Muckenhoupt–Wheeden conjectures for sparse operators

We provide an explicit example of a pair of weights and a dyadic sparse operator 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 the sparse operator is not bounded on the same spaces. Our construction al...

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Bibliographic Details
Published in:Archiv der Mathematik 2017-07, Vol.109 (1), p.49-58
Main Authors: Hoang, Cong, Moen, Kabe
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Operators
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Summary:We provide an explicit example of a pair of weights and a dyadic sparse operator 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 the sparse operator is not bounded on the same spaces. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-017-1046-z