Using Bellman function techniques, we obtain the optimal dependence of the operator norms in of the Haar multipliers on the corresponding or characteristic of the weight , for . These results can be viewed as particular cases of estimates on homogeneous spaces , for a doubling positive measure and , of the weighted dyadic square function . We show that the operator norms of such square functions in are bounded by a linear function of the characteristic of the weight , where the constant depends only on the doubling constant of the measure . We also show an inverse estimate for . Both results are known when . We deduce both estimates from an estimate for the Haar multiplier on when , which mirrors the estimate for in when . The estimate for the Haar multiplier adapted to the measure, , 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 , since the particular case , , correspond to the estimates for the Haar multipliers 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