A sharp equivalence between $H^\infty$ functional calculus and square function estimates

TL;DR

This paper establishes a precise equivalence between the boundedness of the $H^$ functional calculus for an operator and the validity of certain square function estimates, refining previous results on analytic semigroups.

Contribution

It proves that square function estimates imply a bounded $H^$ functional calculus for some angle less than , improving the known threshold.

Findings

Square function estimates imply bounded $H^$ calculus for some  angle

Refinement of previous results on analytic semigroups

Establishes a sharp equivalence between two key operator properties

Abstract

Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp} \lesssim \norm{x} and \bignorm{\bigl(\int_{0}^{\infty}|A^{*}^{1/2} T_t^*(y)|^2\, dt\,\bigr)^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in Lp', then A admits a bounded H^{\infty}(\Sigma_\theta) functional calculus for any \theta>\frac{\pi}{2}. We show that this actually holds true for some \theta<\frac{\pi}{2}.