Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups

We study vector-valued Littlewood–Paley–Stein theory for semigroups ¹T_t º_{t >0} of regular contractions on L_p./ for a fixed 1 < p < 1. We prove that if a Banach space X is of martingale cotype q, then there is a constant C such that (^Z₀¹ t _@t^@ P_t .f / _X^{q d}_t^t)1=q_{Lp ./} C kf k_{Lp .IX} /; 8f 2 L_p.I X /; where ¹P_t º_{t >0} is the Poisson semigroup subordinated to ¹T_t º_t >0. Let L^P_c;q;p.X / be the least constant C, and let M_c;q.X / be the martingale cotype q constant of X . We show L^P_c;q;p.X / . max.p^1=q; p⁰/M_c;q.X /: Moreover, the order max.p^1=q; p⁰/ is optimal as p ! 1 and p ! 1. If X is of martingale type q, the reverse inequality holds. If additionally ¹T_t º_{t >0} is analytic on L_p.I X /, the semigroup ¹P_t º_t >0 in these results can be replaced by ¹T_t º_{t >0} itself. Our new approach is built on holomorphic functional calculus. Compared with the previous approaches, ours is more powerful in several aspects: (a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; (b) it yields the optimal orders of growth on p for most of the relevant constants; (c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood–Paley–Stein inequalities for symmetric submarkovian semigroups are better than those of Stein.