Heat kernel and maximal L-p-L-q estimates: the non-autonomous case

Consider the non-autonomous initial value problem u'(t) + A(t)u(t) = f(t), u(0) = 0, where - A(t) is for each t is an element of [0, T], the generator of a bounded analytic semigroup on L-2 (Omega). We prove maximal L-p - L-q a priori estimates for the solution of the above equation provided the sem igroups Tt are associated to kernels which satisfies an upper Gaussian boun d and {A(t), t is an element of [0, T]} fulfills a Acquistapace-Terreni com mutator condition. (C) Academie des Sciences/Elsevier, Paris.