Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences∗
نویسندگان
چکیده
This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
منابع مشابه
Improved interpolation inequalities on the sphere
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
متن کاملSharp interpolation inequalities for discrete operators and applications
We consider interpolation inequalities for imbeddings of the l2-sequence spaces overd-dimensional lattices into the l∞ 0 spaceswritten as interpolation inequality between the l2-norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlson’s inequalities and spectral ...
متن کاملMoser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere
We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CRpluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CR-pluriharmonic functions on the sphere, and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequali...
متن کاملWeighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy – entropy productio...
متن کاملSpectral estimates on the sphere
In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on Lp norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Eucli...
متن کامل