Poisson integrals and Riesz transforms for Hermite function expansions with weights

Abstract

We consider expansions with respect to the multi-dimensional Hermite functions which are eigenfunctions of the harmonic oscillator L=− Δ+| x| 2. For the heat-diffusion and Poisson semigroups corresponding to a self-adjoint extension L of L we investigate their boundary behaviour and mapping properties. All this is done for functions from L p ( w), 1⩽ p<∞, w∈ A p . Then Riesz transforms and conjugate Poisson integrals are considered. The Riesz transforms occur to be Calderón–Zygmund operators hence their mapping properties follow by using results from a general theory.