2015/08/26 by Sandra Pott, Pott, Sandra, Andrei Stoica +1
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1508.06408
openalex publication_date 2015/08/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 the 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. 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. Hytönen to extend the result to general Calderón-Zygmund operators.