Poincar\'e type inequality for Dirichlet spaces and application to the uniqueness set

TL;DR

This paper extends Poincaré's capacitary inequality for Dirichlet spaces and applies it to characterize uniqueness sets on the unit circle, advancing understanding of boundary behavior in these function spaces.

Contribution

It introduces a generalized Poincaré-type inequality for Dirichlet spaces and uses it to analyze the properties of uniqueness sets on the unit circle.

Findings

Extended Poincaré inequality for Dirichlet spaces

Characterization of uniqueness sets on the unit circle

New tools for boundary analysis in Dirichlet spaces

Abstract

We give an extension of Poincar\'e's type capacitary inequality for Dirichlet spaces and provide an application to study the uniqueness sets on the unit circle for these spaces.