Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group

: In this article, we establish some reverse weighted Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. We then show the existence of extremal functions for the above inequalities by combining the subcritical approach and the renormalization method.


Introduction
Stein and Weiss [34]  (1.1) where < < +∞ p q 1 , , and α β , , and λ satisfy It is known as the Stein-Weiss inequality.A fundamental task in understanding Stein-Weiss inequality (1.1) is to study the sharp constant and existence of extremals.For < ′ q p and ≥ α β , 0, Lieb [30] proved the existence of extremals for the inequality (1.1) by using rearrangement inequalities.Moreover, he pointed out that the extremals cannot be expected to exist when ′ = = p q 2, = − λ n 1, = α 0, and = β 1, see also [27].In the general case of ′ = p q, Beckner [2,3] obtained the sharp constant of inequality (1.1).The Stein-Weiss inequality (1.1) contains the following HLS inequality ( = = α β 0) established in [26,33]: , and + + = 2 p q λ n 1 1 . Lieb [30] proved the existence of extremal functions for equation (1.2) by using rearrangement inequality.He also classified extremal functions and computed the sharp constant in the case of ′ = p 2, or = q 2, or ′ = = − p q n n λ 2 2 . The HLS inequality was intensively studied since it plays, important role in many geometric problems by virtue of the complete knowledge of sharp constants and extremal functions.For example, the existence of extremal functions of HLS inequality (1.2) was considered on the Euclidean space [4,18,19,30], on Riemannian manifolds [24], and on the Heisenberg group [17].
In a ground-breaking work [20], Frank and Lieb obtained the sharp constant and the explicit form of extremal functions for Folland-Stein inequality on the Heisenberg group [17].We recall some basic results concerning the Heisenberg group.
and the distance between The natural dilation of the Heisenberg group is given by ( ) (ℓ ℓ ) ℓ = δ u z t , 2 for any ℓ > 0. With this norm, the ball centered at ∈ u n with radius > R 0 is defined by . We also have the following triangle inequality: n (1.3)Throughout this article, the homogeneous dimension of n is denoted by = + Q n 2 2. The Frank-Lieb inequality then states that for < < λ Q with the equality if and only if for some ∈  c c , , ℓ > 0, and ∈ a n (unless ≡ f 0 or ≡ g 0).Here, H is defined by More recently, Hang and Wang [25] gave a simpler proof of inequality (1.4), which bypasses the sophisticated proof for the existence of minimizers and Hersch-type arguments.
The | | u weighted HLS inequality (namely Stein-Weiss inequality) on the Heisenberg group was established by Han et al. [22].Precisely, it states , , , , where , and α and β satisfy Moreover, they proved the following | | z weighted HLS inequality on the Heisenberg group: , , , , , where α and β are related via the following: for some < < λ Q 0 .Recently, Han and Zhang [23] established the reverse version of inequality (1.4), i.e., for < λ 0 and , there exists a sharp constant N λ Q , such that (1.7) The sharp constant satisfies and the volume of B 1 is defined as (see [12]): Different from the proof of [20], Han and Zhang proved the existence of extremal functions for inequality (1.7) by using the subcritical approach and renormalization method.
In the last two decades, various extensions of HLS inequality have been investigated.We focus on the typical examples of reverse HLS inequalities with weights.Chen et al. [6] established the following reverse Stein-Weiss inequality: where p q α β , , , , and λ satisfy They obtained the existence of extremal functions for inequality (1.8).Later, Chen et al. [5] established reverse Stein-Weiss inequalities on the upper half space and proved the existence of their extremal functions.Dai et al. [8] obtained the sharp reversed HLS inequality with an extended kernel on the upper half space, and computed the sharp constants.More recently, Dou et al. [13] proved the reversed HLS inequality with vertical weights on the upper half space and discussed the extremal functions by the renormalization method.For more results about (weighted) HLS-type inequalities, we refer the reader to [9,10,14,15,21,31,32] and the references therein.In contrast, not much is known about reverse HLS-type inequalities on the Heisenberg group or the CR sphere.The main motivation of this article is to introduce the reverse HLS inequality with weights on the Heisenberg group.We first obtain the following reverse HLS-type inequality with | | z weights on n .Assume that α β λ p , , , , and q satisfy where .
Define operators By duality and the reversed Hölder's inequality, we readily check that the inequality (1.10) is equivalent to the following two inequalities.
Corollary 1.2.Let α β λ p , , , , and q satisfy equation (1.9).Then, there exists a constant ( )> C Q α β λ p , , , , 0 depending on Q α β λ , , , , and p such that holds for any nonnegative function We also obtain an improved form of inequality (1.10) without assumptions 2 , < p 0 and < q 1, and p q , satisfy + = + . Then, there exists a positive constant such that for any nonnegative functions ( ) ∈ f L p n and ( ) ∈ g L q n , we have if one of the following conditions is satisfied: Moreover, for case ( ) i , S α β λ , , satisfies and for case ( ) ii , S α β λ , , satisfies where A 1 and A 2 are defined as in Theorem 1.1.
To prove Theorems 1.1 and 1.3, we adopt the standard approach, based on the reverse integral version of Hardy inequalities on the Heisenberg group (see [29]).The reverse integral version of Hardy inequalities on the upper half space was established by Chen et al. [5] and Tao [35].Once we establish inequality (1.10), it is natural to consider the attainability of sharp constant N α β λ , , .For the case , we consider the following minimizing problem: We obtain the following result for the extremal problem of inequality (1.10), which, to the best of our knowledge, is new.
where C α β λ , , is a positive constant depending on Q α β , , , and λ.Hence, the existence of extremal functions for inequality (1.10) is in fact the existence problem of equation (1.15).
p q 2, Beckner [1] obtained the sharp constant of inequality (1.6) and the non- existence of optimizers.Recently, in general case < ′ q p , Chen et al. [7] proved the existence of extremal functions for inequality (1.5) under assumption + ≥ α β 0. This result is more general than Lieb's results (see [30]) in the sense that they removed the restriction of ≥ α β , 0. Lieb [30] used the rearrangement inequality to prove the existence of extremals for inequality (1.1) in Euclidean spaces.Due to the absence of rearrangement inequality in the Heisenberg group, Chen et al. [7] obtained the existence of the maximizers for inequalities (1.5) and (1.6) by employing the concentration compactness principle.
In this article, we focus on the less investigated inequality (1.10) and the existence of extremal functions.We verify the existence of minimizers for inequality (1.10) by combining the subcritical approach (see [10,16]) Reversed HLS inequalities with weights on the Heisenberg group  5 and the renormalization method (see [13,23,36]).We cannot simply follow the same line of [13,16,23,36] to prove the existence of extremal functions for inequality (1.10) due to the loss of translation invariance of reverse weighted HLS inequality and the complex structure of the Heisenberg group.To overcome these difficulties, we use the subcritical approach to construct the extremal sequences of inequality (1.15), then combine the renormalization method and rotation invariance to obtain the expected attainability of the best constant.We believe this method can also be applied to obtain the existence of extremal function for several other weighted integral inequalities on the Heisenberg group.
We also investigate the reverse Stein-Weiss inequality in the Heisenberg group.
. Then, there exists a positive constant such that for any nonnegative functions ( ) ∈ f L p n and ( ) ∈ g L q n , we have , , , , if one of the following conditions is satisfied: Moreover, for case and for case ( ) where where ω n is defined as Theorem 1.1.
Note that for = + Q n 2 2, the reverse Stein-Weiss inequality on the homogeneous group established by Kassymov et al. [28] also contains the case of Heisenberg group.Different from the study by Kassymov et al. [28], we obtain the upper and lower bounds of constant N Q α β λ p , , , , for inequality (1.16).Similar to Theorem 1.1, we can also prove Theorem 1.6.Hence, we omit the details of the proof of Theorem 1.6.
Define operators holds for any nonnegative function This article is organized as follows: Section 2 is devoted to establishing the reverse weighted HLS inequality (1.10); in Section 3, combining the subcritical approach with the renormalization method, we prove the existence of extremal function of the reverse weighted HLS inequality (1.10).
Throughout the article, positive constants are denoted by c and C (with subscript in some cases) and are allowed to vary within a single line or formula.

The reverse weighted HLS inequality on the Heisenberg group
In this section, we prove the reverse weighted HLS inequalities on the Heisenberg group.The following lemma is the reverse integral version of Hardy inequalities on general homogeneous groups in the study by Kassymov et al. [29], which will be crucial in our proof of Theorem 1.2.
Lemma 2.1.[29] Let be a homogeneous group of homogeneous dimension Q. Assume that ( ) ∈ q 0, 1 and ′ < p 0. Suppose that ≥ W U , 0are locally integrable functions on .Then, the inequality holds for some ( ) ′ > C p q , 0 1 and all nonnegative measurable functions f if and only if (2.2) In addition, inequality and all nonnegative measurable functions f if and only if (2.4) 3) has the following relation to A 2 : Reversed HLS inequalities with weights on the Heisenberg group  7 , the homogeneous group contains the Heisenberg group n .Hence, Lemma 2.1 also holds on the Heisenberg group n .
Proof of Theorem 1.1.By reversed Hölder's inequality, we have It is easy to see that equation (1.12) is equivalent to Therefore, we obtain Similarly, one has Combining equations (2.5) and (2.6), we have Hence, it suffices to show that We divide into two cases to discuss. ( Then, for any > λ 0, we obtain Therefore, we have z βq in equation (2.1), it follows from equation (2.7) that where N α β λ , , can be seen as ( ) ′ C p q , 1 in equation (2.1).Now, we verify the condition (2.2).Since < − ′ β n q 2 , we have That is, Using polar decomposition, we have (2.9) where ( ) 2 2 is the n 2 -Lebesgue measure of the unitary Euclidean sphere in n 2 .
This completes the proof of Theorem 1.1.□ Proof of Theorem 1.3.From the proof process of Theorem 1.1, we immediately prove Theorem 1.3.

Existence of extremal functions for reverse HLS inequality with horizontal weights
This section is devoted to proving the existence of extremal functions for inequality (1.10) by combining the subcritical approach and the renormalization method.
To better illustrate our results, we first introduce some well-known notations.The sphere . On the sphere, the distance function is defined as follows: The standard Euclidean volume element of + n 2 1 will be denoted by ξ d .The Heisenberg group n is conformally equivalent to given by and with the inverse The Jacobian determinant of this transformation is given by , which implies that Reversed HLS inequalities with weights on the Heisenberg group  11 Under Cayley transform, a simple computation gives where For any , there exist corresponding functions respectively.
, it is easy to verify that the inequality (1.10) is equivalent to the following inequality on sphere where C α β λ , , is a positive constant depending on Q α β , , , and λ.We define the extremal function of inequality (3.1) as Then, we have , respectively.

Subcritical reverse weighted HLS inequality on CR sphere
The aim of this subsection is to establish the subcritical reverse weighted HLS inequality on + n 2 1 .We also prove the existence of the corresponding extremal functions, which is inspired by [10,13,16].
depending on Q α β λ , , , , and p such that for any nonnegative functions For ∈ + η B n 1 , we introduce the operator We can easily see that the inequality (3.2) is equivalent to , , , , for any nonnegative function Now we consider the extremal problem of inequality (3.2): By equations (3.2) and (3.4), we know that . The following proposition will be used in the proof of Theorem 1.4.

Proposition 3.3. ( )
i There exists a pair of nonnegative functions and n n α λ β , the functions F and G satisfy the Euler-Lagrange equations , , , , (3.5)

( )
iii There exists some constant Furthermore, is attained by a pair of nonnegative functions We choose a pair of nonnegative minimizing sequence { and We divide the proof of part (i) into three steps.
Step 1.Our first step is to claim that Reversed HLS inequalities with weights on the Heisenberg group  13 Then, using reversed Hölder's inequality, we obtain , , 4 Since ′ < ′ < q q 0 β , we know that for some constant > M 0 (to be determined later), , where . By equation (3.3) and reversed Hölder's inequality, we deduce . Using equations (3.8) and (3.9), we have which helps to conclude that Hence, there exists > ε 0 0 such that for any j, we can find two points 0 .Then, we have From this, we know that ‖ ‖ ( ) Step 2. There exist two subsequences of { } F j p and { } G j p (still denoted by { } F j p and { } G j p ) and two nonnegative functions Without loss of generality, we may assume that ≤ p q.By equation (3.7), we infer that there exist two subsequences of { } F j p and { } G j p (still denoted by { } F j p and { } G j p ) and two nonnegative functions , we immediately derive equation (3.10).
Step 3: We show that By equation (3.7) and the interpolation inequality, we obtain It follows that Then, for any fixed and for any fixed Similar to the arguments in the study by Dou et al. [11], we know that the above convergences are uniformly convergent for all ∈ + ξ η , n 2 1 .Therefore, for any > ε 0 small enough, there exists ∈ j N 0 such that for all > j j 0 , where in the last inequality, we have used Hölder's inequality.Noting that ( ) Reversed HLS inequalities with weights on the Heisenberg group  15 we arrive at From the above convergence and equation (3.11), we obtain Again by Hölder's inequality, we have We can then readily check that Combining the preceding estimate with equation (3.4) gives Therefore, we deduce that ( (ii).We show that F and G satisfy Euler-Lagrange equation (3.5).
To see this, we first prove > F 0 and > G 0 a.e. on where in the last equality, we have used the mean value theorem.Then, applying equation (3.12) and Fatou's lemma, we have Now we claim ( ) > F ξ 0 a.e. on + n 2 1 .If not, then for any > ε 0, there exists Then, using equation (3.12), we derive This gives us a contradiction when > ε 0 is sufficiently small.Again by repeating the whole procedure above for G, we have > G 0 a.e. on , is a pair of solutions of (3.5).(iii).We prove ( .Similarly, for some constant . For any given Using the boundedness of G and equation (3.13), we conclude that for any given Reversed HLS inequalities with weights on the Heisenberg group  17 where 2 .This implies that W is Hölder continuous on

Minimizer of reverse weighted HLS inequality on CR sphere
In this subsection, we prove Theorem 3.1.The method we shall use is similar to that used by Dou et al. [13], but we have to set up a framework to fit the complex structure of Heisenberg group.We first give the following lemma, which is crucial in our proof.
2 1 be defined as in Proposition 3.3.Then, there holds and the minimizer pair , and by reversed Hölder's inequality, we find , , This implies be a pair of minimizing sequence of N α β λ , , .It follows (3.17) Then, by equation (3.17) and dominated convergence theorem, we obtain Letting → ∞ k , we arrive at Therefore, by equations (3.16) and (3.18), we have Then, using Hölder's inequality yields , as , .
Hence, we obtain equation (3.15).□ Now we are ready to give the proof of Theorem 3.1.
Proof of Theorem 3.1.We start our proof by letting { 2 1 be a minimizing sequence of N α β λ , , .Then, { } F G , p q satisfies equation (3.5).By the scale and rotation invariance, without loss of generality, we assume that ( ) with ( ) = 1, 0, …,0 .There are two possible cases as follows.Case 1.For some subsequences → p p j α and → q q j β , is uniformly bounded.
By repeating the same argument as equations (3.6) and (3.14), we deduce that { } F p j and { } G q j are uniformly bounded and equicontinuous on + n 2 1 .By equation (3.5), we know that there exists some constant > C 0 (independent of p j and q j ) such that . Then, it follows from Arelà-Ascoli theorem that there exist two subsequences of { } F p j and { } G q j (still denoted by { } F p j and { } G q j ) and two nonnegative functions Then, combining equation (3.5) with Lemma 3.4 gives as → +∞ j .It follows that F and G are minimizers.Case 2. For any subsequences → p p j α and → q q j β , ( ) . Without loss of generality, we assume that ( ) → +∞ F p j .

Case (i).
( ) . Then, there exist two subsequences of p j and q j (still denoted by p j and q j ) such that ( ) → +∞ F p j and , and noting that ‖ ‖ , we derive From this and equation (3.5), we have , , , , Then, applying Cayley transform and dilation on n gives . 1 (3.20) Now we take ℓ ℓ = j such that ℓ ( ℓ ) ( and let (3.21) By a direct computation, one finds that Φ j and Ψ j satisfy the following renormalized equations: , and for ≤ p q j j , uniformly for any as .
We claim that there exists a constant ≥ Once the inequality (3.24) holds and by equations (1.3) and (3.22) and polar decomposition, we find where ω n is defined as in Theorem 1.
This implies Then, there exist > R 1 large enough and a measurable set E such that On the one hand, for any ∈ u n , we have It follows that there exists a constant ≥ Similarly, there exists a constant ≥ C 1 2

such that
Reversed HLS inequalities with weights on the Heisenberg group  21 On the other hand, we know that To simplify, we write ( ) = u ˜1, 0, …,0 .Then, combining equations (3.26) and (3.27) gives .Then, there exist two subsequences of p j and q j (still denoted by p j and q j ) such that ( ) → +∞ F p j and . This implies that → +∞ ∈ + G max ξ q n j 2 1 . Similar to Case (i), we know that Case (ii) does not hold.

Case (iii).
( ) . Then, there exist two subsequences of p j and q j (still denoted by p j and q j ) such that ( ) → +∞   Thus, ( ) u Φ j is uniformly bounded in ( ) B R 0, 0 .Similarly, ( ) u Ψ j is uniformly bounded in ( ) B R 0, 0 .A similar argument as equation (3.14) also shows that ( ) u Φ j and ( ) u Ψ j are equicontinuous in ( ) B R 0, 0 .By Arzelà-Ascoli theorem, we know that there exist two subsequences of { } Φ j and { } Ψ j (still denoted by { } Φ j and { } Ψ j ) and two functions Φ and Ψ with lower bound > C 0 such that Φ and Ψ Ψ as uniformly in 0, .

7 )
Indeed, we know from equation (1.11) that there exist constants >

+ n 2 1
, and hence F is at least Hölder continuous on + n 2 1 .Repeating the above argument shows that G is at least Hölder continuous on + n 2 1 .□

.
By repeating the similar argument as in Case (i), we choose { } F G , j j defined as in equation (3.21), and hence F j and G j satisfy equation (3.22).Moreover,

1 .
Therefore, ( ) u Ψ j has uniformly lower bound > c 0Repeating the arguments as equation(3.24),we also deduce that there exist constants , using equations (3.28) and (3.31) and the lower bound of ( )

j j 0
Since R 0 is arbitrary, we can conclude that ( ) haveReversed HLS inequalities with weights on the Heisenberg group  23 of minimizers of sharp constant N α β λ , , .This completes the proof of Theorem 3.1.□ Proof of Theorem 1.4.Theorem 1.4 is an easy consequence of Theorem 3.1.□ 16 is easy to verify that the inequality (1.16) is equivalent to the following two inequalities.
1 n This contradicts equation (3.23), and hence Case (i) is impossible.In the last part of Case (i), we prove that the inequality (3.24) holds.Using equations (3.19) and (3.21) gives