A version of Burkholder’s theorem for operator-weighted spaces

Abstract

Abstract. Let W be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space H. We prove that if the dyadic martingale transforms are uniformly bounded on L 2 R (W ) for each dyadic grid in R, then the Hilbert transform is bounded on L 2 R (W ) as well, thus providing an analogue of Burkholder's theorem for operator-weighted L 2 -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into "dyadic shifts".

Bounds for the Hilbert transform with matrix A2 weights

“…They showed W satisfying (3) no longer implies that the Hilbert transform or Haar multipliers are bounded on L 2 (W ). In [31], Petermichl-Pott proved a form of Burkholder's Theorem, connecting the boundedness of the Haar multipliers with that of the Hilbert transform on operator weighted L 2 spaces. In [15,33], Pott and Katz-Pereyra both provided interesting sufficient conditions for the Hilbert transform to be bounded on operator weighted L 2 , but to the best of the authors' knowledge, necessary and sufficient conditions have proved elusive.…”