No abstract

No abstract

We propose a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions. The scheme is supported by numerous examples concerning, in particular, the classical orthogonal expansions in Hermite, Laguerre, and Jacobi polynomials. A general case of expansions associated to a regular or singular SturmLiouville problem is also discussed.

No 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 α j , 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.