Abstract

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.

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