Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials

We prove Strichartz estimates for the Schrödinger equation perturbed with a small time dependent potential V(t,x) belonging to the weak Lebesgue space L^∞_tL^(n/2,∞)_x. We also consider the heat equation perturbed with a time independent singular potential V(x)∈L^(n/2,∞) and we prove both the maximum principle and the Strichartz estimates under a suitable smallness condition on the negative part of the potential.