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

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Simone Di Marino , Gareth Speight ·

YOU? · · 2015 · Open Access · · DOI: https://doi.org/10.1090/s0002-9939-2015-12641-x · OA: W2060148849

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.