Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications

Lam, Nguyen; Lu, Guozhen

doi:10.1016/j.aim.2012.09.004

Cited by 125 publications

(75 citation statements)

References 47 publications

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“…This condition is equivalent to asking that, outside a fixed ball, ν is absolutely continuous with respect to the Lebesgue measure, with bounded Radon-Nikodym derivative. An example is the singular measure dν(x) = |x| (σ−1)n dx considered in [LL1], [LL2], [AY] and other papers.…”

Section: Inequalities For More General Borel Measuresmentioning

confidence: 99%

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

Fontana

Morpurgo

2018

Nonlinear Analysis

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We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R n . As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W α, n α (R n ), 0 < α < n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.

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Section: Inequalities For More General Borel Measuresmentioning

confidence: 99%

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

Fontana

Morpurgo

2018

Nonlinear Analysis

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“…

Wang and Ye conjectured in [22]:Let Ω be a regular, bounded and convex domain in R 2 . There exists a finite constant C(Ω) > 0 such thatThe main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in R 2 via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant).We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space Bby using the method employed earlier by Lam and the first author [9,10], where H denotes the closure of C ∞ 0 (B) with respect to the normUsing this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].

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confidence: 63%

A sharp Trudinger–Moser inequality on any bounded and convex planar domain

Yang

2016

Calc. Var.

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Abstract. Wang and Ye conjectured in [22]:Let Ω be a regular, bounded and convex domain in R 2 . There exists a finite constantwhereThe main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in R 2 via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant).We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space B = {z = x + iy : |z| = x 2 + y 2 < 1}:by using the method employed earlier by Lam and the first author [9,10], where H denotes the closure of C ∞ 0 (B) with respect to the norm(1 − |z| 2 ) 2 dxdy. Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].

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“…We remark as a borderline case of the sharp Sobolev inequality on the Heisenberg group, the sharp Trudinger-Moser inequality on finite domains was established by Cohn and Lu [13] and by Lam and Lu [14] for critical Trudinger-Moser inequality and by Lam, Lu and Tang [15] for the subcritical Trudinger-Moser inequality on the entire Heisenberg group.…”

Section: Introductionmentioning

confidence: 76%

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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Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications

Cited by 125 publications

References 47 publications

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

A sharp Trudinger–Moser inequality on any bounded and convex planar domain

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

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