Bounds for the Hilbert Transform with Matrix $A_2$ Weights
Abstract
Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(\mathbb{R},\mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(\mathbb{R},\mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(\mathbb{R},W,\mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{\frac{3}{2}} \log [W]_{A_2}$.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2014
- DOI:
- 10.48550/arXiv.1402.3886
- arXiv:
- arXiv:1402.3886
- Bibcode:
- 2014arXiv1402.3886B
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- 42A50
- E-Print:
- 20 pages. v3: Revised to address referee comments and include additional references. v4: Grant information added. v5: Revised to address referee comments and include additional references