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

TL;DR

This paper establishes $L^p$ bounds for Stein's square functions linked to Fourier-Bessel expansions, introduces transference results to Hankel transforms, and confirms the optimal $p$ range for boundedness.

Contribution

It provides new $L^p$ estimates for Fourier-Bessel Stein's square functions and develops transference techniques from discrete to continuous Bessel settings.

Findings

Proved $L^p$ estimates for Fourier-Bessel Stein's square functions.

Established transference results from Fourier-Bessel series to Hankel transforms.

Confirmed the sharp $p$ range for $L^p$-boundedness of Fourier-Bessel Stein's square functions.

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.