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Higher Riesz transforms and imaginary powers associated to the harmonic oscillator Krzysztof Stempak ∗ and Jose ́ Luis Torrea† 15.11.2003 We consider higher order Riesz transforms for the multi-dimensional Hermite function ex-pansions. The Riesz transforms occur to be Calderón-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 oscilla-tor by using tools developed for studying Riesz transforms. 2000 Mathematics Subject Classification. Primary 42C10; Secondary 42B20. Key words and phrases. Multi-dimensional Hermite expansions, higher order Riesz transforms, Calderón-Zygmund operators, conjugate Poisson integrals, imaginary powers of harmonic oscilla-tor