On the failure of lower square function estimates in the non-homogeneous weighted setting

  • K. Domelevo
  • P. Ivanisvili
  • S. Petermichl
  • S. Treil
  • A. Volberg


We show that the classical \(A_{\infty }\) condition is not sufficient for a lower square function estimate in the non-homogeneous weighted \(L^2\) space. We also show that under the martingale \(A_2\) condition, an estimate holds true, but the optimal power of the characteristic jumps from 1 / 2 to 1 even when considering the classical \(A_2\) characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete time non-homogeneous martingale transforms. Last, we give a sharp \(A_{\infty }\) estimate for the n-adic homogeneous case, growing with n.


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Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA
  3. 3.Department of MathematicsBrown UniversityProvidenceUSA
  4. 4.Department of MathematicsMichigan State UniversityEast LansingUSA

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