Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

We prove two extrapolation results for singular integral operators with operator-valued kernels, and we apply these results in order to obtain the following extrapolation of L^p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L^p-maximal regularity for some p ∈ (1, ∞), then it has E_w-maximal regularity for every rearrangement invariant Banach function space E with Boyd indices 1 < p_E ≤ q_E < ∞ and every Muckenhoupt weight w ∈ A_pE . We prove a similar result for nonautonomous Cauchy problems on the line.