Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities

Abstract

In this paper, we consider two limiting cases ($α\rightarrow n$ and $α\rightarrow 0 $) of the recent affine HLS inequalities by Haddad and Ludwig. As $α\rightarrow n$, the affine logarithmic HLS inequality is established, which is stronger than the logarithmic HLS inequality by Carlen and Loss from 1992 and Beckner from 1993. As $α\rightarrow 0$, an affine version of Beckner's logarithmic Sobolev inequality is established, which is also a limiting case of the affine fractional $L^2$ Sobolev inequalities. The affine logarithmic Sobolev inequality is stronger than the original version by Beckner from 1995.