Weighted martingale multipliers in the non-homogeneous setting and outer measure spaces

We investigate the unconditional basis property of martingale differences in weighted spaces L2(wdν) in the non-homogeneous situation, that is when the reference measure ν is not doubling. Specifically, we prove that finiteness of the quantity [w]A2=supI⁡〈w〉I〈w−1〉I, defined through averages 〈⋅〉I relative to the reference measure ν, implies that Haar subspaces form an unconditional basis in the weighted space L2(wdν). Moreover, we prove that the unconditional basis constant of this system grows at most linearly in [w]A2. The problem is reduced to the sharp weighted estimates of the so-called Haar multipliers. Even in the classical case of the standard dyadic lattice in Rd with Lebesgue reference measure our result is new in that our estimates are independent of the dimension n. Our approach combines the technique of outer measure spaces with the Bellman function argument.