Boundedness of iterated spherical average on modulation spaces

TL;DR

This paper establishes the precise conditions under which the iterated spherical average operator, combined with the Laplacian, is bounded between different modulation spaces, advancing understanding in harmonic analysis and approximation theory.

Contribution

It provides necessary and sufficient conditions for the boundedness of the operator $ riangle (A_1)^N$ on modulation spaces, a novel result in harmonic analysis.

Findings

Derived conditions for boundedness between modulation spaces.

Extended understanding of spherical averages in harmonic analysis.

Connected spherical averages with approximation theory via Laplacian.

Abstract

The spherical average $A_{1}(f)$ and its iteration $(A_{1})^{N}$ are important operators in harmonic analysis and probability theory. Also $\Delta (A_{1})^{N}$ is used to study the $K$ functional in approximation theory, where $\Delta $ is the Laplace operator. In this paper, we obtain the sufficient and necessary conditions to ensure the boundedness of $\Delta (A_{1})^{N}$ from the modulation space $M_{p_{1},q_{1}}^{s_{1}}$ to the modulation space $M_{p_{2},q_{2}}^{s_{2}}$ for $1\leq p_{1},p_{2},q_{1},q_{2}\leq \infty $ and $s_{1},s_{2}\in \mathbb{R}$.

Equations

$$A_1(f)(x)={\int }_{{\mathbb{S}}^{n-1}}f(x-y^{\prime })d\sigma (y^{\prime }).$$

$${\Vert \Delta (A_1{)}^N(f)\Vert }_{L^p({\mathbb{R}}^n)}⪯\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$$

$${\Vert \Delta (A_1{)}^N(f)\Vert }_{L^1({\mathbb{R}}^n)}⪯\Vert f{\Vert }_{L^1({\mathbb{R}}^n)}.$$

$${\Vert \Delta (A_1{)}^N(f)\Vert }_{L^1({\mathbb{R}}^n)}⪯\Vert f{\Vert }_{L^1({\mathbb{R}}^n)}$$

$$(p,q,s)\in [1,+\infty )\times [1,+\infty )\times \mathbb{R}.$$

$$\widehat{e^{i|D|}f}=e^{i|\xi |}\hat{f}$$

$$p_1\le p_2\text{and}{s}_1\ge s_2+\sigma .$$

$$p_1\le p_2\text{and}{s}_1+\frac{n}{q_1}>s_2+\sigma +\frac{n}{q_2}.$$

$$a_1(f)(x)=\frac{1}{2}(f(x+1)+f(x-1)).$$

$$\sum _{k\in Z^n}\phi (\xi -k)=1,{\phi }_k(\xi ):=\phi (\xi -k)$$

$${\square }_k:={\mathcal{F}}^{-1}{\phi }_k\mathcal{F}.$$

$$M_{p,q}^s(R^n):=\left\{f\in S^{\prime }:\Vert f{\Vert }_{M_{p,q}^s(R^n)}={\left(\sum _{k\in Z^n}⟨k{⟩}^{sq}\Vert {\square }_{k}f{\Vert }_{L^p({\mathbb{R}}^n)}^q\right)}^{\frac{1}{q}}<\infty \right\},$$

$$J_{\sigma }=(I-△{)}^{\frac{\tau }{2}}:M_{p,q}^s\to M_{p,q}^{s-\tau }$$

$$M_{p_1,q_1}^{s_1}\subset M_{p_2,q_2}^{s_2},\text{if}{s}_1\ge s_2,0<p_1\le p_2,0<q_1\le q_2$$

$$M_{p_1,q_1}^{s_1}\subset M_{p_2,q_2}^{s_2},\text{if}{s}_1>s_2,q_1>q_2,s_1-s_2>n/q_2-n/q_1.$$

$$\widehat{m(D)f}(\xi )=m(\xi )\hat{f}(\xi ).$$

$$K(x)=(m(\xi ){)}^{\vee }(x)={\int }_{{\mathbb{R}}^n}m(\xi )e^{i\xi \cdot x}d\xi .$$

$$\Vert m(D)f{\Vert }_{L^p}⪯\Vert (m(\xi ){)}^{\vee }(x){\Vert }_{L^1}\Vert f{\Vert }_{L^p}$$

$${\Vert (m(\xi ){)}^{\vee }(x)\Vert }_{L^p}⪯\sum _{|\gamma |\le \left[n(\frac{1}{p}-\frac{1}{2})\right]+1}\Vert {\partial }^{\gamma }m(\xi ){\Vert }_{L^2}.$$

$$\widehat{\Delta (A_1{)}^{N}f}\simeq |\xi {|}^2(V_{\frac{n-2}{2}}(|\xi |){)}^N\hat{f}(\xi )$$

$$V_{\delta }(r)=\frac{J_{\delta }(r)}{r^{\delta }}$$

$$J_{\delta }(r)=\sqrt{\frac{2}{\pi r}}cos\left(r-\frac{\delta \pi }{2}-\frac{\pi }{4}\right)+\sum _{j=1}^L{a}_j{e}^{ir}{r}^{-\frac{1}{2}-j}+\sum _{j=1}^L{b}_j{e}^{-ir}{r}^{-\frac{1}{2}-j}+E(r)$$

$$\left|E^{(k)}(r)\right|⪯r^{-\frac{1}{2}-L-1}$$

$${\Vert {\square }_k\Delta (A_1{)}^{N}f\Vert }_{L^{p_2}({\mathbb{R}}^n)}⪯⟨k{⟩}^{\sigma }\Vert {\square }_{k}f{\Vert }_{L^{p_2}({\mathbb{R}}^n)}.$$

$${\Omega }_k(x)={\int }_{{\mathbb{R}}^n}{\phi }_k(\xi )|\xi {|}^2{\left(V_{\frac{n-2}{2}}(|\xi |)\right)}^N{e}^{i\xi \cdot x}d\xi .$$

$$\sum _{k\in Z^n}{\square }_k=I$$

$${\Vert {\square }_k\Delta (A_1{)}^{N}f\Vert }_{L^{p_2}}$$

$$\sum _{l\in {\mathbb{Z}}^n,|l-k|\le k_0(n)}{\Vert {\left({\phi }_l(\xi )|\xi {|}^2(V_{\frac{n-2}{2}}(|\xi |){)}^N\right)}^{\vee }\Vert }_{L^1}$$

$${\Lambda }_k:=\{l\in {\mathbb{Z}}^n:|l-k|\le k_0(n)\}$$

Full text

This work is supported by the NSF of China (Grant No.11801518) and NSFZJ (  No.LQ18A010005).

E-mail addresses: huangqiang0704@163.com(Q.Huang), fan@uwm.edu(D.Fan)

1. Introduction

Let $\mathbb{S}^{n-1}$ be the unit sphere in the Euclidean space $\mathbb{R}^{n},$ $n\geq 2.$ We equip it with the normalized surface Lebesgue measure $d\sigma(y^{\prime})$. The average operator of functions $f$ on the unit sphere is defined as

[TABLE]

This operator has a profound background in harmonic analysis, dating back to early 1970’s (see [16],[15]). Moreover, it is closely related to the study of random walks in high dimensional spaces, which is originated by Pearson [13] about 120 years ago. An $N$-steps uniform walk in $\mathbb{R}^{n}$ starts at the origin and consists of $N$ independent steps of length 1, each of which is taken into a uniformly random direction. It is known that the probability density function $p_{N}(\frac{n-2}{2},x)$ of such a random walk is the Fourier inverse of $(A_{1})^{N}$ (see [3]), where $(A_{1})^{N}$ denotes the $N$ iteration of $A_{1}.$

The operator $A_{1}$ also plays a significant role in the approximation theory (see [1]). Let $\Delta$ be the Laplacian. In order to obtain some equivalent forms of the K-functional in $L^{p}(\mathbb{R}^{n})$ spaces, Belinsky, Dai and Ditzian in [1] study the iterates $(A_{1})^{N}$ for positive integers $N$ and obtain the following theorem.

**Theorem A **([1]) Let $1\leq p\leq\infty$, $n\geq 2$ and $N>\frac{2(n+2)}{n-1}$. The inequality

[TABLE]

holds for all $f\in L^{p}(\mathbb{R}^{n})$.

Theorem A then raised the following question.

Question 1 ([1]): Find the smallest positive integer $N$ to guarantee the inequality

[TABLE]

This question was addressed by Fan and Zhao in [6] using the well known estimates of wave operators (see [11][14]), and recently the question was completely solved by Fan, Lou and Wang in [5] in the following theorem.

**Theorem B **([5]). Let $n\neq 3,5,$ and $N$ be positive integers. The inequality

[TABLE]

holds if and only if $N>\frac{n+3}{n-1}$.

Let $n=3,5,$ *and $N$ be positive integers. The inequality   *

[TABLE]

holds if and only if $N\geq\frac{n+3}{n-1}$.

The aim of this article is to explore the behaves of $\Delta(A_{1})^{N}$ on the modulation spaces $M_{p,q}^{s},$ where

[TABLE]

We recall that the modulation space $M_{p,q}^{s}$ was introduced by Feichtinger in [7] and his initial aim was to measure smoothness of a function or distribution in a way different from $L^{p}$ spaces. Nowadays, spaces $M_{p,q}^{s}$ are recognized as a useful tool for studying functional analysis and pseudo-differential operators (see [2][4][17]). The original definition of the modulation space in [7] is based on the short-time Fourier transform and window function. In [10], Wang and Hudizk gave an equivalent definition of the discrete version on modulation spaces by employing the frequency-uniform-decomposition. Later, people found that the space $M_{p,q}^{s}$, with this discrete version, is a good working frame to study boundedness of some operators and certain Cauchy problems of nonlinear partial differential equations (see [12][19][8][9]). For example, the wave operator

[TABLE]

is bounded in $L^{p}$ spaces if and only if $p=2$ when $n\geq 2.$ However, $e^{i|D|}$ is bounded on modulation space $M_{p,q}^{s}$ for any $p,q\in[1,+\infty)$ and $s\in\mathbb{R}$.

Motivated by these works, in this paper, we study boundedness of $\Delta(A_{1})^{N}$ on modulation spaces and give the sufficient and necessary conditions on the boundedness of $\Delta(A_{1})^{N}$ from $M_{p_{1},q_{1}}^{s_{1}}$ to $M_{p_{2},q_{2}}^{s_{2}}$ for $1\leq p_{1},p_{2},q_{1},q_{2}\leq\infty$, $s_{1},s_{2}\in\mathbb{R}$. The following theorem is our main result.

Theorem 1.1**.**

Let $\sigma=2-\frac{n-1}{2}N$ and $1\leq p_{i},q_{i}\leq\infty$ $s_{i}\in R$ for $i=1,2$. When $q_{1}\leq q_{2},$ the iterated spherical average $\Delta(A_{1})^{N}$ is bounded from $M_{p_{1},q_{1}}^{s_{1}}(R^{n})$ to $M_{p_{2},q_{2}}^{s_{2}}(R^{n})$ if and only if

[TABLE]

When $q_{1}>q_{2},$ the iterated spherical average $\Delta(A_{1})^{N}$ is bounded from $M_{p_{1},q_{1}}^{s_{1}}(R^{n})$ to $M_{p_{2},q_{2}}^{s_{2}}(R^{n})$ if and only if

[TABLE]

Remark 1**.**

In above theorem, we can see that the smallest iterate step N which ensures $\Delta(A_{1})^{N}$ is bounded on modulation spaces $M_{p,q}^{s}(\mathbb{R}^{n})$ for all $(p,q,s)\in[1,+\infty)\times[1,+\infty)\times\mathbb{R}$ is $\frac{4}{n-1}$, which is smaller than that in $L^{p}(\mathbb{R}^{n})$ spaces (see Theorem B). Moreover, our theorem finds the sufficiency and necessity for the boundedness of $\Delta(A_{1})^{N}$ from $M_{p_{1},q_{1}}^{s_{1}}$ to $M_{p_{2},q_{2}}^{s_{2}}$ on full ranges of $1\leq p_{1},p_{2},q_{1},q_{2}\leq\infty$ and $s_{1},s_{2}\in\mathbb{R}$.

Remark 2**.**

When $n=1$, the average of sphere is reduced to

[TABLE]

Clearly, $a_{1}(f)$ and its iterates $a_{1}^{N}(f)$ are not in general smoother than $f(x)$. However, with the increase of the dimension of space, the spherical average $A_{1}(f)$ shares more regularity than $f(x)$. Actually, our result also reflects this interesting phenomenon. If we choose $n=1$ in Theorem 1.1, by the isomorphism property of modulation spaces (see Proposition 2.1 ), $(A_{1})^{N}$ is bounded form $M_{p,q}^{s_{1}}(\mathbb{R}^{1})$ to $M_{p,q}^{s_{2}}(\mathbb{R}^{1})$ if and only if $s_{2}\geq s_{1}$ for any iterate steps $N$. However, when $n\geq 2$ in Theorem 1.1, we can gain $\frac{n-1}{2}$ units of regularity in each iterate step of $A_{1}$.

The sufficiency part of the proof for Theorem 1.1 is somewhat routine with the help of Bernstein’s multiplier theorem. The necessity part of the proof is quite involved. Based on the structure of $M_{p,q}^{s}$ and asymptotic form of the Fourier transform of $\Delta(A_{1})^{N},$ we construct a sequence of functions $\left\{f_{k_{j},\lambda}\right\}$ to achieve the necessary conditions.

This paper is organized as follows. In Section 2, we will introduce some preliminary knowledge which includes some properties of modulation spaces and some useful lemmas. The proof of Theorem 1.1 will be presented in Section 3.

Throughout this paper, we use the inequality $A\preceq B$ to mean that there is a positive number $C$ independent of all main variables such that $A\leq CB$, and use the notation $A\simeq B$ to mean $A\preceq B$ and $B\preceq A$.

2. Preliminaries and Lemmas

In this section, we give the definition and discuss some basic properties of modulation spaces. Also, we will prove some estimates and lemmas which will be used in our proof.

**Definition 2.1 **(Modulation spaces) Let $\varphi(\xi)$ be a smooth function satisfying $\varphi(\xi)\equiv 1$ for $x\in\{x\in\mathbb{R}^{n}:|\xi|<\frac{1}{2}\}$, $\mathrm{supp}\varphi\subset\{\xi\in\mathbb{R}^{n}:|\xi|<\frac{3}{2}\}$ and $\{\varphi_{k}\}$ be a partition of the unity satisfying the following conditions:

[TABLE]

for any $\xi\in R^{n}$. And let

[TABLE]

With this frequency-uniform decomposition operator, we define the modulation spaces  $M_{p,q}^{s}(R^{n}),$ for $0<p,q\leq\infty,$ $-\infty<s<\infty,$ by

[TABLE]

where $\langle k\rangle=\sqrt{1+|k|^{2}}$. See [10] for details.

**Proposition 2.1 **(Isomorphism, see [10]) Let $0<p,q\leq\infty,s,\tau\in\mathbb{R}$.

[TABLE]

is an isomorphic mapping, where  $I$  is the identity mapping and  $\Delta$  is the Laplacian.

**Proposition 2.2 **(Embedding, see [10]) For $0<p_{i},q_{i}\leq\infty,s_{i}\in\mathbb{R}$ ($i=1,2$), we have

[TABLE]

[TABLE]

The Fourier multiplier is a linear operator $m(D)$ whose action on a test function $f$ is formally defined by

[TABLE]

The function $m(\xi)$ is called the symbol or multiplier of $m(D)$. Up to a constant multiple, $m(D)$ is a convolution operator with the kernel

[TABLE]

By the Young inequality, we have

[TABLE]

for any $1\leq p\leq\infty$. We will use the following Bernstein multiplier theorem to estimate $\|(m(\xi))^{\vee}(x)\|_{L^{1}}$.

Lemma 2.1**.**

(Bernstein’s multiplier theorem, see [18]) Assume $0<p\leq 2$ and $\partial^{\gamma}m(\xi)\in L^{2}$ for all multi-indices $\gamma$ with $|\gamma|\leq\left[n(\frac{1}{p}-\frac{1}{2})\right]+1$. We have

[TABLE]

By checking the Fourier transform (see [15]), we have that

[TABLE]

where

[TABLE]

and $J_{\delta}(r)$ is the Bessel function of order $\delta$. Recall the following asymptotic form of $J_{\delta}(r)$.

Lemma 2.2**.**

([15]) Let $r>1$ and $\delta>-\frac{1}{2}$. For any positive integer $L$ and $r\in[1,\infty)$, we have

[TABLE]

where $a_{j}$ and $b_{j}$ are constants for all $j$, and $E(r)$ is a $C^{\infty}$ function satisfying

[TABLE]

for any $k=0,1,2...$

3. Proof of Theorem 1.1

We start with showing the sufficiency of Theorem 1.1. By the definition of modulation spaces, we need to estimate $\left\|\square_{k}\Delta(A_{1})^{N}f\right\|_{L^{p_{2}}(\mathbb{R}^{n})}$ and obtain the following lemma.

Lemma 3.1**.**

Let $1\leq p_{2}\leq\infty$ and $\sigma=2-\frac{n-1}{2}N$. Then

[TABLE]

Proof: For $\forall k\in\mathbb{Z}^{n}$, $\square_{k}\Delta(A_{1})^{N}f$ is a convolution operator $m_{k}(D)(f)=\Omega_{k}(x)\ast f$, where

[TABLE]

By the almost orthogonality of unit decomposition, there exists an integer $k_{0}(n)$ which depends only on $n$ such that $\varphi_{l}(\xi)\varphi_{k}(\xi)=0$ when $|l-k|\geq k_{0}(n)$. Since

[TABLE]

where $I$ is the identity operator, Young’s inequality and Minkowski’s inequality yield

[TABLE]

So, it suffices to estimate

[TABLE]

for every $k\in\mathbb{Z}^{n}.$ Notice that the cardinality of

[TABLE]

is uniformly finite for all $k\in\mathbb{Z}^{n}$, and $\langle l\rangle\simeq\langle k\rangle$ when $l\in\Lambda_{k}$. Therefore, we only need to estimate the $L^{1}$ norm

[TABLE]

for $\langle l\rangle\simeq\langle k\rangle$.

When $|k|<100$, by the well known formula ([15])

[TABLE]

we have that $|\Omega_{l}(x)|\preceq 1$ for $|x|<100$.

On the other hand, when $|x|\geq 100$, without loss of generality, we may assume $|x_{1}|\geq\frac{|x|}{n}$. By the derivative formula of $V_{\delta}(t)$

[TABLE]

and taking integration by part on $\xi_{1}$ variable in (7), we obtain that

[TABLE]

for $|x|\geq 100$. This estimate implies that $\|\Omega_{l}(x)\|_{L^{1}}\preceq 1$ when $|k|<100$, since $\langle l\rangle\simeq\langle k\rangle$.

Next, we study the case $|k|\geq 100$. Choosing $L=1$ in Lemma 2.2, we have the following asymptotic form of $V_{\delta}(r)$

[TABLE]

for $|r|>1$.

Therefore, when $|k|>100$ and $\langle l\rangle\simeq\langle k\rangle$, we have

[TABLE]

for $\xi\in\mathrm{supp}\varphi_{l}(\xi)$. Now, by the chain rule and the derivative formula of $V_{\delta}(t)$ , we obtain

[TABLE]

By the asymptotic form of $V_{\delta}(r)$, we obtain that

[TABLE]

for $\xi\in\mathrm{supp}\varphi_{l}(\xi)$.

Thus, $V_{\delta}(|\xi|)^{N}$ and $\frac{\partial}{\partial\xi_{i}}\left(V_{\delta}(|\xi|)\right)^{N}$ share the same upper bound which is $\langle l\rangle^{(-\delta-\frac{1}{2})N}$, for any $\delta$ and $\xi\in\mathrm{supp}\varphi_{l}(\xi)$. By Lemma 2.1 (Bernstein’s multiplier theorem) and the fact $\partial^{\gamma}(|\xi|^{2})\preceq|\xi|^{2-|\gamma|}$ for $|\gamma|\leq 2$ and $\partial^{\gamma}(|\xi|^{2})=0$ for $|\gamma|>2$, we have that

[TABLE]

for $l\in\Lambda_{k}$.

Combining all above estimates, by the definition of modulation spaces, we have that

[TABLE]

By the embedding properties of modulation spaces (Proposition 2.2), we can easily obtain that

[TABLE]

when

[TABLE]

or

[TABLE]

The sufficiency of Theorem 1.1 is proved.

Turn to prove the necessity part of Theorem 1.1. We need the following lemma.

Lemma 3.2**.**

Let $1\leq p\leq\infty$. These exists a constant $\rho=\rho(n)>0$ which depends only on $n$ and a subsequence $\{k_{j}\}\subseteq\mathbb{Z}^{n}$ such that

[TABLE]

where $\{g_{k_{j}}(x)\}$ is a sequence of Schwartz function with $\mathrm{supp}\widehat{g_{k_{j}}}(\xi)\subset\{\xi\in\mathbb{R}^{n}:|\xi-k_{j}|\leq\rho\}$.

Proof: By the same method as in Lemma 3.1, it is easy to get

[TABLE]

for all $k\in\mathbb{Z}^{n}$. Thus, we only need to prove the inverse inequality. By Lemma 2.2, we have

[TABLE]

for $|r|>1$. We consider

[TABLE]

in every semiperiod $r-\frac{n\pi}{4}+\frac{3\pi}{4}\in[j\pi,(j+1)\pi]$, $(j=1,2,...).$

Choosing $\varepsilon_{0}=\sin(0.07)$, we have

[TABLE]

for $r-\frac{n\pi}{4}+\frac{3\pi}{4}\in[j\pi+0.07,j\pi+\pi-0.07]$, which is equivalent to

[TABLE]

By Lemma 3.3 (the lemma will be proved later), for every $j\in\mathbb{N}^{+}$, the set

[TABLE]

is not empty. So, there exists a subsequence of integer $\{k_{j}\}$, such that $k_{j}\in\Lambda_{1,j}$ and

[TABLE]

Moreover,

[TABLE]

which means that

[TABLE]

for

[TABLE]

and

[TABLE]

For the remainder $O(r^{-\frac{n+1}{2}})$ in the expansion of $V_{\frac{n-2}{2}}(r)$, it is obvious that, when $r$ is large enough,

[TABLE]

Let $\epsilon,\rho=\frac{\varepsilon_{0}}{4}$. We obtain that there exist some constants $\epsilon,\rho>0$ and a subsequence $\{k_{j}\}\subseteq\mathbb{Z}^{+}$ such that

[TABLE]

for $\xi\in\{\xi:|\xi-k_{j}|\leq\rho\}$ when $j$ is large enough. Moreover the subsequence $\{k_{j}\}\subseteq\mathbb{Z}^{n}$ satisfies

[TABLE]

when the positive integer $j$ is large enough.

Therefore, when $\xi\in\{\xi:|\xi-k_{j}|\leq\rho\}$ and $N\in\mathbb{Z}^{+}$, we have

[TABLE]

Using the chain rule and the derivative formula of $V_{\delta}(t)$,

[TABLE]

By the asymptotic form of $V_{\delta}(r)$ and (17), we have

[TABLE]

As a result, $V_{\frac{n-2}{2}}(|\xi|)^{-N}$ and $\frac{\partial}{\partial\xi_{i}}\left(V_{\frac{n-2}{2}}(|\xi|)\right)^{-N}$ share the same upper bound which is $\langle k_{j}\rangle^{(\frac{n-1}{2})N}$, for $\xi\in\{\xi:|\xi-k_{j}|\leq\rho\}.$

Let $\eta(\xi)$ be a smooth function with $\mathrm{supp}~{}{\eta_{k}(\xi)}\subset\{\xi:|\xi|\leq 2\rho\}$ and $\eta_{k}(\xi)\equiv 1$ for $\xi\in\{\xi:|\xi|\leq\rho\}$. We define

[TABLE]

Notice that $\rho=\frac{\sin(0.07)}{4}<\frac{1}{4}$. Moreover, for the partition of the unity $\{\varphi_{k}\}$(see Definition 2.1), we have that

[TABLE]

with

[TABLE]

Therefore, by the Bernstein multiplier theorem ( Lemma 2.1) and (17), we obtain that

[TABLE]

Combining above estimate with (15), Lemma 3.2 is proved.

Next, we first verify the condition

[TABLE]

and

[TABLE]

for $\forall q_{1},q_{2}\in[1,+\infty)$. Let $f(x)$ be a nonzero Schwartz function with $\mathrm{supp}\widehat{f}(\xi)\subset\{\xi:|\xi|<\frac{1}{2}\}$. Define

[TABLE]

for $\lambda\in(0,\rho]$, where $\rho$ and $\{k_{j}\}$ are defined in Lemma 3.2. By the definition of $f_{k_{j},\lambda}(x)$, we have

[TABLE]

and

[TABLE]

Then, by Lemma 3.2, we have

[TABLE]

On the other hand

[TABLE]

By the assumption that $\Delta(A_{1})^{N}$ is bounded from $M_{p_{1},q_{1}}^{s_{1}}$ to $M_{p_{2},q_{2}}^{s_{2}}$, we have that

[TABLE]

for all $|k_{j}|$ sufficiently large and $0<\lambda\leq\rho$. Fix $k_{j}$ and let $\lambda$ go to 0. We have

[TABLE]

Thus, the condition $p_{1}\leq p_{2}$ must be hold. Moreover, when $\lambda$ is fixed and $k_{j}$ goes to infinity, we have

[TABLE]

which yields $s_{2}+\sigma\leq s_{1}$.

For the condition of $q$, we first establish the following lemma.

Lemma 3.3**.**

For $j\in\mathbb{N}^{+}$, define

[TABLE]

and

[TABLE]

When $j$ is big enough, we have

[TABLE]

where $C(n)$ is a positive constant depends only n.

Proof: The proofs for $n=2$ and $n>2$ share the same idea. We prove only the case $n=2$ explicitly and leave the proof of another case to the reader.

By symmetry, we only need to consider the case $\{(x,y)\in\mathbb{R}^{2}:x,y\geq 0\}.$ For $j\in\mathbb{Z}^{+}$, we define

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Moreover, for $r,a>0$, $0\leq y\leq r$, we define an auxiliary function

[TABLE]

Taking derivative we know that

[TABLE]

and $f_{r,a}(y)$ is a monotone increasing function.

Then, for any $(x_{0},y_{0})\in I_{x>y}$, we have

[TABLE]

Therefore,

[TABLE]

It is obvious to see that $\lim\limits_{j\rightarrow+\infty}\max\limits_{(x_{0},y_{0})\in I_{x>y}}f_{j\pi,0.07}(y_{0})=\frac{0.14}{\sqrt{2}}<1$. Thus, for any $(x_{0},y_{0})\in I_{x>y}$ we have

[TABLE]

when $j$ is big enough.

On the other hand, for any $(x_{0},y_{0})\in II$, We have

[TABLE]

By monotonicity of $f_{r,a}(y)$,

[TABLE]

Thus, for every $(x_{0},y_{0})\in I_{x>y}\bigcap\mathbb{Z}^{2}$, we have

[TABLE]

and

[TABLE]

Combing all above analysis, we have

[TABLE]

Now, we consider the domain $III_{x>y}$. By the same argument, for any $(x_{0},y_{0})\in III_{x>y}$, we have

[TABLE]

It is easy to see

[TABLE]

Thus, for any $(x_{0},y_{0})\in III_{x>y}$ we have

[TABLE]

when $j$ is big enough. Moreover, it is obvious

[TABLE]

when $j\geq 3$. So, for every $(x_{0},y_{0})\in III_{x>y}$,

[TABLE]

when $j\geq 3$. By (23), we can also obtain

[TABLE]

On the other hand, for $I_{y\geq x}$ and $III_{y\geq x}$, by the same method on the auxiliary function

[TABLE]

we can obtain that

[TABLE]

and

[TABLE]

Combing all above estimates, we have

[TABLE]

Next, we prove the necessary conditions when $q_{1}\leq q_{2}$. By (2.1) and above analysis, the boundedness of $\Delta(A_{1})^{N}$ from $M_{p_{1},q_{1}}^{s_{1}}$ to $M_{p_{2},q_{2}}^{s_{2}}$ must hold for $p_{1}\leq p_{2},s_{2}+\sigma\leq s_{1}$ and $q_{1}\leq q_{2}$. Therefore, we only need to consider the case $q_{1}>q_{2}$. Let $M$ be a large positive number. Define

[TABLE]

where $a_{j}>0$ are constants to be chosen later and $f_{k_{j},\rho}(x)$ are defined in (19) with all $k_{j}$ satisfy

[TABLE]

for some $L\in\mathbb{N^{+}}.$

By (20) (21) and the almost orthogonality of $\{\varphi_{k}\}$, we have

[TABLE]

and

[TABLE]

By the assumption that $\Delta(A_{1})^{N}$ is bounded form $M_{p_{1},q_{1}}^{s_{1}}$ to $M_{p_{2},q_{2}}^{s_{2}}$, we have

[TABLE]

By choosing $a_{j}=\langle k_{j}\rangle^{\frac{s_{1}q_{1}-(s_{2}+\sigma)q_{2}}{q_{1}-q_{2}}}$, we obtain

[TABLE]

By the assumption $q_{1}>q_{2}$, the above series converges as $M\rightarrow+\infty$. By Lemma 3.3, we have

[TABLE]

Therefore, it must yield

[TABLE]

which is equivalent to $s_{1}+\frac{n}{q_{1}}>s_{2}+\sigma+\frac{n}{q_{2}}$. Theorem 1.1 is proved.