Estimates for some convolution operators with singularities in their kernels on a sphere and their applications

We study convolution operators, whose kernels have singularities on the unit sphere. For these operators we obtain $H^p$-$H^q$ estimates, where $p$ is less than or equals $q$, and prove their sharpness. To this end, we develop a new method that uses special representations for the symbol of such an operator as the sum of some oscillatory integrals and applies the stationary phase method and A. Miyachi results for model oscillating multipliers. Moreover, we also obtain estimates from $L^p$ to $BMO$ and those from $BMO$ to $BMO$.