Sharp Bounds for Fractional Conjugate Hardy Operator on Higher-Dimensional Product Spaces

Abstract

In this paper, we obtain the sharp bound for fractional conjugate Hardy operator on higher-dimensional product spaces from to the space and to the space . More generally, the operator norm of the fractional Hardy operator on higher-dimensional product spaces from to is obtained.

1. Introduction

Let f be a nonnegative integrable function on . Define the fractional conjugate Hardy operator on higher-dimensional product spaces bywhere , , and 0 ≤ β_i < n_i with i = 1, 2, …, m.

Operator (1) is a generalization of classical Hardy-type operators. In 1925, Hardy first gave the following definition:for x > 0.

Hardy [1] presented the classical Hardy inequalities on :with 1 < p ≤ ∞, where the constants and p are the best. Later, Christ and Grafakos [2] extended this result to higher-dimensional setting.

In 1930, Bliss [3] proved Hardy inequality with power weight as

Inequality (4) can be stated as Theorem 1.

Theorem 1. Let f be a nonnegative integrable function on and 1 < p < q < ∞. The inequalityholds with the sharp constantwhere B(·, ·) is a beta function.

Inequality (5) turns into equality with the above constant if and only if

The fractional Hardy operator on can be shown as

Theorem 1 shows thatwhere β = 1/p − 1/q.

Lu-Zhao [4] and Persson-Samko [5] extended those one-dimension results to higher-dimension results. For general situations, results could be found in [6–8].

Then, it is natural to consider the Hardy-type operator on product spaces, which can be defined aswhere .

He et al. [9] had already proved the boundedness of the fractional Hardy operator on higher-dimensional product spaces.

We will discuss the conjugate situation defined as (1), and the state main theorem as follows.

Theorem 2. Let 1 < p < q < ∞. Set 0 < β_i < n_i and with 1 ≤ i ≤ m. Note n = (n₁, n₂, …, n_m). If , then we havewhereis sharp and C_pq is the sharp constant in (5).
In 2016, Gao et al. [10] proved the weak-type estimate for with n₁ = ⋯ = n_m = n, β₁ = ⋯ = β_m = β, and m = 1. Their result is as follows.

Theorem 3. Let f be a nonnegative integrable function on . Set 0 ≤ β < n and . Then, we haveWe will extent (13) to higher-dimensional product spaces.

Theorem 4. Set 0 < β_i < n_i, for i = 1, 2, …, m. Let . If , then we havewhere 1 is the sharp bound.

Next, we will generalize the results into mixed norm, which will be mentioned in Section 4.

Theorem 5. Suppose P = (p₁, …, p_m) ≥ 1 (i.e., for every i = 1, 2, …, p_i ≥ 1) and Q = (q₁, …, q_m) ≥ 1. Set with 0 < β_i < n_i, for i = 1, …, m. We further suppose 1 < p_i < ∞ when i ∈ I ⊂ {1, …, m}. Then, operator is bounded from to . Moreover, there holdswhereis sharp and is defined as in (6).

2. Preliminaries

First, we can use the following lemma to reduce the dimension. This lemma is from [9].

Lemma 1. Suppose , with . Definewhere andThen, we have

Equality (20) can be proved by basic integral transformation. Inequality (21) is based on Minkowski’s inequality.

Remark 1. Lemma 1 provides the idea that we only need to consider on radial functions. That is also the basic idea when we handle Hardy-type operators.
Notice that Theorem 2 with m = 1 recovers part of result in [10]. Next, we will give a simple proof.

Theorem 6. Let 1 < p < q < ∞, 0 < β < n, and . If , then we havewhereis sharp and C_pq is defined as in (6).

Proof. From Lemma 1, it suffices to prove the result when f is a nonnegative, radial, smooth function with compact support on . Using the polar coordinate transformation, inequality (22) is equivalent toLet  = f(t⁻¹)t^−β−1. Then, (24) can be written asDefine α₀ = −n − 1 and ω₀ = p(1 + β) − (1 + n). We have ω₀ < p − 1 and .
LetSo we have  =  and so that  = . It follows thatNotice thatWe haveUsing Bliss’s result, we conclude thatThe sharp bound can be reached if and only if

3. Main Result

In this section, we will give the proof of Theorems 2 and 4.

Proof of Theorem 2. Without loss of generality, we only need to discuss the case for m = 2. The case for m > 2 is also true by the same method.
Assume that f is a nonnegative integrable function. Fixing variable x₁, we defineSoBy simple integral estimate, we haveApplying generalized Minkowski’s inequality, we haveIt implies thatOn the other hand, by Lemma 1, we only need to prove the situation when f is a nonnegative radial smooth function with compact support so that we can separate the variable. Using the similar method of Theorem 6, it is easy to find that the best constant can be reached whenwhere with i = 1, 2. We finish the Proof of Theorem 2.

Proof of Theorem 4. Without loss of generality, we only discuss the case m = 2. The case m > 2 is the same.
When m = 2, operator can be written asUsing Theorem 3 and Fubini’s theorem, it implies thatWe conclude thatTherefore, we haveOn the other hand, defineLetWe haveIf |x_i| ≥ 1, we haveIf |x_i| < 1, we haveSince that when |x_i| < 1 and ϵ is small enough,We obtain thatSetNotice that when , . If ϵ_i is small enough, tends to zero. Hence, when ϵ₁ is small enough, we obtain thatUsing the same method for x₂, we obtain thatLet ϵ₁ ⟶ 0⁺ and ϵ₂ ⟶ 0⁺, it implies thatWe finish the Proof of Theorem 4.

4. Case for Mixed Norm

The mixed norm space was first defined in [11] by Benedek and Panzone and received much concern such as [12]. In 2018, Wei and Yan [7] defined a more general mixed norm space called as weak and strong mixed-norm space. We list its definition for completeness.

Definition 1. Let (X_i, S_i, μ_i) be n given, totally σ–finite measure spaces, for 1 ≤ i ≤ n. P = x(p₁, p₂, …, p_n) is a given n–tuple with 1 ≤ p_i ≤ ∞. The set I satisfies I ⊂ {1, …, n}. A function f(x₁, x₂, …, x_n) measurable in the product spaces is said to belong to the space if the number obtained after subsequently taking successfully the mixed norm where we take p_i–norm for i ∈ I and weak p_j–norm for j ∈{1, …, n}\I. In natural order, it is finite. The number so obtained will be denoted by , finite or not.
We give some necessary remarks for the space :(1)  If the set I = {1, …, n}, we call strong mixed norm space, which is also denoted by L^P(X) or (2)  If the set I is empty, we call weak mixed norm space, which is also denoted by wL^P(X) or (3)  The spaces is a quasi-normed space for P ≥ 1For more properties, we refer readers to [7].
There is a basic lemma which plays an important role in the proof of our main theorems.

Lemma 2. Let (X, S, μ) be defined as in the above definitions. If p_n ≥ ⋯ ≥ p₁ ≥ 1 and , then . Moreover, there holds

Lemma 2 is a direct generalization of Minkowski’s inequality. For the proof of the lemma, readers can refer to [7]. It is not hard to see that Fubini’s theorem is a special case of Lemma 2. With all these, we can describe Hardy conjugate operator’s mixed norm.

Theorem 7. Suppose P = (p₁, …, p_m) ≥ 1 and Q = (q₁, …, q_m) ≥ 1. Set 0 < β_i < n and , for i = 1, …, m. We further suppose 1 < p_i < ∞ when i ∈ I ⊂{1, …, m}. Then, operator is bounded from to . Moreover, there holdswhereis sharp and is defined as in (6).

Proof. Without loss of generality, we only discuss the case m = 2 and I = {1}.
Notice that . Applying Theorem 2, we obtain thatUsing Fubini’s theorem, it is easy to findUsing Theorem 3, the following inequality holds:Combining above estimates, we obtain the desired result.
On the other hand, we take f₁ the sharp function on given in Theorem 2 and f₂ the sharp function on given in [10]. DefineBy the definition of norm of general operator, we haveCombining those, we finish the proof of Theorem 7.

Data Availability

The data used to support the study are available upon request to the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11561062 and 11871452).