Riesz transforms for multi-dimensional Laguerre function expansions

Abstract

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , the appropriately defined Riesz–Laguerre transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz–Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy–Riemann type and to recover the Riesz–Laguerre transforms on the boundary.