Sobolev and Hardy–Littlewood–Sobolev inequalities

Abstract

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.

Remainder terms for several inequalities on some groups of Heisenberg-type

“…Similar result for limit case -Beckner-Onofri and Logarithmic Hardy-Littlewood-Sobolev inequalities are also given in [DJ14] in case s = n = 2. We state on the sphere S n and denote − = 1 |S n | S n , then there exists a positive constant α, s.t., if − e f = 1, then…”

“…In a different way, Dolbeault pointed out in [Dol11] that in s = 2 case, the duality of Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities are greatly related to a fast diffusion equation and use that diffusion flow to compare the remainder terms of the two dual inequalities for n ≥ 5. Later Jin and Xiong [JX13] and Dobeault and Jankowiak [DJ14] extended the result respectively to the case s ∈ (0, 2), n ≥ 2, n > 2s and the case s = 2, n ≥ 3. Actually, we have for s = 2, n ≥ 3, p = q ′ = 2n n+s , there exists a positive constant α, s.t.,…”

“…The global estimate tells that the sharp constant, denoted by C sharp , is bigger than C, i.e., C sharp ≥ C, while the local estimate tells C sharp ≤ Q+4+s Q+4−s C. However, we expect a fast diffusion method like in [DJ14] may improve a bit the lower bound strictly to be C sharp > C.…”

“…Now we want to compare the dual remainder terms of FS and HLS inequalities by a constant multiple, both globally and locally. The similar problem on Euclidean spaces has been studied in [JX13,DJ14] and is now extended to the Heisenberg group for total range of s. …”

“…We also want to compare the remainder terms of the dual inequalities similar to provious works on Euclidean spaces like that in [DJ14].…”