THE p-WEAK GRADIENT DEPENDS ON p Article Swipe
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Simone Di Marino
,
Gareth Speight
·
YOU?
·
· 2016
· Open Access
·
· DOI: https://doi.org/10.1090/s0002-9939-2015-12641-x
· OA: W2907833659
YOU?
·
· 2016
· Open Access
·
· DOI: https://doi.org/10.1090/s0002-9939-2015-12641-x
· OA: W2907833659
Given α > 0, we construct a weighted Lebesgue measure on \mathbb{R}^n for which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt A_p 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 \mathbb{R}.
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