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We derive the sharp constants for the inequalities on the Heisenberg group $\mathbb{H}^n$ whose analogues on Euclidean space $\mathbb{R}^n$ are the well known Hardy-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 $\mathbb{H}^n$ are obtained. The methodology is completely different from that used to obtain the $\mathbb{R}^n$ inequalities and can be (and has been) used to give a new, rearrangement free, proof of the HLS inequalities.
@article{10_4007_annals_2012_176_1_6,
author = {Rupert L. Frank and Elliott H. Lieb},
title = {Sharp constants in several inequalities on the {Heisenberg} group},
journal = {Annals of mathematics},
pages = {349--381},
publisher = {mathdoc},
volume = {176},
number = {1},
year = {2012},
doi = {10.4007/annals.2012.176.1.6},
mrnumber = {2925386},
zbl = {1252.42023},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.6/}
}
TY - JOUR AU - Rupert L. Frank AU - Elliott H. Lieb TI - Sharp constants in several inequalities on the Heisenberg group JO - Annals of mathematics PY - 2012 SP - 349 EP - 381 VL - 176 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.6/ DO - 10.4007/annals.2012.176.1.6 LA - en ID - 10_4007_annals_2012_176_1_6 ER -
%0 Journal Article %A Rupert L. Frank %A Elliott H. Lieb %T Sharp constants in several inequalities on the Heisenberg group %J Annals of mathematics %D 2012 %P 349-381 %V 176 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.6/ %R 10.4007/annals.2012.176.1.6 %G en %F 10_4007_annals_2012_176_1_6
Rupert L. Frank; Elliott H. Lieb. Sharp constants in several inequalities on the Heisenberg group. Annals of mathematics, Tome 176 (2012) no. 1, pp. 349-381. doi: 10.4007/annals.2012.176.1.6
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