$L^p$--boundedness of Stein's square functions associated to Fourier--Bessel expansions

Abstract

In this paper we prove L p estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of p for the L p -boundedness of Fourier-Bessel Stein's square functions from the corresponding property for Hankel-Stein square functions. Finally, we deduce L p estimates for Fourier-Bessel multipliers from that ones we have got for our Stein square functions.

Variation operators for semigroups associated with Fourier-Bessel expansions

“…The harmonic analysis associated with {φ ν n } n∈N -expansions started in the celebrated paper by Muckenhoupt and Stein [36]. In the last decade harmonic analysis operators (Riesz transforms, Littlewood-Paley functions, multipliers and transplantation operators) in the {φ ν n } n∈N -setting have been studied in [2,6,7,12,13,14,15,16,39].…”