Variation for singular integrals on Lipschitz graphs: $L^p$ and endpoint estimates

Abstract:

Let $1\leq n<d$ be integers and let $\mu$ denote the $n$-dimensional Hausdorff measure restricted to an $n$-dimensional Lipschitz graph in $\mathbb {R}^d$ with slope strictly less than $1$. For $\rho >2$, we prove that the $\rho$-variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in $L^{p}(\mu )$ for $1<p<\infty$, from $L^1(\mu )$ to $L^{1,\infty }(\mu )$, and from $L^\infty (\mu )$ to $BMO(\mu )$. Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in $\mathbb {R}^d$ to $L^{1,\infty }(\mu )$.