Estimates for some convolution operators with singularities of their kernels on spheres

In the framework of Hardy spaces $H^p$, we study multidimensional convolution operators whose kernels have power-type singularities on a finite union of spheres in $\mathbb R^n$. Necessary and sufficient conditions are obtained for such operators to be bounded from $H^p$ to $H^q$, $0<p\leq q<\infty$, from $H^p$ to BMO, and from BMO to BMO.