Bounds for Calderón–Zygmund operators with matrix A2 weights
(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)
- author
- Pott, Sandra LU and Stoica, Andrei LU
- organization
- publishing date
- 2017-08-01
- 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
- 2025-01-07 21:47:19
@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}}, keywords = {{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}}, doi = {{10.1016/j.bulsci.2017.07.001}}, volume = {{141}}, year = {{2017}}, }