A matrix weighted bilinear Carleson lemma and maximal function

A matrix weighted bilinear Carleson lemma and maximal function 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

A matrix weighted bilinear Carleson lemma and maximal function

, Volume 9 (3) – Jun 27, 2019
18 pages

/lp/springer-journals/a-matrix-weighted-bilinear-carleson-lemma-and-maximal-function-Xb5B1w1Zm8
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00331-9
Publisher site
See Article on Publisher Site

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. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jun 27, 2019

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