JournalsrmiVol. 25, No. 3pp. 799–840

Haar multipliers meet Bellman functions

  • María Cristina Pereyra

    University of New Mexico, Albuquerque, United States
Haar multipliers meet Bellman functions cover
Download PDF

Abstract

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in L2(R)L^2(\mathbb{R}) of the Haar multipliers TwtT_w^t on the corresponding RH2dRH^d_2 or A2dA^d_2 characteristic of the weight ww, for t=1,±1/2t=1,\pm 1/2. These results can be viewed as particular cases of estimates on homogeneous spaces L2(vdσ)L^2(vd\sigma), for σ\sigma a doubling positive measure and vA2d(dσ)v\in A^d_2(d\sigma), of the weighted dyadic square function SσdS_{\sigma}^d. We show that the operator norms of such square functions in L2(vdσ)L^2(v d\sigma) are bounded by a linear function of the A2d(dσ)A^d_2(d\sigma ) characteristic of the weight vv, where the constant depends only on the doubling constant of the measure σ\sigma. We also show an inverse estimate for SσdS_{\sigma}^d. Both results are known when dσ=dxd\sigma=dx. We deduce both estimates from an estimate for the Haar multiplier (Tvσ)1/2(T_v^{\sigma})^{1/2} on L2(dσ)L^2(d\sigma) when vA2d(dσ)v\in A_2^d(d\sigma), which mirrors the estimate for Tw1/2T_w^{1/2} in L2(R)L^2(\mathbb{R}) when wA2dw\in A^d_2. The estimate for the Haar multiplier adapted to the σ\sigma measure, (Tvσ)1/2(T_v^{\sigma})^{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\sigma, since the particular case dσ=dxd\sigma=dx, v=wv=w, correspond to the estimates for the Haar multipliers Tw1/2T^{1/2}_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