On the Functions of Marcinkiewicz Integrals along Surfaces of Revolution on Product Domains via Extrapolation

Abstract

In this paper, we establish certain Lp bounds for several classes of rough Marcinkiewicz integrals over surfaces of revolution on product spaces. By using these bounds and using an extrapolation argument, we obtain the Lp boundedness of these Marcinkiewicz integrals under very weak conditions on the kernel functions. Our results represent natural extensions and improvements of several known results on Marcinkiewicz integrals.

On Certain Rough Marcinkiewicz Integral Operators with Grafakos-Stefanov Kernels on Product Spaces

“…The study of the boundedness of M ℧ started by Ding in [11] and then continued by many authors. For a sample of past studies and for more information about the applications and development of the operator M ℧ , we refer the readers to consult [14][15][16][17][18][19][20][21] and the references therein.…”

“…In light of the assumptions imposed on Φ in [23] and of the results in [10] concerning the boundedness of Marcinkiewicz operator M ϕ ℧ in the one parameter setting whenever ℧ ∈ G α S m−1 as well as of the results in [21] concerning the boundedness of Marcinkiewicz operator M ℧,Φ whenever…”