Sharp bounds for $t$-Haar multipliers on $L^2$

We show that if a weight $w\in C_{2t}^d$ and there is $q>1$ such that $w^{2t}\in A_q^d$, then the $L^2$-norm of the $t$-Haar multiplier of complexity $(m,n)$ associated to $w$ depends on the square root of the $C_{2t}^d$-characteristic of $w$ times the square root $A_q^d$-characteristic of $w^{2t}$ % raised to the power $(p-1)\mathrm{/}2$ times a constant that depends polynomially on the complexity. In particular, if $w\in C_{2t}^d\cap A_{\mathrm{\infty }}^d$ then $w^{2t}\in A_q^d$ for some $q>1$.