Muckenhoupt-Wheeden conjectures for sparse operators

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 $L^{p^{\prime }}(u^{1-p^{\prime }})$ to $L^{p^{\prime }}(v^{1-p^{\prime }})$ 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.

An updated version as suggested by the referee

References in corpus (1)

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