Bounds for Calder\'on-Zygmund operators with matrix A2A_2 weights

It is well-known that dyadic martingale transforms are a good model for Calder\'on-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 $A_2$ matrix weight, then the weighted $L^2$-norm of a Calder\'on-Zygmund operator with cancellation has the same dependence on the $A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calder\'on-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 Calder\'on-Zygmund operators with even kernel. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hyt\"onen to extend the result to general Calder\'on-Zygmund operators.Comment: arXiv admin note: text overlap with arXiv:1310.786

Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO

We study boundedness properties of a class of multiparameter paraproducts on the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO space BMO_d(T^N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO([0,1]^2).Comment: 24 page

Carleson measure and balayage

The balayage of a Carleson measure lies of course in the space of functions of bounded mean oscillation (BMO). We show that the converse statement is false. We also make a two-sided estimate of the Carleson norm of a positive measure in terms of <i>certain</i> balayages

Sharp Bekolle estimates for the Bergman projection

We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Bekolle constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Perez.Comment: 12 pages, 1 figur

On Laplace-Carleson embedding theorems

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. We also consider little Hankel operators on these Bergman spaces. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.Comment: 26 pages, 1 figur

Matrix weights: On the way to the linear bound

In recent years, the attempts to prove sharp bounds for Calderon-Zygmund operators on weighted $L^p$ spaces in terms of the $A_p$-characteristic of the weight has been an im- portant driving force in Harmonic Analysis. After the work of many authors, this culminated with the proof of the conjectured linear bound for p = 2 for all Calderon-Zygmund operators by Tuomas Hyt\”onen in 2010. Recently, the question of the validity of the linear bound for all Calderon-Zygmund operators in the matrix-weighted setting has attracted some interest. In the talk, I want to present the reduction of this question to the case of Haar multipliers and dyadic paraproducts. I also want to talk about the remaining obstacles, some of which have recently been resolved, and focus on the matrix techniques being used. This is joint work with Andrei Stoica