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

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