Quantitative estimates for bounded holomorphic semigroups

TL;DR

This paper provides quantitative bounds for bounded holomorphic semigroups on Banach spaces and applies these results to improve vector-valued Littlewood--Paley--Stein inequalities for symmetric diffusion semigroups.

Contribution

It introduces new quantitative bounds for holomorphic semigroups and extends recent results in vector-valued harmonic analysis.

Findings

Established explicit bounds for holomorphic semigroups.

Derived improved quantitative Littlewood--Paley--Stein inequalities.

Enhanced understanding of semigroup behavior in Banach spaces.

Abstract

In this paper we revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood--Paley--Stein theory for symmetric diffusion semigroups.