Boundedness of 𝑝-Adic Hardy Operators and Their Commutators on 𝑝-Adic Central Morrey and BMO Spaces

Abstract

We obtain the sharp bounds of p-adic Hardy operators on p-adic central Morrey spaces and p-adic 𝜆-central BMO spaces, respectively. We also establish the 𝜆-central BMO estimates for commutators of p-adic Hardy operators on p-adic central Morrey spaces.

1. Introduction

In the past decades, the field of 𝑝-adic numbers has been intensively used in theoretical and mathematical physics (see [1–9] and references therein). As a consequence, new mathematical problems have emerged, among which we refer to [10, 11] for Riesz potentials [12–16], for 𝑝-adic pseudodifferential equations, and so forth. In the past few years, there is an increasing interest in the study of harmonic analysis on 𝑝-adic field and their various generalizations and the related theory of operators and spaces; see, for example [17–27].

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 𝛾 is an integer and integers 𝑚, 𝑛 are indivisible by 𝑝, then |𝑥|_𝑝=𝑝^−𝛾. It is easy to see that the norm satisfies the following properties: 𝑥𝑦_𝑝=|𝑥|_𝑝𝑦_𝑝,𝑥+𝑦_𝑝≤max|𝑥|_𝑝,𝑦_𝑝. (1) Moreover, if |𝑥|_𝑝≠|𝑦|_𝑝, then |𝑥+𝑦|_𝑝=max{|𝑥|_𝑝,|𝑦|_𝑝}. It is well known that ℚ_𝑝 is a typical model of non-Archimedean local fields. From the standard 𝑝-adic analysis [7], we see that any nonzero 𝑝-adic number 𝑥∈ℚ_𝑝 can be uniquely represented in the canonical series as follows: 𝑥=𝑝^𝛾∞_𝑗=0𝑎_𝑗𝑝^𝑗,𝛾=𝛾(𝑥)∈ℤ, (2) where 𝑎_𝑗 are integers, 0≤𝑎_𝑗≤𝑝−1, and 𝑎₀≠0. The series (2) converges in the 𝑝-adic norm since |𝑎_𝑗𝑝^𝑗|_𝑝=𝑝^−𝑗.

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. (3) Denote by 𝐵_𝛾(𝑎)={𝑥∈ℚ^𝑛_𝑝∶|𝑥−𝑎|_𝑝≤𝑝^𝛾} the ball with center at 𝑎∈ℚ^𝑛_𝑝 and radius 𝑝^𝛾 and by 𝑆_𝛾(𝑎)∶={𝑥∈ℚ^𝑛_𝑝∶|𝑥−𝑎|_𝑝=𝑝^𝛾} the sphere with center at 𝑎∈ℚ^𝑛_𝑝 and radius 𝑝^𝛾, 𝛾∈ℤ. It is clear that 𝑆_𝛾(𝑎)=𝐵_𝛾(𝑎)⧵𝐵_𝛾−1(𝑎), and 𝐵_𝛾(𝑎)=_𝑘≤𝛾𝑆_𝑘(𝑎),_{+∞𝛾=−∞}𝐵_𝛾(𝑎)=^+∞_𝛾=−∞𝑆_𝛾(𝑎)=ℚ^𝑛_𝑝. (4) We set 𝐵_𝛾(0)=𝐵_𝛾 and 𝑆_𝛾(0)=𝑆_𝛾.

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, (5) 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 (6) for any 𝑎∈ℚ^𝑛_𝑝. For a more complete introduction to the 𝑝-adic field, see [27] or [7].

The well-known Hardy’s integral inequality [28] tells us that, for 1<𝑞<∞, 𝐻𝑓_{𝐿𝑞(ℝ+)}≤𝑞𝑓𝑞−1^{𝐿𝑞(ℝ+)}, (7) where the classical Hardy operator is defined by (𝑥)1𝐻𝑓∶=𝑥^𝑥₀𝑓(𝑡)𝑑𝑡 (8) 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 (9)

Faris [29] introduced the following 𝑛-dimensional Hardy operator, for nonnegative function 𝑓 on ℝ^𝑛, (𝑥)1ℋ𝑓∶=Ω_𝑛|𝑥|^𝑛_{|𝑦|<|𝑥|}𝑓𝑦𝑑𝑦, (10) where Ω_𝑛 is the volume of the unit ball in ℝ^𝑛. Christ and Grafakos [30] obtained that the norm of ℋ on 𝐿^𝑞(ℝ^𝑛) is ‖ℋ‖_𝐿^𝑞_(ℝ^𝑛_)→𝐿^𝑞_(ℝ^𝑛₎=𝑞,𝑞−1 (11) which is the same as that of the 1-dimensional Hardy operator.

In [31], Fu et al. obtained the precise norm of 𝑚-linear Hardy operators on weighted Lebesgue spaces and central Morrey spaces. Fu et al. [32] introduced 𝑝-adic Hardy operators and got the sharp estimates of 𝑝-adic Hardy operators on 𝑝-adic weighted Lebesgue spaces. Moreover, they proved that the commutators generated by the 𝑝-adic Hardy operators and the central BMO functions are bounded on 𝑝-adic weighted Lebesgue spaces and 𝑝-adic Herz spaces see; [33] for more information about Herz spaces. Ren and Tao [34] Yu and Lu [35] studied the boundedness of commutators of Hardy type on some spaces.

Inspired by these results, in this paper we will establish the sharp estimates of 𝑝-adic Hardy operators on 𝑝-adic central Morrey and 𝜆-central BMO spaces. Furthermore, we will discuss the boundedness for commutators of 𝑝-adic Hardy operators and 𝜆-central BMO functions on 𝑝-adic central Morrey spaces.

Definition 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}, (12) where 𝐵(0,|𝑥|_𝑝) is a ball in ℚ^𝑛_𝑝 with center at 0∈ℚ^𝑛_𝑝 and radius |𝑥|_𝑝.

Morrey [36] introduced the 𝐿^𝑞,𝜆(ℝ^𝑛) spaces to study the local behavior of solutions to second-order elliptic partial differential equations. The 𝑝-adic Morrey space is defined as follows.

Definition 2. Let 1≤𝑞<∞ and let 𝜆≥−1/𝑞. The 𝑝-adic Morrey space 𝐿^𝑞,𝜆(ℚ^𝑛_𝑝) is defined by 𝐿^𝑞,𝜆ℚ^𝑛_𝑝=𝑓∈𝐿^𝑞_locℚ^𝑛_𝑝∶𝑓_{𝐿𝑞,𝜆(ℚ𝑛𝑝)}<∞, (13) where 𝑓_{𝐿𝑞,𝜆(ℚ𝑛𝑝)}=sup_{𝑎∈ℚ𝑛𝑝,𝛾∈ℤ}1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞. (14)

Remark 3. It is clear that 𝐿^{𝑞,−1/𝑞}(ℚ^𝑛_𝑝)=𝐿^𝑞(ℚ^𝑛_𝑝), 𝐿^𝑞,0(ℚ^𝑛_𝑝)=𝐿^∞(ℚ^𝑛_𝑝).

For some recent developments of Morrey spaces and their related function spaces on ℝ^𝑛, we refer the reader to [37]. In 2000, Alvarez et al. [38] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced 𝜆-central bounded mean oscillation spaces and central Morrey spaces, respectively. Next, we introduce their 𝑝-adic versions.

Definition 4. Let 𝜆∈ℝ and let 1<𝑞<∞. The 𝑝-adic central Morrey space 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) is defined by 𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}∶=sup_𝛾∈ℤ1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞<∞, (15) where 𝐵_𝛾=𝐵_𝛾(0).

Remark 5. It is clear that 𝐵̇^{𝑞,−1/𝑞}ℚ^𝑛_𝑝=𝐿^𝑞ℚ^𝑛_𝑝. (16) When 𝜆<−1/𝑞, the space 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) reduces to {0}; therefore, we can only consider the case 𝜆≥−1/𝑞. If 1≤𝑞₁<𝑞₂<∞, by Hölder’s inequality, 𝐵̇^𝑞2,𝜆ℚ^𝑛_𝑝⊂𝐵̇^𝑞1,𝜆ℚ^𝑛_𝑝 (17) for 𝜆∈ℝ.

Definition 6. Let 𝜆<1/𝑛 and let 1<𝑞<∞. The space CBMO^𝑞,𝜆(ℚ^𝑛_𝑝) is defined by the condition 𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}∶=sup_𝛾∈ℤ1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓(𝑥)−𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞<∞, (18) where 𝑓_𝐵𝛾=(1/|𝐵_𝛾|_𝐻)∫_𝐵𝛾𝑓(𝑥)𝑑𝑥.

Remark 7. When 𝜆=0, the space CBMO^𝑞,𝜆(ℚ^𝑛_𝑝) is just CBMO^𝑞(ℚ^𝑛_𝑝), which is defined in [32]. If 1≤𝑞₁<𝑞₂<∞, by Hölder’s inequality, CBMO^𝑞2,𝜆ℚ^𝑛_𝑝⊂CBMO^𝑞1,𝜆ℚ^𝑛_𝑝 (19) for 𝜆∈ℝ. By the standard proof as that in ℝ^𝑛, we can see that 𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}∼sup_𝛾∈ℤinf_𝑐∈ℂ1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}|𝑓(𝑥)−𝑐|^𝑞𝑑𝑥^1/𝑞. (20)

Remark 8. The formulas (18) and (15) yield that 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) is a Banach space continuously included in CBMO^𝑞,𝜆(ℚ^𝑛_𝑝).

In Section 2, we obtain the sharp estimates of 𝑝-adic Hardy operators on 𝑝-adic central Morrey spaces and 𝑝-adic 𝜆-central BMO spaces. Analogous result is also established for 𝑝-adic Morrey spaces. In Section 3, we discuss the boundedness of commutators generated by 𝑝-adic Hardy operators and 𝑝-adic 𝜆-central BMO functions on 𝑝-adic central Morrey spaces.

We should note that in Euclidean space, when estimating the Hardy operator, one usually discusses its restriction on radical functions. However, on 𝑝-adic field, we will consider its restriction on the functions 𝑓 with 𝑓(𝑥)=𝑓(|𝑥|_𝑝⁻¹) instead.

Throughout this paper the letter 𝐶 will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.

2. Sharp Estimates of 𝑝-Adic Hardy Operator

We get the following precise norms of 𝑝-adic Hardy operators on 𝑝-adic central Morrey spaces and 𝑝-adic 𝜆-central BMO spaces.

Theorem 9. Let 1<𝑞<∞ and let 𝜆≥−1/𝑞. Then ℋ^𝑝_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝̇𝐵)→𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (21)

Theorem 10. Let 1<𝑞<∞ and let −1/𝑞<𝜆<1/𝑛. Then ℋ^𝑝_{𝐶𝐵𝑀𝑂𝑞,𝜆(ℚ𝑛𝑝)→𝐶𝐵𝑀𝑂𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (22)

Corollary 11. Let 1<𝑞<∞. Then ℋ^𝑝_{𝐶𝐵𝑀𝑂𝑞(ℚ𝑛𝑝)→𝐶𝐵𝑀𝑂𝑞(ℚ𝑛𝑝)}=1. (23)

Let ℒ^𝑞,𝜆(ℚ^𝑛_𝑝) denote the subspace of 𝐿^𝑞,𝜆(ℚ^𝑛_𝑝) consisting of all functions 𝑓∈𝐿^𝑞,𝜆(ℚ^𝑛_𝑝) with 𝑓(𝑥)=𝑓(|𝑥|_𝑝⁻¹). We obtain the sharp estimate of 𝑝-adic Hardy operator from ℒ^𝑞,𝜆(ℚ^𝑛_𝑝) to 𝐿^𝑞,𝜆(ℚ^𝑛_𝑝).

Theorem 12. Suppose that 1<𝑞<∞, −1/𝑞<𝜆<0. Then ℋ^𝑝 maps ℒ^𝑞,𝜆(ℚ^𝑛_𝑝) to 𝐿^𝑞,𝜆(ℚ^𝑛_𝑝) with norm ℋ^𝑝_{ℒ𝑞,𝜆(ℚ𝑛𝑝)→𝐿𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (24)

Proof of Theorem 9. When 𝜆=−1/𝑞, 𝐵̇^{𝑞,−1/𝑞}(ℚ^𝑛_𝑝)=𝐿^𝑞(ℚ^𝑛_𝑝), by Corollary 2.2 in [32], ℋ^𝑝_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝̇𝐵)→𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1−1/𝑞)}. (25)
When 𝜆>−1/𝑞, we first claim that the operator ℋ^𝑝 and its restriction to the subset of 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝), which consist of functions 𝑔 satisfying 𝑔(𝑥)=𝑔(|𝑥|_𝑝⁻¹), have the same operator norm on 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝).
In fact, for 𝑓∈𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝), set 𝑔(𝑥)=11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉𝑑𝜉,𝑥∈ℚ^𝑛_𝑝. (26) It is easy to see that 𝑔 satisfies that 𝑔(𝑥)=𝑔(|𝑥|_𝑝⁻¹) and ℋ^𝑝𝑔=ℋ^𝑝𝑓. By Hölder’s inequality, for 𝛾∈ℤ, we have 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑔^𝑞=1𝑑𝑥𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉𝑑𝜉^𝑞≤1𝑑𝑥𝐵_𝛾𝐻^1+𝜆𝑞×_𝐵𝛾11−𝑝^−𝑛𝑞_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉^𝑞×𝑑𝜉_|𝜉|𝑝=1𝑑𝜉^𝑞/𝑞′=1𝑑𝑥𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉^𝑞=1𝑑𝜉𝑑𝑥𝐵_𝛾𝐻^1+𝜆𝑞1−𝑝^{−𝑛𝐵𝛾}|𝑥|_𝑝^−𝑛𝑑𝑥_{|𝑦|𝑝=|𝑥|𝑝}𝑓𝑦^𝑞=1𝑑𝑦𝐵_𝛾𝐻^1+𝜆𝑞1−𝑝^{−𝑛𝐵𝛾}𝑓𝑦^𝑞𝑑𝑦_{|𝑥|𝑝=|𝑦|𝑝}|𝑥|_𝑝^−𝑛=1𝑑𝑥𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓𝑦^𝑞𝑑𝑦. (27) Therefore, 𝑔_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}≤𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}. (28) Consequently, ℋ^𝑝𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}≤ℋ^𝑝𝑔_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}𝑔_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}, (29) which implies the claim. In the following, without loss of generality, we may assume that 𝑓∈𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) satisfies 𝑓(𝑥)=𝑓(|𝑥|_𝑝⁻¹). Then by Minkowski’s inequality, we have 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}ℋ^𝑝𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓(𝑡)𝑑𝑡^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾𝐵0}𝑓|𝑥|_𝑝⁻¹𝑦𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾𝐵0}𝑓𝑦_𝑝⁻¹𝑥𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤_𝐵01𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓𝑦_𝑝⁻¹𝑥^𝑞𝑑𝑥^1/𝑞≤𝑑𝑦_𝐵01𝐵_𝛾𝐻^{1+𝜆𝑞𝐵(0,𝑝𝛾|𝑦|𝑝)}𝑓(𝑧)^𝑞𝑦_𝑝^−𝑛𝑑𝑧^1/𝑞≤𝑑𝑦_𝐵0𝑦_𝑝^𝑛𝜆𝑞𝐵0,𝑝^𝛾𝑦_𝑝𝐻^{1+𝜆𝑞𝐵(0,𝑝𝛾|𝑦|𝑝)}𝑓(𝑧)^𝑞𝑑𝑧^1/𝑞≤𝑓𝑑𝑦_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)𝐵0}𝑦_𝑝^𝑛𝜆=𝑓𝑑𝑦_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)∞𝑘=0|𝑦|𝑝=𝑝−𝑘}𝑝^{−𝑘𝑛𝜆}=𝑑𝑦1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}. (30) Thus, ℋ^𝑝𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}≤1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}. (31)
On the other hand, take 𝑓₀(𝑥)=|𝑥|_𝑝^𝑛𝜆. Then 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}|𝑥|_𝑝^𝑛𝜆𝑞𝑑𝑥=𝑝^{𝛾−𝑛𝛾(1+𝜆𝑞)}_{𝑘=−∞𝑆𝑘}𝑝^{𝑛𝑘𝜆𝑞}=𝑑𝑥1−𝑝^−𝑛𝑝^{𝛾−𝑛𝛾(1+𝜆𝑞)}_𝑘=−∞𝑝^{𝑛𝑘(1+𝜆𝑞)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆𝑞)}, (32) where the series converges due to 𝜆>−1/𝑞. Thus, 𝑓₀∈𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) since ℋ^𝑝𝑓₀(𝑥)=1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}|𝑡|_𝑝^𝑛𝜆𝑑𝑡=|𝑥|_𝑝^𝑛𝜆_𝐵0𝑦_𝑝^𝑛𝜆=𝑑𝑦1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓₀(𝑥). (33) Therefore, ℋ^𝑝_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝̇𝐵)→𝑞,𝜆(ℚ𝑛𝑝)}≥1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (34) Then (31) and (34) imply that ℋ^𝑝_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝̇𝐵)→𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (35)

Proof of Theorem 10. As in the proof of Theorem 9, we first show that the operator ℋ^𝑝 and its restriction to the subset of CBMO^𝑞,𝜆(ℚ^𝑛_𝑝) consisting of functions 𝑔 with 𝑔(𝑥)=𝑔(|𝑥|_𝑝⁻¹) have the same operator norm on CBMO^𝑞,𝜆(ℚ^𝑛_𝑝).
In fact, set 𝑔(𝑥)=11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉𝑑𝜉,𝑥∈ℚ^𝑛_𝑝. (36) Then 𝑔(𝑥)=𝑔(|𝑥|_𝑝⁻¹) and ℋ^𝑝𝑔=ℋ^𝑝𝑓. By change of variable, we get 𝑔_𝐵𝛾=1𝐵_𝛾𝐻^𝐵_𝛾𝑔(𝑥)=1𝑑𝑥𝐵_𝛾𝐻^𝐵_𝛾11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉=1𝑑𝜉𝑑𝑥𝐵_𝛾𝐻1−𝑝^−𝑛𝐵_𝛾𝑑𝑥_{|𝑦|𝑝=|𝑥|𝑝}𝑓𝑦|𝑥|_𝑝^−𝑛=1𝑑𝑦𝐵_𝛾𝐻1−𝑝^−𝑛𝐵_𝛾𝑓𝑦𝑑𝑦_{|𝑥|𝑝=|𝑦|𝑝}|𝑥|_𝑝^−𝑛=1𝑑𝑥𝐵_𝛾𝐻^𝐵_𝛾𝑓𝑦𝑑𝑦=𝑓_𝐵𝛾. (37) Using Minkowski’s inequality and (37), we have 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑔−𝑔_𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑔−𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^1+𝜆𝑞×_𝐵𝛾11−𝑝^−𝑛_|𝜉|𝑝=1𝑓|𝑥|_𝑝⁻¹𝜉𝑑𝜉−𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞≤11−𝑝^−𝑛1𝐵_𝛾𝐻^1+𝜆𝑞×_{𝐵𝛾|𝜉|𝑝=1}𝑓|𝑥|_𝑝⁻¹𝜉−𝑓_𝐵𝛾𝑑𝜉^𝑞𝑑𝑥^1/𝑞≤11−𝑝^−𝑛×_|𝜉|𝑝=11𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓|𝑥|_𝑝⁻¹𝜉−𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1𝑑𝜉1−𝑝^−𝑛_|𝜉|𝑝=11𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓(𝑥)−𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞≤𝑓𝑑𝜉_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (38) Therefore, 𝑔_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}≤𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (39) We conclude that ℋ^𝑝𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}≤ℋ^𝑝𝑔_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}𝑔_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (40)
In the following, without loss of generality, we may assume that 𝑓∈CBMO^𝑞,𝜆(ℚ^𝑛_𝑝) with 𝑓(𝑥)=𝑓(|𝑥|_𝑝⁻¹). By Fubini theorem, we have ℋ^𝑝𝑓_𝐵𝛾=1𝐵_𝛾𝐻^𝐵_𝛾ℋ^𝑝𝑓(𝑥)=1𝑑𝑥𝐵_𝛾𝐻^𝐵_𝛾^𝐵₀𝑓|𝑥|_𝑝⁻¹𝑦=𝑑𝑦𝑑𝑥_𝐵01𝐵_𝛾𝐻^𝐵_𝛾𝑓|𝑥|_𝑝⁻¹𝑦=𝑑𝑥𝑑𝑦_𝐵01𝐵_𝛾𝐻^𝐵_𝛾𝑓𝑦_𝑝⁻¹𝑥=𝑑𝑥𝑑𝑦_𝐵01𝐵0,𝑦_𝑝𝑝^𝛾_𝐻^{𝐵(0,|𝑦|}_𝑝^𝑝𝛾)𝑓(𝑧)=𝑑𝑧𝑑𝑦_𝐵0𝑓_{𝐵(0,|𝑦|𝑝𝑝𝛾)}𝑑𝑦. (41) Then by Minkowski's inequality, we get 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}ℋ^𝑝𝑓(𝑥)−ℋ^𝑝𝑓_𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾𝐵0}𝑓|𝑥|_𝑝⁻¹𝑦−𝑓_{𝐵(0,|𝑦|𝑝𝑝𝛾)}𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤_𝐵01𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓|𝑥|_𝑝⁻¹𝑦−𝑓_{𝐵(0,|𝑦|𝑝𝑝𝛾)}^𝑞𝑑𝑥^1/𝑞=𝑑𝑦_𝐵01𝐵0,𝑦_𝑝𝑝^𝛾_𝐻^1+𝜆𝑞×_{𝐵(0,|𝑦|𝑝𝑝𝛾)}𝑓(𝑧)−𝑓_{𝐵(0,|𝑦|𝑝𝑝𝛾)}^𝑞𝑑𝑧^1/𝑞𝑦_𝑝^𝑛𝜆≤𝑓𝑑𝑦_{CBMO𝑞,𝜆(ℚ𝑛𝑝)𝐵0}𝑦_𝑝^𝑛𝜆=𝑑𝑦1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (42) Namely, ℋ^𝑝𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}≤1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (43)
On the other hand, take 𝑓₀(𝑥)=|𝑥|_𝑝^𝑛𝜆. By Remark 8 and (32), for 𝜆>−1/𝑞, we have 𝑓₀∈𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝)⊂CBMO^𝑞,𝜆(ℚ^𝑛_𝑝). Then by (33), we get ℋ^𝑝𝑓_0𝐵𝛾=1𝐵_𝛾𝐻^𝐵_𝛾ℋ^𝑝𝑓₀(𝑥)=1𝑑𝑥𝐵_𝛾𝐻^𝐵_𝛾1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓₀(𝑥)=𝑑𝑥1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_0𝐵𝛾. (44) Therefore, 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}ℋ^𝑝𝑓₀(𝑥)−ℋ^𝑝𝑓_0𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)𝑞}𝑓₀−𝑓_0𝐵𝛾^𝑞𝑑𝑥^1/𝑞=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}𝑓₀−𝑓_0𝐵𝛾^𝑞𝑑𝑥^1/𝑞. (45) We arrive at ℋ^𝑝𝑓_{0CBMO𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{0CBMO𝑞,𝜆(ℚ𝑛𝑝)}. (46)
As a result, ℋ^𝑝_{CBMO𝑞,𝜆(ℚ𝑛𝑝)→CBMO𝑞,𝜆(ℚ𝑛𝑝)}≥1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}. (47) Then Theorem 10 follows from (43) and (47).

Proof of Theorem 12. Let 𝑓∈ℒ^𝑞,𝜆(ℚ^𝑛_𝑝). Then 𝑓(𝑥)=𝑓(|𝑥|_𝑝⁻¹). Using Minkowski’s inequality, we have 1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}ℋ^𝑝𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)𝐵0}𝑓|𝑥|_𝑝⁻¹𝑦𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)𝐵0}𝑓𝑦_𝑝⁻¹𝑥𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤_𝐵01𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}𝑓𝑦_𝑝⁻¹𝑥^𝑞𝑑𝑥^1/𝑞≤𝑑𝑦_𝐵01𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵(𝑎|𝑦|𝑝−1,𝑝𝛾|𝑦|𝑝)}𝑓(𝑧)^𝑞𝑦_𝑝^−𝑛𝑑𝑧^1/𝑞≤𝑑𝑦_𝐵0𝑦_𝑝^𝑛𝜆𝑞𝐵𝑎𝑦_𝑝⁻¹,𝑝^𝛾𝑦_𝑝𝐻^1+𝜆𝑞×_{𝐵(𝑎|𝑦|𝑝−1,𝑝𝛾|𝑦|𝑝)}𝑓(𝑧)^𝑞𝑑𝑧^1/𝑞≤𝑓𝑑𝑦_{𝐿𝑞,𝜆(ℚ𝑛𝑝)𝐵0}𝑦_𝑝^𝑛𝜆𝑑𝑦=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{𝐿𝑞,𝜆(ℚ𝑛𝑝)}. (48)
On the other hand, as in the proof of Theorem 9, we take 𝑓₀(𝑥)=|𝑥|_𝑝^𝑛𝜆, and we only need to show that 𝑓₀∈𝐿^𝑞,𝜆(ℚ^𝑛_𝑝). Consider the following.
(I) If |𝑎|_𝑝>𝑝^𝛾 and 𝑥∈𝐵_𝛾(𝑎), then |𝑥|_𝑝=max{|𝑥−𝑎|_𝑝,|𝑎|_𝑝}>𝑝^𝛾. Since −1/𝑞≤𝜆<0, we have 1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}|𝑥|_𝑝^𝑛𝜆𝑞<1𝑑𝑥𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}𝑝^{𝛾𝑛𝜆𝑞}𝑑𝑥=1. (49)
(II) If |𝑎|_𝑝≤𝑝^𝛾 and 𝑥∈𝐵_𝛾(𝑎), then |𝑥|_𝑝≤max{|𝑥−𝑎|_𝑝,|𝑎|_𝑝}≤𝑝^𝛾; therefore, 𝑥∈𝐵_𝛾. Recall that two balls in ℚ^𝑛_𝑝 are either disjoint or one is contained in the other (cf. page 21 in [39]). So we have 𝐵_𝛾(𝑎)=𝐵_𝛾; thus, 1𝐵_𝛾(𝑎)_𝐻^{1+𝜆𝑞𝐵𝛾(𝑎)}|𝑥|_𝑝^𝑛𝜆𝑞=1𝑑𝑥𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}|𝑥|_𝑝^𝑛𝜆𝑞𝑑𝑥=𝑝^{𝛾−𝛾𝑛(1+𝜆𝑞)}_{𝑘=−∞𝑆𝑘}𝑝^{𝑘𝑛𝜆𝑞}𝑑𝑥=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆𝑞)}. (50) From the previous discussion, we can see that 𝑓₀∈𝐿^𝑞,𝜆(ℚ^𝑛_𝑝). Then by (33), ℋ^𝑝𝑓_{0𝐿𝑞,𝜆(ℚ𝑛𝑝)}=1−𝑝^−𝑛1−𝑝^{−𝑛(1+𝜆)}𝑓_{0𝐿𝑞,𝜆(ℚ𝑛𝑝)}. (51) This completes the proof.

3. Boundedness for Commutators of 𝑝-Adic Hardy Operators on 𝑝-Adic Central Morrey Spaces

The boundedness of commutators is an active topic in harmonic analysis due to its important applications. For example, it can be applied to characterizing some function spaces [40]. In this section, we consider the boundedness for commutators generated by ℋ^𝑝 and 𝜆-central BMO functions on 𝑝-adic central Morrey spaces.

Definition 13. Let 𝑏∈CBMO^𝑞,𝜆(ℚ^𝑛_𝑝). The commutator of ℋ^𝑝 is defined by ℋ^𝑝_𝑏𝑓∶=𝑏ℋ^𝑝𝑓−ℋ^𝑝𝑏𝑓 (52) for some suitable functions 𝑓.

Theorem 14. Let 1<𝑞₁<∞,𝑞^′₁<𝑞₂<∞, 𝑛(1/𝑞₂−1/𝑞^′₁)<𝑛/𝑞₁, 1/𝑞=1/𝑞₁+1/𝑞₂, 𝜆₁>−1/𝑞, 0≤𝜆₂<1/𝑛, and 𝜆=𝜆₁+𝜆₂. If 𝑏∈𝐶𝐵𝑀𝑂^𝑞2,𝜆2(ℚ^𝑛_𝑝); then ℋ^𝑝_𝑏 is bounded from 𝐵̇^𝑞1,𝜆1(ℚ^𝑛_𝑝) to 𝐵̇^𝑞,𝜆(ℚ^𝑛_𝑝) and satisfies ℋ^𝑝_𝑏𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}≤𝐶‖𝑏‖_{𝐶𝐵𝑀𝑂}^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)𝑓_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}. (53)

Before the proof of this theorem, we need the following calculations.

Lemma 15. Suppose that 𝑏∈𝐶𝐵𝑀𝑂^𝑞,𝜆(ℚ^𝑛_𝑝) and 𝑗,𝑘∈ℤ, 𝜆≥0. Then, 𝑏_𝐵𝑗−𝑏_𝐵𝑘≤𝑝^𝑛𝑗−𝑘‖𝑏‖_{𝐶𝐵𝑀𝑂}^𝑞,𝜆_(ℚ^𝑛_𝑝)max𝐵_𝑗^𝜆_𝐻,𝐵_𝑘^𝜆_𝐻. (54)

Proof. Without loss of generality, we may assume that 𝑘>𝑗. Recall that 𝑏_𝐵𝑖=(1/|𝐵_𝑖|_𝐻)∫_𝐵𝑖𝑏(𝑥)𝑑𝑥. By Hölder's inequality, we have 𝑏_𝐵𝑖+1−𝑏_𝐵𝑖≤1𝐵_{𝑖𝐻𝐵𝑖}𝑏(𝑥)−𝑏_𝐵𝑖+11𝑑𝑥≤𝐵_{𝑖𝐻𝐵𝑖+1}𝑏(𝑥)−𝑏_𝐵𝑖+1≤1𝑑𝑥𝐵_{𝑖𝐻𝐵𝑖+1}𝑏(𝑥)−𝑏_𝐵𝑖+1^𝑞𝑑𝑥^1/𝑞𝐵_𝑖+1𝐻^1−1/𝑞≤𝐵_𝑖+1𝐻^1+𝜆𝐵_𝑖𝐻‖𝑏‖_CBMO^𝑞,𝜆_(ℚ^𝑛_𝑝)=𝑝^𝑛𝐵_𝑖+1^𝜆_𝐻‖𝑏‖_CBMO^𝑞,𝜆_(ℚ^𝑛_𝑝). (55)
Therefore, 𝑏_𝐵𝑗−𝑏_𝐵𝑘≤^𝑘−1_𝑖=𝑗𝑏_𝐵𝑗+1−𝑏_𝐵𝑗≤𝑝^𝑛‖𝑏‖_CBMO^𝑞,𝜆_(ℚ^𝑛_𝑝)^𝑘−1_𝑖=𝑗𝐵_𝑗+1^𝜆_𝐻≤𝑘−𝑗𝑝^𝑛‖𝑏‖_CBMO^𝑞,𝜆_(ℚ^𝑛_𝑝)𝐵_𝑘^𝜆_𝐻. (56)

Proof of Theorem 14. Assume that 𝑓∈𝐵̇^𝑞1,𝜆1(ℚ^𝑛_𝑝). Fix 𝛾∈ℤ, and by Minkowski’s inequality, we have 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}ℋ^𝑝_𝑏𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞=1𝐵_𝛾𝐻^1+𝜆𝑞×_𝐵𝛾1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓𝑦𝑏(𝑥)−𝑏𝑦𝑑𝑦^𝑞𝑑𝑥^1/𝑞≤1𝐵_𝛾𝐻^1+𝜆𝑞×_𝐵𝛾1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓𝑦𝑏(𝑥)−𝑏_𝐵𝛾𝑑𝑦^𝑞𝑑𝑥^1/𝑞+1𝐵_𝛾𝐻^1+𝜆𝑞×_𝐵𝛾1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓𝑦𝑏𝑦−𝑏_𝐵𝛾𝑑𝑦^𝑞𝑑𝑥^1/𝑞∶=𝐼+𝐽. (57)
In the following, we will estimate 𝐼 and 𝐽, respectively. For 𝐼, since ℋ^𝑝 is bounded from 𝐿^𝑟(ℚ^𝑛_𝑝) to 𝐿^𝑟(ℚ^𝑛_𝑝), 1<𝑟<∞ [32], then, by Hölder's inequality (𝑞/𝑞₁+𝑞/𝑞₂=1), we get 𝐼≤𝐵_𝛾𝐻^{−1/𝑞−𝜆}_𝐵𝛾𝑏(𝑥)−𝑏_𝐵𝛾^𝑞2𝑑𝑥^1/𝑞2×_𝐵𝛾ℋ^𝑝𝑓(𝑥)^𝑞1𝑑𝑥^1/𝑞1≤𝐵_𝛾^{−1/𝑞1−𝜆1}_𝐻‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)×_ℚ𝑛𝑝ℋ^𝑝𝑓𝜒_𝐵𝛾(𝑥)^𝑞1𝑑𝑥^1/𝑞1≤𝐶𝐵_𝛾^{−1/𝑞1−𝜆1}_𝐻‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_{𝑝)𝐵𝛾}𝑓(𝑥)^𝑞1𝑑𝑥^1/𝑞1≤𝐶‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)𝑓_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}. (58)
Next, let us estimate 𝐽 as follows: 𝐽^𝑞≤1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}1|𝑥|^𝑛_{𝑝𝐵(0,|𝑥|𝑝)}𝑓𝑦𝑏𝑦−𝑏_𝐵𝛾𝑑𝑦^𝑞≤1𝑑𝑥𝐵_𝛾𝐻^1+𝜆𝑞×^𝛾_{𝑘=−∞𝑆𝑘}𝑝^{−𝑘𝑛𝑞}_𝐵𝑘𝑓𝑦𝑏𝑦−𝑏_𝐵𝛾𝑑𝑦^𝑞=𝑑𝑥1−𝑝^−𝑛𝐵_𝛾𝐻^1+𝜆𝑞×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)𝑘}_{𝑗=−∞𝑆𝑗}𝑓𝑦𝑏𝑦−𝑏_𝐵𝛾𝑑𝑦^𝑞=1−𝑝^−𝑛𝐵_𝛾𝐻^1+𝜆𝑞×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)𝑘}_{𝑗=−∞𝑆𝑗}𝑓𝑦𝑏𝑦−𝑏_𝐵𝑗𝑑𝑦^𝑞+1−𝑝^−𝑛𝐵_𝛾𝐻^1+𝜆𝑞×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)𝑘}_{𝑗=−∞𝑆𝑗}𝑓𝑦𝑏_𝐵𝑗−𝑏_𝐵𝛾𝑑𝑦^𝑞∶=𝐽₁+𝐽₂. (59)
For 𝐽₁, by Hölder’s inequality (1/𝑞₁+1/𝑞₂+1−1/𝑞=1) and the fact that 𝜆₁>−1/𝑞₁, 𝜆₂>−1/𝑞₂, we have 𝐽₁≤1−𝑝^−𝑛𝐵_𝛾𝐻^{𝛾1+𝜆𝑞𝑘=−∞}𝑝^{𝑘𝑛(1−𝑞)}×^𝑘_𝑗=−∞𝐵_𝑗𝐻^1−1/𝑞_𝐵𝑗𝑓𝑦^𝑞1𝑑𝑦^1/𝑞1×_𝐵𝑗𝑏𝑦−𝑏_𝐵𝑗^𝑞2𝑑𝑦^1/𝑞2𝑞≤1−𝑝^−𝑛𝐵_𝛾𝐻^{𝛾1+𝜆𝑞𝑘=−∞}𝑝^{𝑘𝑛(1−𝑞)}×^𝑘_𝑗=−∞𝐵_𝑗𝐻^1+𝜆‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)𝑓_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}^𝑞≤1−𝑝^−𝑛𝐵_𝛾𝐻^1+𝜆𝑞‖𝑏‖^𝑞_{CBMO𝑞22,𝜆}𝑓^𝑞_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)𝑘}_𝑗=−∞𝑝^{𝑗𝑛(1+𝜆)𝑞}=𝐶𝐵_𝛾𝐻^1+𝜆𝑞‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}×𝑓^𝑞_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1+𝜆𝑞)}=𝐶‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}𝑓^𝑞_{̇𝐵𝑞11,𝜆}. (60) For 𝐽₂, by Lemma 15 and Hölder’s inequality, we obtain 𝐽₂=1−𝑝^−𝑛𝐵_𝛾𝐻^{𝛾1+𝜆𝑞𝑘=−∞}𝑝^{𝑘𝑛(1−𝑞)𝑘}_{𝑗=−∞𝑆𝑗}𝑓𝑦𝑏_𝐵𝑗−𝑏_𝐵𝛾𝑑𝑦^𝑞≤𝐶𝐵_𝛾𝐻^{𝛾1+𝜆𝑞𝑘=−∞}𝑝^{𝑘𝑛(1−𝑞)}×^𝑘_𝑗=−∞𝛾−𝑗‖𝑏‖_CBMO^𝑞₂₂^,𝜆𝐵_𝛾^𝜆2_𝐻𝐵𝑗𝑓𝑦𝑑𝑦^𝑞≤𝐶𝐵_𝛾^1+𝜆1𝑞_𝐻‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)}×^𝑘_𝑗=−∞𝛾−𝑗𝐵_𝑗^1−1/𝑞1_𝐻𝐵𝑗𝑓𝑦^𝑞1𝑑𝑦^1/𝑞1𝑞≤𝐶𝐵_𝛾^1+𝜆1𝑞_𝐻‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}𝑓^𝑞_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)𝑘}_𝑗=−∞𝛾−𝑗𝐵_𝑗^1+𝜆1_𝐻^𝑞=𝐶𝐵_𝛾^1+𝜆1𝑞_𝐻‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}𝑓^𝑞_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}×^𝛾_𝑘=−∞𝑝^{𝑘𝑛(1−𝑞)}𝛾−𝑘^𝑞𝐵_𝑘^(1+𝜆1_𝐻^)𝑞=𝐶‖𝑏‖^𝑞_{CBMO𝑞22,𝜆(ℚ𝑛𝑝)}𝑓^𝑞_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}. (61) Notice that since 𝜆₁>−1/𝑞₁, then 1+𝜆₁𝑞₁>0 and 1+𝜆₁>0.
The above estimates imply that 1𝐵_𝛾𝐻^{1+𝜆𝑞𝐵𝛾}ℋ^𝑝_𝑏𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞≤𝐶‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)𝑓_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}. (62) Consequently, ℋ^𝑝_𝑏𝑓_{̇𝐵𝑞,𝜆(ℚ𝑛𝑝)}≤𝐶‖𝑏‖_CBMO^𝑞₂₂^,𝜆_(ℚ^𝑛_𝑝)𝑓_{̇𝐵𝑞11,𝜆(ℚ𝑛𝑝)}. (63) Theorem 14 is proved.

Acknowledgment

This work was partially supported by NSF of China (Grant nos. 11271175, 11301248, 11171345, and 10901076), NSF of Shandong Province (Grant no. ZR2010AL006) and AMEP of Linyi University.