Sharp Estimates of 𝑚-Linear 𝑝-Adic Hardy and Hardy-Littlewood-Pólya Operators

Abstract

The sharp estimates of the 𝑚-linear 𝑝-adic Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces with power weights are obtained in this paper.

1. Introduction

In recent years, 𝑝-adic numbers are widely used in theoretical and mathematical physics (cf. [1–8]), such as string theory, statistical mechanics, turbulence theory, quantum mechanics, and so forth.

For a prime number 𝑝, let ℚ_𝑝 be the field of 𝑝-adic numbers. It is defined as the completion of the field of rational numbers ℚ with respect to the non-Archimedean 𝑝-adic norm |⋅|_𝑝. This norm is defined as follows: |0|_𝑝=0; If any nonzero rational number 𝑥 is represented as 𝑥=𝑝^𝛾(𝑚/𝑛), where 𝑚 and 𝑛 are integers which are not divisible by 𝑝 and 𝛾 is an integer, then |𝑥|_𝑝=𝑝^−𝛾. It is not difficult to show that the norm satisfies the following properties: 𝑥𝑦_𝑝=|𝑥|_𝑝𝑦_𝑝,𝑥+𝑦_𝑝≤max|𝑥|_𝑝,𝑦_𝑝. (1.1) From the standard 𝑝-adic analysis [6], we see that any nonzero 𝑝-adic number 𝑥∈ℚ_𝑝 can be uniquely represented in the canonical series 𝑥=𝑝^𝛾∞_𝑗=0𝑎_𝑗𝑝^𝑗,𝛾=𝛾(𝑥)∈ℤ, (1.2) where 𝑎_𝑗 are integers, 0≤𝑎_𝑗≤𝑝−1, 𝑎₀≠0. The series (1.2) converges in the 𝑝-adic norm because |𝑎_𝑗𝑝^𝑗|_𝑝=𝑝^−𝑗. Denote by ℚ^∗_𝑝=ℚ_𝑝⧵{0} and ℤ_𝑝={𝑥∈ℚ_𝑝∶|𝑥|_𝑝≤1}.

The space ℚ^𝑛_𝑝 consists of points 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), where 𝑥_𝑗∈ℚ_𝑝,𝑗=1,2,…,𝑛. The 𝑝-adic norm on ℚ^𝑛_𝑝 is |𝑥|_𝑝∶=max_{1≤𝑗≤𝑛}𝑥_𝑗𝑝,𝑥∈ℚ^𝑛_𝑝. (1.3) Denote by 𝐵_𝛾(𝑎)=𝑥∈ℚ^𝑛_𝑝∶|𝑥−𝑎|_𝑝≤𝑝^𝛾, (1.4) the ball with center at 𝑎∈ℚ^𝑛_𝑝 and radius 𝑝^𝛾, and 𝑆_𝛾(𝑎)∶=𝑥∈ℚ^𝑛_𝑝∶|𝑥−𝑎|_𝑝=𝑝^𝛾=𝐵_𝛾(𝑎)⧵𝐵_𝛾−1(𝑎). (1.5) Since ℚ^𝑛_𝑝 is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure 𝑑𝑥 on ℚ^𝑛_𝑝, which is unique up to positive constant multiple and is translation invariant. We normalize the measure 𝑑𝑥 by the equality 𝑑𝑥=𝐵_𝐵0(0)0(0)_𝐻=1, (1.6) where |𝐸|_𝐻 denotes the Haar measure of a measurable subset 𝐸 of ℚ^𝑛_𝑝. By simple calculation, we can obtain that 𝐵_𝛾(𝑎)_𝐻=𝑝^𝛾𝑛,𝑆_𝛾(𝑎)_𝐻=𝑝^𝛾𝑛1−𝑝^−𝑛, (1.7) for any 𝑎∈ℚ^𝑛_𝑝. For a more complete introduction to the 𝑝-adic field, see [6] or [9].

The space ℚ^𝑛_𝑝×ℚ^𝑛_𝑝×⋯×ℚ^𝑛_𝑝𝑚 consists of points (𝑦₁,𝑦₂,…,𝑦_𝑚), where 𝑦_𝑖=(𝑦_𝑖1,𝑦_𝑖2,…,𝑦_𝑖𝑛)∈ℚ^𝑛_𝑝, 𝑖=1,2,…,𝑚. The 𝑝-adic norm of 𝑚-tuple (𝑦₁,𝑦₂,…,𝑦_𝑚) is 𝑦₁,𝑦₂,…,𝑦_𝑚𝑝∶=max_{1≤𝑖≤𝑚}𝑦_𝑖𝑝. (1.8)

Recently, 𝑝-adic analysis has received a lot of attention due to its application in mathematical physics. There are numerous papers on 𝑝-adic analysis, such as [10, 11] about Riesz potentials, [12–16] about 𝑝-adic pseudodifferential equations, and so forth. The harmonic analysis on 𝑝-adic field has been drawing more and more concern (cf. [17–21] and references therein).

The well-known Hardy’s integral inequality [22] tells us that for 1<𝑞<∞, 𝐻𝑓_{𝐿𝑞(ℝ+)}≤𝑞𝑓𝑞−1^{𝐿𝑞(ℝ+)}, (1.9) where the classical Hardy operator is defined by (𝑥)1𝐻𝑓∶=𝑥^𝑥₀𝑓(𝑡)𝑑𝑡, (1.10) for nonnegative integral function 𝑓 on ℝ⁺, and the constant 𝑞/(𝑞−1) is the best possible. Thus the norm of Hardy operator on 𝐿^𝑞(ℝ⁺) is ‖𝐻‖_𝐿^𝑞_(ℝ⁺_)→𝐿^𝑞_(ℝ⁺₎=𝑞.𝑞−1 (1.11)

Faris [23] introduced the following 𝑛-dimensional Hardy operator, for nonnegative function 𝑓 on ℝ^𝑛, (𝑥)1ℋ𝑓∶=Ω_𝑛|𝑥|^𝑛_{|𝑦|<|𝑥|}𝑓𝑦𝑑𝑦,𝑥∈ℝ^𝑛⧵{0}, (1.12) where Ω_𝑛 is the volume of the unit ball in ℝ^𝑛. Christ and Grafakos [24] obtained that the norm of ℋ on 𝐿^𝑞(ℝ^𝑛) is ‖ℋ‖_𝐿^𝑞_(ℝ^𝑛_)→𝐿^𝑞_(ℝ^𝑛₎=𝑞,𝑞−1 (1.13) which is the same as that of the 1-dimension Hardy operator. In [25], Fu et al. introduced the 𝑚-linear Hardy operator, which is defined by ℋ^𝑚𝑓₁,…,𝑓_𝑚(𝑥)=1Ω_𝑚𝑛|𝑥|^𝑚𝑛_{|(𝑦1,…,𝑦𝑚)|<|𝑥|}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (1.14) where 𝑥∈ℝ^𝑛⧵{0} and 𝑓₁,…,𝑓_𝑚 are nonnegative locally integrable functions on ℝ^𝑛. And they obtained the precise norms of ℋ^𝑚 on Lebesgue spaces with power weight. The authors of [26] also got the best constants of 𝑚-linear Hilbert, Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces.

The study of multilinear averaging operators in Euclidean spaces is a byproduct of the recent interest in multilinear singular integral operator theory. This subject was established by Coifman and Meyer [27] in 1975. In this article, we consider the sharp estimates of 𝑚-linear 𝑝-adic Hardy and Hardy-Littlewood-Pólya operators. In contrast with [25], we use a new technique in calculations based on the feature of 𝑝-adic field, and Theorem 3.1 is also new. They cannot be obtained immediately by [25]. In [28], we defined the 𝑝-adic Hardy operator.

Definition 1.1. For a function 𝑓 on ℚ^𝑛_𝑝, we define the 𝑝-adic Hardy operator as follows ℋ^𝑝𝑓(𝑥)=1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓(𝑡)𝑑𝑡,𝑥∈ℚ^𝑛_𝑝⧵{0}, (1.15) where 𝐵(0,|𝑥|_𝑝) is a ball in ℚ^𝑛_𝑝 with center at 0∈ℚ^𝑛_𝑝 and radius |𝑥|_𝑝.

It is obvious that |ℋ^𝑝𝑓|≤ℳ^𝑝𝑓, where ℳ^𝑝 is the Hardy-Littlewood maximal operator [17] defined by ℳ^𝑝𝑓(𝑥)=sup_𝛾∈ℤ1𝐵_𝛾(𝑥)_𝐻^𝐵_𝛾^(𝑥)𝑓𝑦𝑑𝑦,𝑓∈𝐿¹_locℚ^𝑛_𝑝. (1.16) The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. The boundedness of ℳ^𝑝 on 𝐿^𝑞(ℚ^𝑛_𝑝) has been solved (see, e.g., [9]). But the best estimate of ℳ^𝑝 on 𝐿^𝑞(ℚ^𝑛_𝑝),𝑞>1, even that of Hardy-Littlewood maximal operator on Euclidean spaces ℝ^𝑛 is very difficult to obtain. Instead, we have obtained the sharp estimates of ℋ^𝑝 (and 𝑝-adic Hardy-Littlewood-Pólya operator) elsewhere.

Definition 1.2. Let 𝑚 be a positive integer and 𝑓₁,…,𝑓_𝑚 be nonnegative locally integrable functions on ℚ^𝑛_𝑝. The 𝑚-linear𝑝-adic Hardy operator is defined by ℋ^𝑝_𝑚𝑓₁,…,𝑓_𝑚(𝑥)=1|𝑥|_𝑝^𝑚𝑛_{|(𝑦1,…,𝑦𝑚)|𝑝≤|𝑥|𝑝}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (1.17) where 𝑥∈ℚ^𝑛_𝑝⧵{0}.

The Hardy-Littlewood-Pólya’s linear operator [26] is defined by 𝑇𝑓(𝑥)=^∞₀𝑓𝑦max𝑥,𝑦𝑑𝑦. (1.18) In [26], the authors obtained that the norm of Hardy-Littlewood-Pólya’s operator on 𝐿^𝑞(ℝ⁺) (see also [22, page 254]), 1<𝑞<∞, is ‖𝑇‖_𝐿^𝑞_(ℝ⁺_)→𝐿^𝑞_(ℝ⁺₎=𝑞².𝑞−1 (1.19)

We define the 𝑝-adic Hardy-Littlewood-Pólya operator as (see [28]) 𝑇^𝑝(𝑥)=_ℚ𝑝𝑓𝑦max|𝑥|_𝑝,𝑦_𝑝𝑑𝑦,𝑥∈ℚ^∗_𝑝. (1.20)

Definition 1.3. Let 𝑚 be a positive integer and 𝑓₁,…,𝑓_𝑚 be nonnegative locally integrable functions on ℚ^𝑛_𝑝. The 𝑚-linear 𝑝-adic Hardy-Littlewood-Pólya operator is defined by 𝑇^𝑝_𝑚𝑓₁,…,𝑓_𝑚(𝑥)=_ℚ𝑝⋯_ℚ𝑝𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚max|𝑥|_𝑝,𝑦_1𝑝,…,𝑦_𝑚𝑝^𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚,𝑥∈ℚ^∗_𝑝. (1.21)

We obtain the sharp estimates of the 𝑚-linear 𝑝-adic Hardy operator on Lebesgue spaces with power weights in Section 2. In Section 3, we get the best estimate of 𝑚-linear 𝑝-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.

In the following sequel, for 𝑘∈ℤ, we denote 𝐵_𝑘={𝑥∈ℚ^𝑛_𝑝∶|𝑥|_𝑝≤𝑝^𝑘} and 𝑆_𝑘={𝑥∈ℚ^𝑛_𝑝∶|𝑥|_𝑝=𝑝^𝑘}.

2. Sharp Estimates of 𝑚-Linear 𝑝-Adic Hardy Operator

Theorem 2.1. Let 𝑚∈ℤ⁺, 𝑓_𝑖∈𝐿^𝑞𝑖(|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥), 1<𝑞_𝑖<∞, 𝑖=1,2,…,𝑚, 1≤𝑞<∞, 1/𝑞=∑^𝑚_𝑖=1(1/𝑞_𝑖), 𝛼_𝑖<𝑞𝑛(1−(1/𝑞_𝑖)) and 𝛼=∑^𝑚_𝑖=1𝛼_𝑖. Then ℋ^𝑝𝑓₁,…,𝑓_{𝑚𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤𝐶_ℋ𝑓_{1𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_𝑑𝑥)⋯𝑓_{𝑚𝐿𝑞𝑚(|𝑥|}^𝛼_𝑚^𝑞_𝑚𝑝^/𝑞_𝑑𝑥), (2.1) where 𝐶_ℋ=1−𝑝^−𝑛𝑚∏^𝑚_𝑖=11−𝑝^{𝑛/𝑞𝑖+(𝛼𝑖/𝑞)−𝑛} (2.2) is the best constant.

When 𝛼=0, we get the sharp estimates of the 𝑚-linear 𝑝-adic Hardy operator on Lebesgue spaces.

Corollary 2.2. Let 𝑚∈ℤ⁺, 1<𝑞_𝑖<∞, 𝑖=1,2,…,𝑚, 1≤𝑞<∞ and 1/𝑞=∑^𝑚_𝑖=11/𝑞_𝑖. Then ℋ^𝑝_{𝑚𝐿𝑞1(ℚ𝑛𝑝)×⋯×𝐿𝑞𝑚(ℚ𝑛𝑝)→𝐿𝑞(ℚ𝑛𝑝)}=1−𝑝^−𝑛𝑚∏^𝑚_𝑖=11−𝑝^{(𝑛/𝑞𝑖)−𝑛}. (2.3)

Proof of Theorem 2.1. Since the proof of the case when 𝑚=1 is similar to and even simpler than that of the case when 𝑚≥2, for simplicity, we will only give the proof of case when 𝑚≥2. To make the proof clearer, we will discuss it in two parts.(I) When 𝑚=2
Firstly, we claim that the operator ℋ^𝑝 and its restriction to the functions 𝑔 satisfying 𝑔(𝑥)=𝑔(|𝑥|_𝑝⁻¹) have the same operator norm on 𝐿^𝑞(|𝑥|^𝛼_𝑝𝑑𝑥). In fact, set 𝑔_𝑖(𝑥)=11−𝑝^−𝑛_{|𝜉𝑖|𝑝=1}𝑓_𝑖|𝑥|_𝑝⁻¹𝜉_𝑖𝑑𝜉_𝑖,𝑥∈ℚ^𝑛_𝑝,𝑖=1,2. (2.4) It’s clear that 𝑔_𝑖(𝑥)=𝑔_𝑖(|𝑥|_𝑝⁻¹), 𝑖=1,2, and ℋ^𝑝₂𝑔₁,𝑔₂(𝑥)=1|𝑥|_𝑝^2𝑛|(𝑦₁^,𝑦₂^)|_𝑝^≤|𝑥|_𝑝𝑔₁𝑦₁𝑔₂𝑦₂𝑑𝑦₁𝑑𝑦₂=11−𝑝^−𝑛21|𝑥|_𝑝^2𝑛|(𝑦₁^,𝑦₂^)|_𝑝^≤|𝑥|_𝑝²_𝑖=1^|𝜉_𝑖^|_𝑝⁼¹𝑓_𝑖𝑦_𝑖𝑝⁻¹𝜉_𝑖𝑑𝜉_𝑖𝑑𝑦₁𝑑𝑦₂=11−𝑝^−𝑛21|𝑥|_𝑝^2𝑛|(𝑦₁^,𝑦₂^)|_𝑝^≤|𝑥|_𝑝²_𝑖=1^|𝑧_𝑖^|_𝑝^=|𝑦_𝑖^|_𝑝𝑓_𝑖𝑧_𝑖𝑦_𝑖𝑝^−𝑛𝑑𝑧_𝑖𝑑𝑦₁𝑑𝑦₂=11−𝑝^−𝑛21|𝑥|_𝑝^2𝑛|(𝑧₁^,𝑧₂^)|_𝑝^≤|𝑥|_𝑝²_𝑖=1^|𝑦_𝑖^|_𝑝^=|𝑧_𝑖^|_𝑝𝑦_𝑖𝑝^−𝑛𝑑𝑦_𝑖𝑓₁𝑧₁𝑓₂𝑧₂𝑑𝑧₁𝑑𝑧₂=1|𝑥|_𝑝^2𝑛|(𝑧₁^,𝑧₂^)|_𝑝^≤|𝑥|_𝑝𝑓₁𝑧₁𝑓₂𝑧₂𝑑z₁𝑑𝑧₂=ℋ^𝑝₂𝑓₁,𝑓₂(𝑥). (2.5)
By Hölder's inequality, we get 𝑔_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥)=_ℚ𝑛𝑝11−𝑝^−𝑛_{|𝜉𝑖|𝑝=1}𝑓_𝑖|𝑥|_𝑝⁻¹𝜉_𝑖𝑑𝜉_𝑖^𝑞𝑖|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥^1/𝑞𝑖≤_ℚ𝑛𝑝11−𝑝^{−𝑛𝑞𝑖}_{|𝜉𝑖|𝑝=1}𝑓_𝑖|𝑥|_𝑝⁻¹𝜉_𝑖^𝑞𝑖𝑑𝜉_{𝑖|𝜉𝑖|𝑝=1}𝑑𝜉_𝑖^𝑞𝑖−1|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥^1/𝑞𝑖=_ℚ𝑛𝑝11−𝑝^−𝑛_{|𝜉𝑖|𝑝=1}𝑓_𝑖|𝑥|_𝑝⁻¹𝜉_𝑖^𝑞𝑖𝑑𝜉_𝑖|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥^1/𝑞𝑖=11−𝑝^{−𝑛1/𝑞𝑖ℚ𝑛}_𝑝^|𝑧_𝑖^|_𝑝^=|𝑥|_𝑝𝑓_𝑖𝑧_𝑖^𝑞𝑖𝑑𝑧_𝑖|𝑥|^{(𝛼𝑖𝑞𝑖}_𝑝^{/𝑞)−𝑛}𝑑𝑥^1/𝑞𝑖=11−𝑝^{−𝑛1/𝑞𝑖ℚ𝑛}_𝑝^|𝑥|_𝑝^=|𝑧_𝑖^|_𝑝|𝑥|^{(𝛼𝑖𝑞𝑖}_𝑝^{/𝑞)−𝑛}𝑑𝑥𝑓_𝑖𝑧_𝑖^𝑞𝑖𝑑𝑧_𝑖^1/𝑞𝑖=𝑓_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥),𝑖=1,2. (2.6) Therefore, ℋ^𝑝₂𝑓₁,𝑓_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}𝑓_{1𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_𝑑𝑥)𝑓_{2𝐿𝑞2(|𝑥|}^𝛼₂^𝑞_2𝑝^/𝑞_𝑑𝑥)≤ℋ^𝑝₂𝑔₁,𝑔_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}𝑔_{1𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_𝑑𝑥)𝑔_{2𝐿𝑞2(|𝑥|}^𝛼₂^𝑞_2𝑝^/𝑞_𝑑𝑥), (2.7) which implies the claim. In the following, without loss of generality, we may assume that 𝑓_𝑖∈𝐿^𝑞𝑖(|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥), 𝑖=1,2, which satisfy that 𝑓_𝑖(𝑥)=𝑓_𝑖(|𝑥|_𝑝⁻¹), 𝑖=1,2.
By changing of variables 𝑦_𝑖=|𝑥|_𝑝⁻¹𝑧_𝑖, 𝑖=1,2, we have ℋ^𝑝₂𝑓₁,𝑓_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=_ℚ𝑛𝑝1|𝑥|_𝑝^2𝑛|(𝑦₁^,𝑦₂^)|_𝑝^≤|𝑥|_𝑝𝑓₁(𝑦₁)𝑓₂(𝑦₂)𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞=_{ℚ𝑛𝑝|(𝑧1,𝑧2)|𝑝≤1}𝑓₁|𝑥|_𝑝⁻¹𝑧₁𝑓₂|𝑥|_𝑝⁻¹𝑧₂𝑑𝑧₁𝑑𝑧₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞=_{ℚ𝑛𝑝|(𝑧1,𝑧2)|𝑝≤1}𝑓₁𝑧_1𝑝⁻¹𝑥𝑓₂𝑧_2𝑝⁻¹𝑥𝑑𝑧₁𝑑𝑧₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞. (2.8) Then using Minkowski's integral inequality and Hölder's inequality ((𝑞/𝑞₁)+(𝑞/𝑞₂)=1), we get ℋ^𝑝₂𝑓₁,𝑓_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤_{|(𝑧1,𝑧2)|𝑝≤1ℚ𝑛𝑝}𝑓₁𝑧_1𝑝⁻¹𝑥𝑓₂𝑧_2𝑝⁻¹𝑥^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞𝑑𝑧₁𝑑𝑧₂≤_{|(𝑧1,𝑧2)|𝑝}²_{≤1𝑖=1ℚ𝑛𝑝}𝑓_𝑖(𝑧_𝑖𝑝⁻¹𝑥)^𝑞𝑖|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥^1/𝑞𝑖𝑑𝑧₁𝑑𝑧₂=_{|(𝑧1,𝑧2)|𝑝}²_≤1𝑖=1𝑧_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂²_𝑖=1𝑓_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥). (2.9)
By calculation, we have _{|(𝑧1,𝑧2)|𝑝}²_≤1𝑖=1𝑧_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂=_{|𝑧1|𝑝≤1|𝑧2|𝑝≤|𝑧1|𝑝}²_𝑖=1𝑧_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂+_{|𝑧2|𝑝≤1|𝑧1|𝑝<|𝑧2|𝑝}²_𝑖=1𝑧_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂=_{|𝑧1|𝑝≤1}𝑧₁^{−(𝑛/𝑞1)−(𝛼1}_𝑝^{/𝑞)log𝑝𝑧1𝑝}_{𝑘=−∞𝑆𝑘}𝑧₂^{−(𝑛/𝑞2)−(𝛼2}_𝑝^/𝑞)𝑑𝑧₂𝑑𝑧₁+_{|𝑧2|𝑝≤1}𝑧₂^{−(𝑛/𝑞2)−(𝛼2}_𝑝^{/𝑞)log𝑝𝑧2𝑝−1}_{𝑘=−∞𝑆𝑘}𝑧₁^{−(𝑛/𝑞1)−(𝛼1}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂=1−𝑝^−𝑛1−𝑝^{(𝑛/𝑞2)+(𝛼2/𝑞)−𝑛}_{|𝑧1|𝑝≤1}𝑧_1𝑝^{−(𝑛/𝑞)−(𝛼/𝑞)+𝑛}𝑑𝑧₁+1−𝑝^−𝑛𝑝^{(𝑛/𝑞1)+(𝛼1/𝑞)−𝑛}1−𝑝^{(𝑛/𝑞1)+(𝛼1/𝑞)−𝑛}_{|𝑧2|𝑝≤1}𝑧_2𝑝^{−(𝑛/𝑞)−(𝛼/𝑞)+𝑛}𝑑𝑧₂=1−𝑝^−𝑛2∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)−𝑛}. (2.10) Therefore, ℋ^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤1−𝑝^−𝑛2∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)−𝑛}. (2.11)
Now let us prove that our estimate is sharp. For 0<𝜖<1 and |𝜖|_𝑝>1, we take 𝑓^𝜖_𝑖𝑥_𝑖=0,𝑥_𝑖𝑝𝑥<1,_{𝑖−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)𝑝},𝑥_𝑖𝑝≥1,𝑖=1,2. (2.12) Then by calculation, we have 𝑓^𝜖₁^𝑞1_{𝐿𝑞1𝑥||𝛼1𝑞1𝑝/𝑞𝑑𝑥}=𝑓^𝜖₂^𝑞2_{𝐿𝑞2𝑥||𝛼2𝑞2𝑝/𝑞𝑑𝑥}=1−𝑝^−𝑛1−𝑝^−𝜖𝑞2. (2.13)
It is clear that when |𝑥|_𝑝<1, ℋ^𝑝₂(𝑓^𝜖₁,𝑓^𝜖₂)(𝑥)=0. But when |𝑥|_𝑝≥1, ℋ^𝑝₂𝑓^𝜖₁,𝑓^𝜖₂(𝑥)=|𝑥|^{−(𝑛/𝑞)−(𝛼/𝑞)−(𝑞2}_𝑝^𝜖/𝑞)_{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝑥|𝑝,|𝑦2|𝑝≥1/|𝑥|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦₂. (2.14)
Since |𝜖|_𝑝>1, we get ℋ^𝑝₂𝑓^𝜖₁,𝑓^𝜖_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=_{|𝑥|𝑝≥1}|𝑥|^{−(𝑛/𝑞)−(𝛼/𝑞)−(𝑞2}_𝑝^𝜖/𝑞)×_{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝑥|𝑝,|𝑦2|𝑝≥1/|𝑥|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞≥_{|𝑥|𝑝≥|𝜖|𝑝}|𝑥|^{−(𝑛/𝑞)−(𝛼/𝑞)−(𝑞2}_𝑝^𝜖/𝑞)×_{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝜖|𝑝,|𝑦2|𝑝≥1/|𝜖|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞=_{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝜖|𝑝,|𝑦2|𝑝≥1/|𝜖|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦_{2|𝑥|𝑝≥|𝜖|𝑝}|𝑥|^{−𝑛−𝜖𝑞2}_𝑝𝑑𝑥^1/𝑞=_{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝜖|𝑝,|𝑦2|𝑝≥1/|𝜖|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦₂|𝜖|^−𝜖𝑞2_𝑝^2/𝑞_𝑖=1𝑓^𝜖_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥). (2.15)
By the same calculation as that in (2.10), we obtain that _{|(𝑦1,𝑦2)|𝑝≤1,|𝑦1|𝑝≥1/|𝜖|𝑝,|𝑦2|𝑝≥1/|𝜖|𝑝}²_𝑖=1𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁𝑑𝑦₂=1−𝑝^−𝑛21−𝑝|𝜖|_𝑝^{(𝑛/𝑞)+(𝛼/𝑞)+(𝑞2𝜖/𝑞)−2𝑛}∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)+(𝑞2𝜖/𝑞𝑖)−𝑛}. (2.16) Therefore, |𝜀|^−𝜖𝑞2_𝑝^/𝑞1−𝑝^−𝑛21−𝑝|𝜖|_𝑝^{(𝑛/𝑞)+(𝛼/𝑞)+(𝑞2𝜖/𝑞)−2𝑛}∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)+(𝑞2𝜖/𝑞𝑖)−𝑛}≤ℋ^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}. (2.17)
Now take 𝜖=𝑝^−𝑘, 𝑘=1,2,3,…. Then |𝜖|_𝑝=𝑝^𝑘>1. Letting 𝑘 approach to ∞, then 𝜖 approaches to 0 and |𝜖|^−𝜖𝑞2_𝑝^/𝑞 approaches to 1. Since 𝛼_𝑖<𝑞𝑛(1−(1/𝑞_𝑖)), 𝑖=1,2, we have 1−𝑝^−𝑛2∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)−𝑛}≤ℋ^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}. (2.18) Then (2.11) and (2.18) imply that ℋ^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=1−𝑝^−𝑛2∏²_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝛼𝑖/𝑞)−𝑛}. (2.19)(II) When 𝑚≥3
The proof of the upper bound in this case is similar to that of the previous case, and we can obtain that ℋ^𝑝_𝑚𝑓₁,…,𝑓_{𝑚𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤𝐶_ℋ𝑓_{1𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_𝑑𝑥)⋯𝑓_{𝑚𝐿𝑞𝑚(|𝑥|}^𝛼_𝑚^𝑞_𝑚𝑝^/𝑞_𝑑𝑥), (2.20) where 𝐶_ℋ=_{|(𝑧1,…,𝑧𝑚)|𝑝}^𝑚_≤1𝑘=1𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚. (2.21)
Let 𝐷₁=𝑧₁,…,𝑧_𝑚∈ℚ^𝑛_𝑝×⋯×ℚ^𝑛_𝑝∣𝑧_1𝑝≤1,𝑧_𝑘𝑝≤𝑧_1𝑝,𝐷,1<𝑘≤𝑚_𝑖=𝑧₁,…,𝑧_𝑚∈ℚ^𝑛_𝑝×⋯×ℚ^𝑛_𝑝∣𝑧_𝑖𝑝≤1,𝑧_𝑗𝑝<𝑧_𝑖𝑝,𝑧_𝑘𝑝≤𝑧_𝑖𝑝,𝐷,1≤𝑗<𝑖<𝑘≤𝑚_𝑚=𝑧₁,…,𝑧_𝑚∈ℚ^𝑛_𝑝×⋯×ℚ^𝑛_𝑝∣𝑧_𝑚𝑝≤1,𝑧_𝑗𝑝<𝑧_𝑚𝑝,1≤𝑗<𝑚. (2.22) It is clear that ^𝑚_𝑗=1𝐷_𝑗=𝑧₁,…,𝑧_𝑚∈ℚ^𝑛_𝑝×⋯×ℚ^𝑛_𝑝∣𝑧₁,…,𝑧_𝑚𝑝≤1, (2.23) and 𝐷_𝑖∩𝐷_𝑗=∅. Then 𝐶_ℋ=^𝑚_{𝑗=1𝐷𝑗}^𝑚_𝑘=1𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚∶=^𝑚_𝑗=1𝐼_𝑗. (2.24)
Now let us calculate 𝐼_𝑗, 𝑗=1,2,…,𝑚, respectively, 𝐼₁=_𝐷1^𝑚_𝑘=1𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚=_{|𝑧1|𝑝≤1}𝑧₁^{−(𝑛/𝑞1)−(𝛼1}_𝑝^/𝑞)𝑚_{𝑘=2|𝑧𝑘|𝑝≤|𝑧1|𝑝}𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧_𝑘𝑑𝑧₁=_{|𝑧1|𝑝≤1}𝑧₁^{−(𝑛/𝑞1)−(𝛼1}_𝑝^/𝑞)𝑚_𝑘=2^{log𝑝𝑧1𝑝}_{𝑗=−∞𝑆𝑗}𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧_𝑘𝑑𝑧₁=1−𝑝^{−𝑛𝑚−1}∏^𝑚_𝑘=21−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛}_{|𝑧1|𝑝≤1}𝑧_1𝑝^{−(𝑛/𝑞)−(𝛼/𝑞)+(𝑚−1)𝑛}𝑑𝑧₁=1−𝑝^−𝑛𝑚1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏^𝑚_𝑘=21−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛}. (2.25)
Similarly, for 𝑖=2,…,𝑚−1, we have 𝐼_𝑖=_𝐷𝑖^𝑚_𝑘=1𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚=_{|𝑧𝑖|𝑝≤1}𝑧_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖}_𝑝^{/𝑞)𝑖−1}_{𝑗=1|𝑧𝑗|𝑝<|𝑧𝑖|𝑝}𝑧_𝑗^{−(𝑛/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗×^𝑚_{𝑘=𝑖+1|𝑧𝑘|𝑝≤|𝑧𝑖|𝑝}𝑧_𝑘^{−(𝑛/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧_𝑘𝑑𝑧_𝑖=1−𝑝^{−𝑛𝑚−1}∏^𝑖−1_𝑗=1𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}∏_{1≤𝑘≤𝑚,𝑘≠𝑖}1−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛}_{|𝑧𝑖|𝑝≤1}𝑧_𝑖𝑝^{−(𝑛/𝑞)−(𝛼/𝑞)+(𝑚−1)𝑛}𝑑𝑧_𝑖=1−𝑝^−𝑛𝑚∏^𝑖−1_𝑗=1𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏_{1≤𝑘≤𝑚,𝑘≠𝑖}1−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛},𝐼_𝑚=_{|𝑧𝑚|𝑝≤1}𝑧_𝑚^{−(𝑛/𝑞𝑚)−(𝛼𝑚}_𝑝^{/𝑞)𝑚−1}_{𝑗=1|𝑧𝑗|𝑝<|𝑧𝑚|𝑝}𝑧_𝑗^{−(𝑛/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗𝑑𝑧_𝑚=1−𝑝^−𝑛𝑚∏^𝑚−1_𝑗=1𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏^𝑚−1_𝑗=11−𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}. (2.26) Therefore, 𝐶_ℋ=1−𝑝^−𝑛𝑚1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏^𝑚_𝑘=21−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛}+^𝑚−1_𝑖=21−𝑝^−𝑛𝑚∏^𝑚−1_𝑗=1𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏^𝑚−1_𝑘=11−𝑝^{(𝑛/𝑞𝑘)+(𝛼𝑘/𝑞)−𝑛}+1−𝑝^−𝑛𝑚∏^𝑚−1_𝑗=1𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}1+𝑝^{(𝑛/𝑞)+(𝛼/𝑞)−𝑚𝑛}∏^𝑚−1_𝑗=11−𝑝^{(𝑛/𝑞𝑗)+(𝛼𝑗/𝑞)−𝑛}=1−𝑝^−𝑛𝑚∏^𝑚_𝑖=11−𝑝^{(𝑛/𝑞𝑖)+(𝑛𝛼𝑖/𝑞)−𝑛}. (2.27)
To show that 𝐶_ℋ is the best constant, we should prove that it is also the lower bound of the norm of ℋ^𝑝_𝑚 from 𝐿^𝑞1(|𝑥|^𝛼1𝑞1_𝑝^/𝑞𝑑𝑥)×⋯×𝐿^𝑞𝑚(|𝑥|^{𝛼𝑚𝑞𝑚}_𝑝^/𝑞𝑑𝑥) to 𝐿^𝑞(|𝑥|^𝛼_𝑝𝑑𝑥). For 0<𝜖<1 and |𝜖|_𝑝>1, we take 𝑓^𝜖_𝑖𝑥_𝑖=0,𝑥_𝑖𝑝𝑥<1,_{𝑖−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞𝑚𝜖/𝑞𝑖)𝑝},𝑥_𝑖𝑝≥1,𝑖=1,2,…,𝑚. (2.28) By simple calculation, we have 𝑓^𝜖₁^𝑞1_{𝐿𝑞1𝑥||𝛼1𝑞1𝑝/𝑞𝑑𝑥}=⋯=𝑓^𝜖_𝑚^𝑞𝑚_{𝐿𝑞𝑚|𝑥|𝛼𝑚𝑞𝑚𝑝/𝑞𝑑𝑥}=1−𝑝^−𝑛1−𝑝^{−𝜖𝑞𝑚}. (2.29) And when |𝑥|_𝑝<1, ℋ^𝑝_𝑚(𝑓^𝜖₁,…,𝑓^𝜖_𝑚)(𝑥)=0. But when |𝑥|_𝑝≥1, ℋ^𝑝_𝑚𝑓^𝜖₁,…,𝑓^𝜖_𝑚𝑥)=|𝑥|_{−(𝑛/𝑞)−(𝛼/𝑞)−(}(_{𝑞𝑚𝑝𝜖/𝑞)|(𝑦1,…,𝑦𝑚)|𝑝≤1,|𝑦1|𝑝≥1/|𝑥|𝑝,…,|𝑦𝑚|𝑝≥1/|𝑥|𝑝𝑚𝑖=1}𝑦_𝑖^{−(𝑛/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝𝑑𝑦₁⋯𝑑𝑦_𝑚. (2.30) Then by the similar discussion to that in previous case, we can obtain that ℋ^𝑝_{𝑚𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_{𝑑𝑥)×⋯×𝐿𝑞𝑚(|𝑥|}^𝛼_𝑚^𝑞_𝑚𝑝^/𝑞_{𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≥𝐶_ℋ. (2.31) Theorem 2.1 is established by (2.20), (2.27), and (2.31).

3. Sharp Estimate of 𝑚-Linear 𝑝-Adic Hardy-Littlewood-Pólya Operator

We get the following best estimate of 𝑚-linear 𝑝-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.

Theorem 3.1. Let 𝑚∈ℤ⁺, 𝑓_𝑖∈𝐿^𝑞_𝑖(|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥), 1<𝑞_𝑖<∞, 𝑖=1,2,…,𝑚, 1≤𝑞<∞, 1/𝑞=∑^𝑚_𝑖=11/𝑞_𝑖, 𝛼_𝑖<𝑞(1−(1/𝑞_𝑖)) and 𝛼=∑^𝑚_𝑖=1𝛼_𝑖. Then 𝑇^𝑝_𝑚𝑓₁,…,𝑓_{𝑚𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=𝐶_𝑇𝑓_{1𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_𝑑𝑥)⋯𝑓_{𝑚𝐿𝑞𝑚(|𝑥|}^𝛼_𝑚^𝑞_𝑚𝑝^/𝑞_𝑑𝑥), (3.1) where 𝐶_𝑇=1−𝑝^−1𝑚1−𝑞^−𝑚1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏^𝑚_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼i/𝑞)−1}, (3.2) is the best constant.

In particular, when 𝛼=0, we obtain the norm of the 𝑚-linear 𝑝-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces.

Corollary 3.2. Let 𝑚∈ℤ⁺, 1<𝑞_𝑖<∞, 𝑖=1,2,…,𝑚, 1≤𝑞<∞ and 1/𝑞=∑^𝑚_𝑖=11/𝑞_𝑖. Then 𝑇^𝑝_{𝑚𝐿𝑞1(ℚ𝑝)×⋯×𝐿𝑞𝑚(ℚ𝑝)→𝐿𝑞(ℚ𝑝)}=1−𝑝^−1𝑚1−𝑞^−𝑚1−𝑝^−1/𝑞∏^𝑚_𝑖=11−𝑝^{(1/𝑞𝑖)−1}. (3.3)

Proof of Theorem 3.1. Just as the proof of Theorem 2.1, we will only give the proof of case when 𝑚≥2.(I) Case 𝑚=2
By definition and the change of variables 𝑦_𝑖=𝑥𝑧_𝑖, 𝑖=1,2, we have 𝑇^𝑝₂𝑓₁,𝑓_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=_{ℚ𝑝ℚ𝑝}𝑓₁𝑦₁𝑓₂𝑦₂max|𝑥|_𝑝,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞≤_{ℚ𝑝ℚ𝑝}𝑓₁𝑦₁𝑓₂𝑦₂max|𝑥|_𝑝,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞=_{ℚ𝑝ℚ𝑝}𝑓₁𝑥𝑧₁𝑓₂𝑥𝑧₂max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞. (3.4)
By Minkowski’s integral inequality and Hölder's inequality ((𝑞/𝑞₁)+(𝑞/𝑞₂)=1), we get 𝑇^𝑝₂𝑓₁,𝑓_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤_{ℚ𝑝ℚ𝑝}𝑓₁𝑥𝑧₁𝑓₂𝑥𝑧₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞1max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂≤_ℚ𝑝²_{𝑖=1ℚ𝑝}𝑓_𝑖𝑥𝑧_𝑖^𝑞𝑖|𝑥|^{𝛼𝑖𝑞𝑖}_𝑝^/𝑞𝑑𝑥^1/𝑞𝑖1max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂≤_ℚ𝑝∏²_𝑖=1𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂²_𝑖=1𝑓_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥). (3.5)
By calculation, we have _ℚ𝑝∏²_𝑖=1𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂=_{|𝑧1|𝑝≤1|𝑧2|𝑝}²_≤1𝑖=1𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂+_{|𝑧1|𝑝>1|𝑧2|𝑝≤|𝑧1|𝑝}𝑧₁^{−(1/𝑞1)−(𝛼1}_𝑝^/𝑞)−2𝑧₂^{−(1/𝑞2)−(𝛼2}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂+_{|𝑧2|𝑝>1|𝑧1|𝑝<|𝑧2|𝑝}𝑧₁^{−(1/𝑞1)−(𝛼1}_𝑝^/𝑞)𝑧₂^{−(1/𝑞2)−(𝛼2}_𝑝^/𝑞)−2𝑑𝑧₁𝑑𝑧₂∶=𝐿₀+𝐿₁+𝐿₂. (3.6)
By definition, 𝐿₀=_{|𝑧1|𝑝≤1|𝑧2|𝑝}²_≤1𝑖=1𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)𝑑𝑧₁𝑑𝑧₂=1−𝑝⁻¹²1−𝑝^{(1/𝑞1)+(𝛼1/𝑞)−1}1−𝑝^{(1/𝑞2)+(𝛼2/𝑞)−1},𝐿₁=_{|𝑧1|𝑝>1|𝑧2|𝑝≤|𝑧1|𝑝}𝑧₁^{−(1/𝑞1)−(𝛼1}_𝑝^/𝑞)−2𝑧₂^{−(1/𝑞2)−𝛼2}_𝑝^/𝑞𝑑𝑧₁𝑑𝑧₂=1−𝑝⁻¹1−𝑝^{(1/𝑞2)+(𝛼2/𝑞)−1}_{|𝑧1|𝑝>1}𝑧_1𝑝^{−(1/𝑞)−(𝛼/𝑞)−1}𝑑𝑧₁=1−𝑝⁻¹²𝑝^{−(1/𝑞)−(𝛼/𝑞)}1−𝑝^{(1/𝑞2)+(𝛼2/𝑞)−1}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}. (3.7) Similarly, 𝐿₂=_{|𝑧2|𝑝>1|𝑧1|𝑝<|𝑧2|𝑝}𝑧₁^{−(1/𝑞1)−(𝛼1}_𝑝^/𝑞)𝑧₂^{−(1/𝑞2)−(𝛼2}_𝑝^/𝑞)−2𝑑𝑧₁𝑑𝑧₂=1−𝑝⁻¹²𝑝^{(1/𝑞1)+(𝛼1/𝑞)−1}𝑝^{−(1/𝑞)−(𝛼/𝑞)}1−𝑝^{(1/𝑞1)+(𝛼1/𝑞)−1}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}. (3.8)
Substituting (3.7) and (3.8) into (3.6), we get _ℚ𝑝∏²_𝑖=1𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^/𝑞)max1,𝑧_1𝑝,𝑧_2𝑝²𝑑𝑧₁𝑑𝑧₂=1−𝑝⁻¹²1−𝑞⁻²1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏²_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)−1}. (3.9) Then (3.5) and (3.9) imply that 𝑇^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤1−𝑝⁻¹²1−𝑝⁻²1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏²_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)−1}. (3.10)
On the other hand, for 0<𝜖<1 and |𝜖|_𝑝>1, we take 𝑓^𝜖_𝑖𝑥_𝑖=0,𝑥_𝑖𝑝𝑥<1,_{𝑖−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)𝑝},𝑥_𝑖𝑝≥1,𝑖=1,2. (3.11) Then 𝑓^𝜖₁^𝑞1_{𝐿𝑞1𝑥||𝛼1𝑞1𝑝/𝑞𝑑𝑥}=𝑓^𝜖₂^𝑞2_{𝐿𝑞2𝑥||𝛼2𝑞2𝑝/𝑞𝑑𝑥}=1−𝑝⁻¹1−𝑝^−𝜖𝑞2. (3.12)
Since |𝜖|_𝑝>1, we have 𝑇^𝑝₂𝑓^𝜖₁,𝑓^𝜖_{2𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=_{ℚ𝑝ℚ𝑝}𝑓^𝜖₁𝑥₁𝑓^𝜖₂𝑥₂max|𝑥|_𝑝,𝑥_1𝑝,𝑥_2𝑝²𝑑𝑥₁𝑑𝑥₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞≥_{|𝑥|𝑝≥1|𝑥1|𝑝≥1|𝑥2|𝑝≥1}∏²_𝑖=1𝑥_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max|𝑥|_𝑝,𝑥_1𝑝,𝑥_2𝑝²𝑑𝑥₁𝑑𝑥₂^𝑞|𝑥|^𝛼_𝑝𝑑𝑥^1/𝑞=_{|𝑥|𝑝≥1|𝑦1|𝑝≥1/|𝑥|𝑝|𝑦2|𝑝≥1/|𝑥|𝑝}∏²_𝑖=1|𝑦_𝑖|^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max1,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^{−1−𝑞2𝜖}_𝑝𝑑𝑥^1/𝑞≥_{|𝑥|𝑝≥|𝜖|𝑝|𝑦1|𝑝≥1/|𝜖|𝑝|𝑦2|𝑝≥1/|𝜖|𝑝}∏²_𝑖=1𝑦_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max1,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂^𝑞|𝑥|^{−1−𝑞2𝜖}_𝑝𝑑𝑥^1/𝑞=²_𝑖=1𝑓^𝜖_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥)|𝜖|^−𝑞2_𝑝^𝜖/𝑞_{|𝑦1|𝑝≥1/|𝜖|𝑝|𝑦2|𝑝≥1/|𝜖|𝑝}∏²_𝑖=1𝑦_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max1,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂. (3.13) Therefore, 𝑇^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≥|𝜖|^−𝑞2_𝑝^𝜖/𝑞_{|𝑦1|𝑝≥1/|𝜖|𝑝|𝑦2|𝑝≥1/|𝜖|𝑝}∏²_𝑖=1𝑦_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max1,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂. (3.14)
As the calculation of (3.6)–(3.8), we obtain that _{|𝑦1|𝑝≥1/|𝜖|𝑝|𝑦2|𝑝≥1/|𝜖|𝑝}∏²_𝑖=1|𝑦_𝑖|^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝max1,𝑦_1𝑝,𝑦_2𝑝²𝑑𝑦₁𝑑𝑦₂=1−𝑝⁻¹²∏²_𝑖=11−(𝑝|𝜖|_𝑝)^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)+(𝑞2𝜖/𝑞𝑖)−1}∏²_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)+(𝑞2𝜖/𝑞𝑖)−1}+1−𝑝⁻¹²𝑝^{−(1/𝑞)−(𝛼/𝑞)−(𝑞2𝜖/𝑞)}1−𝑝^{(1/𝑞2)+(𝛼2/𝑞)+𝜖−1}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)−(𝑞2𝜖/𝑞)}+1−𝑝⁻¹²𝑝^{(1/𝑞1)+(𝛼1/𝑞)+(𝑞2𝜖/𝑞1)−1}𝑝^{−(1/𝑞)−(𝛼/𝑞)−(𝑞2𝜖/𝑞)}1−𝑝^{(1/𝑞1)+(𝛼1/𝑞)+(𝑞2𝜖/𝑞1)−1}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)−(𝑞2𝜖/𝑞)}. (3.15)
Now take 𝜖=𝑝^−𝑘,𝑘∈ℤ⁺ and let 𝑘 approach to ∞, then by (3.9), (3.14), (3.15), and the fact that 𝛼_𝑖<𝑞𝑛(1−(1/𝑞_𝑖)), 𝑖=1,2, we have 𝑇^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≥1−𝑝⁻¹²1−𝑞⁻²1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏²_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)−1}. (3.16) Then by (3.10) and (3.16), we get 𝑇^𝑝_{2𝐿𝑞1(|𝑥|𝛼1𝑞1𝑝/𝑞𝑑𝑥)×𝐿𝑞2(|𝑥|𝛼2𝑞2𝑝/𝑞𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}=1−𝑝⁻¹²1−𝑞⁻²1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏²_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)−1}. (3.17)(II) Case 𝑚≥3
The upper bound estimate for the norm can be obtained by the same way as that when 𝑚=2, and we can obtain that 𝑇^𝑝_𝑚𝑓₁,…,𝑓_{𝑚𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≤𝐶_𝑇^𝑚_𝑖=1𝑓_{𝑖𝐿𝑞𝑖(|𝑥|}^𝛼_𝑖^𝑞_𝑖𝑝^/𝑞_𝑑𝑥), (3.18) where 𝐶_𝑇=_{|(𝑧1,…,𝑧𝑚)|𝑝≤1}∏^𝑚_𝑗=1𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)max1,𝑧_1𝑝,…,𝑧_𝑚𝑝^𝑚𝑑𝑧₁⋯𝑑𝑧_𝑚. (3.19)
Let 𝐸₀=ℤ_𝑝×⋯×ℤ_𝑝,𝐸₁=𝑧₁,…,𝑧_𝑚∈ℚ_𝑝×⋯×ℚ_𝑝∣𝑧_1𝑝>1,𝑧_𝑘𝑝≤𝑧_1𝑝,𝐸,1<𝑘≤𝑚_𝑖=𝑧₁,…,𝑧_𝑚∈ℚ_𝑝×⋯×ℚ_𝑝∣𝑧_𝑖𝑝>1,𝑧_𝑗𝑝<𝑧_𝑖𝑝,𝑧_𝑘𝑝≤𝑧_𝑖𝑝,𝐸,1≤𝑗<𝑖<𝑘≤𝑚_𝑚=𝑧₁,…,𝑧_𝑚∈ℚ_𝑝×⋯×ℚ_𝑝∣𝑧_𝑚𝑝>1,𝑧_𝑗𝑝<𝑧_𝑚𝑝,1≤𝑗<𝑚. (3.20) Obviously, ^𝑚_𝑘=1𝐸_𝑘=ℚ_𝑝×⋯×ℚ_𝑝,𝐸_𝑖∩𝐸_𝑗=∅,𝑖≠𝑗,1≤𝑖,𝑗≤𝑚. (3.21) Then 𝐶_𝑇=^𝑚_{𝑘=0𝐸𝑘}∏^𝑚_𝑗=1|𝑧_𝑗|^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)max1,𝑧_1𝑝,…,𝑧_𝑚𝑝^𝑚𝑑𝑧₁⋯𝑑𝑧_𝑚∶=^𝑚_𝑘=0𝐽_𝑘. (3.22)
Now let us calculate 𝐽_𝑘,𝑘=0,1,…,𝑚, respectively, 𝐽₀=_ℤ𝑝⋯_ℤ𝑝^𝑚_𝑗=1𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚=^𝑚_{𝑗=1ℤ𝑝}𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗=^𝑚_𝑗=1⁰_{𝑘=−∞𝑆𝑘}|𝑧_𝑗|^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗=1−𝑝^−1𝑚∏^𝑚_𝑗=11−𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1},𝐽₁=_{|𝑧1|𝑝>1|𝑧2|𝑝≤|𝑧1|𝑝}⋯_{|𝑧𝑚|𝑝≤|𝑧1|𝑝}𝑧₁^{−(1/𝑞1)−(𝛼1}_𝑝^{𝑚/𝑞)−𝑚}_𝑗=2𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧₁⋯𝑑𝑧_𝑚=_{|𝑧1|𝑝>1}|𝑧₁|^{−(1/𝑞1)−(𝛼1}_𝑝^{/𝑞)−𝑚𝑚}_{𝑗=2|𝑧𝑗|𝑝≤|𝑧1|𝑝}|𝑧_𝑗|^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗𝑑𝑧₁=_{|𝑧1|𝑝>1}|𝑧₁|^{−(1/𝑞1)−(𝛼1}_𝑝^{𝑚/𝑞)−𝑚}_𝑗=21−𝑝⁻¹𝑧₁^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)+11−𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}𝑑𝑧₁=1−𝑝^−1𝑚−1∏^𝑚_𝑗=21−𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1|𝑧}₁^|_𝑝^>1𝑧_1𝑝^{−(1/𝑞)−(𝛼/𝑞)−1}𝑑𝑧₁=1−𝑝^−1𝑚−1𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏^𝑚_𝑗=21−𝑝^{(1/𝑞j)+(𝛼𝑗/𝑞)−1}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}. (3.23) Similar to 𝐽₁, for 1<𝑖<𝑚, it is true that 𝐽_𝑖=_{|𝑧𝑖|𝑝>1}𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^{/𝑞)−𝑚𝑖−1}_{𝑗=1|𝑧𝑗|𝑝<|𝑧𝑖|𝑝}𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗×^𝑚_{𝑘=𝑖+1|𝑧𝑘|𝑝≤|𝑧𝑖|𝑝}𝑧_𝑘^{−(1/𝑞𝑘)−(𝛼𝑘}_𝑝^/𝑞)𝑑𝑧_𝑘𝑑𝑧_𝑖=1−𝑝^−1𝑚−1∏^𝑖−1_𝑗=1𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}∏_{1≤𝑘≤𝑚,𝑘≠𝑖}1−𝑝^{(1/𝑞𝑘)+(𝛼𝑘/𝑞)−1}_{|𝑧𝑖|𝑝>1}𝑧_𝑖𝑝^{−(1/𝑞)−(𝛼/𝑞)−1}𝑑𝑧_𝑖=1−𝑝^−1𝑚∏^𝑖−1_𝑗=1𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}𝑝^{−(1/𝑞)−(𝛼/𝑞)}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏_{1≤𝑘≤𝑚,𝑘≠𝑖}1−𝑝^{(1/𝑞𝑘)+(𝛼𝑘/𝑞)−1},𝐽_𝑚=_{|𝑧𝑚|𝑝>1}𝑧_𝑖^{−(1/𝑞𝑖)−(𝛼𝑖}_𝑝^{/𝑞)−𝑚𝑚−1}_{𝑗=1|𝑧𝑗|𝑝<|𝑧𝑚|𝑝}𝑧_𝑗^{−(1/𝑞𝑗)−(𝛼𝑗}_𝑝^/𝑞)𝑑𝑧_𝑗𝑑𝑧_𝑚=1−𝑝^−1𝑚−1∏^𝑚−1_𝑗=1𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}∏^𝑚−1_𝑗=11−𝑝^{(1/𝑞𝑘)+(𝛼𝑘/𝑞)−1|𝑧}_𝑚^|_𝑝^>1𝑧_𝑚𝑝^{−(1/𝑞)−(𝛼/𝑞)−1}𝑑𝑧_𝑚=1−𝑝^−1𝑚∏^𝑚−1_𝑗=1𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}𝑝^{−(1/𝑞)−(𝛼/𝑞)}1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏^𝑚−1_𝑗=11−𝑝^{(1/𝑞𝑗)+(𝛼𝑗/𝑞)−1}. (3.24) Consequently, we have 𝐶_𝑇=^𝑚_𝑘=0𝐽_𝑘=1−𝑝^−1𝑚1−𝑞^−𝑚1−𝑝^{−(1/𝑞)−(𝛼/𝑞)}∏^𝑚_𝑖=11−𝑝^{(1/𝑞𝑖)+(𝛼𝑖/𝑞)−1}. (3.25)
To obtain that 𝐶_𝑇 is also the lower bound, for 0<𝜖<1 and |𝜖|_𝑝>1, we define 𝑓^𝜖_𝑖=0,𝑥_𝑖𝑝<1,|𝑥_𝑖|^{−(1/𝑞𝑖)−(𝛼𝑖/𝑞)−(𝑞2𝜖/𝑞𝑖)}_𝑝,𝑥_𝑖𝑝≥1,𝑖=1,2,…,𝑚. (3.26) By the similar discussion to that in Case 𝑚=2, we can also get that 𝑇^𝑝_{𝑚𝐿𝑞1(|𝑥|}^𝛼₁^𝑞_1𝑝^/𝑞_{𝑑𝑥)×⋯×𝐿𝑞𝑚(|𝑥|}^𝛼_𝑚^𝑞_𝑚𝑝^/𝑞_{𝑑𝑥)→𝐿𝑞(|𝑥|𝛼𝑝𝑑𝑥)}≥𝐶_𝑇. (3.27) Combining (3.18) with (3.27), we complete the proof of Theorem 3.1.

Acknowledgments

This paper was partially supported by NSF of China (Grant nos. 10871024 and 10901076) and NSF of Shandong Province (Grant nos. Q2008A01 and ZR2010AL006).