Failure of the linear $A_2$ bound for the strong maximal operator

Abstract

We show that the strong maximal operator on ${\mathbb{R}}^2$ does not satisfy a linear weighted $L^2$ estimate in terms of the rectangular $A_2$ characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by $[w{]}_{A_2^{\mathrm{str}}}\sqrt{\log [w{]}_{A_2^{\mathrm{str}}}}$. The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.

arXiv comment9 pages, 1 figure