A variation norm Carleson theorem

@article{Oberlin2009AVN,
  title={A variation norm Carleson theorem},
  author={Richard Oberlin and Andreas Seeger and Terence Tao and Christoph Thiele and James Wright},
  journal={Journal of the European Mathematical Society},
  year={2009},
  volume={14},
  pages={421-464}
}
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 to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by… 

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References

SHOWING 1-10 OF 43 REFERENCES

BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR

The authors prove L bounds in the range 1 < p < ∞ for a maximal dyadic sum operator on R. This maximal operator provides a discrete multidimensional model of Carleson’s operator. Its boundedness is

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 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

Ergodic theory and translation-invariant operators.

  • A. Calderón
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1968
It has been known for a long time that there is a close connection between the ergodic theorem and certain facts about functions on the real line, such as the maximal theory of Hardy and Littlewood

On A littlewood-paley type inequality

The following is proved: If u is a function harmonic in the unit ball B C R N and if 0 < p < 1, then the inequality ∫ ∂B u*(y) p dσ ≤ C p,N (|u(0)| p +∫ B (1-|x| p-1 |∇u(x)| p dV(x)) holds, where u*

A Carleson type theorem for a Cantor group model of the scattering transform

We consider a basic d-adic model for the scattering transform on the line. We prove L2 bounds for this scattering transform and a weak L2 bound for a Carleson type maximal operator (theorem 1.4). The

A proof of boundedness of the Carleson operator

We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson’s theorem on almost everywhere convergence of Fourier

A counterexample to a multilinear endpoint question of Christ and Kiselev

Christ and Kiselev have established that the generalized eigenfunctions of one-dimensional Dirac operators with $L^p$ potential $F$ are bounded for almost all energies for $p < 2$. Roughly speaking,

Breaking the duality in the return times theorem

We prove Bourgain's Return Times Theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence

On the convergence of Fourier series

We define the space Bp={f:(−π,π]→R,   f(t)=∑n=0∞cnbn(t),   ∑n=0∞|cn|<∞}. Each bn is a special p-atom, that is, a real valued function, defined on (−π,π], which is either b(t)=1/2π or