Weighted discrete Hardy inequalities on trees and applications

TL;DR

This paper develops new weighted discrete Hardy inequalities on trees and applies them to establish improved inequalities and solvability results for weighted Sobolev spaces on H"older domains, broadening the scope of applicable weights.

Contribution

It introduces a novel approach using weighted discrete Hardy inequalities on trees to derive improved inequalities and solvability conditions for weighted Sobolev spaces.

Findings

Weaker conditions on weight exponents compared to existing literature.

Establishment of weighted Hardy inequalities on trees.

Applications to divergence, Poincaré, and Korn inequalities.

Abstract

In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on H\"older-$\alpha$ domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation's solvability, and the improved Poincar\'e, the fractional Poincar\'e, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature.