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

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}.