| Title:
|
Further generalized versions of Ilmanen's lemma on insertion of $C^{1,\omega}$ or $C^{1,\omega}_{\text{\rm loc}}$ functions (English) |
| Author:
|
Kryštof, Václav |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
445-455 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The author proved in 2018 that if $ G $ is an open subset of a Hilbert space, $ f_1,f_2\colon G\to\mathbb{R} $ continuous functions and $ \omega $ a nontrivial modulus such that $ f_1\leq f_2 $, $ f_1 $ is locally semiconvex with modulus $ \omega $ and $ f_2 $ is locally semiconcave with modulus $ \omega $, then there exists $ f\in C^{1,\omega}_{\text{loc}}(G) $ such that $ f_1\leq f\leq f_2 $. This is a generalization of Ilmanen's lemma (which deals with linear modulus and functions on an open subset of $ \mathbb{R}^{n} $). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $ L^p $ spaces, $ p\in[2,\infty) $. We also prove a ``global" version of Ilmanen's lemma (where a $ C^{1,\omega} $ function is inserted between functions on an interval $ I\subset\mathbb{R} $). (English) |
| Keyword:
|
Ilmanen's lemma |
| Keyword:
|
$ C^{1,\omega} $ function |
| Keyword:
|
semiconvex function with general modulus |
| MSC:
|
26B25 |
| idZBL:
|
Zbl 07511572 |
| idMR:
|
MR4405815 |
| DOI:
|
10.14712/1213-7243.2021.031 |
| . |
| Date available:
|
2022-02-21T13:25:17Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149368 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[9] Kryštof V.: Generalized versions of Ilmanen lemma: insertion of $ C^{1,\omega} $ or $ C^{1,\omega}_{ loc} $ functions.Comment. Math. Univ. Carolin. 59 (2018), no. 2, 223–231. MR 3815687 |
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