Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

In this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space H1((0,∞)d,γα)$H^1((0,\infty)^d,\gamma _\alpha)$ to L1((0,∞)d,γα)$L^1((0,\infty)^d,\gamma _\alpha)$ and also from L∞((0,∞)d,γα)$L^\infty ((0,\infty)^d,\gamma _\alpha)$ to the space of functions of bounded mean oscillation BMO((0,∞)d,γα)$\textup {BMO}((0,\infty)^d,\gamma _\alpha)$, with respect to the probability measure dγα(x)=∏j=1d2Γ(αj+1)xj2αj+1e−xj2dxj$d\gamma _\alpha (x)=\prod _{j=1}^d\frac{2}{\Gamma (\alpha _j+1)} x_j^{2\alpha _j+1} \text{e}^{-x_j^2} dx_j$ on (0,∞)d$(0,\infty)^d$ when α=(α1,⋯,αd)$\alpha =(\alpha _1, \dots,\alpha _d)$ is a multi‐index in −12,∞d$\left(-\frac{1}{2},\infty \right)^d$. 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.