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

##### 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.)
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