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

@article{Mas2011VariationFS,
title={Variation for singular integrals on Lipschitz graphs: \$L^\{p\}\$ and endpoint estimates},
author={Albert Mas},
journal={Transactions of the American Mathematical Society},
year={2011},
volume={365},
pages={5759-5781}
}
• A. Mas
• Published 4 October 2011
• Mathematics
• Transactions of the American Mathematical Society
Let 0 2, we prove that the r-variation and oscillation for Calder\'on-Zygmund singular integrals with odd kernel are bounded operators in L^p(H) for 1<p finite, from L^1(H) to weak-L^1(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 R^d to weak-L^1(H).
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## References

SHOWING 1-10 OF 16 REFERENCES

Given a doubling measure $\mu$ on ${\mathbb R}^d$, it is a classical result of harmonic analysis that Calderon-Zygmund operators which are bounded in $L^2(\mu)$ are also of weak type (1,1). Recently
• Mathematics
• 2011
For 0 2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded
• Mathematics
• 2000
It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra
• Mathematics
• 2009
By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues
• Mathematics
• 2011
We prove that, for ρ>2, the ρ‐variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in Lp for 1
• Mathematics
• 2008
We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,Tt) and f∈Lp(X,μ), there is a
converge almost surely for N -+ co, assuming f a function of class L~(~, ~). Here and in the sequel, one denotcs by ~ a probability measure and by T a measure-preserving automorphism. The natural
• Mathematics
• 1993
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant
• Mathematics
Ergodic Theory and Dynamical Systems
• 1998
In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise
• Mathematics
• 2004
These notes are the lecture notes of a series of talks given at the Universidad de Sevilla in December 2003. We survey some results of CalderónZygmund theory with non doubling measures, and we apply