This report is based on a talk given by the author in the Laurent Schwartz seminar at IHÉS, Paris, on February 16, 2016. This involves joint works with Michael Christ and Heping Liu [CLZ16a, CLZ16b, LZ15]. We review several sharp Hardy-Littlewood-Sobolev-type inequalities (HLS) on I-type groups (rank one), which is a special class of H-type groups, using the symmetrization-free method of Frank and Lieb, who proved the sharp HLS on the Heisenberg group in a seminal paper [FL12b]. We give the sharp HLS both on the compact and noncompact pictures. The “unique” extremal function, as expected, can only be constant function on the sphere. Their dual form, a sharp conformally invariant inequality involving an intertwining operator (“fractional subLaplacian”), and the right endpoint case, a Log-Sobolev inequality, are also obtained. Besides, some stability and dual type improvements are also discussed. A positivity-type restriction on the singular exponent is required in the cases with centres of high dimensions, which bring extra difficulty. The conformal symmetry of the inequalities, zero center-mass technique, estimates involving meticulous computation of eigenvalues of singular kernels, compactness and local stability play a critical role in the argument.
@article{SLSEDP_2015-2016____A11_0,
author = {An Zhang},
title = {Sharp {Hardy-Littlewood-Sobolev} inequalities on~a~class of {H-type} groups},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:11},
pages = {1--8},
publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
year = {2015-2016},
doi = {10.5802/slsedp.98},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.98/}
}
TY - JOUR AU - An Zhang TI - Sharp Hardy-Littlewood-Sobolev inequalities on a class of H-type groups JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:11 PY - 2015-2016 SP - 1 EP - 8 PB - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.98/ DO - 10.5802/slsedp.98 LA - en ID - SLSEDP_2015-2016____A11_0 ER -
%0 Journal Article %A An Zhang %T Sharp Hardy-Littlewood-Sobolev inequalities on a class of H-type groups %J Séminaire Laurent Schwartz — EDP et applications %Z talk:11 %D 2015-2016 %P 1-8 %I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.98/ %R 10.5802/slsedp.98 %G en %F SLSEDP_2015-2016____A11_0
An Zhang. Sharp Hardy-Littlewood-Sobolev inequalities on a class of H-type groups. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 11, 8 p. doi : 10.5802/slsedp.98. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.98/
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