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 is an element of L-infinity (R x R, L(H)) (H being a Hilbert space ) which admits a holomorphic extension to a suitable sector of C acts as a bounded operator on L-2(R, H). By showing that maximal L-p-regularity for t he non-autonomous parabolic equation u'(t) + A(t)u(t) = f(t), u(0) = 0 is i ndependent of p is an element of (1, infinity), we obtain as a consequence a maximal L-p ([0; T],H)-regularity result for solutions of the above equat ion.