<i>L^p</i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains

Abstract

In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate Lp bounds of these Maximal operators under the assumption Ω∈Lq(Sm−1×Sn−1) for some q>1, and then we employ these bounds along with Yano’s extrapolation argument to obtain the Lp boundedness of the aforementioned integral operators under a weaker condition in which Ω belongs to either the space Bq(0,2τ′−1)(Sm−1×Sn−1) or to the space L(logL)2/τ′(Sm−×Sn−1). Our results extend and improve many previously known results.

Keywords

• product spaces
• surfaces of revolution
• rough kernels
• Maximal integrals