Given α > 0, we construct a weighted Lebesgue measure on Rnfor which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt Ap weight for p > 1 + α. In particular, the p-weak gradient is trivial for small p but nontrivial 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 R.

The p-weak gradient depends on p

di Marino S.;Speight G.
2015

Abstract

Given α > 0, we construct a weighted Lebesgue measure on Rnfor which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt Ap weight for p > 1 + α. In particular, the p-weak gradient is trivial for small p but nontrivial 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 R.
2015
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/79428
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