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arXiv · 1311.4171

The p-Weak Gradient Depends on p

Simone Di Marino·Gareth Speight
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Abstract

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial 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 the real line.

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Simone Di Marino, Gareth Speight. 2014-08-28. The p-Weak Gradient Depends on p. https://arxiv.org/abs/1311.4171

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