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

Let H-alpha be the modified Hankel transform H-alpha(f,x) = integral(0)(infinity) Jalpha(xt)/(xt)(alpha) f(t) t(2alpha+1) dt, defined for suitable functions and extended to some L-p((0,infinity), x(2alpha+1)) spaces. Given delta > 0, let M-alpha(delta) be the Bochner-Riesz operator for the Hankel transform. Also, we take the following generalization H-alpha(k)(f,x) = integral(0)(infinity) J(alpha+k)(xt)/(xt)(alpha) f(t) t(2alpha+1), dt, k = 0,1,2,... for the Hankel transform, and define M-alpha,k(delta) as M(alpha,k)(delta)f = H-alpha(k) ((1-x(2))(+)(delta) H(alpha)(k)f), k = 0,1,2,... (thus, in particular, M-alpha(delta) = M-alpha,0(delta)). In the paper, we study the uniform boundedness of {M-alpha,k(delta)}(kis an element ofN) in L-p((0,infinity),x(2alpha+1)) spaces when alpha greater than or equal to 0. We found that, for delta > (2alpha+1)/2 (the critical index), the uniform boundedness of {M-alpha,k(delta)}(k=0)(infinity) is satisfied for every p in the range 1 less than or equal to p less than or equal to infinity. And, for 0 less than or equal to delta less than or equal to (2alpha+1)/2 , the uniform boundedness happens if and only if 4(alpha+1)/2alpha+3+2delta < p < 4(alpha+1)/2alpha+1-2delta. In the paper, the case delta = 0 (the corresponding generalization of the chi([0,1])-multiplier for the Hankel transform) is previously analyzed; here, for alpha> -1 . For this value of delta, the uniform boundedness of {M-alpha,k(0)}(k=0)(infinity) is related to the convergence of Fourier-Neumann series.