A note on generalized Marcinkiewics integrals associated to surface of revolution

Abstract

Along the surface of revolution, this article gives appropriate L^p estimates for generalized Marcinkiewicz integral operators with kernels in L^q(S^n-1), q > 1. Through an extrapolation argument, we establish the L^p boundedness of the generalized Marcinkiewicz integral operators to these bounds under weaker kernel assumptions. This not only enlarges the applicability of the operators but also indicates the functional relationships among various integral transforms. Finally, our findings are a contribution to the ongoing research on harmonic analysis and applications of partial differential equations.