# Haar multipliers meet Bellman functions

### María Cristina Pereyra

University of New Mexico, Albuquerque, United States

## Abstract

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in $L_{2}(R)$ of the Haar multipliers $T_{w}$ on the corresponding $RH_{2}$ or $A_{2}$ characteristic of the weight $w$, for $t=1,±1/2$. These results can be viewed as particular cases of estimates on homogeneous spaces $L_{2}(vdσ)$, for $σ$ a doubling positive measure and $v∈A_{2}(dσ)$, of the weighted dyadic square function $S_{σ}$. We show that the operator norms of such square functions in $L_{2}(vdσ)$ are bounded by a linear function of the $A_{2}(dσ)$ characteristic of the weight $v$, where the constant depends only on the doubling constant of the measure $σ$. We also show an inverse estimate for $S_{σ}$. Both results are known when $dσ=dx$. We deduce both estimates from an estimate for the Haar multiplier $(T_{v})_{1/2}$ on $L_{2}(dσ)$ when $v∈A_{2}(dσ)$, which mirrors the estimate for $T_{w}$ in $L_{2}(R)$ when $w∈A_{2}$. The estimate for the Haar multiplier adapted to the $σ$ measure, $(T_{v})_{1/2}$, is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all $σ$, since the particular case $dσ=dx$, $v=w$, correspond to the estimates for the Haar multipliers $T_{w}$ proven to be sharp.

## Cite this article

María Cristina Pereyra, Haar multipliers meet Bellman functions. Rev. Mat. Iberoam. 25 (2009), no. 3, pp. 799–840

DOI 10.4171/RMI/584