Muckenhoupt-Wheeden conjectures for sparse operators

TL;DR

This paper constructs specific weight examples demonstrating that the boundedness of the Hardy-Littlewood maximal function does not imply the boundedness of sparse operators, highlighting nuanced differences in weighted inequalities.

Contribution

It provides explicit counterexamples showing the failure of certain weighted inequalities for sparse operators despite the boundedness of classical maximal functions.

Findings

Counterexamples for weighted inequalities of sparse operators

Distinction between maximal function and sparse operator boundedness

Failure of weak-type endpoint for certain weights

Abstract

We provide an example of a pair of weights $(u,v)$ for which the Hardy-Littlewood maximal function is bounded from $L^p(v)$ to $L^p(u)$ and from $L^{p'}(u^{1-p'})$ to $L^{p'}(v^{1-p'})$ while a dyadic sparse operator is not bounded on the same domain and range. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.