The $p$-weak gradient depends on $p$
Copyright policy )The $p$-weak gradient depends on $p$
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.
13 pages. Updated version generalizes the construction to R^n. Accepted for publication in Proceedings of the American Mathematical Society
- University of Genoa Italy
- French National Centre for Scientific Research France
- University of Paris-Saclay France
- Département de Mathématiques France
- University of Paris-Sud France
Mathematics - Functional Analysis, Mathematics - Metric Geometry, FOS: Mathematics, 49J52, 46G05, 30L99, Metric Geometry (math.MG), Functional Analysis (math.FA)
Mathematics - Functional Analysis, Mathematics - Metric Geometry, FOS: Mathematics, 49J52, 46G05, 30L99, Metric Geometry (math.MG), Functional Analysis (math.FA)
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