=0}^{\infty }\sum \limits _{n_{2}=0}^{\infty }A_{n_{1},n_{2}}\left( x_{1},x_{2}\right) $ be the corresponding Fourier series for $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $. We define the partial sums of Fourier series of $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) ,$ $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $ as

\begin{eqnarray*} S_{m,\circ }\left( f\right) \left( x_{1},x_{2}\right) & =& \sum \limits _{n_{1}=0}^{m}\sum \limits _{n_{2}=0}^{\infty }A_{n_{1},n_{2}}\left( x_{1},x_{2},f\right) , \\ S_{\circ ,n}\left( f\right) \left( x_{1},x_{2}\right) & =& \sum \limits _{n_{1}=0}^{\infty }\sum \limits _{n_{2}=0}^{n}A_{n_{1},n_{2}}\left( x_{1},x_{2},f\right) , \\ S_{m_{1},m_{2}}\left( f\right) \left( x_{1},x_{2}\right) & =& \sum \limits _{n_{1}=0}^{m_{1}}\sum \limits _{n_{2}=0}^{m_{2}}A_{n_{1},n_{2}}\left( x_{1},x_{2},f\right) . \end{eqnarray*}

Define the partial de la Valleè Poussin means of $f$ as

\begin{align} V_{m,\circ }\left( f\right) \left( x_{1},x_{2}\right) =& \tfrac {1}{m+1}\sum \limits _{k=m}^{2m-1}S_{k,\circ }\left( f\right) , \label{u1} \\ V_{\circ ,n}\left( f\right) \left( x_{1},x_{2}\right) =& \tfrac {1}{n+1}\sum \limits _{l=n}^{2n-1}S_{\circ ,l}\left( f\right) , \\ V_{m_{1},m_{2}}\left( f\right) \left( x_{1},x_{2}\right) =& \tfrac {1}{\left( n+1\right) \left( m+1\right) }\sum \limits _{k=m_{1}}^{2m_{1}-1}\sum \limits _{l=m_{2}}^{2m_{2}-1}S_{k,l}\left( f\right) . \label{u2} \end{align}

Lemma 4

[ 2 ] If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, then

\begin{align*} \left\{ \left\Vert S_{m,\circ }\left( f\right) \right\Vert _{p,\omega },\left\Vert S_{\circ ,n}\left( f\right) \right\Vert _{p,\omega },\left\Vert S_{m_{1},m_{2}}\left( f\right) \right\Vert _{p,\omega }\right\} & \lesssim \left\Vert f\right\Vert _{p,\omega }, \\ \left\{ \left\Vert V_{m,\circ }\left( f\right) \right\Vert _{p,\omega },\left\Vert V_{\circ ,n}\left( f\right) \right\Vert _{p,\omega },\left\Vert V_{m_{1},m_{2}}\left( f\right) \right\Vert _{p,\omega }\right\} & \lesssim \left\Vert f\right\Vert _{p,\omega }, \\ \left\Vert f-W_{m_{1},m_{2}}f\right\Vert _{p,\omega }& \lesssim Y_{m_{1},m_{2}}\left( f\right) _{p,\omega } \end{align*}

where $W_{m_{1},m_{2}}f\left( x_{1},x_{2}\right) :=\left( V_{m_{1},\circ }\left( f\right) +V_{\circ ,m_{2}}\left( f\right) -V_{m_{1},m_{2}}\left( f\right) \right) \left( x_{1},x_{2}\right) $ with all constants depending only on $\left[ \omega \right] _{A_{p}}$ and $p$.

By Theorem 6 of [ 13 ]

\begin{equation} \left\Vert f-C_{m_{1},m_{2}}^{\alpha }f\right\Vert _{p,\omega }\rightarrow 0 \label{oar} \end{equation}

as $m_{1},m_{2}\rightarrow \infty $ where $C_{m_{1},m_{2}}^{\alpha }f$ is $\alpha $th Cesàro mean of $f$. From this we can deduce that $C\left( \mathbb {T}^{2}\right) $ is a dense subset of $L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ for $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $. Then $Y_{m_{1},m_{2}}\left( f\right) _{p,\omega }\lesssim \left\Vert f-C_{m_{1},m_{2}}^{\alpha }f\right\Vert _{p,\omega }\rightarrow 0$ and $Y_{m_{1},m_{2}}\left( f\right) _{p,\omega }\rightarrow 0 $ as $m_{1},m_{2}\rightarrow \infty $.

Let $W_{p,\omega }^{r,s}$, $r,s\in \mathbb {N}$, (respectively $W_{p,\omega }^{r,\circ };\quad W_{p,\omega }^{\circ ,s}$) denote the collection of functions $f\in L^{1}\left( \mathbb {T}^{d}\right) $ such that $f^{\left( r,s\right) }\in L_{\omega }^{p}\left( \mathbb {T}^{d}\right) $ (respectively $f^{\left( r,\circ \right) }\in L_{\omega }^{p}\left( \mathbb {T}^{d}\right) ;$ $f^{\left( \circ ,s\right) }\in L_{\omega }^{p}\left( \mathbb {T}^{d}\right) $).

The following inequalities can be obtained by the method given in [ 2 ] . For $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $r\in \mathbb {N}$, there exist constants depending only on $\left[ \omega \right] _{A_{p}}$ and $p,r$ so that

(i) (Jackson inequalities of Favard type)

\begin{align} Y_{m_{1},m_{2}}\left( g_{1}\right) _{p,\omega }& \lesssim \tfrac {1}{\left( m_{1}+1\right) ^{2r}}\left\Vert g_{1}^{\left( 2r,\circ \right) }\right\Vert _{p,\omega },\quad g_{1}\in W_{p,\omega }^{2r,\circ }, \label{fax} \\ Y_{m_{1},m_{2}}\left( g_{2}\right) _{p,\omega }& \lesssim \tfrac {1}{\left( m_{2}+1\right) ^{2r}}\left\Vert g_{2}^{\left( \circ ,2r\right) }\right\Vert _{p,\omega },\quad g_{2}\in W_{p,\omega }^{\circ ,2r}, \label{fay} \\ Y_{m_{1},m_{2}}\left( g\right) _{p,\omega }& \lesssim \tfrac {1}{\left( m_{1}+1\right) ^{2r}\left( m_{2}+1\right) ^{2r}}\left\Vert g^{\left( 2r,2r\right) }\right\Vert _{p,\omega },\quad g\in W_{p,\omega }^{2r,2r}. \label{faxy} \end{align}

(ii) if $\delta _{1},\delta _{2}{\gt}0$ then

\begin{eqnarray} \Omega _{r}\left( g_{1},\delta ,\cdot \right) _{p,\omega } & \lesssim & \delta ^{2}\Omega _{r-1}\left( g_{_{1}}^{\left( 2,\circ \right) },\delta ,\cdot \right) _{p,\omega },\quad g_{1}\in W_{p,\omega }^{2,\circ }, \label{inv1} \\ \Omega _{r}\left( g_{2},\cdot ,\xi \right) _{p,\omega } & \lesssim & \xi ^{2}\Omega _{r-1}\left( g_{_{2}}^{\left( \circ ,2\right) },\cdot ,\xi \right) _{p,\omega },\quad g_{2}\in W_{p,\omega }^{\circ ,2}, \label{inv2} \\ \Omega _{r}\left( g,\delta _{1},\delta _{2}\right) _{p,\omega } & \lesssim & \delta _{1}^{2}\delta _{2}^{2}\Omega _{r-1}\left( g^{\left( 2,2\right) },\delta _{1},\delta _{2}\right) _{p,\omega },\quad g\in W_{p,\omega }^{2,2}. \label{inv3} \end{eqnarray}

and hence

\begin{eqnarray*} \Omega _{r}\left( f,\delta ,\cdot \right) _{p,\omega } & \lesssim & \delta ^{2r}\left\Vert f^{\left( 2r,\circ \right) }\right\Vert _{p,\omega }, \\ \Omega _{r}\left( f,\cdot ,\xi \right) _{p,\omega }& \lesssim & \xi ^{2r}\left\Vert f^{\left( \circ ,2r\right) }\right\Vert _{p,\omega }, \\ \Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega } & \lesssim & \delta _{1}^{2r}\delta _{2}^{2r}\left\Vert f^{\left( 2r,2r\right) }\right\Vert _{p,\omega }\text{.} \end{eqnarray*}

Definition 5

The mixed $K$-functional is defined as

\begin{align*} & K(f,\delta _{1},\delta _{2},p,\omega ,r,s):= \\ & :=\underset {g_{1},g_{2},g}{\inf }\Big\{ \! \left\Vert f\! -\! g_{1}\! -\! g_{2}\! -\! g\right\Vert _{p,\omega }\! \! +\delta _{1}^{r}\left\Vert \tfrac {\partial ^{r}g_{1}}{\partial x^{r}}\right\Vert _{p,\omega } \! \! +\delta _{2}^{s}\left\Vert \tfrac {\partial ^{s}g_{2}}{\partial y^{s}}\right\Vert _{p,\omega }\! \! +\delta _{1}^{r}\delta _{2}^{s}\left\Vert \tfrac {\partial ^{r+s}g}{\partial x^{r}\partial y^{s}}\right\Vert _{p,\omega }\Big\} \end{align*}

where the infimum is taken for all $g_{1},$ $g_{2},$ $g$ so that $g_{1}\in W_{p,\omega }^{r,\circ },$ $g_{2}\in W_{p,\omega }^{\circ ,s},g\in W_{p,\omega }^{r,s}$ where $r,s\in \mathbb {N}$, $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $.

(iii) If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ and $r\in \mathbb {N}$, then there exist constants depending only on Muckenhoupt’s constant $\left[ \omega \right] _{A_{p}}$ of $\omega $ and $p,r$ so that the equivalence

\begin{equation} \Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }\approx K(f,\delta _{1},\delta _{2},p,\omega ,2r) \label{real} \end{equation}

and the properties

\begin{align*} \Omega _{r}\left( f,\lambda \delta _{1},\eta \delta _{2}\right) _{p,\omega }& \lesssim \left( 1+\lfloor \lambda \rfloor \right) ^{2r}\left( 1+\lfloor \eta \rfloor \right) ^{2r}\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }, \\ \frac{\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,\omega }}{\delta _{1}^{2r}\delta _{2}^{2r}}& \lesssim \frac{\Omega _{r}\left( f,t_{1},t_{2}\right) _{p,\omega }}{t_{1}^{2r}t_{2}^{2r}},\qquad 0{\lt}t_{i}\leq \delta _{i};\quad i=1,2, \end{align*}

hold for $\delta _{1},\delta _{2}{\gt}0$, where $\lfloor x\rfloor :=\max \left\{ z\in \mathbb {Z}:z\leq x\right\} .$

(iv) (11) can be refined by the inequality (19) below [ 2 ] . If $1{\lt}p{\lt}\infty ,$ $\omega \in A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $ and $r\in \mathbb {N}$, then there exists $C_{\left[ \omega \right] _{A_{p}},p,r}$ depending only on Muckenhoupt’s constant $\left[ \omega \right] _{A_{p}}$ of $\omega $ and $p,r$ so that

\begin{equation} Y_{m_{1},m_{2}}\left( f\right) _{p,\omega }\leq C_{\left[ \omega \right] _{A_{p}},p,r}\Omega _{r}\left( f,\tfrac {1}{m_{1}},\tfrac {1}{m_{2}}\right) _{p,\omega } \label{JT} \end{equation}

where $m_{1},m_{2}\in \mathbb {N}$.

3 Proof of Theorem 1

Let $\omega _{r}\left( \cdot ,\delta _{1},\delta _{2}\right) _{p}$, $1\leq p\leq \infty $, be the usual nonweighted mixed modulus of smoothness:

\begin{equation*} \omega _{r}\left( g,\delta _{1},\delta _{2}\right) _{p}:=\underset {0\leq h_{1}\leq \delta _{1},0\leq h_{2}\leq \delta _{2}}{\sup }\left\Vert \left( \mathbb {I}-T_{h_{1},\circ }\right) ^{r}\left( \mathbb {I}-T_{\circ ,h_{2}}\right) ^{r}g\right\Vert _{p}\text{,\quad }g\in L^{p}\left( \mathbb {T}^{2}\right) \text{,} \end{equation*}

where $T_{h_{1},\circ }g\left( x_{1},x_{2}\right) :=g(x_{1}+h_{1},x_{2})$; $T_{\circ ,h_{2}}g\left( x_{1},x_{2}\right) :=g(x_{1},x_{2}+h_{2})$. From [ 25 ] ($1\leq p{\lt}\infty $) and [ 6 ] ($p=\infty $) and (18) there exist positive constants, depending only $r,p$, such that

\begin{equation} \omega _{2r}\left( g,\delta _{1},\delta _{2}\right) _{p}\approx \Omega _{r}\left( g,\delta _{1},\delta _{2}\right) _{p} \label{gec} \end{equation}

holds for $1\leq p\leq \infty $ and $g\in L^{p}\left( \mathbb {T}^{2}\right) . $

Theorem 2.5 of [ 26 ] give that: Let $r\in \mathbb {N},$ $p\in \left[ 1,\infty \right] .$

(a) If $f\in L^{p}\left( \mathbb {T}^{2}\right) $, then there exists $\psi \in \Phi _{r,r}$ such that

\begin{equation} \omega _{r}\left( f,t_{1},t_{2}\right) _{p}\approx \psi \left( t_{1},t_{2}\right) \label{w1} \end{equation}

holds for all $t_{1},t_{2}\in \left( 0,\infty \right) \times \left( 0,\infty \right) $ with equivalence constants depending only on $r$.

(b) If $\psi \in \Phi _{r,r}$ then there exist $f_{0}\in L^{p}\left( \mathbb {T}^{2}\right) $ and the positive real numbers $t_{0},t_{3}$ such that

\begin{equation} \omega _{r}\left( f_{0},\delta _{1},\delta _{2}\right) _{p}\approx \psi \left( \delta _{1},\delta _{2}\right) \label{w2} \end{equation}

holds for all $\delta _{1},\delta _{2}\in \left( 0,t_{0}\right) \times \left( 0,t_{3}\right) $ with equivalence constants depending only on $r.$

Proof â–¼

[Proof of Theorem 1] (i) Note that if $F\in C\left( \mathbb {T}^{2}\right) $ then from (7)

\begin{equation} \left\Vert \bigtriangledown _{h_{1},h_{2}}^{r,r}F\right\Vert _{p,w}\leq C_{p, \left[ w\right] _{A_{p}}}\left\Vert \bigtriangledown _{h_{1},h_{2}}^{r,r}F\right\Vert _{C\left( \mathbb {T}^{2}\right) }. \label{emb} \end{equation}

Using Theorem 2.5 (A) of [ 26 ] , (7), (20), (18), (23) there exists $\psi \in \Phi _{2r}$ such that

\begin{eqnarray*} \Omega _{r}\left( F,\delta _{1},\delta _{2}\right) _{p,w} & \leq & C_{p,\left[ w\right] _{A_{p}}}\Omega _{r}\left( F,\delta _{1},\delta _{2}\right) _{\infty }\leq C_{p,\left[ w\right] _{A_{p}}}\omega _{2r}\left( F,\delta _{1},\delta _{2}\right) _{\infty } \\ & \leq & C_{r,p,\left[ w\right] _{A_{p}}}\psi \left( \delta _{1},\delta _{2}\right) . \end{eqnarray*}

If $p\in \left( 1,\infty \right) $, $A_{p}\left( \mathbb {T}^{2},\mathbb {J}\right) $, $f\in L_{\omega }^{p}\left( \mathbb {T}^{2}\right) $, then, by (11), for any $\varepsilon {\gt}0$ there exists $F\in C\left( \mathbb {T}^{2}\right) $ such that $\Vert f-F\Vert _{p,w}{\lt}\varepsilon $. Thus

\begin{eqnarray*} \Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,w} & \leq & \Omega _{r}\left( f-F,\delta _{1},\delta _{2}\right) _{p,w}+\Omega _{r}\left( F,\delta _{1},\delta _{2}\right) _{p,w} \\ & \leq & C_{r,p,\left[ w\right] _{A_{p}}}\Vert f-F\Vert _{p,w}+C_{r,p,\left[ w\right] _{A_{p}}}\psi \left( \delta _{1},\delta _{2}\right) . \end{eqnarray*}

Letting $\varepsilon \rightarrow 0^{+}$ we get

\begin{equation*} \Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,w}\leq C_{r,p,\left[ w\right] _{A_{p}}}\psi \left( \delta _{1},\delta _{2}\right) . \end{equation*}

On the other hand, from (18), (20), (7) and Theorem 2.5 (A) of [ 26 ]

\begin{equation*} \psi \left( \delta _{1},\delta _{2}\right) \leq C_{r,p,\left[ w\right] _{A_{p}}}\omega _{2r}\left( f,\delta _{1},\delta _{2}\right) _{1}\leq C_{r,p,\left[ w\right] _{A_{p}}}\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,w} \end{equation*}

and the equivalence (2) is established.

(ii) For the equivalence (3) let $\psi \in \Phi _{2r}.$ By Theorem 2.5 (B) and Remark 2.7 (1) of [ 26 ] there exist $f\in L^{\infty }\left( \mathbb {T}^{2}\right) $ and the positive real numbers $t_{0},t_{3}$ such that

\begin{equation*} \omega _{2r}\left( f,\delta _{1},\delta _{2}\right) _{p}\approx \psi \left( \delta _{1},\delta _{2}\right) ,\quad p=1,\infty \end{equation*}

holds for all $\delta _{1},\delta _{2}\in \left( 0,t_{0}\right) \times \left( 0,t_{3}\right) $ with equivalence constants depending only on $r$. Then by (18), (20) we get

\begin{eqnarray*} \psi \left( \delta _{1},\delta _{2}\right) & \leq & C_{r}\omega _{2r}\left( f,\delta _{1},\delta _{2}\right) _{1}\leq C_{r}\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{1}\leq C_{r,p,\left[ w\right] _{A_{p}}}\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{p,w} \\ & \leq & C_{r,p,\left[ w\right] _{A_{p}}}\Omega _{r}\left( f,\delta _{1},\delta _{2}\right) _{\infty }\leq C_{r,p,\left[ w\right] _{A_{p}}}\omega _{2r}\left( f,\delta _{1},\delta _{2}\right) _{\infty } \\ & \leq & C_{r,p,\left[ w\right] _{A_{p}}}\psi \left( \delta _{1},\delta _{2}\right) \end{eqnarray*}

for all $\delta _{1},\delta _{2}\in \left( 0,t_{0}\right) \times \left( 0,t_{3}\right) $.

Proof â–¼

Acknowledgements

The author wish to express his sincere gratitude to the referee(s) for his/her valuable suggestions.

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This work is licensed under a Creative Commons Attribution 4.0 International License.

Characterization of mixed modulus of smoothness in weighted \(L^{p}\) spaces\(^{\ast ,a}\)

1 Introduction

2 Preliminaries

3 Proof of Theorem 1

Bibliography