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

Abstract

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 $${p \in (1,\infty )}$$ , then it has $${\mathbb{E}_w}$$ -maximal regularity for every rearrangement invariant Banach function space $${\mathbb{E}}$$ with Boyd indices $${1 < p_\mathbb{E} \leq q_\mathbb{E} < \infty}$$ and every Muckenhoupt weight $${w \in A_{p \mathbb{E}}}$$ . We prove a similar result for nonautonomous Cauchy problems on the line.