Riesz transform characterization of H^1 spaces associated with certain Laguerre expansions

TL;DR

This paper characterizes the Hardy space associated with certain Laguerre expansions using the Riesz transform, establishing an equivalence between membership in the space and integrability of the function and its Riesz transform.

Contribution

It introduces an atomic Hardy space linked to Laguerre functions and proves its characterization via the Riesz transform, extending harmonic analysis tools to this setting.

Findings

H^1_{at}(X) is characterized by the Riesz transform Rf.

The space H^1_{at}(X) is a subspace of L^1((0,∞), x^α dx).

The Riesz transform provides an equivalent norm for H^1_{at}(X).

Abstract

For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\frac{d^2}{dx^2}f-\frac{alpha}{x}\frac{d}{dx}f+x^2 f. We define an atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\sqrt{\pi}\frac{d}{dx}L^{-1/2}f in the sense that f\in H^1_{at}(X) if and only if f,Rf \in L^1((0,infty),x^alpha dx).