Bounds for Calderón–Zygmund operators with matrix A_{2} weights
(2017) In Bulletin des Sciences Mathematiques 141(6). p.584614 Abstract
It is wellknown 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 vectorvalued functions. We prove that if W is an A_{2} matrix weight, then the weighted L^{2}norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A_{2} characteristic of W as the weighted L^{2}norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrixweighted Calderón–Zygmund operators on the A_{2} characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof... (More)
It is wellknown 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 vectorvalued functions. We prove that if W is an A_{2} matrix weight, then the weighted L^{2}norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A_{2} characteristic of W as the weighted L^{2}norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrixweighted Calderón–Zygmund operators on the A_{2} 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 matrixweighted 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
 20170801
 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
 GauthierVillars
 external identifiers

 scopus:85029511965
 wos:000412966000004
 ISSN
 00074497
 DOI
 10.1016/j.bulsci.2017.07.001
 language
 English
 LU publication?
 yes
 id
 338cd201e0f84d24a1fe9850384b7f2b
 date added to LUP
 20171003 09:35:09
 date last changed
 20180116 13:20:25
@article{338cd201e0f84d24a1fe9850384b7f2b, abstract = {<p>It is wellknown 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 vectorvalued 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 matrixweighted 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 matrixweighted 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 = {00074497}, keyword = {Bellman function,Calderón–Zygmund operator,Carleson embedding theorem,Martingale transform,Matrix A weights,Weighted L spaces}, language = {eng}, month = {08}, number = {6}, pages = {584614}, publisher = {GauthierVillars}, 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}, }