Stein–Weiss inequalities with the fractional Poisson kernel

Chen, Lu; Liu, Zhao; Lu, Guozhen; Tao, Chunxia

doi:10.4171/rmi/1167

Cited by 15 publications

(6 citation statements)

References 39 publications

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“…We mention that the Stein-Weiss inequalities with fractional Poisson kernels have been established recently by Chen, Liu, Lu and Tao [7] using the Hardy-Littlewood-Sobolev inequalities proved by Chen, Lu and Tao [8].…”

Section: Introductionmentioning

confidence: 88%

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Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

Chen

Tao

2019

Journal of Functional Analysis

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In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1 and the Stein-Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein-Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein-Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein-Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).

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Section: Introductionmentioning

confidence: 88%

“…Thanks to (u, v) ∈ L p 0 +1 (R m+n ) × L p 2 +1 (R m+n ) and integral inequality (1.11), we derive that In virtue of the conditions (u, v) ∈ L p 1 +1 (R m+n ) × L p 2 +1 (R m+n ), we can choose sufficiently negative ν such that 7) which implies that H u ν and H v ν must be must be empty sets.…”

Section: The Proof Of Theorem 16mentioning

confidence: 99%

Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

Chen

Tao

2019

Journal of Functional Analysis

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“…Different from Dou et al [21], Chen et al [12] derived the Hardy-Littlewood-Sobolev inequality to all critical index for β = 1. Furthermore, Chen et al [11] extended it to the weighted Hardy-Littlewood-Sobolev inequality.…”

Section: Introductionmentioning

confidence: 99%

Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

Liu

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel (0.1) \begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation} for any nonnegative functions $f\in L^{p}(\partial \mathbb {R}_+^{n})$ , $g\in L^{q'}(\mathbb {R}_+^{n})$ and $p,\,\ q'\in (1,\,\infty )$ , $\beta \geq 0$ , $\alpha +\beta >1$ such that $\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$ . We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

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“…More recently, the authors have also proved some relevant works (see [10] and [12]) on the Hardy-Littlewood-Sobolev inequality and the Stein-Weiss inequality with fractional Poisson kernel on the upper half space, which was motivated by the work [28].…”

Section: Introductionmentioning

confidence: 99%

Reverse Stein–Weiss inequalities and existence of their extremal functions

Chen

Liu

et al. 2018

Trans. Amer. Math. Soc.

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In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in R n :We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere S n by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities.

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Stein–Weiss inequalities with the fractional Poisson kernel

Cited by 15 publications

References 39 publications

Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

Reverse Stein–Weiss inequalities and existence of their extremal functions

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