Unique continuation for the heat operator with potentials in weak spaces

TL;DR

This paper establishes strong unique continuation for the heat operator with potentials in weak Lorentz spaces, extending previous results and using Carleman estimates to handle potentials in $L^ abla_t L^{d/2, abla}_x$.

Contribution

It proves the strong unique continuation property for the heat operator with potentials in weak Lorentz spaces, filling a gap left open in prior research.

Findings

Proves strong unique continuation for potentials in $L^ abla_t L^{d/2, abla}_x$.

Uses Carleman estimates in Lorentz spaces.

Extends previous results by Escauriaza and Vega.

Abstract

We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.