Negative powers of Laguerre operators (Q2880084)

The classical Hardy-Littlewood-Sobolev theorem concerns \(L^{p}-L^{q}\) estimates for the negative powers of the Laplacian. \textit{E. M. Stein} and \textit{G. Weiss} [J. Math. Mech. 7, 503--514 (1958; Zbl 0082.27201)] extended this theorem to weighted spaces. Recently in [``Sobolev spaces associated to the harmonic oscillator'', Proc. Indian Acad. Sci., Math. Sci. 116, No. 3, 337--360 (2006; Zbl 1115.46025)], \textit{B. Bongioanni} and \textit{J. L. Torrea} obtained \(L^{p}-L^{q}\) estimates for the negative powers of the harmonic oscillator.NEWLINENEWLINENEWLINEIn the paper under review, the authors obtain weighted \(L^{p}-L^{q}\) estimates for Laguerre differential operators in \(\mathbb{R}^{d},\;d\geq1\), for two cases: the case of Laguerre functions expansions of Hermite type and the case of Laguerre functions expansions of convolution type. The proof in the first case is straightforward and follows from Bongioanni and Torrea's result and its extension to weighted \(L^{p}\) spaces. In the second case, the result is proved first for Laguerre operators \(L_{\alpha}\) with half-integer type indices \(\alpha\) and then is extended by interpolation to \(\alpha \in [-1/2, \infty)^{d}\). The interpolation argument is based on a tool developed by the authors and called the convexity principle. In the final part of the paper, \(L^{p}-L^{q}\) estimates for the Dunkl harmonic oscillator in the context of a finite reflection group acting on \(\mathbb{R}^{d}\) and isomorphic to \(\mathbb{Z}^{d}_{2}\) are obtained.