Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Abstract

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , α i ∉ ( − 1 / 2 , 1 / 2 ) , the appropriately defined Riesz transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α ∈ ( − 1 / 2 , 1 / 2 ) , by means of a local Calderón–Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy–Riemann type equations and to recover the Riesz–Laguerre transforms on the boundary. The two specific values of α, ( − 1 / 2 , … , − 1 / 2 ) and ( 1 / 2 , … , 1 / 2 ) , are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.