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