Advanced

Bounds for Calderón–Zygmund operators with matrix A2 weights

Pott, Sandra LU and Stoica, Andrei LU (2017) In Bulletin des Sciences Mathematiques 141(6). p.584-614
Abstract

It is well-known that dyadic martingale transforms are a good model for Calderón–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A2 matrix weight, then the weighted L2-norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A2 characteristic of W as the weighted L2-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón–Zygmund operators on the A2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof... (More)

It is well-known that dyadic martingale transforms are a good model for Calderón–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A2 matrix weight, then the weighted L2-norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A2 characteristic of W as the weighted L2-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón–Zygmund operators on the A2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderón–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Bellman function, Calderón–Zygmund operator, Carleson embedding theorem, Martingale transform, Matrix A weights, Weighted L spaces
in
Bulletin des Sciences Mathematiques
volume
141
issue
6
pages
31 pages
publisher
Gauthier-Villars
external identifiers
  • scopus:85029511965
  • wos:000412966000004
ISSN
0007-4497
DOI
10.1016/j.bulsci.2017.07.001
language
English
LU publication?
yes
id
338cd201-e0f8-4d24-a1fe-9850384b7f2b
date added to LUP
2017-10-03 09:35:09
date last changed
2018-01-16 13:20:25
@article{338cd201-e0f8-4d24-a1fe-9850384b7f2b,
  abstract     = {<p>It is well-known that dyadic martingale transforms are a good model for Calderón–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A<sub>2</sub> matrix weight, then the weighted L<sup>2</sup>-norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A<sub>2</sub> characteristic of W as the weighted L<sup>2</sup>-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón–Zygmund operators on the A<sub>2</sub> characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderón–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.</p>},
  author       = {Pott, Sandra and Stoica, Andrei},
  issn         = {0007-4497},
  keyword      = {Bellman function,Calderón–Zygmund operator,Carleson embedding theorem,Martingale transform,Matrix A weights,Weighted L spaces},
  language     = {eng},
  month        = {08},
  number       = {6},
  pages        = {584--614},
  publisher    = {Gauthier-Villars},
  series       = {Bulletin des Sciences Mathematiques},
  title        = {Bounds for Calderón–Zygmund operators with matrix A<sub>2</sub> weights},
  url          = {http://dx.doi.org/10.1016/j.bulsci.2017.07.001},
  volume       = {141},
  year         = {2017},
}