Weighted inequalities for the Bochner–Riesz means related to the Fourier–Bessel expansions

Abstract

We prove weighted inequalities for the Bochner-Riesz means for Fourier-Bessel series with more general weights w(x) than previously considered power weights. These estimates are given by using the local A p theory and Hardy's inequalities with weights. Moreover, we also obtain weighted weak type (1, 1) inequalities. The case when w(x) = x a is sketched and follows as a corollary of the main result.

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].…”

On derivatives, Riesz transforms and Sobolev spaces for Fourier-Bessel expansions

“…Harmonic analysis related to these expansions has been widely developed over many decades, and a good deal of this activity falls on the 21th century. For instance, convergence of Fourier-Bessel series in various senses was studied by Wing [57], Hochstadt [34], Benedek and Panzone [2,3,4], Markett [39], Guadalupe et al [33], Balodis and Córdoba [1], Stempak [52], and Ciaurri and Roncal [19,20,22]. Transplantation problems related to Fourier-Bessel expansions were investigated by Gilbert [31], Stampak [51], and Ciaurri and Stempak [24,25].…”

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

“…Weighted L p -inequalities for L ν -Bochner-Riesz means were obtained by Ciaurri and Roncal ([14] and [15]). L p -boundedness properties for the maximal operator associated to {R α,Lν t } t>0 were established in [16].…”