A note on the $L^p$ integrability of a class of Bôchner-Riesz kernels

Abstract

For a general compact variety $\Gamma$ of arbitrary codimension, one can consider the $L^p$ mapping properties of the Bôchner-Riesz multiplier $$m_{\Gamma ,\alpha }(\zeta )=\mathrm{dist}(\zeta ,\Gamma {)}^{\alpha }\varphi (\zeta )$$ where $\alpha >0$ and $\varphi$ is an appropriate smooth cut-off function. Even for the sphere $\Gamma ={\mathbb{S}}^{N-1}$, the exact $L^p$ boundedness range remains a central open problem in Euclidean Harmonic Analysis. In this paper we consider the $L^p$ integrability of the Bôchner-Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of $L^p$ integrability of the kernels differs substantially from the $L^p$ boundedness range of the corresponding Bôchner-Riesz multiplier operator.

arXiv comment19 pages