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

Zequn Wang , Mingquan Wei , Qianjun He, and Dunyan Yan 1 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, Henan, China School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China Correspondence should be addressed to Zequn Wang; wangzequn17@mails.ucas.ac.cn

Operator (1) is a generalization of classical Hardy-type operators. In 1925, Hardy first gave the following definition: for x > 0.
In 1930, Bliss [3] proved Hardy inequality with power weight as Inequality (4) can be stated as eorem 1.
Inequality (5) turns into equality with the above constant if and only if e fractional Hardy operator on L q (G) can be shown as where β � 1/p − 1/q. Lu-Zhao [4] and Persson-Samko [5] extended those onedimension results to higher-dimension results. For general situations, results could be found in [6][7][8].
en, it is natural to consider the Hardy-type operator on product spaces, which can be defined as where x � (x 1 , . . . , x m ) ∈ R n 1 × · · · × R n m . He et al. [9] had already proved the boundedness of the fractional Hardy operator on higher-dimensional product spaces.
where 1 is the sharp bound.
Next, we will generalize the results into mixed norm, which will be mentioned in Section 4.
is sharp and C p i q i is defined as in (6).

Preliminaries
First, we can use the following lemma to reduce the dimension. is lemma is from [9].
en, 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 H * β 1 ,β 2 ,...,β m on radial functions. at is also the basic idea when we handle Hardy-type operators.
Notice that eorem 2 with m � 1 recovers part of result in [10]. Next, we will give a simple proof. where is 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 R n . Using the polar coordinate transformation, inequality (22) is equivalent to en, (24) can be written as (26) Notice that We have Journal of Function Spaces Using Bliss's result, we conclude that e sharp bound can be reached if and only if

Main Result
In this section, we will give the proof of eorems 2 and 4.
Proof of eorem 2. Without loss of generality, we only need to discuss the case for m � 2. e case for m > 2 is also true by the same method. Assume that f is a nonnegative integrable function. Fixing variable x 1 , we define So By simple integral estimate, we have Applying generalized Minkowski's inequality, we have It implies that On 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 eorem 6, it is easy to find that the best constant can be reached when where ω n i � p i (1 + β i ) − (1 + n i ) with i � 1, 2. We finish the Proof of eorem 2.

□
Proof of eorem 4. Without loss of generality, we only discuss the case m � 2. e case m > 2 is the same.

Journal of Function Spaces 5
Let We have Since that when |x i | < 1 and ϵ is small enough, We obtain that Notice that when If ϵ i is small enough, C ϵ i tends to zero. Hence, when ϵ 1 is small enough, we obtain that Using the same method for x 2 , we obtain that Let ϵ 1 ⟶ 0 + and ϵ 2 ⟶ 0 + , it implies that We finish the Proof of eorem 4.

Case for Mixed Norm
e 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(p 1 , p 2 , . . ., p n ) is a given n-tuple with 1 ≤ p i ≤ ∞. e set I satisfies I ⊂ {1, . . ., n}. A function f(x 1 , x 2 , . . ., x n ) measurable in the product spaces (X, S, μ) � ( n i�1 X i , n i�1 S i , n i�1 μ i ) is said to belong to the space L P I (X) 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. e number so obtained will be denoted by ‖f‖ P I , finite or not.
We give some necessary remarks for the space L P I (X): (1) If the set I � {1, . . ., n}, we call L P I (X) strong mixed norm space, which is also denoted by L P (X) or L p 1 ,...,p n (X) (2) If the set I is empty, we call L P I (X) weak mixed norm space, which is also denoted by wL P (X) or wL p 1 ,...,p n (X) (3) e spaces L P I (X) is a quasi-normed space for P ≥ 1 For more properties, we refer readers to [7]. ere 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 ≥ · · · ≥ p 1 ≥ 1 and f ∈ L p n ,...,p 1 (X), then f ∈ L p 1 ,...,p n (X). Moreover, there holds ‖f‖ L p 1 ,...,p n (X) ≤ ‖f‖ L p n ,...,p 1 (X) . (53) 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.
en, operator H * β 1 ,...,β m is bounded from L P (R n 1 × · · · × R n m ) to L Q I (R n 1 × · · · × R n m ). Moreover, there holds is sharp and C p i q i is defined as in (6).
On the other hand, we take f 1 the sharp function on R n 1 given in eorem 2 and f 2 the sharp function on R n 2 given in [10]. Define By the definition of norm of general operator, we have Combining those, we finish the proof of eorem 7. □

Data Availability
e data used to support the study are available upon request to the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest.