The p-weak gradient depends on p

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}.