Space-like quantitative uniqueness for parabolic operators

TL;DR

This paper establishes sharp quantitative vanishing order estimates for solutions to parabolic equations with potentials, generalizing and sharpening previous results through a new Carleman estimate.

Contribution

It introduces a new quantitative Carleman estimate to obtain sharp vanishing order bounds for parabolic equations, extending prior work to a time-specific setting.

Findings

Sharp maximal vanishing order at a fixed time level.

A new Carleman estimate tailored for parabolic operators.

Generalization of Donnelly-Fefferman and Bakri's results.

Abstract

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C{^1}$ potential $V$. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.