Linear Poisson Equations with Potential on Riemann Surfaces

TL;DR

This work analyzes the Poisson equation with a potential, $\Delta u = g u + f$, on a Riemann surface $M$ endowed with an isoperimetric inequality, with data in the Zygmund space $L\log L$ and related Hardy spaces. It proves an interior $C^0$-estimate for $u$ in terms of the $L^1$-norm of $u$ and the $L\log L$-norm of $f$, via geometric-analytic bounds adapted to $M$ and a Moser-style iteration adapted to these data. Building on this, the authors derive Harnack inequalities for solutions, including cases where $f\in L\log L$ and $f\in H^1$, and further obtain a global $C$-norm bound on $M$ in terms of $\|f\|^*_{L\log L}$. The results extend classical interior estimates to the setting of Riemann surfaces with isoperimetric control and low-regularity data, integrating tools from geometric analysis, Hardy spaces, and nonlinear potential theory to yield robust a priori bounds.

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.

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof