Bounds for the Hilbert Transform with Matrix $A_2$ Weights

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 $[W]_{A_2}^{frac{3}{2}} log [W]_{A_2}$.

Gene network reconstruction from transcriptional dynamics under kinetic model uncertainty: a case for the second derivative

Motivation: Measurements of gene expression over time enable the reconstruction of transcriptional networks. However, Bayesian networks and many other current reconstruction methods rely on assumptions that conflict with the differential equations that describe transcriptional kinetics. Practical approximations of kinetic models would enable inferring causal relationships between genes from expression data of microarray, tag-based and conventional platforms, but conclusions are sensitive to the assumptions made. Results: The representation of a sufficiently large portion of genome enables computation of an upper bound on how much confidence one may place in influences between genes on the basis of expression data. Information about which genes encode transcription factors is not necessary but may be incorporated if available. The methodology is generalized to cover cases in which expression measurements are missing for many of the genes that might control the transcription of the genes of interest. The assumption that the gene expression level is roughly proportional to the rate of translation led to better empirical performance than did either the assumption that the gene expression level is roughly proportional to the protein level or the Bayesian model average of both assumptions. Availability: http://www.oisb.ca points to R code implementing the methods (R Development Core Team 2004). Supplementary information: http://www.davidbickel.com