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Davide Di Giorgio, Alessandra Lunardi, Roland Schnaubelt, Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line, Proceedings of the London Mathematical Society, Volume 91, Issue 3, November 2005, Pages 703–737, https://doi.org/10.1112/S0024611505015406
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Abstract
We study the operator Lu(t):= u′(t) − A(t) u(t) on Lp (R; X) for sectorial operators A(t), with t ∈ R, on a Banach space X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreni conditions, and have the property of maximal Lp-regularity. We establish optimal regularity on the time interval R showing that L is closed on its minimal domain. We further give conditions for ensuring that L is a semi-Fredholm operator. The Fredholm property is shown to persist under A(t)-bounded perturbations, provided they are compact or have small A(t)-bounds. We apply our results to parabolic systems and to generalized Ornstein–Uhlenbeck operators. 2000 Mathematics Subject Classification 35K20, 35K90, 47A53.