Hausdorff measure of critical set for Luzin N condition

Abstract

It is well-known that there is a Sobolev homeomorphism f is an element of W-1,W-p ([-1, 1](n), [-1, 1](n)) for any p < n which maps a set C of zero Lebesgue n-dimensional measure onto a set of positive measure. We study the size of this critical set C and characterize its lower and upper bounds from the perspective of Hausdorff measures defined by a general gauge function. (C) 2020 Elsevier Inc.

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Keywords

• Luzin condition
• Hausdorff measure
• Gauge function