Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

TL;DR

This paper characterizes Hardy spaces associated with Laguerre polynomial expansions using maximal functions and proves boundedness of a heat semigroup maximal operator on these spaces.

Contribution

It introduces a new atomic Hardy space related to Laguerre expansions and provides maximal function characterizations and boundedness results.

Findings

Characterization of Hardy space via local maximal functions

Boundedness of heat semigroup maximal operator on Hardy space

Extension of Hardy space theory to non-doubling measures

Abstract

In this paper we introduce the atomic Hardy space $\mathcal{H}^1((0,\infty),\gamma_\alpha)$ associated with the non-doubling probability measure $d\gamma_\alpha(x)=\frac{2x^{2\alpha+1}}{\Gamma(\alpha+1)}e^{-x^2}dx$ on $(0,\infty)$, for ${\alpha>-\frac12}$. We obtain characterizations of $\mathcal{H}^1((0,\infty),\gamma_\alpha)$ by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from $\mathcal{H}^1((0,\infty),\gamma_\alpha)$ into $L^1((0,\infty),\gamma_\alpha)$.