Maximal Singular Integral Operators Along Surfaces

Let b y be a bounded radial function and y = γ1 y γ2 y γm y , where each γj y j = 1 m is a real-valued radial function. For x y ∈ n and x ∗ ∈ m, we define the maximal singular integral along the surface y y by T∗f x x ∗ = sup ε>0 y >ε f x − y x ∗ − y b y y −n y dy Suppose that is an H1 function on the sphere Sn−1 satisfying Sn−1 x dσ x = 0. We prove that T∗ is bounded on Lp n+m 1 < p < ∞, provided the lower dimensional maximal function M g x1 x ∗ = sup k∈ 2−k 2k+1 2k g x1 − t x ∗ − t dt is bounded on Lp m+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].