Journal of Inequalities and Applications
Volume 7 (2002), Issue 6, Pages 759-777

An uiniform boundedness for Bochner–Riesz operators related to the Hankel transform

Óscar Ciaurri and Juan L. Varona

Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, Logroño 26004, Spain

Received 14 June 2001; Revised 3 October 2001

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.


Let α be the modified Hankel transform α(f,x)=0Jα(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 αk(f,x)=0Jα+k(xt)(xt)αf(t)t2α+1dt,k=0,1,2 for the Hankel transform, and define Mα,kδ as Mα,kδf=αk((1x2)+δαkf),k=0,1,2, (thus, in particular, Mαδ=Mα,0δ). In the paper, we study the uniform boundedness of {Mα,kδ}kN in Lp((0,),x2α+1) spaces when α0. 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 1p. 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+12δ. 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.