Linear Poisson Equations with Potential on Riemann Surfaces

TL;DR

This paper investigates interior estimates and inequalities for solutions to a linear Poisson equation with potential on Riemann surfaces, focusing on cases where the data belongs to the Zygmund space and the surface satisfies the isoperimetric inequality.

Contribution

It provides new interior estimates, Harnack inequalities, and global bounds for solutions with data in the Zygmund space on Riemann surfaces.

Findings

Established interior estimates for solutions

Derived Harnack inequalities for the equation

Provided global estimates under isoperimetric conditions

Abstract

We study interior estimates for solutions of the linear Poisson equation: $$ \triangle u = g u + f $$ where $g$ and $f$ belong to the Zygmund space $L\ln L$ on a Riemann surface $M$ satisfying the isoperimetric inequality. As applications, we derive corresponding interior estimates, Harnack inequalities, and a global estimate.