The p-weak gradient depends on p

Given α > 0, we construct a weighted Lebesgue measure on ℝn for which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt Ap weight for p > 1 + α. In particular, the p-weak gradient is trivial for small p but nontrivial for large p. This answers an...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2015-12, Vol.143 (12), p.5239-5252
Main Authors: DI MARINO, SIMONE, SPEIGHT, GARETH
Format: Article
Language:English
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Summary:Given α > 0, we construct a weighted Lebesgue measure on ℝn for which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt Ap weight for p > 1 + α. In particular, the p-weak gradient is trivial for small p but nontrivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on ℝ.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-2015-12641-X