Heat kernels and maximal L^p-L^q estimates for parabolic evolution equations

Let A be the generator of an analytic semigroup T on L²(Ω), where Ω is a homogeneous space with doubling property. We prove maximal L^p - L^q a-priori estimates for the solution of the parabolic evolution equation u′(t) = Au(t) + f(t), u(0) = 0 provided T may be represented by a heat-kernel satisfying certain bounds (and in particular a Gaussian bound).