Abstract: Let μ be a locally uniformly α-dimensioual measure on Rⁿ and P_t(fdμ) be the Able Poisson means of Hermite expansions for f∈L^P(dμ), it is studied that the asymptotis properties of P_t(fdμ) as t→1_. Analogue of Wiener's theorem is obtained.Author also establishs the boundedness of the α-dimensional maximal conjugate Poisson integral operators from L^P(dμ) to the Lebesgue p-power integrable function spaces L^P(dx),and this derives directly the boundedness of Riesz transforms.

Key words: Fractal measurest Hermite expansions, Riesz transforms