# Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere

@article{Branson2007MoserTrudingerAB, title={Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere}, author={Thomas P. Branson and Luigi Fontana and Carlo Morpurgo}, journal={Annals of Mathematics}, year={2007}, volume={177}, pages={1-52} }

We derive sharp Moser-Trudinger inequalities on the CR sphere. The rst type is in the Adams form, for powers of the sublaplacian and for general spectrally dened operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CRpluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss.

#### 98 Citations

A Derivation of the Sharp Moser–Trudinger–Onofri Inequalities from the Fractional Sobolev Inequalities

- Mathematics
- Peking Mathematical Journal
- 2018

We derive the sharp Moser–Trudinger–Onofri inequalities on the standard n-sphere and CR $$(2n+1)$$(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all $$n\ge 1$$n≥1. On the… Expand

Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth

- Mathematics
- 2015

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces… Expand

Moser-Trudinger inequality on conformal discs

- Mathematics
- 2009

We prove that a sharp Moser–Trudinger inequality holds true on a conformal disc if and only if the metric is bounded from above by the Poincare metric. We also derive necessary and sufficient… Expand

Moser-Trudinger and Adams type inequalities and their applications

- Mathematics
- 2014

MOSER-TRUDINGER AND ADAMS TYPE INEQUALITIES AND THEIR APPLICATIONS by NGUYEN LAM August 2014 Advisor: Dr. Guozhen Lu Major: Mathematics Degree: Doctor of Philosophy In this dissertation, we study… Expand

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

- Mathematics
- 2016

The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a… Expand

SHARP TRUDINGER-MOSER INEQUALITIES WITH HOMOGENEOUS WEIGHTS

- 2019

We investigate sharp Trudinger-Moser type inequalities with the homogeneous weight satisfying a natural curvature-dimension bound condition. Also we study the optimal versions of these inequalities… Expand

Sharp Moser-Trudinger inequalities for the Laplacian without boundary conditions

- Mathematics
- 2011

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of W2,n/2(Bn) into the exponential class, where Bn is the unit ball of Rn. The corresponding sharp results for the… Expand

The Moser-Trudinger-Onofri inequality

- Mathematics
- 2014

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev… Expand

Small-Time Asymptotics for Subelliptic Hermite Functions on SU(2) and the CR Sphere

- Mathematics
- 2017

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1.… Expand

Strichartz estimates for the Schrödinger equation for the sublaplacian on complex spheres

- Mathematics, Physics
- 2012

We study the nonlinear Schrodinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates… Expand

#### References

SHOWING 1-10 OF 97 REFERENCES

Best constants for Moser-Trudinger inequalities on the Heisenberg group

- Mathematics
- 2001

A Moser-Trudinger inequality (with sharp constant) is proven for convolution type potentials defined on stratified groups. When combined with a new representation formula for functions defined on the… Expand

Fundamental solution for the Q-Laplacian and sharp Moser–Trudinger inequality in Carnot groups

- Mathematics
- 2003

Abstract For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q -Laplacian u Q and a constant a Q >0 such that exp(− a Q u Q ) is a… Expand

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

- Mathematics
- 2004

The main results of this paper concern sharp constants for the Moser-Trudinger inequalities on spheres in complex space C n . We derive Moser-Trudinger inequalities for smooth functions and… Expand

Adams inequalities on measure spaces

- Mathematics
- 2009

Abstract In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of R n . The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In… Expand

Competing symmetries, the logarithmic HLS inequality and Onofri's inequality onsn

- Mathematics
- 1992

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckner's generalization of Onofri's inequality… Expand

Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality

- Mathematics
- 1993

where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains… Expand

Extremal metrics of zeta function determinants on 4-manifolds

- Mathematics
- 1995

In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe… Expand

The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions onSn

- Mathematics
- 1996

The functional estimated sharply by the logarithmic Hardy-Littlewood-Sobolev inequality onSn, for evenn≥2, is linked to the spectrum of the Paneitz operators. From the point of view of conformal… Expand

Sharp inequalities, the functional determinant, and the complementary series

- Mathematics
- 1995

Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional… Expand

Sobolev spaces and the Cayley transform

- Mathematics
- 2005

The generalised Cayley transform C from an Iwasawa N-group into the corresponding real unit sphere S induces isomorphisms between suitable Sobolev spaces H α (S) and H α (N). We study the… Expand