Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, Logroño 26004, Spain
Copyright © 2002 Óscar Ciaurri and Juan L. Varona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let ℋα be the modified Hankel transform ℋα(f,x)=∫0∞Jα(xt)(xt)αf(t)t2α+1dt,
defined for suitable functions and extended to some Lp((0,∞),x2α+1) spaces. Given δ>0, let
Mαδ be the Bochner–Riesz operator for the Hankel transform. Also, we take the following generalization ℋαk(f,x)=∫0∞Jα+k(xt)(xt)αf(t)t2α+1dt,
k=0,1,2… for the Hankel transform, and define Mα,kδ as Mα,kδf=ℋαk((1−x2)+δℋαkf),
k=0,1,2,… (thus, in particular,
Mαδ=Mα,0δ). In the paper, we study the uniform boundedness of {Mα,kδ}k∈N in Lp((0,∞),x2α+1)
spaces when α≥0. We found that, for
δ>(2α+1)/2 (the critical index), the uniform boundedness of
{Mα,kδ}k=0∞ is satisfied for every p in the range 1≤p≤∞. And, for 0<δ≤(2α+1)/2 the uniform boundedness happens if and only if
4(α+1)2α+3+2δ<p<4(α+1)2α+1−2δ. In the paper, the case δ=0 (the corresponding generalization of the χ[0,1]-multiplier for the Hankel transform) is previously analyzed; here, for α>−1. For this value of δ, the uniform
boundedness of {Mα,k0}k=0∞ is related to the convergence of Fourier–Neumann series.