An uniform boundedness for Bochner-Riesz operators related to the Hankel transform. (English) Zbl 1040.42007

Ciaurri, Óscar; Varona, Juan L.

J. Inequal. Appl. 7, No. 6, 759-777 (2002).

The authors consider a generalization of the Hankel transform \[ {\mathcal H}^k_\alpha (f,x)=x^{-\alpha} \int^\infty_0{\mathcal I}_{\alpha+k} (xt)f(t) t^{\alpha+1}dt, \] where \(k\) is a nonnegative integer. Let \(\delta>0\) and \(M^\delta_{\alpha,k}\) be the generalized Bochner-Riesz operator for the Hankel transform defined by means of \[ M^\delta_{ \alpha,k} f={\mathcal H}^k_\alpha \bigl((1-x^2)^\delta_+ {\mathcal H}^k_\alpha f\bigr). \] The authors prove:
– For \(\delta >\alpha + 1/2\) the sequence \(\{M^\delta_{\alpha,k}\}^\infty_{k=0}\) is uniformly bounded for every \(p\) with \(1\leq p\leq\infty\).
– For \(0< \delta\leq\alpha +1/2\) the sequence \(\{M^\delta_{\alpha,k}\}_{k=0}^\infty\) is uniformly bounded if and only if \[ \frac {4(\alpha+1)} {2\alpha+3+2 \delta}<p <\frac {4(\alpha+1)} {2\alpha+1-2 \delta}. \] An application to the convergence of Fourier-Neumann series is given.

Reviewer: H.-J. Glaeske (Jena)

MSC:

42A45	Multipliers in one variable harmonic analysis
42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A15	Special integral transforms (Legendre, Hilbert, etc.)

Keywords:

uniform boundedness; Hankel transform; Bochner-Riesz operator; Fourier-Neumann series