The $p$-Weak Gradient Depends on $p$

Di Marino Simone - Speight Gareth

year: 2014
journal: Proceedings of the American Mathematical Society
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 $\mathbb{R}$.

The paper is available on the cvgmt preprint server.