Optimal complexity dependence for balanced non-doubling Haar-shift bounds

Determine the optimal dependence on the total complexity parameter N in the quantitative weighted norm estimate for Haar shifts with coefficients bounded by 1 acting on balanced, non-doubling measures.

Background

The paper proves a sharp mixed $A_p$–$A_\infty$ estimate for Haar shifts of total complexity at most $N$ over balanced non-doubling measures. The estimate has the correct powers of the weight characteristics and answers an earlier quantitative question concerning the weighted theory in this setting.

The dependence of the implicit constant on $N$ is not optimized by the result. The authors explicitly identify determining this dependence as an unresolved issue, leaving open the problem of finding the best possible complexity growth.