The 𝑇1 theorem for martingales

Abstract

The T 1 T1 theorem of David and Journé gives necessary and sufficient conditions that a singular integral operator be bounded from L 2 ( R n ) {L^2}({R^n}) to L 2 ( R n ) {L^2}({R^n}) . In this paper, the definition of singular integral operator is extended to the setting of operators on L 2 ( Ω ) {L^2}(\Omega ) where Ω \Omega denotes Wiener space. The main theorem is that the T 1 T1 theorem holds in this new setting.