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ID 52074
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Author
Yamagishi, Hiroyuki
Watanabe, Kohtaro
Kametaka, Yoshinori
Abstract
We have obtained the best constant of the following Lp Sobolev inequality sup 0≤y≤1| u(j)(y)| ≤C (∫ 01 | u(M)(x)| p dx)1/p , where u is a function satisfying u(M) ∈ Lp(0, 1), u(2i)(0) = 0 (0 ≤i ≤ [(M − 1)/2]) and u(2i+1)(1) = 0 (0 ≤ i ≤ [(M − 2)/2]), where u(i) is the abbreviation of (d/dx)iu(x). In [9], the best constant of the above inequality was obtained for the case of p = 2 and j = 0. This paper extends the result of [9] under the conditions p > 1 and 0 ≤ j ≤ M −1. The best constant is expressed by Bernoulli polynomials.
Keywords
L<sup>p</sup> Sobolev inequality
Best constant
Green function
Reproducing kernel
Bernoulli polynomial
Hölder inequality
Published Date
2014-01
Publication Title
Mathematical Journal of Okayama University
Volume
volume56
Issue
issue1
Publisher
Department of Mathematics, Faculty of Science, Okayama University
Start Page
145
End Page
155
ISSN
0030-1566
NCID
AA00723502
Content Type
Journal Article
language
英語
Copyright Holders
Copyright©2014 by the Editorial Board of Mathematical Journal of Okayama University
File Version
publisher
Refereed
True
Submission Path
mjou/vol56/iss1/11