Calderón commutators and the Cauchy integral on Lipschitz curves revisited II: The Cauchy integral and its generalizations
type: Item
instance of: scholarly article
title: Calderón commutators and the Cauchy integral on Lipschitz curves revisited II: The Cauchy integral and its generalizations
zbMATH Open document ID: 1306.42027
DOI: 10.4171/RMI/808
author: Camil Muscalu
published in: Revista Matemática Iberoamericana
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.3850
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 6353618
zbMATH Keywords: Cauchy integral
zbMATH Keywords: Lipschitz curves
zbMATH Keywords: Calderón commutators
zbMATH Keywords: multilinear Fourier multipliers
zbMATH Keywords: Littlewood-Paley projections
zbMATH Keywords: logarithmic estimates
zbMATH Keywords: polynomial upper bounds
MaRDI profile type: Publication
arXiv ID: 1201.3850
cites work: COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
cites work: A BMO estimate for multilinear singular integrals
cites work: On Commutators of Singular Integrals and Bilinear Singular Integrals
cites work: Q4338833
cites work: L'intégrale de Cauchy définit un opératuer borne sur \(L^ 2 \)pour les courbes lipschitziennes
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty\)
review text: This is the second paper of three ones, whose aim is to give new proofs to the well-known theorems of Calderón, Coifman, McIntosh and Meyer, concerning \(L^p\) estimates for the Calderón's commutators and the Cauchy integral on Lipschitz curves. This one describes the case of the Cauchy integral on Lipschitz curves and its generalizations. More precisely, the author treats the Cauchy integral \(C_\Gamma(f)\) defined by NEWLINE\[NEWLINEC_\Gamma(f)(x) =\mathrm{p.v.}\,\int_{\mathbb R}\frac{f(y)}{(x-y)+i(A(x)-A(y))}\,dy,NEWLINE\]NEWLINE where \(A'=a\in L^\infty(\mathbb R)\). As is well-known, standard arguments reduce the \(L^p\) boundedness of this operator to the problem of proving polynomial bounds for the associated \(d\)-th Calderón commutators defined by NEWLINE\[NEWLINEC_d(f)(x) =\mathrm{p.v.}\,\int_{\mathbb R}\frac{(A(x)-A(y))^d}{(x-y)^{d+1}}f(y)\,dy.NEWLINE\]NEWLINE As is known, this can be seen as a \(d+1\)-linear Fourier multiplier with singular symbol. \(C_d\) has \(d+1\) natural adjoints \(C_d^{*i}\) \((i=1,\dots,d+1)\), i.e. NEWLINE\[NEWLINE\int_{\mathbb R}C_d(f_1,\dots,f_{d+1})(x)f_{d+2}(x)\,dx=\int_{\mathbb R} C_d^{*i}(f_1,\dots,f_{i-1},f_{i+1},\dots,f_{d+2})(x)f_{i}(x)\,dx.NEWLINE\]NEWLINE Let \(C_d^{*i}=C_d\). In this paper, the author proves that for \(1\leq i\leq d+2\), NEWLINE\[NEWLINE(*)\quad \|C_d^{*i}(f_1,\dots,f_{d+1})\|_{p}\leq C(d)C(\ell )C(p_1)\cdots C(p_{d+1}) \|f_1\|_{p_1}\cdots\|f_{d+1}\|_{p_{d+1}},NEWLINE\]NEWLINE where \(1\leq p_1,\dots,p_{d+1}\leq \infty\), \(1\leq p<\infty\), \(1/p=1/p_1+\cdots+1/p_{d+1}\), \(\ell\) is the number of indices for which \(pI\neq\infty\), \(C(d)\) grows at most polynomially in \(d\) and \(C_{p_i}=1\) if \(p_i=\infty\).NEWLINENEWLINEHe proves (\(*\)) by developing the ideas for treating multilinear Fourier multiplier theory developed by him, Lacey, Pipher, Tao, Thiele, i.e. by decomposing and discretizing \(C_d^{*i}\) and reducing it to a finite and well localized model operator. He also discuss several gen…
description: scientific article; zbMATH DE number 6353618