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

Pott, S. (2007) A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform. Studia Mathematica, 182(2), pp. 99-111. (doi: 10.4064/sm182-2-1)

Abstract

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space H. We show that if W and its inverse W⁻¹ both satisfy a matrix reverse Holder property introduced in [2], then the weighted Hilbert transform H : L²_W(R,H) → L²_W(R,H) and also all weighted dyadic martingale transforms T_σ: L²_W(R,H) → L²_W(R,H) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Pott, Dr Sandra
• Authors: Pott, S.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Studia Mathematica
• Publisher: Polska Akademia Nauk
• ISSN: 0039-3223
• ISSN (Online): 1730-6337
• Copyright Holders: Copyright © 2007 Polska Akademia Nauk
• First Published: First published in Studia Mathematica 182(2):99-111
• Publisher Policy: Reproduced with the permission of the publisher