  For a general compact variety $\Gamma$ of arbitrary codimension, one can
consider the $L^p$ mapping properties of the B\^ochner-Riesz multiplier $$
m_{\Gamma, \alpha}(\zeta) \ = \ {\rm dist}(\zeta, \Gamma)^{\alpha} \phi(\zeta)
$$ where $\alpha > 0$ and $\phi$ 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 B\^ochner-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
B\^ochner-Riesz multiplier operator.
