$BMO$ Spaces for Laguerre Expansions

Abstract

Let $\{\varphi_n^{\alpha}\}_{n \in \mathbb{N}}$ be the Laguerre functions of Hermite type with index $\alpha$. These are eigenfunctions of the Laguerre differential operator $L_\alpha = \dfrac{1}{2}(-\frac{d^2}{dy^2} + y^2 + \frac{1}{y^2}(\alpha^2-\frac{1}{4}))$. We define and study a $BMO$-type space $BMO_{L_{\alpha}}$, which is identified as the dual space of the Hardy-type space associated with $L_{\alpha}$. We characterize $BMO_{L_{\alpha}}$ by Carlesson measures related to appropriate square functions. Finally, we prove the boundedness on this space of the fractional integral operator and the Riesz transform related to $L_{\alpha}$.