JournalsrmiVol. 25, No. 3pp. 799–840

Haar multipliers meet Bellman functions

  • María Cristina Pereyra

    University of New Mexico, Albuquerque, United States
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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