Calculus of Variations and Geometric Measure Theory
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S. Di Marino - G. Speight

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

created by dimarino on 18 Nov 2013
modified by speight on 16 May 2017


Published Paper

Inserted: 18 nov 2013
Last Updated: 16 may 2017

Journal: Proceedings of the American Mathematical Society
Volume: 143
Number: 5239-5252
Year: 2014

The new version generalizes the construction to $\mathbb{R}^{n}$ as suggested by a helpful referee. We also fix some minor typos, clarify terminology and update the references.


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 $\mathbb{R}$.

Tags: GeMeThNES
Keywords: weighted Sobolev spaces, Lipschitz functions, Weak Gradient


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