Haar multipliers meet Bellman functions

摘要: Using Bellman function techniques, we obtain the optimal dependence of operator norms in $L^2(\mathbb{R})$ Haar multipliers $T_w^t$ on corresponding $RH^d_2$ or $A^d_2$ characteristic weight $w$, for $t=1,\pm 1/2$. These results can be viewed as particular cases estimates homogeneous spaces $L^2(vd\sigma)$, $\sigma$ a doubling positive measure and $v\in A^d_2(d\sigma)$, weighted dyadic square $S_{\sigma}^d$. We show that such functions $L^2(v d\sigma)$ are bounded by linear $A^d_2(d\sigma )$ $v$, where constant depends only $\sigma$. also an inverse estimate Both known when $d\sigma=dx$. deduce both from multiplier $(T_v^{\sigma})^{1/2}$ $L^2(d\sigma)$ A_2^d(d\sigma)$, which mirrors $T_w^{1/2}$ $w\in A^d_2$. The adapted to measure, $(T_v^{\sigma})^{1/2}$, is proved using functions. sharp sense rates cannot improved expected hold all $\sigma$, since case $d\sigma=dx$, $v=w$, correspond $T^{1/2}_w$ proven sharp.