Uniformly convex and smooth Banach spaces and Lp-boundedness properties of Littlewood-Paley and area functions associated with semigroups

TL;DR

This paper characterizes uniformly convex and smooth Banach spaces through Lp-boundedness of Littlewood-Paley and area functions linked to heat semigroups, including non-Markovian cases like Hermite and Laguerre operators.

Contribution

It introduces new characterizations of Banach space geometry using Lp-boundedness of specific harmonic analysis functions associated with semigroups.

Findings

Characterizations of Banach spaces via Littlewood-Paley functions

Lp-boundedness properties for heat semigroups with fractional derivatives

Applicability to Hermite and Laguerre semigroups without Markovian property

Abstract

In this paper we obtain new characterizations of the uniformly convex and smooth Banach spaces. These characterizations are established by using Lp-boundedness properties of Littlewood-Paley functions and area integrals associated with heat semigroups and involving fractional derivatives. Our results apply for instance by considering the heat semigroups defined by Hermite and Laguerre operators that do not satisfy the Markovian property.