Bounds on the Walsh model for $M^{q,*}$ Carleson and related operators

  title={Bounds on the Walsh model for \$M^\{q,*\}\$ Carleson and related operators},
  author={Richard Oberlin},
  journal={Revista Matematica Iberoamericana},
  • R. Oberlin
  • Published 5 October 2011
  • Mathematics
  • Revista Matematica Iberoamericana
We prove an extension of the Walsh-analog of the Carleson-Hunt theorem, where the $L^\infty$ norm defining the Carleson maximal operator has been replaced by an $L^q$ maximal-multiplier-norm. Additionally, we consider certain associated variation-norm estimates. 

Estimates for Compositions of Maximal Operators with Singular Integrals

  • R. Oberlin
  • Mathematics
    Canadian Mathematical Bulletin
  • 2013
Abstract. We prove weak-type $\left( 1,\,1 \right)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta *\Psi $ where $\Delta

Variational bounds for a dyadic model of the bilinear Hilbert transform

We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase

A variable coefficient multi-frequency lemma

We show a variable coefficient version of Bourgain's multi-frequency lemma. It can be used to obtain major arc estimates for a discrete Stein--Wainger type operator considered by Krause

Modulation invariant operators

The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations,


We prove a variation norm Carleson theorem for Walsh-Fourier series of func- tions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion



The Walsh model for $M_2^*$ Carleson

We study the Walsh model of a certain maximal truncation of Carleson's operator, related to the Return Times Theorem from Ergodic Theory.

New uniform bounds for a Walsh model of the bilinear Hilbert transform

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We

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

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

Endpoint multiplier theorems of Marcinkiewicz type

We establish sharp (H1,L1,q) and local (L logrL,L1,q) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map

Improved Range in the Return Times Theorem

  • C. Demeter
  • Mathematics
    Canadian Mathematical Bulletin
  • 2012
Abstract We prove that the Return Times Theoremholds true for pairs of ${{L}^{p}}\,-\,{{L}^{q}}$ functions, whenever $\frac{1}{p}\,+\,\frac{1}{q}\,<\,\frac{3}{2}$ .

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

A Calderon Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain

We introduce a Calderon Zygmund decomposition such that the bad function has vanishing integral against a number of pure frequencies. Then we prove a variation norm variant of a maximal inequality

Issues related to Rubio de Francia's Littlewood--Paley Inequality: A Survey

Rubio de Francia's Littlewood Paley inequality is an extension of the classical Littlewood Paley inequality to one that holds for a decomposition of frequency space into arbitrary disjoint intervals.

Return times in dynamical systems

Dans cette these, deux aspects asymptotiques des temps de retour et d'entree sont etudies: les taux locaux de temps de retour, et les lois limites des k-iemes temps de retour et d'entree. Dans le