Best constants for Moser-Trudinger inequalities on the Heisenberg group

Cohn, William S.; Lu, Guozhen

doi:10.1512/iumj.2001.50.2138

Cited by 128 publications

(84 citation statements)

References 18 publications

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“…The calculations are similar in spirit as those in [CoLu1], with some added complications given that we are now working on the sphere rather than H n . The smoothness hypothesis on g can be also relaxed a little to a Dini-type condition such as that of [CoLu1].…”

Section: Then There Existsmentioning

confidence: 99%

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

Branson¹,

2013

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

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Section: Then There Existsmentioning

confidence: 99%

“…The smoothness hypothesis on g can be also relaxed a little to a Dini-type condition such as that of [CoLu1]. More detailed work on this will appear in [FM], where the proof of the sharpness statement appears as an immediate application of general "abstract" theorems on measure spaces.…”

Section: Then There Existsmentioning

confidence: 99%

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

Branson¹,

2013

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“…Another reason to believe the correctness of H is that the endpoint case, with |u −1 v| −λ replaced by log |u −1 v|, has recently been settled [BFM07], and the function H with λ = 0 turns out to be the optimizer there, too. Some other recent, related works on sharp constants are [CL01], [CL04].…”

Section: Introductionmentioning

confidence: 99%

Sharp constants in several inequalities on the Heisenberg group

Frank¹,

Lieb²

2012

Ann. Math.

128

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We derive the sharp constants for the inequalities on the Heisenberg group H n whose analogues on Euclidean space R n are the well knownHardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities sharp constants for the analogues of the Onofri and log-Sobolev inequalities on H n are obtained. The methodology is completely different from that used to obtain the R n inequalities and can be (and has been) used to give a new, rearrangement free, proof of the HLS inequalities.

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“…Before we come to the proof of Lemma 2.1, we need to use the following result proved by Cohn and Lu [8]: Proposition 2.4 Let ω 2n−1 = 2π n Γ(n) be the surface area of the unit sphere in C n = R 2n and, for β > −2n, let…”

Section: A Representation Formula and Global Poincaré Inequality On H Nmentioning

confidence: 99%

“…It adapts ideas from our earlier paper by Cohn and Lu [8] on representation formulas for functions with compact support. Though we do not assume here that the functions f under consideration have compact support, we are able to carry out similar calculations to those given in [9] by carefully using the fundamental theorem of calculus.…”

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confidence: 98%

Global Poincaré Inequalities on the Heisenberg Group and Applications

Dong

Sun

2006

Acta Math Sinica

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Let f be in the localized nonisotropic Sobolev space W 1,p loc (H n ) on the n-dimensional Heisenberg group H n = C n × R, where 1 ≤ p < Q and Q = 2n + 2 is the homogeneous dimension of H n . Suppose that the subelliptic gradient is gloablly L p integrable, i.e., Ê H n |∇ H n f | p du is finite. We prove a Poincaré inequality for f on the entire space H n . Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C ∞ 0 (H n ) under the norm ofWe will also prove that the best constants and extremals for such Poincaré inequalities on H n are the same as those for Sobolev inequalities on H n . Using the results of Jerison and Lee on the sharp constant and extremals for L 2 to L 2Q Q−2 Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H n when p = 2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H n .

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Best constants for Moser-Trudinger inequalities on the Heisenberg group

Cited by 128 publications

References 18 publications

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

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

Sharp constants in several inequalities on the Heisenberg group

Global Poincaré Inequalities on the Heisenberg Group and Applications

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