Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

We show that, for negative generators of analytic semigroups, a bounded H∞‐calculus self‐improves to an Rs‐bounded H∞‐calculus in an appropriate scale of Lp‐spaces if the semigroup satisfies suitable generalized Gaussian estimates. As application of our result we obtain that large classes of differential operators have an Rs‐bounded H∞‐calculus.