Poisson integrals and Riesz transforms for Hermite function expansions with weights

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 Lp(w), 1⩽p<∞, w∈Ap. 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.