Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

Abstract

In this paper, we give a criterion to prove boundedness results for several operators from the Hardy-type space $H^1({(0,\infty )}^d,{\gamma }_{\alpha })$ to $L^1({(0,\infty )}^d,{\gamma }_{\alpha })$ and also from $L^{\infty }({(0,\infty )}^d,{\gamma }_{\alpha })$ to the space of functions of bounded mean oscillation $BMO({(0,\infty )}^d,{\gamma }_{\alpha })$, with respect to the probability measure $d{\gamma }_{\alpha }\left(x\right)={\prod }_{j=1}^d\frac{2}{\Gamma ({\alpha }_j+1)}{x}_j^{2{\alpha }_j+1}{e}^{-x_j^2}d{x}_j$ on ${(0,\infty )}^d$ when $\alpha =({\alpha }_1,\cdots ,{\alpha }_d)$ is a multi-index in ${\left(-,\frac{1}{2},,,\infty \right)}^d$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.