APDE seminar@UPV/EHU: An optimal fractional Hardy inequality on the discrete half-line

Abstract

Hardy type inequalities have been a widely studied field since the first proofs in the early 20th century. The interest in these inequalities comes from their many applications in functional analysis, PDEs, spectral theory, or probability. 

In this context, we prove a discrete fractional Hardy’s inequality. In particular, a Hardy inequality concerning the fractional Laplacian in the discrete half-line with a weight that is optimal, in the sense that it cannot be substituted by any pointwise larger weight and its decay is as slow as possible. The strategy of the proof relies mainly on spectral properties of the aforementioned operator and criticality theory for graph Laplacians. As an immediate application, we derive some unique continuation results for positive Schrödinger operators involving the fractional Laplacian on the discrete half-line. Joint work with Ujjal Das.