On Riesz transforms for Laguerre expansions
Given a multi-index \(\alpha=(\alpha_1,\ldots,\alpha_d) \in (-1,\infty)^d\), the \(d\)-dimensional Laguerre polynomials of type \(\alpha\) are \[ L_k^{\alpha}(x) = \prod_{i=1}^{d} L_{k_i}^{\alpha_i}(x_i), \quad k \in {\mathbb N}^d, \quad x \in {\mathbb R}^d_+, \] where \[ L_{k_i}^{\alpha_i}(x_i) = \frac{1}{(k_i)!} e^{x_i} x_i^{-\alpha_i}. \] The Laguerre differential operator is \[ {\mathcal L}^{\alpha} = - \sum_{i=1}^{d} \left[ x_i\frac{\partial^2}{\partial x_i^2} + (\alpha_i+1-x_i)\frac{\partial}{\partial x_i} \right]. \] Consider in \({\mathbb R}^d_+\) the measure \(d\mu_{\alpha} = \prod_{i=1}^d x_i^{\alpha_i} e^{-x_i} dx\). Set \[ \text{grad}_{\alpha} = \left(\sqrt{x_1}\frac{\partial}{\partial x_1}, \ldots, \sqrt{x_d}\frac{\partial}{\partial x_d} \right). \] The Riesz-Laguerre transform \({\mathcal R}^{\alpha} = (R_1^{\alpha},\ldots,R_d^{\alpha})\) is defined by \[ {\mathcal R}^{\alpha} = \text{grad}_{\alpha} ({\mathcal L}^{\alpha})^{-1/2} \Pi_0, \] where \(\Pi_0\) is the orthogonal projection onto the orthogonal complement of the eigenspace of \({\mathcal L}^{\alpha}\) corresponding to the eigenvalue 0. It is proved that if \(1 < p < \infty\) and \(\alpha \in \left[-\frac{1}{2},\infty\right)\) then \[ \| R_i^{\alpha}f\| _{L^p({\mathbb R}^d_+,d\mu_{\alpha})} \leq c_p \| f\| _{L^p({\mathbb R}^d_+,d\mu_{\alpha})} \] with a constant \(c_p\) depending neither on \(d\) nor on \(\alpha\). This is an extension of a result by \textit{C. E. Gutiérrez, A. Incognito} and \textit{J. L. Torrea} [Houston J. Math. 27, No. 3, 579--592 (2001; Zbl 1004.42017)] where the boundedness were proved for a discrete set of half-integer multi-indices \(\alpha\). As a consequence, boundedness and convergence results are obtained for the corresponding conjugate Poisson integral.
scientific article; zbMATH DE number 2117853