Unique continuation for the heat operator with potentials in weak spaces

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 $V\in L_t^{\infty }{L}_x^{d/2,\infty }$, 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.

The current paper is the last section of the paper Hermite spectral projection operator (arXiv:2006.11762v2). The previous paper will remain unpublished

References in corpus (1)

• On the boundary convergence of solutions to the Hermite-Schrödinger equation