Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform

Abstract

Proved are transference results that show connections between: a) multipliers for the Fourier-Bessel series and multipliers for the Hankel transform; b) maximal operators defined by Fourier-Bessel multipliers and maximal operators given by Hankel transform multipliers; c) Fourier-Bessel transplantation and Hankel transform transplantation. In some way the connections described in a) and b) can be seen as multi-dimensional extensions of the classical results of Igari, and Kenig and Tomas for the one dimensional Fourier transform. We prove our results for the non-modified Hankel transform in the power weight setting, and this allows to translate them also to the context of the modified Hankel transform. Together with Gilbert's transplantation theorem, our transference shows that harmonic analysis results for the Hankel transform of arbitrary order are consequences of corresponding results for the cosine expansions.