Theory of Markov Semigroups
and Schrödinger Operators

Abstract:

The aim of the talk is to present results from the preprint “Critical discrete Hardy inequalities in $L^p$,” coauthored with K. Bogdan, K. Kaleta, and S. Gupta. In this work, we establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p \in (1,\infty)$. Our approach uses Sobolev–Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

Abstract:

In the first part of the talk, we discuss optimal discrete $p$-Hardy inequalities of higher order for general $p>1$, i.e., optimal Hardy-type inequalities for integer powers of the discrete $p$-Laplacian on the half-line. The second part then focuses on recent developments in the context of Hardy inequalities for $p=2$ and arbitrary positive powers of the discrete Laplacian on the half-line.