On subordinated semigroups and Hardy spaces associated to fractional powers of operators

TL;DR

This paper investigates the properties of fractional power subordinated semigroups generated by a positive self-adjoint operator on a metric measure space, establishing boundedness and characterizations of associated Hardy spaces.

Contribution

It provides new results on the boundedness and Hardy space characterizations for fractional powers of operators under various conditions.

Findings

Weak type (1,1) boundedness of the maximal operator

Characterizations of Hardy spaces via area integral and square function

Extension of semigroup results to L^p spaces

Abstract

Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $\sigma$-finite metric measure space. When $\alpha \in (0,1)$, the subordinated semigroup $\{\exp(-tL^{\alpha}):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and extended to $L^p(X)$. We prove various results about the semigroup $\{\exp(-tL^{\alpha}):t \in \mathbb{R}^+\}$, under different assumptions on $L$. These include the weak type $(1,1)$ boundedness of the maximal operator $f \mapsto \sup _{t\in \mathbb{R}^+}\exp(-tL^{\alpha})f$ and characterisations of Hardy spaces associated to the operator $L$ by the area integral and vertical square function.