*Published Paper*

**Inserted:** 18 nov 2013

**Last Updated:** 24 jul 2018

**Journal:** Proceedings of the American Mathematical Society

**Volume:** 143

**Number:** 5239-5252

**Year:** 2014

The new version generalizes the construction to $\mathbb{R}^{n}$ as suggested by a helpful referee. We also fix some minor typos, clarify terminology and update the references.

**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.

**Tags:**
GeMeThNES

**Keywords:**
weighted Sobolev spaces, Lipschitz functions, Weak Gradient

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