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

We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions ${T_t}_{t>0}$ on $L_p(Omega)$ for a fixed $1<p<infty$. We prove that if a Banach space $X$ is of martingale cotype $q$, then there is a constant $C$ such that $$ left|left(int_0^inftybig|tfrac{partial}{partial t}P_t (f)big|_X^q,frac{dt}tright)^{frac1q}right|_{L_p(Omega)}le C, big|fbig|_{L_p(Omega; X)},, quadforall, fin L_p(Omega; X),$$ where ${P_t}_{t>0}$ is the Poisson semigroup subordinated to ${T_t}_{t>0}$. Let $mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$mathsf{L}^{P}_{c,q, p}(X)lesssim maxbig(p^{frac1{q}},, pbig) mathsf{M}_{c,q}(X).$$ Moreover, the order $maxbig(p^{frac1{q}},, pbig)$ is optimal as $pto1$ and $ptoinfty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally ${T_t}_{t>0}$ is analytic on $L_p(Omega; X)$, the semigroup ${P_t}_{t>0}$ in these results can be replaced by ${T_t}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one 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 the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and ${P_t}_{t>0}$ is the classical Poisson and heat semigroups on $mathbb{R}^d$.