The $p$-weak gradient depends on $p$

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 $\mathbb {R}$.