An uniform boundedness for Bochner-Riesz operators related with the Hankel transform

Resumen

Let Hα be the modified Hankel transform Hα(f,x) = ∫0∞ Jα(xt)/(xt)α f(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)α f(t)t2α+1 dt, k = 0, 1, 2, ... for the Hankel transform, and define Mα,kδ as Mα,kδf = Hαk ((1 - x2)δ+Hαkf), 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 X[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.