Generalized Lebesgue points for Sobolev functions

Nijjwal Karak

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 143-150
  • ISSN: 0011-4642

Abstract

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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 , μ ) 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 ∈ M s , p ( X ) , 0 < s ≤ 1 , 0 < p < 1 , where X is a doubling metric measure space, has generalized Lebesgue points outside a set of ℋ h -Hausdorff measure zero for a suitable gauge function h .

How to cite

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Karak, Nijjwal. "Generalized Lebesgue points for Sobolev functions." Czechoslovak Mathematical Journal 67.1 (2017): 143-150. <http://eudml.org/doc/287898>.

@article{Karak2017,
abstract = {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 $f\in M^\{s,p\}(X)$, $0<s\le 1$, $0<p<1$, 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 $h$.},
author = {Karak, Nijjwal},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sobolev space; metric measure space; median; generalized Lebesgue point},
language = {eng},
number = {1},
pages = {143-150},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized Lebesgue points for Sobolev functions},
url = {http://eudml.org/doc/287898},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Karak, Nijjwal
TI - Generalized Lebesgue points for Sobolev functions
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 143
EP - 150
AB - 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 $f\in M^{s,p}(X)$, $0<s\le 1$, $0<p<1$, 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 $h$.
LA - eng
KW - Sobolev space; metric measure space; median; generalized Lebesgue point
UR - http://eudml.org/doc/287898
ER -

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