| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2017-03-13T12:07:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146045 |
| . |
| Reference:
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