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

2017/05/23 by Komla Domelevo, Paata Ivanisvili, Domelevo, K. +7
Mathematics · #Advanced Harmonic Analysis Research #advanced mathematical theories #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1705.08376

Abstract

We show that the classical A_∞ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted L² space. We also show that under the martingale A₂ condition, an estimate holds true, but the optimal power of the characteristic jumps from 1 / 2 to 1 even when considering the classical A₂ 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 timenon-homogeneous martingale transforms. Last, we give a sharp A_∞ estimate for the n-adic homogeneous case, growing with n.

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