A gradient estimate for harmonic functions sharing the same zeros

TL;DR

This paper establishes a gradient estimate for the ratio of two harmonic functions sharing the same zeros, extending classical results and providing bounds based solely on their common zero set.

Contribution

It introduces a new gradient estimate for harmonic functions with shared zeros, generalizing Li-Yau's estimate and connecting to the boundary Harnack principle.

Findings

Gradient of log |u/v| is bounded in the unit disk based on the zero set Z.

Extends Li-Yau's estimate to harmonic functions with shared zeros.

Provides Hölder estimates for log |u/v| using the boundary Harnack principle.

Abstract

Let u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives H\"older estimates on \log |u/v|.