A Study of Weighted Polynomial Approximations with Several Variables (I)
Abstract: In this paper, we investigate the weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomials. Then we will estimate the degree of approximation.

1. Introduction

Let ${ℝ}^{s}:=ℝ×ℝ×\cdots ×ℝ$ ( $s$ times, $s\ge 1$ integer) be the direct product space, and let $W\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right):={w}_{1}\left({x}_{1}\right){w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)$ , where ${w}_{i}\left({x}_{i}\right)\ge 0$ are even weight functions. We suppose that for every nonnegative integer n,

${\int }_{0}^{\infty }{x}^{n}{w}_{i}\left(x\right)\text{d}x<\infty ,n=0,1,2,\cdots ,i=1,2,\cdots ,s.$

In this paper, we will study to approximate the real-valued weighted function $\left(Wf\right)\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)$ by weighted polynomials $\left(WP\right)\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)$ , where

$P\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)\in {\mathcal{P}}_{n,n,\cdots ,n}\left({ℝ}^{s}\right)$ . Here, ${\mathcal{P}}_{n,n,\cdots ,n}\left({ℝ}^{s}\right)\left(=:{\mathcal{P}}_{n;s}\left({ℝ}^{s}\right)\right)$ means a class of

all polynomials with at most n-degree for each variable ${x}_{i},i=1,2,\cdots ,s$ . We need to define the norms. Let $0 , and let $f:{ℝ}^{s}\to ℝ$ be measurable. Then we define

${‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}:=\left\{\begin{array}{l}{\left[{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{|\left(Wf\right)\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right]}^{1/p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}0

We assume that for $0 the integral is independent of the order of integration with respect to each ${x}_{i},i=1,2,\cdots ,s$ . When ${‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}<\infty$ , we write $Wf\in {L}^{p}\left({ℝ}^{s}\right)$ . If $p=\infty$ , we require that $f$ is continuous and

$li{m}_{|X|\to \infty }\left(Wf\right)\left(X\right)=0$ , where $|X|=|\left({x}_{1},\cdots ,{x}_{s}\right)|=\mathrm{max}|{x}_{i}|;i=1,2,\cdots ,s$ .

Our purpose in this paper is to approximate the weighted function $Wf\in {L}^{p}\left({ℝ}^{s}\right)$ by weighted polynomials $WP;P\in {\mathcal{P}}_{n;s}\left({ℝ}^{s}\right)$ . The paper is arranged as the following. In Section 2, we give the definition of the weights which are treated in this paper. In Section 3, we consider the approximation for the functions in ${L}^{p}\left({ℝ}^{s}\right)$ . In Section 4, we consider a property of higher order derivatives. In Section 5, we estimate the degree of approximations. In Section 6, we consider the approximation for the functions with bounded variation. In Section 7, we consider the approximation of the Lipschitz-type functions. In Section 8, we treat the functions with higher order derivatives.

2. Class of Weight Functions and Preliminaries

Throughout the paper $C,{C}_{1},{C}_{2},\cdots$ denote positive constants independent of $n,x,t$ or polynomials $P\left(x\right)$ . The same symbol does not necessarily denote the same constant in different occurrences. Let $f\left(x\right)~g\left(x\right)$ mean that there exists a constant $C>0$ such that ${C}^{-1}f\left(x\right)\le g\left(x\right)\le Cf\left(x\right)$ holds for all $x\in I$ , where $I\subset ℝ$ is a subset.

We say that $f:ℝ\to \left[0,\infty \right)$ is quasi-increasing if there exists $C>0$ such that $f\left(x\right)\le Cf\left(y\right)$ for $0 . Hereafter we consider following weights.

Definition 2.1. Let $Q:ℝ\to \left[0,\infty \right)$ be a continuous and even function, and satisfy the following properties:

(a) ${Q}^{\prime }\left(x\right)$ is continuous in $ℝ$ , with $Q\left(0\right)=0$ .

(b) ${Q}^{″}\left(x\right)$ exists and is positive in $ℝ\\left\{0\right\}$ .

(c) ${\mathrm{lim}}_{x\to \infty }Q\left(x\right)=\infty .$

(d) The function

$T\left(x\right):=\frac{x{Q}^{\prime }\left(x\right)}{Q\left(x\right)},x\ne 0$

is quasi-increasing in $\left(0,\infty \right)$ , with

$T\left(x\right)\ge \Lambda >1,x\in ℝ\\left\{0\right\}.$

(e) There exists ${C}_{1}>0$ such that

$\frac{{Q}^{″}\left(x\right)}{|{Q}^{\prime }\left(x\right)|}\le {C}_{1}\frac{|{Q}^{\prime }\left(x\right)|}{Q\left(x\right)},a.e.x\in ℝ.$

Then we write $w=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}\right)$ .

Moreover, if there also exists a compact subinterval $J\left(\ni 0\right)$ of $ℝ$ , and ${C}_{2}>0$ such that

$\frac{{Q}^{″}\left(x\right)}{|{Q}^{\prime }\left(x\right)|}\ge {C}_{2}\frac{|{Q}^{\prime }\left(x\right)|}{Q\left(x\right)},a.e.x\in ℝ\J,$

then we write $w=\mathrm{exp}\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ . If $T\left(x\right)$ is bounded, then the weight $w=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ is called a Freud-type weight, and if $T\left(x\right)$ is unbounded, then w is called an Erdös-type weight.

For $w\left(x\right)=exp\left(-Q\left(x\right)\right)\in \mathcal{F}\left({C}^{2}+\right),Q\in {C}^{3}\left(ℝ\\left\{0\right\}\right)$ , if there exists $K>0$ such that for $|x|\ge K$ ,

$|\frac{{Q}^{‴}\left(x\right)}{{Q}^{″}\left(x\right)}|\le C|\frac{{Q}^{″}\left(x\right)}{{Q}^{\prime }\left(x\right)}|,$ (2.1)

and there exist $\lambda ,C>0$ such that for $0<\lambda <\frac{3}{2}$ ,

$\frac{|{Q}^{\prime }\left(x\right)|}{Q{\left(x\right)}^{\lambda }}\le C,$ (2.2)

then we write $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)$ . Furthermore, if

$|\frac{{Q}^{\left(4\right)}\left(x\right)}{{Q}^{\left(3\right)}\left(x\right)}|\le C|\frac{{Q}^{‴}\left(x\right)}{{Q}^{″}\left(x\right)}|~|\frac{{Q}^{″}\left(x\right)}{{Q}^{\prime }\left(x\right)}|$ (2.3)

and the inequality (2.2) with $0<\lambda <\frac{4}{3}$ hold, then we write $w\in {\mathcal{F}}_{\lambda }\left({C}^{4}+\right)$ .

We have some examples satisfying Definition 2.1.

Example 2.2 (cf.   ). (1) If an exponential $Q\left(x\right)$ satisfies

$1<{\Lambda }_{1}\le \frac{{\left(x{Q}^{\prime }\left(x\right)\right)}^{\prime }}{{Q}^{\prime }\left(x\right)}\le {\Lambda }_{2},$

where ${\Lambda }_{i},i=1,2$ are constants, then we call $w=\mathrm{exp}\left(-Q\left(x\right)\right)$ the Freud weight. The class $\mathcal{F}\left({C}^{2}+\right)$ contains the Freud weights.

(2) For $\alpha >1,l\ge 1$ we define

$Q\left(x\right)={Q}_{l;\alpha }\left(x\right)={\mathrm{exp}}_{l}\left({|x|}^{\alpha }\right)-{\mathrm{exp}}_{l}\left(0\right),$

where ${\mathrm{exp}}_{l}\left(x\right)=\mathrm{exp}\left(\mathrm{exp}\left(\mathrm{exp}\cdots \mathrm{exp}x\right)\cdots \right)\left(l\text{times}\right)$ . Moreover, we define

${Q}_{l;\alpha ,m}\left(x\right)={|x|}^{m}\left\{{\mathrm{exp}}_{l}\left({|x|}^{\alpha }\right)-{\alpha }^{*}{\mathrm{exp}}_{l}\left(0\right)\right\},\alpha +m>1,m\ge 0,\alpha \ge 0,$

where ${\alpha }^{*}=0$ if $\alpha =0$ , and otherwise ${\alpha }^{*}=1$ . We note that ${Q}_{l;0,m}$ gives a Freud-type weight, that is, $T\left(x\right)$ is bounded..

(3) We define

${Q}_{\alpha }\left(x\right)={\left(1+|x|\right)}^{{|x|}^{\alpha }}-1,\alpha >1.$

(4) Let $w=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ , and let us define

${\mu }_{+}:=\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{{Q}^{″}\left(x\right)}{{Q}^{\prime }\left(x\right)}/\frac{{Q}^{\prime }\left(x\right)}{Q\left(x\right)},{\mu }_{-}:=\underset{x\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{{Q}^{″}\left(x\right)}{{Q}^{\prime }\left(x\right)}/\frac{{Q}^{\prime }\left(x\right)}{Q\left(x\right)}.$

If ${\mu }_{+}={\mu }_{-}$ , then we say that the weight w is regular. All weights in examples (1), (2) and (3) are regular.

(5) More generally we can give the examples of weights $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)$ . If the weight w is regular and if $Q\in {C}^{3}\left(ℝ\\left\{0\right\}\right)$ satisfies (2.1), then for the regular weights we have $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)$ (see  , Corollary 5.5 (5.8)).

The following fact is very important for our study.

Proposition 2.3 (  , Theorem 4.1 and (4.11)). Let $0<\lambda <3/2$ and $\alpha \in ℝ$ . Then for $w=exp\left(-Q\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)$ , we can construct a new weight

${w}_{\alpha }\in \mathcal{F}\left({C}^{2}+\right)$ such that

${T}_{w}{\left(x\right)}^{\alpha }w\left(x\right)~{w}_{\alpha }\left(x\right)\text{on}ℝ,$

and for some $C\ge 1$ ,

${a}_{n/C}\left({w}_{\alpha }\right)\le {a}_{n}\left(w\right)\le {a}_{Cn}\left({w}_{\alpha }\right)\text{and}{T}_{{w}_{\alpha }}\left(x\right)~{T}_{w}\left(x\right)=T\left(x\right),$

where ${a}_{n}\left({w}_{\alpha }\right)$ and ${a}_{n}\left(w\right)$ are MRS-numbers for the weight ${w}_{\alpha }$ and $w$ , respectively, and ${T}_{{w}_{\alpha }}$ and ${T}_{w}$ are correspond for ${w}_{\alpha }$ or $w$ , respectively.

Let $\left\{{p}_{n}\right\}$ be orthonormal polynomials with respect to a weight w, that is, ${p}_{n}$ is the polynomial of degree n such that

${\int }_{-\infty }^{\infty }{p}_{n}\left(x\right){p}_{m}\left(x\right){w}^{2}\left(x\right)\text{d}x={\delta }_{mn}\left(\text{theKroneckerdelta}\right).$

For $1\le p\le \infty$ , we denote by ${L}^{p}\left(ℝ\right)$ the usual ${L}^{p}$ space on $ℝ$ (here for $p=\infty$ , if $wf\in {L}^{\infty }\left(ℝ\right)$ , then we require $f$ to be continuous, and $wf$ to have limit 0 at $±\infty$ ). For $wf\in {L}^{p}\left(ℝ\right)$ , we set

$\begin{array}{l}{s}_{n}\left(f,x\right):=\underset{k=0}{\overset{n-1}{\sum }}\text{ }\text{ }{b}_{k}\left(f\right){p}_{k}\left(x\right),\\ \text{where}{b}_{k}\left(f\right)={\int }_{-\infty }^{\infty }\text{ }f\left(t\right){p}_{k}\left(t\right){w}^{2}\left(t\right)\text{d}t\end{array}$ (2.4)

for $n\in ℕ$ (the partial sum of Fourier-type series). The de la Vallée Poussin mean of order n is defined by

${v}_{n}\left(f,x\right):=\underset{j=n+1}{\overset{2n}{\sum }}{s}_{j}\left(f,x\right).$ (2.5)

Let $w\in \mathcal{F}\left({C}^{2}+\right)$ . We need the Mhaskar-Rakhmanov-Saff numbers (MRS-numbers) ${a}_{x}$ ;

$x=\frac{2}{\text{π}}{\int }_{0}^{1}\frac{{a}_{x}u{Q}^{\prime }\left({a}_{x}u\right)}{{\left(1-{u}^{2}\right)}^{1/2}}\text{d}u,x>0.$

We easily see

$\underset{x\to \infty }{\mathrm{lim}}{a}_{x}=\infty \text{and}\underset{x\to +0}{\mathrm{lim}}{a}_{x}=0$

and

$\underset{x\to \infty }{\mathrm{lim}}\frac{{a}_{x}}{x}=0\text{and}\underset{x\to +0}{\mathrm{lim}}\frac{{a}_{x}}{x}=\infty .$

For $wf\in {L}_{p}\left(ℝ\right)\text{\hspace{0.17em}}\left(1\le p\le \infty \right)$ , the degree of weighted polynomial approximation is defined by

${E}_{n,p}\left(w;f\right):=\underset{P\in {\mathcal{P}}_{n}}{inf}{‖w\left(f-P\right)‖}_{{L}^{p}\left(ℝ\right)}.$

3. Approximations for Lp-Functions

In this section, we treat the function such as $Wf\in {L}^{p}\left({ℝ}^{s}\right)$ , where $1\le p\le \infty$ ), and if $p=\infty$ , then we suppose that $Wf$ is continuous and

$li{m}_{|X|\to \infty }\left(Wf\right)\left(X\right)=0$ . For any multivariate point $X=\left({x}_{1},\cdots ,{x}_{s}\right)\in {ℝ}^{s}$ , we consider the weights;

$W\left(X\right):={\prod }_{j=1}^{s}{w}_{j}\left({x}_{j}\right)={\prod }_{j=1}^{s}\mathrm{exp}\left(-{Q}_{j}\left({x}_{j}\right)\right)\text{ }.$

As shown under, we will also use $X\left(u\right):=\left({u}_{1},{u}_{2},\cdots ,{u}_{s}\right)$ . Let

${w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)$ , $0<\lambda <3/2$ , $i=1,2,\cdots ,s$ . From Proposition 2.3 we see ${T}_{i}^{1/4}{w}_{i}~{w}_{i,1/4}\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ . Then we admit to write

${T}_{i}^{1/4}{w}_{i}\in \mathcal{F}\left({C}^{2}+\right)$ . For the weight $W$ we construct the modulus of continuity of $f$ . It involves the function

${\Phi }_{t,i}\left({x}_{i}\right):=\sqrt{1-\frac{|{x}_{i}|}{{\sigma }_{i}\left(t\right)}}+\frac{1}{\sqrt{{T}_{i}\left({\sigma }_{i}\left(t\right)\right)}},i=1,2,\cdots ,s,$

where ${\sigma }_{i}\left(t\right)$ is defined by

${\sigma }_{i}\left(t\right):=\mathrm{inf}\left\{{a}_{n}^{\left(i\right)}:\frac{{a}_{n}^{\left(i\right)}}{n}\le t\right\},t>0,$

where ${a}_{n}^{\left(i\right)}$ is the MRS-number for the weight ${w}_{i}\left(x\right)$ . If ${a}_{n}^{\left(i\right)}/n=t$ , then we have ${\sigma }_{i}\left(t\right)={a}_{n}^{\left(i\right)}$ . In the sequel, if $1\le j\le s$ is an integer, then ${‖f‖}_{p;j}$ will denote the ${L}^{p}$ norm of $f$ taken with respect to the j-th variable. This is a function of the remaining $\left(s-1\right)$ variables. For each fixed

${\stackrel{^}{X}}_{j}:=\left({x}_{1},\cdots ,{x}_{j-1},{x}_{j+1},\cdots ,{x}_{s}\right)\in {ℝ}_{j}^{s-1}$ , we write

${f}_{{\stackrel{^}{X}}_{j}}\left(x\right):=f\left({x}_{1},\cdots ,{x}_{j-1},x,{x}_{j+1},\cdots ,{x}_{s}\right),j=1,2,\cdots ,s.$ (3.1)

Using

${\Delta }_{h}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right):={f}_{{\stackrel{^}{X}}_{j}}\left(x+\frac{h}{2}\right)-{f}_{{\stackrel{^}{X}}_{j}}\left(x-\frac{h}{2}\right),$

we define the modulus of continuity. For the Freud-type weight, we define

$\begin{array}{l}{\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};t\right)\\ :={\left(\frac{1}{t}{\int }_{0}^{t}{‖{w}_{j}\left(x\right)\left({\Delta }_{h}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right)}^{1/p}\\ +\underset{{c}_{j}\left(\text{constant}\right)}{\mathrm{inf}}{‖\left({f}_{{\stackrel{^}{X}}_{j}}\left(x\right)-{c}_{j}\right){w}_{j}\left(x\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(4t\right)\right)}.\end{array}$

If ${w}_{j}$ is Erdös-type, then we define

$\begin{array}{l}{\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};t\right)\\ :={\left(\frac{1}{t}{\int }_{0}^{t}{‖{w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right)}^{1/p}\\ +\underset{{c}_{j}\left(\text{constant}\right)}{\mathrm{inf}}{‖\left({f}_{{\stackrel{^}{X}}_{j}}\left(x\right)-{c}_{j}\right){w}_{j}\left(x\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(4t\right)\right)}.\end{array}$

We remark that if ${T}_{j}\left(x\right)$ is bounded, then we see ${\Phi }_{t,j}\left(x\right)~1$ , so we do not need the definition for the Freud-type weight.

Let vn be the de la Vallée Poussin mean opetator, and let ${v}_{n,j}\left(f\right),j=1,2,\cdots ,s$ denote the operation to f with respect to j-th co-ordinate, and ${v}_{n}^{\left[j\right]}$ will denote the operator ${v}_{n}$ applied to f with respect to each of the first j co-ordinates. Clearly,

${v}_{n}^{\left[1\right]}\left(f\right)={v}_{n,1}\left(f\right),{v}_{n}^{\left[j\right]}\left(f\right)={v}_{n}^{\left[j-1\right]}\left({v}_{n,j}\left(f\right)\right),j=2,3,\cdots ,s.$ (3.2)

Let ${a}_{n}^{\left(j\right)}$ be the MRS-number for the weight ${w}_{j}=\mathrm{exp}\left(-{Q}_{j}\right)$ .

First, we consider the following Proposition.

Proposition 3.1 (  , Theorem 3.14). For $1\le p<\infty$ , ${C}_{c}\left({ℝ}^{s}\right)$ is dence in ${L}^{p}\left({ℝ}^{s}\right)$ , where ${C}_{c}\left({ℝ}^{s}\right)$ is a set of all continuous functions with a compact support on ${ℝ}^{s}$ .

From this proposition, for any $\epsilon >0$ there exist a constant $K>0$ and a continuous function ${f}_{K}$ with a compact support ${\left[-K,K\right]}^{s}$ such that

${‖W\left(X\right)\left(f\left(X\right)-{f}_{K}\left(X\right)\right)‖}_{{L}^{p}\left(|X|\le K\right)}<\epsilon .$ (3.3)

Then we give the following assumption:

Assumption 3.2. In (3.3) we suppose that for every co-ordinate ${x}_{j},j=1,2,\cdots ,s$

${‖{w}_{j}\left(x\right)\left({f}_{{\stackrel{^}{X}}_{j}}\left(x\right)-{\left({f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le K\right)}<\epsilon$ (3.4)

holds.

We define a new class of functions ${L}_{p}^{*}\left({ℝ}^{s}\right),1\le p\le \infty$ as follows:

${L}_{W}^{p*}\left({ℝ}^{s}\right):=\left\{f|Wf\in {L}_{p}\left({ℝ}^{s}\right)\text{holds}\left(3.4\right)\right\},$ (3.5)

where if $p=\infty$ , then ${L}_{W}^{p*}\left({ℝ}^{s}\right)={L}_{W}^{p}\left({ℝ}^{s}\right)$ and we suppose that f is continuous and

$\underset{|X|\to \infty }{\mathrm{lim}}W\left(X\right)f\left(X\right)=0$

(we write this fact as $Wf\in {C}_{0}\left({ℝ}^{s}\right)$ ). We state the theorem in this section.

Theorem 3.3. (1) We suppose

${w}_{j}=\mathrm{exp}\left(-{Q}_{j}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right),j=1,2,\cdots ,s$ , and let

${T}_{j}\left({a}_{n}^{\left(j\right)}\right)\le c{\left(\frac{n}{{a}_{n}^{\left(j\right)}}\right)}^{2/3},j=1,2,\cdots ,s.$ (3.6)

Let $n\ge 1,1\le p\le \infty$ . Then we have

$\begin{array}{l}{‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le \underset{j=1}{\overset{s}{\sum }}\text{ }{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right)\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)},\end{array}$ (3.7)

where

${ℝ}_{j}^{s-1}:=\left\{\left({x}_{1},\cdots ,{x}_{j-1},{x}_{j+1},\cdots ,{x}_{s}\right)\right\},$ (3.8)

and ${\prod }_{k=1}^{0}{T}_{k}^{1/4}=1$ . Especially $f\in {L}_{{T}^{〈s〉}W}^{p*}\left({ℝ}^{s}\right)$ , then we have

$\begin{array}{l}\underset{j=1}{\overset{s}{\sum }}\text{ }{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right)\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\to 0\\ \text{as}\text{\hspace{0.17em}}n\to \infty .\end{array}$ (3.9)

(2) We suppose ${w}_{j}=\mathrm{exp}\left(-{Q}_{j}\right)\in \mathcal{F}\left({C}^{2}+\right),j=1,2,\cdots ,s$ , and let (3.6) holds. Let $n\ge 1,1\le p\le \infty$ . Then we have

${‖\frac{W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)}{{\prod }_{k=1}^{s}{T}_{k}^{1/4}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le \underset{j=1}{\overset{s}{\sum }}\text{ }{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}.$ (3.10)

Especially $f\in {L}_{W}^{p*}\left({ℝ}^{s}\right)$ , then we have

$\underset{j=1}{\overset{s}{\sum }}{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\to 0\text{as}\text{\hspace{0.17em}}n\to \infty .$ (3.11)

First we will show (3.7). We need some preliminaries.

Proposition 3.4 (  , Theorem 1). Let $1\le p\le \infty$ .

(1) We assume that $w\in \mathcal{F}\left({C}^{2}+\right)$ satisfies $T\left({a}_{n}\right)\le C{\left(n/{a}_{n}\right)}^{2/3}$ . Then there exists a constant $C>0$ such that when $wg\in {L}_{p}\left(ℝ\right)$ , then

${‖\frac{w{v}_{n}\left(g\right)}{{T}^{1/4}}‖}_{{L}^{p}\left(ℝ\right)}\le C{‖wg‖}_{{L}^{p}\left(ℝ\right)},$

and so,

${‖w{v}_{n}\left(g\right)‖}_{{L}^{p}\left(ℝ\right)}\le CT{\left({a}_{n}\right)}^{1/4}{‖wg‖}_{{L}^{p}\left(ℝ\right)}.$

(2) We assume that $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right)$ satisfies $T\left({a}_{n}\right)\le C{\left(n/{a}_{n}\right)}^{2/3}$ . Then there exists a constant $C>0$ such that if ${T}^{1/4}wg\in {L}^{p}\left(ℝ\right)$ , then

${‖w{v}_{n}\left(g\right)‖}_{{L}^{p}\left(ℝ\right)}\le C{‖{T}^{1/4}wg‖}_{{L}^{p}\left(ℝ\right)}.$

Proposition 3.5 (  , Corollary 6.2 (6.5)). Let $1\le p\le \infty$ .

(1) Let $w\in \mathcal{F}\left({C}^{2}+\right)$ , and $n\ge 1$ be an integer. Then

${‖\frac{w\left(g-{v}_{n}\left(g\right)\right)}{{T}^{1/4}}‖}_{{L}^{p}\left(ℝ\right)}\le C{E}_{n,p}\left(w;g\right),$

where C do not depend on g and n.

(2) Let $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right)$ , and $n\ge 1$ be an integer. Then

${‖w\left(g-{v}_{n}\left(g\right)\right)‖}_{{L}^{p}\left(ℝ\right)}\le C{E}_{n,p}\left({T}^{1/4}w;g\right),$

where C do not depend on g and n.

Proposition 3.6. (  ) Let $w=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ , and let $0 . Then there exist ${n}_{0}\in ℕ$ and positive constants C1, C2 such that for every $wg\in {L}^{p}\left(ℝ\right)$ (and for $p=\infty$ , we require $g$ to be continuous, and $wf$ to vanish at $±\infty$ ) and every $n\ge {n}_{0}$ ,

${E}_{n,p}\left(g;w\right)\le {C}_{1}{\stackrel{¯}{\omega }}_{p}\left(g,w;{C}_{2}\frac{{a}_{n}}{n}\right),$

where ${n}_{0}$ and ${C}_{1},{C}_{2}$ do not depend on $g$ and $n$ , and ${\omega }_{p}^{*}\left(g,w,t\right)$ will be defined in Section 6.

We set

${T}^{〈j〉}:={\prod }_{i=1}^{j}{T}_{i}^{1/4},{ℝ}_{\left(j\right)}:=\left\{{x}_{j}\in ℝ\right\},$

${ℝ}_{\le j}^{j}:=\left\{\left({x}_{1},\cdots ,{x}_{j}\right)\in {ℝ}^{j}\right\},{ℝ}_{j\le }^{s-j+1}:=\left\{\left({x}_{j},\cdots ,{x}_{s}\right)\in {ℝ}^{s-j+1}\right\},$

${ℝ}_{j}^{s-1}:=\left\{\left({x}_{1},\cdots ,{x}_{j-1},{x}_{j+1},\cdots ,{x}_{s}\right)\in {ℝ}^{s-1}\right\},$

$W:={\prod }_{i=1}^{s}{w}_{i},{W}_{j}:={\prod }_{i=1,i\ne j}^{s}{w}_{i},j=1,2,\cdots ,s.$

We need the infinite-finite inequality.

Theorem 3.7 (Infinite-finite inequality). Let $00$ , and let

$P\left(X\right)\in {\mathcal{P}}_{n,\cdots ,n}\left({ℝ}^{s}\right)\left(=:{\mathcal{P}}_{n;s}\left({ℝ}^{s}\right)\right)$ . Then

${‖W\left(X\right)P\left(X\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖W\left(X\right)P\left(X\right)‖}_{{L}^{p}\left(|{x}_{i}|\le {a}_{n}^{\left(i\right)}\left(1-L{\delta }_{n}^{\left(i\right)}\right),i=1,2,\cdots ,s\right)}.$ (3.12)

If $r>1$ , then there exists $\epsilon >0$ such that

${‖W\left(X\right)P\left(X\right)‖}_{{L}^{p}\left({ℝ}_{i,r}^{s}\right)}\le Cexp\left(-{n}^{\epsilon }\right){‖W\left(X\right)P\left(X\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)},$ (3.13)

where ${ℝ}_{i,r}^{s}:=\left\{{x}_{i};|{x}_{i}|\ge {a}_{rn}^{\left(i\right)}\right\}×{ℝ}_{i}^{s-1}$ .

To prove Theorem 3.7 we use the following proposition with $s=1$ .

Proposition 3.8 (  , Theorem 1.9). Let $00$ , and let $P\left(x\right)\in {\mathcal{P}}_{n}\left(ℝ\right)$ . Then

${‖w\left(x\right)P\left(x\right)‖}_{{L}^{p}\left(ℝ\right)}\le C{‖w\left(x\right)P\left(x\right)‖}_{{L}^{p}\left(|x|\le {a}_{n}\left(1-L{\delta }_{n}\right)\right)}.$ (3.14)

If $r>1$ , then there exists $\epsilon >0$ such that

${‖w\left(x\right)P\left(x\right)‖}_{{L}^{p}\left({a}_{rn}\le |x|\right)}\le Cexp\left(-{n}^{\epsilon }\right){‖w\left(x\right)P\left(x\right)‖}_{{L}^{p}\left(|x|\le {a}_{n}\right)}.$ (3.15)

Proof of Theorem 3.7. For the proof of (3.12) we use (3.14). We put A for the left side of the above equation. Let $0 . By repeatedly applying Proposition 3.8 (3.14), we have

$\begin{array}{c}{A}^{p}={\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{|{w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)|}^{p}×\left\{{\int }_{-\infty }^{\infty }{|{w}_{1}\left({x}_{1}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\right\}\text{d}{x}_{2}\cdots \text{d}{x}_{s}\\ \le {C}_{1}{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{|{w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)|}^{p}{\int }_{-{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}^{{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{|{w}_{1}\left({x}_{1}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\\ ={C}_{1}{\int }_{-{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}^{{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{w}_{1}^{p}\left({x}_{1}\right){\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{|{w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{2}\cdots \text{d}{x}_{s}\text{d}{x}_{1}\\ \le {C}_{2}{\int }_{-{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}^{{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{\int }_{-{a}_{n}^{\left(2\right)}\left(1-L{\delta }_{n}^{\left(2\right)}\right)}^{{a}_{n}^{\left(2\right)}\left(1-L{\delta }_{n}^{\left(2\right)}\right)}{w}_{1}^{p}\left({x}_{1}\right){w}_{2}^{p}\left({x}_{2}\right)\\ ×{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{|{w}_{3}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{3}\cdots \text{d}{x}_{d}\text{d}{x}_{1}\text{d}{x}_{2}\\ \le \cdots \\ \le {C}_{s}{\int }_{-{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}^{{a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}\cdots {\int }_{-{a}_{n}^{\left(s\right)}\left(1-L{\delta }_{n}^{\left(s\right)}\right)}^{{a}_{n}^{\left(s\right)}\left(1-L{\delta }_{n}^{\left(s\right)}\right)}{|{w}_{1}\left({x}_{1}\right)\cdots {w}_{s}\left({x}_{s}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\\ ={C}_{s}{‖W\left(X\right)P\left(X\right)‖}_{{L}^{p}\left(|{x}_{i}|\le {a}_{n}^{\left(i\right)}\left(1-L{\delta }_{n}^{\left(i\right)}\right),i=1,2,\cdots ,s\right)}^{p}.\end{array}$

Next, we show the case of $p=\infty$ .

$\begin{array}{c}A=\underset{{x}_{s}\in ℝ}{\mathrm{sup}}\cdots \underset{{x}_{2}\in ℝ}{\mathrm{sup}}|{w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)|\underset{{x}_{1}\in ℝ}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|\\ \le {C}_{1}\underset{{x}_{s}\in ℝ}{\mathrm{sup}}\cdots \underset{{x}_{2}\in ℝ}{\mathrm{sup}}|{w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)|\underset{|{x}_{1}|\le {a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|\\ ={C}_{1}\underset{{x}_{s}\in ℝ}{\mathrm{sup}}\cdots \underset{{x}_{3}\in ℝ}{\mathrm{sup}}|{w}_{3}\left({x}_{3}\right)\cdots {w}_{s}\left({x}_{s}\right)|\underset{|{x}_{1}|\le {a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{\mathrm{sup}}\underset{{x}_{2}\in ℝ}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right){w}_{2}\left({x}_{2}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|\\ \le {C}_{1}{C}_{2}\underset{{x}_{s}\in ℝ}{\mathrm{sup}}\cdots \underset{{x}_{3}\in ℝ}{\mathrm{sup}}|{w}_{3}\left({x}_{3}\right)\cdots {w}_{s}\left({x}_{s}\right)|\\ ×\underset{|{x}_{1}|\le {a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{\mathrm{sup}}\underset{|{x}_{2}|\le {a}_{n}^{\left(2\right)}\left(1-L{\delta }_{n}^{\left(2\right)}\right)}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right){w}_{2}\left({x}_{2}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|\\ \le \cdots \\ \le {C}_{1}{C}_{2}\cdots {C}_{s}\underset{|{x}_{1}|\le {a}_{n}^{\left(1\right)}\left(1-L{\delta }_{n}^{\left(1\right)}\right)}{\mathrm{sup}}\underset{|{x}_{2}|\le {a}_{n}^{\left(2\right)}\left(1-L{\delta }_{n}^{\left(2\right)}\right)}{\mathrm{sup}}\cdots \underset{|{x}_{2}|\le {a}_{n}^{\left(s\right)}\left(1-L{\delta }_{n}^{\left(s\right)}\right)}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right){w}_{2}\left({x}_{2}\right)\\ ×\cdots ×{w}_{s}\left({x}_{s}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|\\ =C\underset{|{x}_{i}|\le {a}_{n}^{\left(i\right)}\left(1-L{\delta }_{n}^{\left(i\right)}\right),i=1,\cdots ,s}{\mathrm{sup}}|{w}_{1}\left({x}_{1}\right){w}_{2}\left({x}_{2}\right)\cdots {w}_{s}\left({x}_{s}\right)P\left({x}_{1},\cdots ,{x}_{s}\right)|.\end{array}$

Similarly, using Proposition 3.8 (3.15), we easily have (3.13). #

Lemma 3.9. Let $1\le p\le \infty$ .

(1) We assume that ${w}_{i}\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ satisfies (3.6). Then there exists a constant $C>0$ such that when $Wh\in {L}_{p}\left({ℝ}^{s}\right)$ ,

${‖\frac{W{v}_{n}^{\left[j\right]}\left(h\right)}{{T}^{〈j〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖Wh‖}_{{L}^{p}\left({ℝ}^{s}\right)},j=1,2,\cdots ,s,$

and

${‖W{v}_{n}^{\left[s\right]}\left(f\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{\prod }_{i=1}^{s}{T}_{i}^{1/4}\left({a}_{n}^{\left(i\right)}\right){‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}.$

(2) We assume that ${w}_{i}\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right),i=1,2,\cdots ,s$ . Let

${T}^{〈s〉}Wh\in {L}^{p}\left({ℝ}^{s}\right)$ , then

${‖W{v}_{n}^{\left[j\right]}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖{T}^{〈j〉}Wh‖}_{{L}^{p}\left({ℝ}^{s}\right)},j=1,2,\cdots ,s.$

Proof. (1) From Theorem 3.4 (1), for $j=1$

${‖\frac{W{v}_{n}^{\left[1\right]}\left(h\right)}{{T}^{〈1〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}={‖{‖\frac{W{v}_{n,1}\left({h}_{{\stackrel{^}{X}}_{1}}\right)}{{T}_{1}^{1/4}}‖}_{{L}^{p}\left({ℝ}_{\le 1}^{1}\right)}‖}_{{L}^{p}\left({ℝ}_{2\le }^{s-1}\right)}\le C{‖Wh‖}_{{L}^{p}\left({ℝ}^{s}\right)}.$

Inductively,

$\begin{array}{c}{‖\frac{W{v}_{n}^{\left[j\right]}\left(h\right)}{{T}^{〈j〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}={‖{‖\frac{W{v}_{n}^{\left[j-1\right]}\left({v}_{n,j}\left(h\right)\right)}{{T}^{〈j〉}}‖}_{{L}^{p}\left({ℝ}_{\le j-1}^{j-1}\right)}‖}_{{L}^{p}\left({ℝ}_{j\le }^{s-j+1}\right)}\\ \le C{‖\frac{W{v}_{n,j}\left({h}_{{\stackrel{^}{X}}_{j}}\right)}{{T}_{j}^{1/4}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖Wh‖}_{{L}^{p}\left({ℝ}^{s}\right)}.\end{array}$

For the second formula, using Theorem 3.7 and the above inequality, we have

$\begin{array}{l}{‖W{v}_{n}^{\left[j\right]}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{\prod }_{i=1}^{j}{T}_{i}^{1/4}\left({a}_{n}^{\left(i\right)}\right){‖\left({\prod }_{k=j+1}^{s}{w}_{k}\right){‖\frac{\left({\prod }_{i=1}^{j}{w}_{i}\right){v}_{n}^{\left[j\right]}\left(h\right)}{{T}^{〈j〉}}‖}_{{L}^{p}\left(|{x}_{i}|\le {a}_{2n}^{\left(i\right)}\right),1\le i\le j}‖}_{{L}^{p}\left({ℝ}_{j+1\le }^{s-j}\right)}\\ \le C{\prod }_{i=1}^{j}{T}_{i}^{1/4}\left({a}_{n}^{\left(i\right)}\right){‖Wh‖}_{{L}^{p}\left({ℝ}^{s}\right)},\\ j=1,2,\cdots ,s.\end{array}$

Similarly we have the following:

(2) From Theorem 3.4 (2) for $j=1$ ,

$\begin{array}{l}{‖W{v}_{n}^{\left[1\right]}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}={‖W{v}_{n,1}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖W{T}_{1}^{1/4}h‖}_{{L}^{p}\left({ℝ}^{s}\right)},\\ j=1,2,\cdots ,s.\end{array}$

Inductively,

$\begin{array}{c}{‖W{v}_{n}^{\left[j\right]}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}={‖W{v}_{n}^{\left[j-1\right]}\left({v}_{n,j}\left(h\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{‖W{T}^{〈j-1〉}{v}_{n,j}\left(h\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{‖W{T}^{〈j〉}h‖}_{{L}^{p}\left({ℝ}^{s}\right)}.\end{array}$ #

Proof of (3.7) in Theorem 3.3. By Proposition 3.5 (2) and Proposition 3.6, we get

${‖{w}_{1}\left(x\right)\left({f}_{{\stackrel{^}{X}}_{1}}-{v}_{n,1}\left({f}_{{\stackrel{^}{X}}_{1}}\right)\right)‖}_{{L}^{p}\left(ℝ\right)}\le C{E}_{n,p}\left({f}_{{\stackrel{^}{X}}_{1}};{T}_{1}^{1/4}{w}_{1}\right)\le {C}_{1}{\stackrel{¯}{\omega }}_{p,1}\left({f}_{{\stackrel{^}{X}}_{1}},{T}_{1}^{1/4}{w}_{1};{c}_{1}\frac{{a}_{n}^{\left(1\right)}}{n}\right),$

where the constant C1 and c1 is independent of ${\stackrel{^}{X}}_{1}$ . Similarly, for $j=1,2,\cdots ,s$ ,

${‖{w}_{j}\left(x\right)\left({f}_{{\stackrel{^}{X}}_{j}}-{v}_{n,j}\left({f}_{{\stackrel{^}{X}}_{j}}\right)\right)‖}_{{L}^{p}\left(ℝ\right)}\le {C}_{j}{\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right).$ (3.16)

Using $f-{v}_{n}^{\left[s\right]}=\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)+\left({v}_{n}^{\left[1\right]}\left(f\right)-{v}_{n}^{\left[2\right]}\left(f\right)\right)+\cdots +\left({v}_{n}^{\left[s-1\right]}\left(f\right)-{v}_{n}^{\left[s\right]}\left(f\right)\right)$ , we get from Lemma 3.9 (2) and (3.2),

$\begin{array}{l}{‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le {‖W\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W\left({v}_{n}^{\left[j-1\right]}\left(f\right)-{v}_{n}^{\left[j\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C\left[{‖W\left(f-{v}_{n,1}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W{v}_{n}^{\left[j-1\right]}\left(f-{v}_{n,j}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\\ \le C\left[{‖W\left(f-{v}_{n,1}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right)\left(f-{v}_{n,j}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\\ \le {C}_{1}{‖{W}_{1}{\stackrel{¯}{\omega }}_{p,1}\left({f}_{{\stackrel{^}{X}}_{1}},{T}_{1}^{1/4}{w}_{1};{c}_{1}\frac{{a}_{n}^{\left(1\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{1}^{s-1}\right)}\\ +\underset{j=2}{\overset{s}{\sum }}{C}_{j}{‖{W}_{j}\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\end{array}$

by (3.16)

$\le \underset{j=1}{\overset{s}{\sum }}{C}_{j}{‖{W}_{j}\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)},$

where ${\prod }_{k=1}^{j-1}{T}_{k}^{1/4}=1$ for $j=1$ . Hence we obtain (3.7).

Proof of (3.10) in Theorem 3.3. By Proposition 3.5 (1) and Proposition 3.6, we get

${‖\frac{{w}_{1}\left(x\right)\left({f}_{{\stackrel{^}{X}}_{1}}-{v}_{n,1}\left({f}_{{\stackrel{^}{X}}_{1}}\right)\right)}{{T}_{1}^{1/4}\left(x\right)}‖}_{{L}^{p}\left(ℝ\right)}\le C{E}_{n,p}\left({f}_{{\stackrel{^}{X}}_{1}};{w}_{1}\right)\le {C}_{1}{\stackrel{¯}{\omega }}_{p,1}\left({f}_{{\stackrel{^}{X}}_{1}},{w}_{1};{c}_{1}\frac{{a}_{n}^{\left(1\right)}}{n}\right),$

where the constant C1 and c1 is independent of ${\stackrel{^}{X}}_{1}$ . Similarly, for $j=1,2,\cdots ,s$ ,

${‖\frac{{w}_{j}\left(x\right)\left({f}_{{\stackrel{^}{X}}_{j}}-{v}_{n,j}\left({f}_{{\stackrel{^}{X}}_{j}}\right)\right)}{{T}_{j}^{1/4}\left(x\right)}‖}_{{L}^{p}\left(ℝ\right)}\le {C}_{j}{\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right).$ (3.17)

Using $f-{v}_{n}^{\left[s\right]}=\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)+\left({v}_{n}^{\left[1\right]}\left(f\right)-{v}_{n}^{\left[2\right]}\left(f\right)\right)+\cdots +\left({v}_{n}^{\left[s-1\right]}\left(f\right)-{v}_{n}^{\left[s\right]}\left(f\right)\right)$ , we get from Lemma 3.9 (1) and (3.2),

$\begin{array}{l}{‖\frac{W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)}{{T}^{〈s〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le {‖\frac{W\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)}{{T}^{〈1〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖\frac{W\left({v}_{n}^{\left[j-1\right]}\left(f\right)-{v}_{n}^{\left[j\right]}\left(f\right)\right)}{{T}^{〈j〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C\left[{‖\frac{W\left(f-{v}_{n,1}\left(f\right)\right)}{{T}_{1}^{1/4}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖\frac{W{v}_{n}^{\left[j-1\right]}\left(\left\{f-{v}_{n,j}\left(f\right)\right\}\right)/{T}_{j}^{1/4}}{{T}^{〈j-1〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\\ \le C\left[{‖\frac{W\left(f-{v}_{n,1}\left(f\right)\right)}{{T}_{1}^{1/4}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖\frac{W\left(f-{v}_{n,j}\left(f\right)\right)}{{T}_{j}^{1/4}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\\ \le {C}_{1}{‖\left({\prod }_{2\le i\le s}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,1}\left({f}_{{\stackrel{^}{X}}_{1}},{w}_{1};{c}_{1}\frac{{a}_{n}^{\left(1\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{1}^{s-1}\right)}\\ +\underset{j=2}{\overset{s}{\sum }}{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\end{array}$

by (3.17)

$\le \underset{j=1}{\overset{s}{\sum }}{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}.$

Hence we obtain (3.10).

To prove (3.9) and (3.11) we need a lemma:

Lemma 3.10. Let $0 , $\sigma \left(t\right)\ge 1$ and $|x|\le \sigma \left(2t\right)$ . We have

${‖\left(wf\right)\left(x±h{\Phi }_{t}\left(x\right)\right)‖}_{{L}^{p}\left(ℝ\right)}~{‖\left(wf\right)\left(x\right)‖}_{{L}^{p}\left(ℝ\right)}.$ (3.18)

Proof. We may show

${‖\left(wf\right)\left(x±h\sqrt{1-\frac{|x|}{\sigma \left(t\right)}}\right)‖}_{{L}^{p}\left(ℝ\right)}~{‖\left(wf\right)\left(x\right)‖}_{{L}^{p}\left(ℝ\right)}.$ (3.19)

Let $x>0$ . If we put

$x±h\sqrt{1-\frac{x}{\sigma \left(t\right)}}=:y,\frac{{a}_{2u}}{2u}=t,\frac{{a}_{v}}{v}=2t,$ (3.20)

we will see

$\frac{1}{2}\le \frac{\text{d}y}{\text{d}x}\le \frac{3}{2}.$ (3.21)

Then we conclude (3.19). Now, from (3.20)

$\frac{\text{d}y}{\text{d}x}=1\mp \frac{h}{2\sqrt{\sigma \left(t\right)}\sqrt{\sigma \left(t\right)-x}}.$

Since ${a}_{2u}/u=2t$ , we see

$\frac{{a}_{v}}{v}=\frac{{a}_{2u}}{u}>\frac{{a}_{u}}{u},$

that is, we have

$\sigma \left(2t\right)={a}_{v}<{a}_{u}<\sigma \left(t\right)={a}_{2u}.$

Then, using (  , Lemmas 3.6, 3.7), we see

$\begin{array}{c}\frac{h}{2\sqrt{\sigma \left(t\right)}\sqrt{\sigma \left(t\right)-x}}\le \frac{t}{2\sqrt{\sigma \left(t\right)}\sqrt{\sigma \left(t\right)-\sigma \left(2t\right)}}\le \frac{\sqrt{{a}_{2u}}}{4u}\frac{1}{\sqrt{{a}_{2u}-{a}_{u}}}\\ \le \frac{\sqrt{{a}_{2u}}}{4u}\sqrt{{C}_{1}\frac{T\left({a}_{2u}\right)}{{a}_{2u}}}\le \frac{1}{4u}\sqrt{{C}_{1}{C}_{2}}{u}^{1-\delta }=\frac{\sqrt{{C}_{1}{C}_{2}}}{4}\frac{1}{{u}^{\delta }}\le \frac{1}{2}\end{array}$

for some $0<\delta \le 1$ and $u$ large enough. We have (3.21). So we conclude (3.19). #

Proof of (3.9). We will estimate

${‖{W}_{j}{T}^{〈j-1〉}{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}.$

To do so we may estimate

${I}_{1}:={\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}.$

For $\epsilon >0$ we take $K>0$ large enough, and then by $f\in {L}_{{T}^{〈s〉}w}^{p*}\left(ℝ\right)$ we can select a continuous function ${f}_{K}$ such that

$\begin{array}{c}{I}_{1}\le {\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{\left(f-{f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}\\ +{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{\left({f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}\\ =A+B.\end{array}$

We note ${w}_{j}\left(x\right)~w\left(x+\left(h/2\right){\Phi }_{t,j}\left(x\right)\right)~w\left(x-\left(h/2\right){\Phi }_{t,j}\left(x\right)\right)$ (see  , Lemma 7). If ${\sigma }_{j}\left(2t\right)\le K$ from our assumption and Lemma 3.10 we have

$\begin{array}{c}A\le C{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right){\left(f-{f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}\\ \le C\epsilon {\left\{\frac{1}{t}{\int }_{0}^{t}\text{ }\text{d}h\right\}}^{1/p}\le C\epsilon .\end{array}$

If ${\sigma }_{j}\left(2t\right)>K$ , then by Lemma 3.10 we see

$\begin{array}{c}A\le C{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right){\left(f-{f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)‖}_{{L}^{p}\left(|x|\le K\right)}^{p}\text{d}h\right\}}^{1/p}\\ +{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right){f}_{{\stackrel{^}{X}}_{j}}\left(x\right)‖}_{{L}^{p}\left(K<|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}\\ \le C\epsilon +{C}_{1}\epsilon \le {C}_{2}\epsilon .\end{array}$

When we take $t>0$ small enough, we see

$\begin{array}{c}B={\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{\left({f}_{K}\right)}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le K\right)}^{p}\text{d}h\right\}}^{1/p}\\ \le {\left\{\frac{1}{t}{\int }_{0}^{t}\epsilon \text{d}h\right\}}^{1/p}\le C\epsilon ,\end{array}$

because of the continuity of ${f}_{K}$ . Therefore we have ${I}_{1}<\epsilon$ . Consequently, we have

$\begin{array}{l}{‖{W}_{j}{T}^{〈j-1〉}{\left\{\frac{1}{t}{\int }_{0}^{t}{‖{T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)\left({\Delta }_{h{\Phi }_{t,j}\left(x\right)}{f}_{{\stackrel{^}{X}}_{j}}\left(x\right)\right)‖}_{{L}^{p}\left(|x|\le {\sigma }_{j}\left(2t\right)\right)}^{p}\text{d}h\right\}}^{1/p}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le C\epsilon {‖{W}_{j}{T}^{〈j-1〉}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\le C\epsilon .\end{array}$ (3.22)

Finally, we see

$\begin{array}{l}{‖{W}_{j}{T}^{〈j-1〉}\underset{{c}_{j}\left(\text{constant}\right)}{inf}{‖\left({f}_{{\stackrel{^}{X}}_{j}}\left(x\right)-{c}_{j}\right){T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)‖}_{{L}^{p}\left(ℝ\\left[-{\sigma }_{j}\left(4t\right),{\sigma }_{j}\left(4t\right)\right]\right)}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le C{‖{W}_{j}{T}^{〈j-1〉}{‖{f}_{{\stackrel{^}{X}}_{j}}\left(x\right){T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)‖}_{{L}^{p}\left(ℝ\\left[-{\sigma }_{j}\left(4t\right),{\sigma }_{j}\left(4t\right)\right]\right)}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le C{w}_{j}^{1/4}\left({\sigma }_{j}\left(4t\right)\right){‖Wf‖}_{{L}^{p}\left({ℝ}_{j}^{s}\right)}.\end{array}$

Here, if we set $4t={a}_{u}/u$ , then we see

${w}_{j}^{1/4}\left({\sigma }_{j}\left(4t\right)\right)=exp\left(-\frac{1}{4}{Q}_{j}\left({a}_{u}\right)\right)~exp\left(-\frac{u}{2\sqrt{T\left({a}_{u}\right)}}\right)\le {\text{e}}^{-{u}^{\delta }}$

for some $0<\delta <1$ , that is,

${w}_{j}^{1/4}\left({\sigma }_{j}\left(4t\right)\right)\le C{\text{e}}^{-{u}^{\delta }}\le C\frac{{a}_{u}}{4u}=Ct.$

Therefore

$\begin{array}{l}{‖{W}_{j}{T}^{〈j-1〉}\underset{{c}_{j}\left(\text{constant}\right)}{inf}{‖\left({f}_{{\stackrel{^}{X}}_{j}}\left(x\right)-{c}_{j}\right){T}_{j}^{1/4}\left(x\right){w}_{j}\left(x\right)‖}_{{L}^{p}\left(ℝ\\left[-{\sigma }_{j}\left(4t\right),{\sigma }_{j}\left(4t\right)\right]\right)}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le Ct.\end{array}$ (3.23)

For given $\epsilon >0$ if we take $K>0$ large enough and then $t>0$ small enough, then by (3.22) and (3.23) we have

${‖{W}_{j}{T}^{〈j-1〉}{\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{T}_{j}^{1/4}{w}_{j},t\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}<\epsilon .$

Consequently, we have (3.9).

Proof of (3.11). If in the proof of (3.9) we set as $T=1$ (constant), then we obtain (3.11). #

Corollary 3.11. We suppose that ${w}_{j}=exp\left(-{Q}_{j}\right)\in \mathcal{F}\left({C}^{2}+\right),j=1,2,\cdots ,s$ are the Freud-type weights. Let $1\le p\le \infty$ . Then we have

$\begin{array}{l}{‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le \underset{j=1}{\overset{s}{\sum }}{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}.\end{array}$

Especially $f\in {L}_{W}^{p*}\left({ℝ}^{s}\right)$ , then we have

$\underset{j=1}{\overset{s}{\sum }}{C}_{j}{‖\left({\prod }_{1\le i\le s,i\ne j}{w}_{i}\right){\stackrel{¯}{\omega }}_{p,j}\left({f}_{{\stackrel{^}{X}}_{j}},{w}_{j};{c}_{j}\frac{{a}_{n}^{\left(j\right)}}{n}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\to 0\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}n\to \infty .$

Corollary 3.12. We suppose

${w}_{j}=exp\left(-{Q}_{j}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right),j=1,2,\cdots ,s$ , and let (3.6) hold. Let $1\le p\le \infty$ . If ${T}^{〈s〉}Wf\in {C}_{0}\left({ℝ}^{s}\right)\cap {L}^{p}\left({ℝ}^{s}\right)$ ), then we have

${‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\to 0\text{as}\text{\hspace{0.17em}}n\to \infty .$

Moreover, we suppose ${w}_{j}=exp\left(-{Q}_{j}\right)\in \mathcal{F}\left({C}^{2}+\right),j=1,2,\cdots ,s$ , and let (3.6) hold. If $Wf\in {C}_{0}\left({ℝ}^{s}\right)\cap {L}^{p}\left({ℝ}^{s}\right)$ , then we have

${‖\frac{W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)}{{T}^{〈s〉}}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\to 0\text{as}\text{\hspace{0.17em}}n\to \infty .$

4. A Property of Higher Order Derivatives

In this section we show an important theorem which is useful in approximation theory. We use the following notations for

${w}_{i}\left({x}_{i}\right)=\mathrm{exp}\left(-{Q}_{i}\left({x}_{i}\right)\right)\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ . Let r be a positive integer, and $|{x}_{i}|\ge \gamma >0$ .

${W}_{0}:={\prod }_{i=1}^{s}{w}_{i,0};{{Q}^{\prime }}_{i}{\left({x}_{i}\right)}^{r}{w}_{i}\left({x}_{i}\right)~{w}_{i,0}\left({x}_{i}\right)=\mathrm{exp}\left(-{Q}_{i,0}\left({x}_{i}\right)\right)\in \mathcal{F}\left({C}^{2}+\right),$

$\begin{array}{l}{W}_{\nu }:={\prod }_{i=1}^{s}{w}_{i,\nu };{{Q}^{\prime }}_{i}{\left({x}_{i}\right)}^{r-\nu }{w}_{i}\left({x}_{i}\right)~{w}_{i,\nu }\left({x}_{i}\right)=\mathrm{exp}\left(-{Q}_{i,\nu }\left({x}_{i}\right)\right)\in \mathcal{F}\left({C}^{2}+\right),\\ \nu =1,2,\cdots ,r.\end{array}$

Then we see

${{Q}^{\prime }}_{i}{\left({x}_{i}\right)}^{r-\nu +1}{w}_{i}\left({x}_{i}\right)~{{Q}^{\prime }}_{i,\nu }\left({x}_{i}\right){w}_{i,\nu }\left({x}_{i}\right).$

Especially, if $\nu =r$ , then

${w}_{i,r}\left({x}_{i}\right)={w}_{i}\left({x}_{i}\right).$

Theorem 4.1. Let ${w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right)\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ , and let $1\le p\le \infty$ . Let a constant $\gamma \ge 0$ be fixed. We suppose that $g:=g\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)$ is absolutely continuous on ${ℝ}^{s}$ and $W{g}^{\left(1,1,\cdots ,1\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ . Then we have

$\begin{array}{l}{‖{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}{w}_{i}\right)g‖}_{{L}^{p}\left(|{x}_{i}|\le \gamma ,i=1,\cdots ,s\right)}\\ \le C\left\{{\int }_{|{x}_{1}|\le \gamma }\cdots {\int }_{|{x}_{s}|\le \gamma }\underset{0\le {j}_{1}\le 1}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}\cdots \underset{0\le {j}_{s}\le 1}{\sum }{|{w}_{s,{j}_{s}}\left({x}_{s}\right)|}^{p}\\ ×{{|{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p},\end{array}$ (4.1)

where we set for each ${j}_{i}=0$ or 1, $i=1,2,\cdots ,s$ ,

${y}_{i}=\left\{\begin{array}{l}\gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{j}_{i}=0;\\ {x}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{j}_{i}=1.\end{array}$

Furthermore, let r be a positive integer, and let for each $i=1,2,\cdots ,s$ , ${w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\subset \mathcal{F}\left({C}^{2}+\right)$ $\left(0<\lambda <3/2\right)$ . We suppose that ${g}^{\left(r-1,r-1,\cdots ,r-1\right)}$ is absolutely continuous and $W{g}^{\left(r,r,\cdots ,r\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ . Then we have

${\left\{{\int }_{{ℝ}^{s}}{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}{|{w}_{s,{j}_{s}}\left({x}_{s}\right)|}^{p}{|{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p}<\infty ,$ (4.2)

for each $0\le {j}_{i}\le r,i=1,2,\cdots ,s$ with

${y}_{i}=\left\{\begin{array}{l}\gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {j}_{i}\le r-1;\\ {x}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{j}_{i}=r,\end{array}$ (4.3)

and

$\begin{array}{l}{‖{\left({\prod }_{i=1}^{s}{{Q}^{\prime }}_{i}\right)}^{r}Wg‖}_{{L}^{p}\left(|{x}_{i}|\ge \gamma ,i=1,\cdots ,s\right)}\\ \le C\left\{{\int }_{|{x}_{1}|\ge \gamma }\cdots {\int }_{|{x}_{s}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}\cdots \underset{0\le {j}_{s}\le r}{\sum }{|{w}_{s,{j}_{s}}\left({x}_{s}\right)|}^{p}\\ ×{{|{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p}<\infty .\end{array}$ (4.4)

Proposition 4.2 (  , Theorem 9, cf.  , Lemma 3.4.4). Let

$w=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ and a constant $\gamma \ge 0$ be fixed.

(a) We have

$|{Q}^{\prime }\left(x\right)w\left(x\right){\int }_{\gamma }^{x}{w}^{-1}\left(t\right)\text{d}t|\le C,|x|\ge \gamma .$

(b) Let $1\le p\le \infty$ , and let r be a positive integer. If g is absolutely continuous, $g\left(\gamma \right)=0$ and $w{g}^{\prime }\in {L}^{p}\left(ℝ\right)$ , then

${‖{Q}^{\prime }wg‖}_{{L}^{p}\left(|x|\ge \gamma \right)}\le C{‖w{g}^{\prime }‖}_{{L}^{p}\left(|x|\ge \gamma \right)}.$

When $w=exp\left(-Q\right)\in {\mathcal{F}}_{\lambda }\left({C}^{r+1}+\right)\subset \mathcal{F}\left({C}^{2}+\right)$ ( $0<\lambda <\left(r+1\right)/r$ ), and ${g}^{\left(r-1\right)}$ is absolutely continuous, ${g}^{\left(j\right)}\left(\gamma \right)=0,j=0,1,\cdots ,r-1$ with $w{g}^{\left(r\right)}\in {L}^{p}\left(ℝ\right)$ , we see

${‖{\left({Q}^{\prime }\right)}^{r}wg‖}_{{L}^{p}\left(|x|\ge \gamma \right)}\le C{‖w{g}^{\left(r\right)}‖}_{{L}^{p}\left(|x|\ge \gamma \right)}.$

Proposition 4.3 (  , Theorem 4.2). Let $w=exp\left(-Q\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\subset \mathcal{F}\left({C}^{2}+\right)$ . Then for $\alpha \in ℝ$ , we can construct a new weight ${w}_{\alpha }\in \mathcal{F}\left({C}^{2}+\right)$ such that

${\left(1+|{Q}^{\prime }\left(x\right)|\right)}^{\alpha }w\left(x\right)~{w}_{\alpha }\left(x\right)=exp\left(-{Q}_{\alpha }\right)$

on $ℝ$ , $\left(1/c\right){a}_{n}\left(w\right)\le {a}_{n}\left({w}_{\alpha }\right)\le c{a}_{n}\left(w\right)$ (c is an absolutely constant) on $ℕ$ and ${T}_{{w}_{\alpha }}\left(x\right)~{T}_{w}\left(x\right)$ hold on $ℝ$ . Furthermore, we see

${Q}_{\alpha }^{\left(j\right)}\left(x\right)~{Q}^{\left(j\right)}\left(x\right)\left(j=0,1\right)\text{for}|x|\ge \gamma >0.$

Proof of Theorem 4.1. For the proof of (4.1) we may put $r=1$ with ${w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right)\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ in the proof of (4.4) below. So we prove only (4.4). We use Proposition 4.2 and 4.3 repeatedly.

$\begin{array}{l}{\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ =\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{{Q}^{\prime }}_{1}^{r}\left({x}_{1}\right){w}_{1}\left({x}_{1}\right)g\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ \le {C}_{1,0}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{{Q}^{\prime }}_{1,1}\left({x}_{1}\right){w}_{1,1}\left({x}_{1}\right)g\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right],\end{array}$

where ${\left({{Q}^{\prime }}_{1}\right)}^{r}{w}_{1}={{Q}^{\prime }}_{1}{\left({{Q}^{\prime }}_{1}\right)}^{r-1}{w}_{1}~{{Q}^{\prime }}_{1}{w}_{1,1}~{{Q}^{\prime }}_{1,1}{w}_{1,1},{w}_{1,1}={\text{e}}^{-{Q}_{1,1}},$

$\begin{array}{l}\le {C}_{1,0}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{{Q}^{\prime }}_{1,1}\left({x}_{1}\right){w}_{1,1}\left({x}_{1}\right)\left(g\left({x}_{1},\cdots ,{x}_{s}\right)-g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{{Q}^{\prime }}_{1,1}\left({x}_{1}\right){w}_{1,1}\left({x}_{1}\right)g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right]\\ \le {C}_{1,1}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{w}_{1,1}\left({x}_{1}\right){g}^{\left(1,0,\cdots ,0\right)}\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +{\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right]\end{array}$

by ${{Q}^{\prime }}_{1,1}{w}_{1,1}~{\left({{Q}^{\prime }}_{1}\right)}^{r}{w}_{1},$

$\begin{array}{l}\le {{C}^{\prime }}_{1,1}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{{Q}^{\prime }}_{1,2}\left({x}_{1}\right){w}_{1,2}\left({x}_{1}\right){g}^{\left(1,0,\cdots ,0\right)}\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ \text{where}{w}_{1,1}~{\left({{Q}^{\prime }}_{1}\right)}^{r-1}w~{{Q}^{\prime }}_{1,2}{w}_{1,2},{w}_{1,2}={\text{e}}^{-{Q}_{1,2}},\\ +{\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right]\end{array}$

$\begin{array}{l}\le {C}_{1,2}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{w}_{1,2}\left({x}_{1}\right){g}^{\left(2,0,\cdots ,0\right)}\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{|{{Q}^{\prime }}_{1}^{r-1}\left({x}_{1}\right){w}_{1}\left({x}_{1}\right){g}^{\left(1,0,\cdots ,0\right)}\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +{\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right]\\ \end{array}$

$\begin{array}{l}\le \cdots \\ \le {C}_{1,r}\left[\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }{|{w}_{1}\left({x}_{1}\right){g}^{\left(r,0,\cdots ,0\right)}\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +\left\{{\int }_{‖{x}_{s}‖\ge \gamma }\cdots {\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{|{{Q}^{\prime }}_{1}\left({x}_{1}\right){w}_{1}\left({x}_{1}\right){g}^{\left(1,0,\cdots ,0\right)}\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\end{array}$

$\begin{array}{l}+\cdots \\ +\left\{{\int }_{‖{x}_{s}‖\ge \gamma }...{\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{|{{Q}^{\prime }}_{1}^{r-1}\left({x}_{1}\right){w}_{1}\left({x}_{1}\right){g}^{\left(1,0,\cdots ,0\right)}\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ +{\left\{{\int }_{‖{x}_{s}‖\ge \gamma }...{\int }_{‖{x}_{1}‖\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left(\gamma ,{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\right]\end{array}$

$\begin{array}{l}={C}_{1,r}\left\{{\int }_{‖{x}_{s}‖\ge \gamma }...{\int }_{‖{x}_{2}‖\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{‖{x}_{1}‖\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right){g}^{\left({j}_{1},0,\cdots ,0\right)}\left({y}_{1},{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p},\end{array}$

where

${y}_{1}=\left\{\begin{array}{l}\gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {j}_{1}\le r-1;\\ {x}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{j}_{1}=r.\end{array}$

We continue this manner with respect to ${x}_{2},{x}_{3},\cdots ,{x}_{s}$ . Then we can easily obtain as follows:

$\begin{array}{l}{\left\{{\int }_{|{x}_{s}|\ge \gamma }\cdots {\int }_{|{x}_{1}|\ge \gamma }{|{\prod }_{i=1}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)g\left({x}_{1},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\\ \le {C}_{1,r}\left\{{\int }_{|{x}_{s}|\ge \gamma }\cdots {\int }_{|{x}_{2}|\ge \gamma }{|{\prod }_{i=2}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{|{x}_{1}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right){g}^{\left({j}_{1},0,\cdots ,0\right)}\left({y}_{1},{x}_{2},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{1}\cdots \text{d}{x}_{s}\right\}}^{1/p}\end{array}$

$\begin{array}{l}\le {C}_{2}\left\{{\int }_{|{x}_{1}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}{\int }_{|{x}_{s}|\ge \gamma }\cdots {\int }_{|{x}_{3}|\ge \gamma }{|{\prod }_{i=3}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\\ ×{{\int }_{|{x}_{2}|\ge \gamma }\underset{0\le {j}_{2}\le r}{\sum }{|{w}_{2,{j}_{2}}\left({x}_{2}\right){g}^{\left({j}_{1},{j}_{2},0,\cdots ,0\right)}\left({y}_{1},{y}_{2},{x}_{3},\cdots ,{x}_{s}\right)|}^{p}\text{d}{x}_{2}\cdots \text{d}{x}_{s}\text{d}{x}_{1}\right\}}^{1/p}\end{array}$

$\begin{array}{l}={C}_{2}\left\{\begin{array}{l}{\int }_{|{x}_{1}|\ge \gamma }{\int }_{|{x}_{2}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}\underset{0\le {j}_{2}\le r}{\sum }{|{w}_{2,{j}_{2}}\left({x}_{2}\right)|}^{p}{\int }_{|{x}_{s}|\ge \gamma }\\ \cdots {\int }_{|{x}_{4}|\ge \gamma }{|{\prod }_{i=4}^{s}\left({{Q}^{\prime }}_{i}^{r}\left({x}_{i}\right){w}_{i}\left({x}_{i}\right)\right)|}^{p}\end{array}\\ ×{{\int }_{|{x}_{3}|\ge \gamma }{|{w}_{3,{j}_{3}}\left({x}_{3}\right){g}^{\left({j}_{1},{j}_{2},{j}_{3},0,\cdots ,0\right)}\left({y}_{1},{y}_{2},{y}_{3},{x}_{4},\cdots ,{x}_{s}\right)|}^{p}×\text{d}{x}_{3}\cdots \text{d}{x}_{s}\text{d}{x}_{2}\text{d}{x}_{1}\right\}}^{1/p}\end{array}$

$\begin{array}{l}\le \cdots \\ \le {C}_{r}\left\{{\int }_{|{x}_{1}|\ge \gamma }\cdots {\int }_{|{x}_{s}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}\cdots \underset{0\le {j}_{s}\le r}{\sum }{|{w}_{s,{j}_{s}}\left({x}_{s}\right)|}^{p}\\ {\underset{}{{}_{}^{}}×{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right){|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p},\end{array}$

where for each $0\le {j}_{i}\le r,i=1,2,\cdots ,s$ . We set (4.3).

Let $W{g}^{\left(r,\cdots ,r\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ . Then we need to show

$\begin{array}{l}A:=\left\{{\int }_{|{x}_{1}|\ge \gamma }\cdots {\int }_{|{x}_{s}|\ge \gamma }\underset{0\le {j}_{1}\le r}{\sum }{|{w}_{1,{j}_{1}}\left({x}_{1}\right)|}^{p}\cdots \underset{0\le {j}_{s}\le r}{\sum }{|{w}_{s,{j}_{s}}\left({x}_{s}\right)|}^{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{{|{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p}<\infty .\end{array}$

We rearrange $\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)$ as ${x}_{{k}_{i}}={y}_{i}={x}_{i},t+1\le {k}_{i}\le s$ if ${j}_{i}=r$ , and as ${x}_{{k}_{i}}={y}_{i}=\gamma <1\le {k}_{i}\le t$ if $0\le {j}_{i}\le r-1$ , where $0\le t\le s$ . Then we set

$f\left({x}_{{k}_{1}},{x}_{{k}_{2}},\cdots ,{x}_{{k}_{s}}\right):=g\left({x}_{1},{x}_{2},\cdots ,{x}_{s}\right)$ . We see

${g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)={f}^{\left({j}_{{k}_{1}},{j}_{{k}_{2}},\cdots ,{j}_{{k}_{s}}\right)}\left(\gamma ,\cdots ,\gamma ,{x}_{t+1},{x}_{t+2},\cdots ,{x}_{s}\right).$

Then we have

$\begin{array}{c}A={‖{\prod }_{i=1}^{t}{w}_{{k}_{i}}{‖{\prod }_{i=t+1}^{s}{w}_{{k}_{i}}{f}^{\left({j}_{{k}_{1}},\cdots ,{j}_{{k}_{s}}\right)}\left(\gamma ,\cdots ,\gamma ,{x}_{t+1},\cdots ,{x}_{s}\right)‖}_{{L}^{p}\left({ℝ}_{t+1\le }^{s-t}\right)}‖}_{{L}^{p}\left({ℝ}_{\le t}^{t}\right)}\\ =:{‖{\prod }_{i=1}^{t}{w}_{{k}_{i}}|{h}^{\left({j}_{{k}_{1}},\cdots ,{j}_{{k}_{t}}\right)}\left(\gamma ,\cdots ,\gamma \right)|‖}_{{L}^{p}\left({ℝ}_{\le t}^{t}\right)}\\ \le C|{h}^{\left({j}_{{k}_{1}},\cdots ,{j}_{{k}_{t}}\right)}\left(\gamma ,\cdots ,\gamma \right)|{‖{\prod }_{i=1}^{t}{w}_{{k}_{i}}‖}_{{L}^{p}\left({ℝ}_{\le t}^{t}\right)}<\infty .\end{array}$ #

We can generalize Theorem 4.1 easily. We give a class of nonnegative integers $\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)$ , and set ${J}_{s}:=\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)$ . For ${r}_{i}\ge 1,i=1,2,\cdots ,s$ we set ${R}_{s}:=\left({r}_{1},{r}_{2},\cdots ,{r}_{s}\right)$ . Then we consider the order as follows:

${K}_{s}:=\left({k}_{1},{k}_{2},\cdots ,{k}_{s}\right)\le {R}_{s}:=\left({r}_{1},{r}_{2},\cdots ,{r}_{s}\right)$

means

${k}_{i}\le {r}_{i}\left(i=1,2,\cdots ,s\right).$

Corollary 4.4. Let ${K}_{s}=\left({k}_{1},{k}_{2},\cdots ,{k}_{s}\right)\le {R}_{s}=\left({r}_{1},{r}_{2},\cdots ,{r}_{s}\right)$ be classes of nonnegative integers, where ${r}_{i}\ge 1,i=1,2,\cdots ,s$ . For each $i=1,2,\cdots ,s$ , we

suppose ${w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{r+1}+\right)\subset \mathcal{F}\left({C}^{2}+\right)$ ( $0<\lambda <\left(r+1\right)/r$ ). If

${g}^{\left({r}_{1}-1,{r}_{2}-1,\cdots ,{r}_{s}-1\right)}$ is absolutely continuous, and $W{g}^{\left({r}_{1},{r}_{2},\cdots ,{r}_{s}\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ , then we see

$\begin{array}{l}{‖{\left({\prod }_{i=1}^{s}{{Q}^{\prime }}_{i}\right)}^{{r}_{i}-{k}_{i}}W{g}^{\left({k}_{1},\cdots ,{k}_{s}\right)}‖}_{{L}^{p}\left(|{x}_{i}|\ge \gamma ,i=1,\cdots ,s\right)}\\ \le C\left\{{\int }_{|{x}_{1}|\ge \gamma }\cdots {\int }_{|{x}_{s}|\ge \gamma }\underset{{k}_{1}\le {j}_{1}\le {r}_{1}}{\sum }{|{w}_{1,{j}_{1}-{k}_{1}}\left({x}_{1}\right)|}^{p}\cdots \underset{{k}_{s}\le {j}_{s}\le {r}_{s}}{\sum }{|{w}_{s,{j}_{s}-{k}_{s}}\left({x}_{s}\right)|}^{p}\\ ×{{|{g}^{\left({j}_{1},{j}_{2},\cdots ,{j}_{s}\right)}\left({y}_{1},{y}_{2},\cdots ,{y}_{s}\right)|}^{p}\text{d}{x}_{s}\cdots \text{d}{x}_{1}\right\}}^{1/p}<\infty ,\end{array}$

where for each $i=1,2,\cdots ,s$ we set

${y}_{i}=\left\{\begin{array}{l}\gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{i}\le {j}_{i}\le {r}_{i}-1;\\ {x}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{j}_{i}={r}_{i}.\end{array}$

We remark that $W{g}^{\left({r}_{1},\cdots ,{r}_{s}\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ means $W{g}^{\left({k}_{1},\cdots ,{k}_{s}\right)}\in {L}^{p}\left({ℝ}^{s}\right)$ for

$0\le {K}_{s}=\left({k}_{1},\cdots ,{k}_{s}\right)\le {R}_{s}=\left({r}_{1},\cdots ,{r}_{s}\right)$ .

5. Degree of Approximation

We define the degree of approximation for $Wf\in {L}^{p}\left({ℝ}^{s}\right)$ as follows:

${E}_{n,p;s}\left(W,f\right):=\underset{P\in {\mathcal{P}}_{n;s}\left({ℝ}^{s}\right)}{\mathrm{inf}}{‖W\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}.$

Using this ${E}_{n,p;s}\left(W,f\right)$ , we can estimate the degree of approximation of $Wf\in {L}^{p}\left({ℝ}^{s}\right)$ from ${\mathcal{P}}_{n;s}\left({ℝ}^{s}\right)$ .

Theorem 5.1. (1) Let ${w}_{i}\in \mathcal{F}\left({C}^{2}+\right)\left(i=1,2,\cdots ,s\right)$ and let

$1\le p\le \infty ,Wf\in {L}^{p}\left({ℝ}^{s}\right)$ . Furthermore, we suppose (3.3). Then we have

${‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}{\left({x}_{i}\right)}^{1/4}}\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{E}_{n,p;s}\left(W,f\right).$

(2) If ${w}_{i}\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(i=1,2,\cdots ,s\right),0<\lambda <3/2$ , and let

${‖\left({\prod }_{i=1}^{s}{T}_{i}^{1/4}\right)Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}<\infty$ , then we have

${‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{E}_{n,p;s}\left(\left({\prod }_{i=1}^{s}{T}_{i}^{1/4}\right)W,f\right).$

(3) Let ${w}_{i}\in \mathcal{F}\left({C}^{2}+\right)\left(i=1,2,\cdots ,s\right)$ and let $1\le p\le \infty ,Wf\in {L}^{p}\left({ℝ}^{s}\right)$ . Then we have

${‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C\left({\prod }_{i=1}^{s}{T}_{i}^{1/4}\left({a}_{n}^{\left(i\right)}\right)\right){E}_{n,p;s}\left(W,f\right).$

(4) Furthermore, let ${w}_{i}\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right),i=1,2,\cdots ,s$ . If $f\in {L}_{{W}^{\delta }}^{p*}\left({ℝ}^{s}\right)$ for some $0<\delta <1$ , then we have

${E}_{n,p;s}\left(W,f\right)\to 0\text{as}n\to \infty .$

Proof. (1) There exists $P\in {\mathcal{P}}_{n}$ such that ${‖W\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{E}_{n,p;s}\left(W,f\right)$ . Therefore, by Lemma 3.9 (1)

$\begin{array}{l}{‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{1/4}}\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ ={‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{1/4}}\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+{‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{1/4}}{v}_{n}^{\left[s\right]}\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{‖W\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{E}_{n,p;s}\left(W,f\right).\end{array}$

(2) We see $W\left(X\right){\prod }_{i=1}^{s}{T}_{i}^{1/4}\left({x}_{i}\right)~\stackrel{˜}{W}\left(X\right)={\prod }_{i=1}^{s}{\stackrel{˜}{w}}_{i}\left({x}_{i}\right)$ ,

${T}_{i}^{1/4}w~{\stackrel{˜}{w}}_{i}\in \mathcal{F}\left({C}^{2}+\right),i=1,2,\cdots ,s$ . Then, there exists $P\in {\mathcal{P}}_{n}$ such that

${‖\stackrel{˜}{W}\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{E}_{n-1,p;s}\left(\stackrel{˜}{W},f\right)$ . Therefore, by Lemma 3.9 (2)

$\begin{array}{l}{‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ ={‖W\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+{‖W{v}_{n}^{\left[s\right]}\left(f-P\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{E}_{n,p;s}\left(\stackrel{˜}{W},f\right)\le C{E}_{n,p;s}\left(\left({\prod }_{i=1}^{s}{T}_{i}^{1/4}\right)W,f\right).\end{array}$

(3) Similarly, we have (3).

(4) It follows from Theorem 3.3. #

Theorem 5.2. Let ${w}_{i}\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(i=1,2,\cdots ,s\right),0<\lambda <3/2$ , and let $1\le p\le \infty$ . Then if $Wf\in {L}_{p}\left({ℝ}^{s}\right)$ , we have

${‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{\left(2{j}_{i}+1\right)/4}}{v}_{n}^{\left[s\right]}{\left(f\right)}^{\left({j}_{1},\cdots ,{j}_{s}\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)},$

and

${‖W{v}_{n}^{\left[s\right]}{\left(f\right)}^{\left({j}_{1},\cdots ,{j}_{s}\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}\left({\prod }_{i=1}^{s}{T}_{i}^{\left(2{j}_{i}+1\right)/4}\left({a}_{n}^{\left(i\right)}\right)\right){‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}.$

Proposition 5.3 (  , Lemma 2.5,  , Corollary 10.2). Let $1\le p\le \infty$ and $w\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right)$ . Then there exists a constant ${C}_{1}={C}_{1}\left(w,p\right)>0$ such that, if $P\in {\mathcal{P}}_{n}\left(ℝ\right)$ ( $n\in ℕ$ ),

${‖\frac{w}{{T}^{j/2}}{P}^{\left(j\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le {C}_{1}{\left(\frac{n}{{a}_{n}}\right)}^{j}{‖wP‖}_{{L}^{p}\left({ℝ}^{s}\right)},j\in ℕ,$

and

${‖w{P}^{\left(j\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le {C}_{1}{\left(\frac{nT{\left({a}_{n}\right)}^{1/2}}{{a}_{n}}\right)}^{j}{‖wP‖}_{{L}^{p}\left({ℝ}^{s}\right)},j\in ℕ.$

Proof of Theorem 5.2. We use Proposition 5.3 and Lemma 3.7 (1).

$\begin{array}{l}{‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{\left(2{j}_{i}+1\right)/4}}{v}_{n}^{\left[s\right]}{\left(f\right)}^{\left({j}_{1},\cdots ,{j}_{s}\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{-1/4}}{v}_{n}^{\left[s\right]}\left(f\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)},\end{array}$

and further, using Theorem A1 (the Markov-Bernstein inequality) in Appendix,

$\begin{array}{l}{‖W{v}_{n}^{\left[s\right]}{\left(f\right)}^{\left({j}_{1},\cdots ,{j}_{s}\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}\le C{\prod }_{i=1}^{s}{\left(\frac{n{T}_{i}^{1/2}\left({a}_{n}^{\left(i\right)}\right)}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{‖W{v}_{n}^{\left[s\right]}\left(f\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{\prod }_{i=1}^{s}{\left(\frac{n{T}_{i}^{1/2}\left({a}_{n}^{\left(i\right)}\right)}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{‖W{v}_{n}^{\left[s\right]}\left(f\right)‖}_{{L}^{p}\left(|{x}_{i}|\le {a}_{n}^{\left(i\right)},i=1,2,\cdots ,s\right)}\\ \le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{\prod }_{i=1}^{s}{T}_{i}^{\left(2{j}_{i}+1\right)/4}\left({a}_{n}^{\left(i\right)}\right){‖\frac{W}{{\prod }_{i=1}^{s}{T}_{i}^{1/4}}{v}_{n}^{\left[s\right]}\left(f\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C{\prod }_{i=1}^{s}{\left(\frac{n}{{a}_{n}^{\left(i\right)}}\right)}^{{j}_{i}}{\prod }_{i=1}^{s}{T}_{i}^{\left(2{j}_{i}+1\right)/4}\left({a}_{n}^{\left(i\right)}\right){‖Wf‖}_{{L}^{p}\left({ℝ}^{s}\right)}.\end{array}$ #

In the rest of only this section, we suppose

$w=\mathrm{exp}\left(-Q\right)={w}_{i}=\mathrm{exp}\left(-{Q}_{i}\right),i=1,2,\cdots ,s,$

so

${a}_{n}={a}_{n}^{\left(i\right)},T={T}_{i},i=1,2,\cdots ,s.$

Let

${W}_{i}:={W}_{i}\left({x}_{1},\cdots ,{x}_{i-1},{x}_{i+1},\cdots ,{x}_{s}\right):={\prod }_{j\ne i,1\le j\le s}{w}_{j}\left({x}_{j}\right),i=1,2,\cdots ,s.$

In (  , Corollary 8) we give the Favard-type inequalities:

Proposition 5.4  . Let $w\in \mathcal{F}\left({C}^{2}+\right)$ , and let $r\ge 0$ be an integer. Let $1\le p\le \infty$ , and let $w{f}^{\left(r\right)}\in {L}_{p}\left(ℝ\right)$ . Then we have

${E}_{p,n}\left(f,w\right)\le C{\left(\frac{{a}_{n}}{n}\right)}^{k}{‖w{f}^{\left(k\right)}‖}_{{L}_{p}\left(ℝ\right)},k=1,2,\cdots ,r,$

and equivalently,

${E}_{p,n}\left(f,w\right)\le C{\left(\frac{{a}_{n}}{n}\right)}^{k}{E}_{p,n-k}\left({f}^{\left(k\right)},w\right).$

The following theorem is a generalization of Proposition 5.4.

Theorem 5.5. We suppose

${w}_{j}=\mathrm{exp}\left(-{Q}_{j}\right)\in {\mathcal{F}}_{\lambda }\left({C}^{3}+\right)\left(0<\lambda <3/2\right),j=1,2,\cdots ,s$ , and let (3.3) satisfy, that is,

${T}_{j}\left({a}_{n}\right)\le c{\left(\frac{n}{{a}_{n}}\right)}^{2/3},j=1,2,\cdots ,s.$

Let $W{f}^{\left(r,r,\cdots ,r\right)}\in {L}_{p}\left({ℝ}^{s}\right)$ for some positive integer r. Then we have

${E}_{n,p;s}\left(W;f\right)\le C{\left(\frac{{a}_{n}}{n}\right)}^{r}{‖{T}^{〈s〉}W{f}^{\left(r,r,\cdots ,r\right)}‖}_{{L}^{p}\left({ℝ}^{s}\right)}.$

Equivalently,

${E}_{n,p;s}\left(W;f\right)\le C{\left(\frac{{a}_{n}}{n}\right)}^{r}{E}_{n-r,p;s}\left({T}^{〈s〉}W;{f}^{\left(r,r,\cdots ,r\right)}\right).$

Proof. Using

$f-{v}_{n}^{\left[s\right]}=\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)+\left({v}_{n}^{\left[1\right]}\left(f\right)-{v}_{n}^{\left[2\right]}\left(f\right)\right)+\cdots +\left({v}_{n}^{\left[s-1\right]}\left(f\right)-{v}_{n}^{\left[s\right]}\left(f\right)\right)$ , we get from Lemma 3.9 (2) and (3.2),

$\begin{array}{c}{E}_{n,p;s}\le {‖W\left(f-{v}_{n}^{\left[s\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le {‖W\left(f-{v}_{n}^{\left[1\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W\left({v}_{n}^{\left[j-1\right]}\left(f\right)-{v}_{n}^{\left[j\right]}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ \le C\left[{‖W\left(f-{v}_{n,1}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W{v}_{n}^{\left[j-1\right]}\left(f-{v}_{n,j}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\\ \le C\left[{‖W\left(f-{v}_{n,1}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}+\underset{j=2}{\overset{s}{\sum }}{‖W\left({\prod }_{k=1}^{j-1}{T}_{i}^{1/4}\right)\left(f-{v}_{n,j}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\right]\end{array}$

We estimate each term. From Proposition 5.4 with the weight ${T}_{j}^{1/4}{w}_{j}$ ,

$\begin{array}{l}{‖W\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right)\left(f-{v}_{n,j}\left(f\right)\right)‖}_{{L}^{p}\left({ℝ}^{s}\right)}\\ ={‖{‖{W}_{i}\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){w}_{j}\left({f}_{{\stackrel{^}{X}}_{j}}-{v}_{n,j}\left({f}_{{\stackrel{^}{X}}_{j}}\right)\right)‖}_{{L}^{p}\left({ℝ}_{\left(j\right)}\right)}‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le C{‖{W}_{i}\left({\prod }_{k=1}^{j-1}{T}_{k}^{1/4}\right){E}_{n,p}\left({T}_{j}^{1/4}{w}_{j},{f}_{{\stackrel{^}{X}}_{j}}\right)‖}_{{L}^{p}\left({ℝ}_{j}^{s-1}\right)}\\ \le C{\left(\frac{{a}_{n}}{n}\right)}^{r}{‖{\prod }_{i}}_{}\end{array}$