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Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces. (English) Zbl 1312.42017

Summary: 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.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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