Positive Criticality and Optimal Hardy Inequality for Fractional Laplacians

TL;DR

This paper characterizes positive critical Hardy weights for Laplacians on graphs, applies the results to fractional Laplacians, and identifies optimal Hardy weights with examples on various graph types.

Contribution

It provides a new characterization of Hardy weights for graph Laplacians and fractional Laplacians, leading to the identification of optimal Hardy weights under certain conditions.

Findings

Characterization of positive critical Hardy weights for Laplacians on graphs

Application to fractional Laplacians on general graphs

Identification of optimal Hardy weights with illustrative examples

Abstract

We characterize positive critical Hardy weights for general Laplacians on weighted graphs. We then apply this result to fractional Laplacians on general graphs and use the characterization to identify an optimal Hardy weight under suitable assumptions. We finally illustrate our results with examples of graphs which arise as Cayley graphs of groups, satisfy curvature assumptions or are fractal graphs.