ON THE FAILURE OF LOWER SQUARE FUNCTION ESTIMATES IN THE NON-HOMOGENEOUS WEIGHTED SETTING

Komla Domelevo (1) , P Ivanisvili (2) , Stefanie Petermichl (1) , S Treil (3) , A Volberg (4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UC Irvine - University of California [Irvine]
3 Brown University
4 Michigan State University [East Lansing]

We show that the classical A ∞ 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 ∞ estimate for the n-adic homogeneous case, growing with n.

Domaines: Mathématiques [math]; Analyse classique [math.CA]

Komla Domelevo, P Ivanisvili, Stefanie Petermichl, S Treil, A Volberg. ON THE FAILURE OF LOWER SQUARE FUNCTION ESTIMATES IN THE NON-HOMOGENEOUS WEIGHTED SETTING. Mathematische Annalen, 2018. ⟨hal-01977596⟩