# 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}
}```
• Published 8 October 2009
• Mathematics
• Journal of the European Mathematical Society
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…
110 Citations
• Mathematics
Analysis & PDE
• 2020
We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates
• Mathematics
• 2018
We study discrete random variants of the Carleson maximal operator. Intriguingly, these questions remain subtle and difficult, even in this setting. Let {Xm} be an independent sequence of {0,1}
• Mathematics
• 2022
. We prove that the weak- 𝐿 𝑝 norms, and in fact the sparse ( 𝑝, 1 ) -norms, of the Carleson maximal partial Fourier sum operator are . ( 𝑝 − 1 ) − 1 as 𝑝 → 1 + . This is an improvement on the
• Mathematics
• 2012
Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and let A_n x denote the ergodic average (1/n) sum_{i 0, the sequence has only finitely many fluctuations greater
• Mathematics
• 2019
In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative \$L_{p}\$-spaces. The main result is a weak \$(1,1)\$ type estimate of this square function. We
• Mathematics
• 2011
Let T : Lp --> Lp be a positive contraction, with p strictly between 1 and infinity. Assume that T is analytic, that is, there exists a constant K such that \norm{T^n-T^{n-1}} < K/n for any positive
In this paper, we complete the study of mapping properties about the difference associated with dyadic differentiation average and dyadic martingale on noncommutative \$L_{p}\$-spaces. To be more
The summability of Fourier transforms can be generalized for higher dimensions basically in two ways. In this chapter, we study the l q -summability of higher dimensional Fourier transforms. As in
• Mathematics
• 2015
Let p ∈ (0, 1], q ∈ (0,∞] and A be a general expansive matrix on ℝn. We introduce the anisotropic Hardy-Lorentz space HAp,q (ℝn) associated with A via the non-tangential grand maximal function and

## References

SHOWING 1-10 OF 43 REFERENCES

• Mathematics
• 2002
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
• 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
• 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
• 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
• Mathematics
• 2007
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*
• Mathematics
• 2003
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
• Mathematics
• 2000
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
• Mathematics
• 2001
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,
• Mathematics
• 2006
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
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