Unique continuation for the heat operator with potentials in weak spaces

Abstract

We prove the strong unique continuation property for the differential inequality

$$|({\partial }_t+\mathrm{\Delta })u(x,t)|\le V(x,t)|u(x,t)|,$$

with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in L_t^{\infty }{L}_x^{[t]d∕2,\infty }$, which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.