Abstract

Let $\nu =({\nu }_1,\dots <!--FUNCTION\; APPLICATION-->,{\nu }_n)\in {(-1,\infty )}^n$, $n\ge 1$, and let ${\mathcal{ℒ}}_{\nu }$ be a self-adjoint extension of the differential operator

on $C_c^{\infty }({ℝ}_{+}^n)$ as the natural domain. We investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator ${\mathcal{ℒ}}_{\nu }$. In the special case of the Riesz transform, we complete the description of the Riesz transform for the full range of $\nu \in {(-1,\infty )}^n$ which significantly improves a result of Nowak and Stempak (J. Funct. Anal. 244:2 (2007), 399–443) for ${\nu }_i\ge -\frac{1}{2},{\nu }_i\notin (-\frac{1}{2},\frac{1}{2})$ for $i=1,\dots <!--FUNCTION\; APPLICATION-->,n$.

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