Paper detail

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

For a general compact variety $Γ$ of arbitrary codimension, one can consider the $L^p$ mapping properties of the Bôchner-Riesz multiplier $$ m_{Γ, α}(ζ) \ = \ {\rm dist}(ζ, Γ)^α ϕ(ζ) $$ where $α> 0$ and $ϕ$ is an appropriate smooth cut-off function. Even for the sphere $Γ= {\mathbb S}^{N-1}$, 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.