Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations

In this paper, we show that a pseudo-differential operator associated to a symbol a ε L^∞(ℝ × ℝ, L(H)) (H being a Hubert space) which admits a holomorphic extension to a suitable sector of ℂ acts as a bounded operator on L²(ℝ, H). By showing that maximal L^p-regularity for the nonautonomous parabolic equation u′(t)+A(t)u(t) = f(t), u(0) = 0 is independent of p ε (1, ∞), we obtain as a consequence a maximal L^p([0, T], H)-regularity result for solutions of the above equation.