Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs (Q2901906)

The authors study \(\rho\)-variation and oscillation for families of singular integral operators defined on Lipschitz graphs. Their results include the \(L^p\)-boundedness of the \(\rho\)-variation and the oscillation for the smooth truncations of the Cauchy transform and the \(n\)-dimensional Riesz transform on Lipschitz graphs, for \(p\in(1,\infty)\) and \(\rho>2\). In particular, the results strengthen the classical theorem on the \(L^2\)-boundedness of the Cauchy transform on Lipschitz graphs of \textit{R. R. Coifman, A. McIntosh} and \textit{Y. Meyer} [Ann. Math. (2) 116, 361--387 (1982; Zbl 0497.42012)].