Sharp constants in several inequalities on the Heisenberg group

By Rupert L. Frank, Elliott H. Lieb. 2010

230 citations recorded by the source corpus.

We derive the sharp constants for the inequalities on the Heisenberg group H^n whose analogues on Euclidean space 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 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38cd89537d4950e3e98004fc07c1b0c3982d1110

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In Applied Mathematics

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Citations in the corpus

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group (2012, unknown): “Lieb studied the optimizers of the (double) weighted fractional integral in Euclidean spaces (Theorem 5.1 in [5]), under a stronger assumption that α, β ≥ 0 instead of α + β ≥ 0.”
An extension problem for the CR fractional Laplacian (2013, background): “The problem of determining the sharp constant for general γ was solved in [26].”
Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group (2015, methodology): “Moreover, we show with detail the weaker Hardy inequality that can be obtained from the HLS inequality in [20].”
A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality (2010, methodology): “Introduction In a recent paper [11] we showed how to compute the sharp constants for the analogue of the Hardy-Littlewood-Sobolev (HLS) inequality on the Heisenberg group.”
Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications ✩ (2012, background): “We can state the result in [32] as the following theorem.”