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.;
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.
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