A strong unique continuation property for the heat operator with Hardy type potential

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality $$\mathrm{\mid }\mathrm{\Delta }u-u_t\mathrm{\mid }\le \frac{M}{\mathrm{\mid }x{\mathrm{\mid }}^2}\mathrm{\mid }u\mathrm{\mid },$$ with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.