Higher Riesz transforms and imaginary powers associated to the harmonic oscillator

We consider higher order Riesz transforms for the multi-dimensional Hermite function expansions. The Riesz transforms occur to be Calderon-Zygmund operators hence their mapping properties follow by using results from a general theory. Then we investigate higher order conjugate Poisson integrals showing that at the boundary they approach appropriate Riesz transforms of a given function. Finally, we consider imaginary powers of the harmonic oscillator by using tools developed for studying Riesz transforms.