Optimal regularity and Fredholm properties of abstract parabolic operators in $L^{p}$ spaces on the real line

• Autores: Davide Di Giorgio, Alessandra Lunardi, Roland Schnaubelt
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 91, Nº 3, 2005, págs. 707-737
• Idioma: inglés
• DOI: 10.1112/s0024611505015406
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• Resumen

  • We study the operator $(\mathcal{L} u)(t) := u'(t) - A(t) u(t)$ on $L^p (\mathbb{R}; X)$ for sectorial operators $A(t)$, with $t \in \mathbb{R}$, on a Banach space $X$ that are asymptotically hyperbolic, satisfy the Acquistapace¿Terreni conditions, and have the property of maximal $L^p$-regularity. We establish optimal regularity on the time interval $\mathbb{R}$ showing that $\mathcal{L}$ is closed on its minimal domain. We further give conditions for ensuring that $\mathcal{L}$ 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.