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

Variation for the Riesz transform and uniform rectifiability

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

Oscillation and variation for the Hilbert transform

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

A variation norm Carleson theorem

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

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

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

A Wiener–Wintner theorem for the Hilbert transform

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

Pointwise ergodic theorems for arithmetic sets

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

Analysis of and on uniformly rectifiable sets

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

ANALYTIC CAPACITY AND CALDERÓN-ZYGMUND THEORY WITH NON DOUBLING MEASURES

  • 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

La variation d'ordre p des semi-martingales

RésuméSoient X une semi-martingale, p un nombre réel positif, S = (ti) une subdivision de l'intervalle $$[0,t],\sum {\left| {X_{t_i + 1} - X_{t_i } } \right|} ^p $$ la somme variationnelle

Oscillation in ergodic theory

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