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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 Itype 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.

Sharp Hardy-Littlewood-Sobolev inequalities on a class of H-type groups