A Maximal Theorem for Holomorphic Semigroups

This has a unique solution v(x, t) = Ttf(x) in the sense of Hille and Phillips [14, p. 622] whenever Tt is a C0–semigroup on X with generator −A; one writes Tt = e−tA [14, p. 321]. In order to ensure that v(x, t) converges μ-almost everywhere to f(x) as t→ 0+, it is often necesary to impose further conditions on f . For any closed linear operator V in X, we recall that the domain of V is the Banach space D(V ) = {f ∈ X : V f ∈ X} with the graph norm ‖f‖D(V ) = ‖f‖Lp + ‖V f‖Lp . For 0 < α < 1, we can define a fractional power A such that D(A) contains D(A), and A−α is bounded whenever A has bounded inverse; see [20, Theorem 2.3.1]. In many cases of interest, Tt extends to a bounded holomorphic semigroup on the † Corresponding author; email: [email protected] § Email: [email protected] MSC 2000 Classification: 47D60, 47D06