Riesz transforms, \(g\)-functions, and multipliers for the Laguerre semigroup (Q2771476)

For any \(\alpha>-1\), the one-dimensional Laguerre polynomials of order \(\alpha\) are NEWLINE\[NEWLINE L^{\alpha}_{k}(y)=\frac 1{k!} e^yy^{-\alpha}\frac{d^k}{dy^k}\left(e^{-y}y^{k+\alpha}\right). NEWLINE\]NEWLINE Let \(d\in \mathbb N^{\ast}\). Given a multi-index \(\alpha=(\alpha_{1},\ldots,\alpha_{d})\) with \(\alpha_{i}>-1\), define, for \(k=(k_{1},\ldots,k_{d})\in \mathbb N^d\) and \(y\in \left]0,+\infty\right[^d\), NEWLINE\[NEWLINE L^{\alpha}_{k}(y)=\prod\limits_{1\leq i\leq d} L^{\alpha_{i}}{k_{i}}(y_{i}). NEWLINE\]NEWLINE The Laguerre differential operator of type \(\alpha\) is defined by NEWLINE\[NEWLINE {\mathcal L}_{\alpha}=\sum\limits_{1\leq i\leq d} y_{i}\frac{\partial^{2}}{\partial y_{i}^{2}}+(\alpha_{i}+1-y_{i})\frac{\partial}{\partial y_{i}}. NEWLINE\]NEWLINE There is a notion of gradient associated with \({\mathcal L}_{\alpha}\). For \(y=(y_{1},\ldots,y_{d})\in \left]0,+\infty\right[^d\), set NEWLINE\[NEWLINE \text{grad}_{\alpha}f(y)=\left(\sqrt{y_{1}}\frac{\partial f}{\partial y_{1}}(y),\ldots,\sqrt{y_{d}}\frac{\partial f}{\partial y_{d}}(y)\right). NEWLINE\]NEWLINE The Riesz transform associated with \({\mathcal L}_{\alpha}\), as usual, is given by NEWLINE\[NEWLINE {\mathcal R}_{\alpha}=\text{grad}_{\alpha}(-{\mathcal L}_{\alpha})^{-1/2}. NEWLINE\]NEWLINE The authors show that \({\mathcal R}_{\alpha}\) is \(L^p\)-bounded for all \(1<p<+\infty\), with constants independent from \(d\). Moreover, \({\mathcal R}_{\alpha}\) is of weak type \((1,1)\). NEWLINENEWLINENEWLINEThe authors also consider \(g\)-functions and multipliers associated with the Laguerre semigroup and the Laguerre differential operator, and prove that they are also \(L^p\)-bounded for all \(1<p<+\infty\). NEWLINENEWLINENEWLINEThe case when \(d=1\) was due to \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 139, 231-242 (1969; Zbl 0175.12602); ibid. 147, 419-431 (1970; Zbl 0191.07602)].