Modulation invariant and multilinear singular integral operators

Michael Christ

Séminaire Bourbaki (2005-2006)

  • Volume: 48, page 295-320
  • ISSN: 0303-1179

Abstract

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In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication by characters exp ( i x ξ ) . I will briefly review some of the conceptual backdrop to these problems, discuss the key concepts which provide the structural basis for the analysis, sketch a proof, and if time permits, mention related unsolved problems. I will attempt to convey an accurate sense of the work, without presenting full details.

How to cite

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Christ, Michael. "Modulation invariant and multilinear singular integral operators." Séminaire Bourbaki 48 (2005-2006): 295-320. <http://eudml.org/doc/252147>.

@article{Christ2005-2006,
abstract = {In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication by characters $\exp (i x \xi )$. I will briefly review some of the conceptual backdrop to these problems, discuss the key concepts which provide the structural basis for the analysis, sketch a proof, and if time permits, mention related unsolved problems. I will attempt to convey an accurate sense of the work, without presenting full details.},
author = {Christ, Michael},
journal = {Séminaire Bourbaki},
keywords = {opérateurs d’intégrale singulière; transformée de Hilbert; opérateurs multilinéaires; invariance par modulation; presque orthogonalité; décomposition de l’espace des phases; coefficients de Fourier localisés; opérateur maximalde somme partielle},
language = {eng},
pages = {295-320},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Modulation invariant and multilinear singular integral operators},
url = {http://eudml.org/doc/252147},
volume = {48},
year = {2005-2006},
}

TY - JOUR
AU - Christ, Michael
TI - Modulation invariant and multilinear singular integral operators
JO - Séminaire Bourbaki
PY - 2005-2006
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 48
SP - 295
EP - 320
AB - In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication by characters $\exp (i x \xi )$. I will briefly review some of the conceptual backdrop to these problems, discuss the key concepts which provide the structural basis for the analysis, sketch a proof, and if time permits, mention related unsolved problems. I will attempt to convey an accurate sense of the work, without presenting full details.
LA - eng
KW - opérateurs d’intégrale singulière; transformée de Hilbert; opérateurs multilinéaires; invariance par modulation; presque orthogonalité; décomposition de l’espace des phases; coefficients de Fourier localisés; opérateur maximalde somme partielle
UR - http://eudml.org/doc/252147
ER -

References

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