Fine properties of functions from Hajłasz–Sobolev classes Mαp, p > 0, I. Lebesgue points
- Authors: Bondarev S.A.1, Krotov V.G.1
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Affiliations:
- Belarusian State University
- Issue: Vol 51, No 6 (2016)
- Pages: 282-295
- Section: Real and Complex Analysis
- URL: https://journals.rcsi.science/1068-3623/article/view/227977
- DOI: https://doi.org/10.3103/S1068362316060029
- ID: 227977
Cite item
Abstract
Let X be a metric measure space satisfying the doubling condition of order γ > 0. For a function f ∈ Llocp(X), p > 0 and a ball B ⊂ X by IB(p)f we denote the best approximation by constants in the space Lp(B). In this paper, for functions f from Hajłasz–Sobolev classes Mαp(X), p > 0, α > 0, we investigate the size of the set E of points for which the limit limr→+0IB(x,r)(p)f = f*(x). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x ∈ E
About the authors
S. A. Bondarev
Belarusian State University
Author for correspondence.
Email: bsa0393@gmail.com
Belarus, Minsk
V. G. Krotov
Belarusian State University
Email: bsa0393@gmail.com
Belarus, Minsk