Sharp Beckner-type inequalities for Cauchy and spherical distributions
Volume 251 / 2020
Studia Mathematica 251 (2020), 219-245
MSC: Primary 60G10, 60-XX; Secondary 58-XX.
DOI: 10.4064/sm180503-17-1
Published online: 17 October 2019
Abstract
Using some harmonic extensions on the upper-half plane, probabilistic representations, and curvature-dimension inequalities with negative dimensions, we obtain some new optimal functional inequalities of Beckner type for Cauchy-type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non-tight optimal Beckner inequalities on the sphere, and this family of inequalities appears to be a new form of the Sobolev inequality.
