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.