Heat kernels and maximal l^p—l^qestimates for parabolic evolution equations

Abstract

Let A be the generator of an analytic semigroup Ton L2(Ω), where Ω is a homogeneous space with doubling property. We prove maximal Lp-Lp a—priori estimates for the solution of the parabolic evolution equation u'(t)=Au(t)+f(t), u(0)=0 provided Tmay be represented by a heat—kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification:35K22, 58D25, 47D06