APDE seminar @UPV/EHU: Unique continuation for the heat operator with potentials in weak spaces

Abstract

This talk concerns the strong unique continuation property for the differential inequality

|(∂/∂t + Δ)u(x, t)| ≤ V(x, t) |u(x, t)|

where the potential $V$ belongs to suitable function spaces.  In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has remained open since the earlier works of Escauriaza and Escauriaza--Vega. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces, which in turn are built on the spectral projection estimates for the Hermite operator.