Maximal Singular Integral Operators Along Surfaces

Abstract

Let b(y) be a bounded radial function and Γ(y)=(γ1(y),γ2(y),…,γm(y)), where each γj(y) (j=1,…,m) is a real-valued radial function. For x,y∈Rn and x*∈Rm, we define the maximal singular integral along the surface (y,Γ(|y|)) byT*f(x,x*)=supε>0∫|y|>εf(x−y,x*−Γ(|y|))b(y)|y|−nΩ(y′)dy.Suppose that Ω is an H1 function on the sphere Sn−1 satisfying ∫Sn−1Ω(x′)dσ(x′)=0. We prove that T* is bounded on Lp(Rn+m),1<p<∞, provided the lower dimensional maximal functionMΓg(x1,x*)=supk∈Z2−k∫2k+12k|g(x1−t,x*−Γ(t))|dtis bounded on Lp(Rm+1) for all p>1. The result is an extension and improvement of the main theorem in [S. Lu, Y. Pan, and D. Yang, Rough singular integrals associated to surfaces of revolution, Proc. Amer. Math. Soc.129 (2001), 2931–2940].