Lp -Boundedness of Stein’s Square Functions Associated with Fourier–Bessel Expansions

Abstract

In this paper we prove Lp estimates for Stein’s square functions associated with 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 Lp-boundedness of Fourier–Bessel Stein’s square functions from the corresponding property for Hankel–Stein square functions. Finally, we deduce Lp estimates for Fourier–Bessel multipliers from that ones we have got for our Stein square functions.