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Hilbert transforms associated with Dunkl-Hermite polynomials. (English) Zbl 1162.42002

Summary: We consider expansions of functions in \(L^p(\mathbb R,|x|^{2k}dx), 1\leq p<+\infty \) with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calderón-Zygmund operators and hence their mapping properties follow from general results.

MSC:

42A50 Conjugate functions, conjugate series, singular integrals
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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