390 research outputs found
A formal system of mathematics based on definitions
We discuss a formal system of mathematics. We use it to construct the natural
numbers.Comment: Updated from "On the first chapter" and "Definitions in mathematics"
39 pages, many tree figure
On the two dimensional Bilinear Hilbert Transform
We investigate the Bilinear Hilbert Transform in the plane and the pointwise
convergence of bilinear averages in Ergodic theory, arising from
actions. Our techniques combine novel one and a half dimensional phase-space
analysis with more standard one dimensional theory.Comment: 46 pages, 0 figure
A T(1) theorem for entangled multilinear dyadic Calder\'{o}n-Zygmund operators
We prove a boundedness criterion for a class of dyadic multilinear forms
acting on two-dimensional functions. Their structure is more general than the
one of classical multilinear Calder\'{o}n-Zygmund operators as several
functions can now depend on the same one-dimensional variable. The study of
this class is motivated by examples related to the two-dimensional bilinear
Hilbert transform and to bilinear ergodic averages. This paper is a sequel to
the prior paper arXiv:1108.0917 by the first author.Comment: 20 page
Singular Brascamp-Lieb inequalities with cubical structure
We prove a singular Brascamp-Lieb inequality, stated in Theorem 1, with a
large group of involutive symmetries.Comment: 16 page
Singular Brascamp-Lieb: a survey
We present an overview of results on multi-linear singular integrals in the
broader context of Brascamp-Lieb inequalities. This elaborates a lecture given
at the inspiring conference on Geometric Aspects of Harmonic Analysis at
Cortona 2018 in honor of Fulvio Ricci.Comment: 21 page
Endpoint bounds for the bilinear Hilbert transform
We study the behavior of the bilinear Hilbert transform at the
boundary of the known boundedness region . A sample of our results
is the estimate
valid for all tuples of sets of finite measure and functions such that , , with the additional restriction that be
supported on a major subset of that depends on .
The double logarithmic term improves over the single logarithmic term obtained
by Bilyk and Grafakos. Whether the double logarithmic term can be removed
entirely, as is the case for the quartile operator discussed by Demeter and the
first author, remains open. We employ our endpoint results to describe the
blow-up rate of weak-type and strong-type estimates for as the
tuple approaches the boundary of . We also discuss
bounds on Lorentz-Orlicz spaces near , improving on results of
Carro, Grafakos, Martell and Soria. The main technical novelty in our article
is an enhanced version of the multi-frequency Calder\'on-Zygmund decomposition
by Nazarov, Oberlin and the second author.Comment: 42 pages, 1 figure, 1 table. Submitte
Maximal multilinear operators
We establish multilinear bounds for a class of maximal multilinear
averages of functions on one variable, reproving and generalizing the bilinear
maximal function bounds of Lacey. As an application we obtain almost everywhere
convergence results for these averages, and in some cases we also obtain almost
everywhere convergence for their ergodic counterparts on a dynamical system.Comment: 55 pages, no figures, submitted, Geom. Func. Ana
Variational estimates for paraproducts
We generalize a family of variation norm estimates of Lepingle with endpoint
estimates of Bourgain and Pisier-Xu to a family of variational estimates for
paraproducts, both in the discrete and the continuous setting. This expands on
work of Friz and Victoir, our focus being on the continuous case and an
expanded range of variation exponents.Comment: 23 page
Multi-linear multipliers associated to simplexes of arbitrary length
In this article we prove that the -linear operator whose symbol is the
characteristic function of the simplex is
bounded from into , generalizing in this
way our previous work on the "bi-est" operator (which corresponds to the case
) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform
(which corresponds to the case ).Comment: 52 pages, 6 figure
estimates for the biest II. The Fourier case
We prove L^p estimates for the "biest", a trilinear multiplier with singular
symbol which arises naturally in the expansion of eigenfunctions of a
Schrodinger operator, and which is also related to the bilinear Hilbert
transform. In a previous paper these estimates were obtained for a simpler
Walsh model for this operator, but in the Fourier case additional complications
arise due to the inability to perfectly localize in both space and frequency.Comment: 30 pages, no figures, submitted, Math. Annale
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