A GENERALIZED H~∞ FUNCTIONAL CALCULUS FOR OPERATORS ON SUBSPACES OF L~p AND APPLICATION TO MAXIMAL REGULARITY

摘要

Let H be a Hilbert space and let A be the closure, which exists, of A directX I_H on L~p(Ω; H). In a recent joint work with F. Lancien, we showed that for any v > θ, the bounded holomorphic functional calculus of A naturally extends to a bounded H~∞(Σ_v; B(H)) functional calculus for A. As a consequence, we could deduce abstract maximal regularity results on spaces of the form L~p(Ω; H), for operators which are the sum of an operator acting on L~p(Ω) and another one acting on H. The purpose of this paper is to extend these results to the case p = 1 and to the situation where L~p(Ω) is replaced by one of its closed subspaces. As a consequence, we get a new class of operators satisfying the L_p-maximal regularity property for the first order Cauchy problem on intervals. As a matter of fact, the present work yields a new proof of Theorem 5.2 in [8] which is somewhat simpler than the original one.