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)}_𝑝𝑦