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

Abstract

We study vector-valued Littlewood–Paley–Stein theory for semigroups $\{T_t{\}}_{t>0}$ of regular contractions 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

where $\{P_t{\}}_{t>0}$ is the Poisson semigroup subordinated to $\{T_t{\}}_{t>0}$. Let ${\mathsf{L}}_{\mathsf{c},q,p}^P(X)$ be the least constant $C$, and let ${\mathsf{M}}_{\mathsf{c},q}(X)$ be the martingale cotype $q$ constant of $X$. We show

Moreover, the order $\max (p^{1/q},p^{\prime })$ is optimal as $p\to 1$ 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 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. 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 or heat semigroup on ${\mathbb{R}}^d$.