The $p$-weak gradient depends on $p$
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- by Simone Di Marino and Gareth Speight PDF
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Abstract:
Given $\alpha >0$, we construct a weighted Lebesgue measure on $\mathbb {R}^{n}$ for which the family of nonconstant curves has $p$-modulus zero for $p\leq 1+\alpha$ but the weight is a Muckenhoupt $A_p$ weight for $p>1+\alpha$. 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 $\mathbb {R}$.References
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Additional Information
- Simone Di Marino
- Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay Cedex, France
- Email: simone.dimarino@sns.it
- Gareth Speight
- Affiliation: Scuola Normale Superiore, Piazza Dei Cavalieri 7, 56126 Pisa, Italy
- MR Author ID: 1003655
- Email: gareth.speight@sns.it
- Received by editor(s): December 4, 2013
- Received by editor(s) in revised form: July 22, 2014, and August 27, 2014
- Published electronically: April 2, 2015
- Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5239-5252
- MSC (2010): Primary 46G05, 49J52, 30L99
- DOI: https://doi.org/10.1090/S0002-9939-2015-12641-X
- MathSciNet review: 3411142