Communications in Mathematical Analysis

Method of rotations for bilinear singular integrals

Geoff Diestel, Loukas Grafakos, Peter Honzik, Zengyan Si, and Erin Terwilleger

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Abstract

Suppose that $\Omega$ lies in the Hardy space $H^1$ of the unit circle $\mathbf S^{1}$ in $\mathbf R^2$. We use the Calderón-Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms to show that the bilinear singular operator with the rough kernel $\mathrm{p.v.} \, \Omega(x/|x|) |x|^{-2}$ is bounded from $L^p(\mathbf R)\times L^q(\mathbf R)$ to $L^r(\mathbf R)$, for a large set of indices satisfying $1/p+1/q=1/r$. We also provide an example of a function $\Omega$ in $L^q(\mathbf S^{ 1})$ with mean value zero to show that the singular integral operator given by convolution with $\mathrm{p.v.} \, \Omega(x/|x|) |x|^{-2}$ is not bounded from $L^{p_1}(\mathbf R)\times L^{p_2} (\mathbf R )$ to $ L^{p}(\mathbf R )$ for $1/2<p<1$, $1<p_1,p_2<\infty$, $1/p_1+1/p_2=1/p$, $1\le q<\infty$, and $1/p+1/q>2.$

Article information

Source
Commun. Math. Anal., Conference 3 (2011), 99-107.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.cma/1298670006

Mathematical Reviews number (MathSciNet)
MR2772055

Zentralblatt MATH identifier
1214.42017

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46B70 47B38

Keywords
Bilinear singular integrals bilinear Hilbert transform Fourier multipliers method of rotations

Citation

Diestel, Geoff; Grafakos, Loukas; Honzik, Peter; Si, Zengyan; Terwilleger, Erin. Method of rotations for bilinear singular integrals. Commun. Math. Anal. (2011), no. 3, 99--107. https://projecteuclid.org/euclid.cma/1298670006


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