Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities

In this paper, we consider two limiting cases ($\alpha\rightarrow n$ and $\alpha\rightarrow 0$) of the recent affine HLS inequalities by Haddad and Ludwig. As $\alpha\rightarrow n$, 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 $\alpha\rightarrow 0$, 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.