Zygmund type and flag type maximal functions, and sparse operators

TL;DR

This paper demonstrates that certain Zygmund and flag maximal functions in Euclidean spaces cannot be controlled by multiparameter sparse operators, highlighting limitations in sparse domination for these structures.

Contribution

It establishes the non-existence of sparse bounds for Zygmund and flag maximal functions, extending understanding of their boundedness properties.

Findings

Zygmund dyadic maximal functions are not dominated by sparse operators.

Flag dyadic maximal functions lack sparse bounds.

Strong dyadic maximal functions do not admit sparse domination.

Abstract

We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.