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Variation for singular integrals on Lipschitz graphs: L^p and endpoint estimates (1110.0633v1)
Published 4 Oct 2011 in math.CA
Abstract: Let 0<n<d be integers and let H denote the n-dimensional Hausdorff measure restricted to an n-dimensional Lipschitz graph in R^d with slope strictly less than 1. For r\>2, we prove that the r-variation and oscillation for Calder\'on-Zygmund singular integrals with odd kernel are bounded operators in Lp(H) for 1<p finite, from L1(H) to weak-L1(H), and from the space of bounded H-measurable functions to BMO(H). Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in Rd to weak-L1(H).