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

We investigate the unconditional basis property of martingale differences in weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity $[w]_{A_2}=\sup_I \, < w>_I < w^{-1}>_I$, defined through averages $ <\cdot >_I$ relative to the reference measure $\nu$, implies that each martingale transform relative to $\nu$ is bounded in $L^2(w\, d\nu)$. Moreover, we prove the linear in $[w]_{A_2}$ estimate of the unconditional basis constant of the Haar system. Even in the classical case of the standard dyadic lattice in $\mathbb{R}^n$, where the results about unconditional basis and linear in $[w]_{A_2}$ estimates are known, our result gives something new, because all the estimates are independent of the dimension $n$. Our approach combines the technique of outer measure spaces with the Bellman function argument.