{A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Sandra Pot¹

¹ Department of Mathematics University of Glasgow Glasgow G12 8QW, UK

Studia Mathematica, Tome 182 (2007) no. 2, pp. 99-111

Résumé

Let $W$ be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space ${\mathcal H}$. We show that if $W$ and its inverse $W^{-1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H:L^2_W({\mathbb R}, {\mathcal H}) \rightarrow L^2_W({\mathbb R}, {\mathcal H})$ and also all weighted dyadic martingale transforms $T_\sigma:L^2_W({\mathbb R}, {\mathcal H}) \rightarrow L^2_W({\mathbb R}, {\mathcal H})$ are bounded.We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

Détail

DOI : 10.4064/sm182-2-1

Keywords: operator weight taking values almost everywhere bounded positive invertible linear operators separable hilbert space mathcal its inverse satisfy matrix reverse lder property introduced christ goldberg weighted hilbert transform mathbb mathcal rightarrow mathbb mathcal weighted dyadic martingale transforms sigma mathbb mathcal rightarrow mathbb mathcal bounded condition necessary boundedness weighted hilbert transform