Functional calculi for linear operators in vector-valued L^p-spaces via the transference principle

Let-A be the generator of a bounded C₀-group or of a positive contraction semigroup, respectively, on L^p(ω,μ,Y), where (ω,μ)is measure space, Y is a Banach space of class HT and 1 <p <∞. If Y = ℂ ℂ, it is shown by means of the transference principle due to Coifman and Weiss that A admits an H^∞-calculus on each double cone C_θ = {λ ε ℂ\{0} : | arg λ ± π/2| <θ}, where θ > 0 and on each sector ∑_θ = {λ ε ℂ\{0} : | arg λ| <θ} with θ > π/2, respectively. Several extensions of these results to the vector-valued case L^p(ω,μ,Y) are presented. In particular, let-A be the generator of a bounded group on a Banach spaces of class HT. Then it is shown that A admits an H_∞-calculus on each double cone C_θ, θ > 0, and that-A₂ admits an H_∞-calculus on each sector ∑_θ;, where θ > 0. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.