*Published Paper*

**Inserted:** 18 nov 2013

**Last Updated:** 16 may 2017

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

**Volume:** 143

**Number:** 5239-5252

**Year:** 2014

**Notes:**

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 $\alpha>0$, we construct a weighted Lebesgue measure on $\mathbb{R}^{n}$ for which the family of non constant 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 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 $\mathbb{R}$.

**Tags:**
GeMeThNES

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

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