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A matrix weighted bilinear Carleson lemma and maximal function

Petermichl, Stefanie; Pott, Sandra LU and Reguera, Maria Carmen (2019) In Analysis and Mathematical Physics
Abstract

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob’s maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight.... (More)

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob’s maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.

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Please use this url to cite or link to this publication:
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organization
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Contribution to journal
publication status
epub
subject
in
Analysis and Mathematical Physics
external identifiers
  • scopus:85068326795
ISSN
1664-2368
DOI
10.1007/s13324-019-00331-9
language
English
LU publication?
yes
id
a6e869aa-d11f-4e88-802d-88b5f601289e
date added to LUP
2019-07-11 16:33:24
date last changed
2019-07-11 16:33:24
@article{a6e869aa-d11f-4e88-802d-88b5f601289e,
  abstract     = {<p>We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob’s maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.</p>},
  author       = {Petermichl, Stefanie and Pott, Sandra and Reguera, Maria Carmen},
  issn         = {1664-2368},
  language     = {eng},
  month        = {06},
  series       = {Analysis and Mathematical Physics},
  title        = {A matrix weighted bilinear Carleson lemma and maximal function},
  url          = {http://dx.doi.org/10.1007/s13324-019-00331-9},
  year         = {2019},
}