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 $$\left\|\left(\int_0^\infty\big\|t\frac{\partial}{\partial t}P_t (f)\big\|_X^q\,\frac{dt}t\right)^{\frac1q}\right\|_{L_p(\Omega)}\le C\, \big\|f\big\|_{L_p(\Omega; X)}\,, \quad\forall\, f\in L_p(\Omega; X),$$ 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 $$\mathsf{L}^{P}_{c,q, p}(X)\lesssim \max\big(p^{\frac1{q}},\, p'\big) \mathsf{M}_{c,q}(X).$$ Moreover, the order $\max\big(p^{\frac1{q}},\, p'\big)$ is optimal as $p\to1$ and $p\to\infty$. 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 $\mathbb{R}^d$.