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

We show that if a weight $w\in C^d_{2t}$ 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 $C^d_{2t}$-characteristic of $w$ times the square root $A^d_q$-characteristic of $w^{2t}$ % raised to the power $(p-1)/2$ times a constant that depends polynomially on the complexity. In particular, if $w\in C^d_{2t}\cap A_{\infty}^d$ then $w^{2t}\in A_q^d$ for some $q>1$.