# A new of looking at distributional estimates; applications for the bilinear Hilbert transform.

Dimitriy Bilyk; Loukas Grafakos

Collectanea Mathematica (2006)

- Volume: 57, Issue: Extra, page 141-169
- ISSN: 0010-0757

## Access Full Article

top## Abstract

top## How to cite

topBilyk, Dimitriy, and Grafakos, Loukas. "A new of looking at distributional estimates; applications for the bilinear Hilbert transform.." Collectanea Mathematica 57.Extra (2006): 141-169. <http://eudml.org/doc/41787>.

@article{Bilyk2006,

abstract = {Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially lntegrable over compact sets. They also provide restricted type endpoint results on products of Lebesgue spaces where one exponent is 1 or the sum of the reciprocal of the exponents is 3/2. The proof of the distributional estimates for the bilinear Hilbert transform rely on an improved energy estimate for characteristic functions with respect to sets of tiles from which appropriate exceptional subsets have been removed.},

author = {Bilyk, Dimitriy, Grafakos, Loukas},

journal = {Collectanea Mathematica},

keywords = {Análisis de Fourier; Operadores multilineales; Transformada de Hilbert; bilinear Hilbert transform; distributional inequality; restricted weak-type inequality},

language = {eng},

number = {Extra},

pages = {141-169},

title = {A new of looking at distributional estimates; applications for the bilinear Hilbert transform.},

url = {http://eudml.org/doc/41787},

volume = {57},

year = {2006},

}

TY - JOUR

AU - Bilyk, Dimitriy

AU - Grafakos, Loukas

TI - A new of looking at distributional estimates; applications for the bilinear Hilbert transform.

JO - Collectanea Mathematica

PY - 2006

VL - 57

IS - Extra

SP - 141

EP - 169

AB - Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially lntegrable over compact sets. They also provide restricted type endpoint results on products of Lebesgue spaces where one exponent is 1 or the sum of the reciprocal of the exponents is 3/2. The proof of the distributional estimates for the bilinear Hilbert transform rely on an improved energy estimate for characteristic functions with respect to sets of tiles from which appropriate exceptional subsets have been removed.

LA - eng

KW - Análisis de Fourier; Operadores multilineales; Transformada de Hilbert; bilinear Hilbert transform; distributional inequality; restricted weak-type inequality

UR - http://eudml.org/doc/41787

ER -