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 $a\in L^{\infty}(\mathbb{R}\times\mathbb{R},\mathcal{L}(H))$ ($H$ being a Hilbert space) which admits a holomorphic extension to a suitable sector of $\mathbb{C}$ acts as a bounded operator on $L^{2}(\mathbb{R},H)$. By showing that maximal $L^{p}$-regularity for the non-autonomous parabolic equation $u'(t) + A(t)u(t) = f(t), u(0)=0$ is independent of $p\in (1,\infty)$, we obtain as a consequence a maximal $L^{p}([0,T],H)$-regularity result for solutions of the above equation.