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Title: Generalized Lebesgue points for Sobolev functions (English)
Author: Karak, Nijjwal
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 67
Issue: 1
Year: 2017
Pages: 143-150
Summary lang: English
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Category: math
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Summary: In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f(x)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function $f\in M^{s,p}(X)$, $0<s\leq 1$, $0<p<1$, where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal {H}^h$-Hausdorff measure zero for a suitable gauge function $h$. (English)
Keyword: Sobolev space
Keyword: metric measure space
Keyword: median
Keyword: generalized Lebesgue point
MSC: 28A78
MSC: 46E35
idZBL: Zbl 06738509
idMR: MR3633003
DOI: 10.21136/CMJ.2017.0405-15
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Date available: 2017-03-13T12:07:39Z
Last updated: 2020-01-05
Stable URL: http://hdl.handle.net/10338.dmlcz/146045
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