The p-Weak Gradient Depends on p
Functional Analysis
2014-08-29 v2 Metric Geometry
Abstract
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.
Cite
@article{arxiv.1311.4171,
title = {The p-Weak Gradient Depends on p},
author = {Simone Di Marino and Gareth Speight},
journal= {arXiv preprint arXiv:1311.4171},
year = {2014}
}
Comments
13 pages. Updated version generalizes the construction to R^n. Accepted for publication in Proceedings of the American Mathematical Society