Riesz transforms and conjugacy for Laguerre function expansions of Hermite type (Q880101)

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index \(\alpha\) are defined and investigated. It is proved that for any multi-index \(\alpha=(\alpha_1,\dots,\alpha_d)\) such that \(\alpha_i\geq -1/2\), \(\alpha_i\notin (-1/2,1/2)\), the appropriately defined Riesz transforms \(R_j^\alpha\), \(j=1,2,\dots,d\), are Calderón--Zygmund operators, hence their mapping properties follow from a general theory. Similar mappings results are obtained on one dimension, without excluding \(\alpha\in (-1/2,1/2)\), 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 \(\alpha\), \((-1/2,\dots,-1/2)\) and \((1/2,\dots,1/2)\), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.