Some s-Numbers of an Integral Operator of Hardy Type on L-p(.) Spaces

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F09%3A00157463" target="_blank" >RIV/68407700:21110/09:00157463 - isvavai.cz</a>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Some s-Numbers of an Integral Operator of Hardy Type on L-p(.) Spaces
Original language description
Let I = [ab], let p : I -> (1, infinity) be either a step-function or strong log- Holder continuous on I, let L-p(.)(I) be the usual space of Lebesgue type with. variable exponent p, and let T : L-p(.)(I) -> L-p(.)(I) be the operator of Hardy type defined by T f(x) = int(x)(a) f (t)dt. For any n is an element of N, let s(n) denote the nth approximation,Gelfand, Kolmogorov or Bernstein number of T. We show that lim(n ->infinity) ns(n) = 1/2 pi integral(I) {p'(t)p(t)(p(t)-1)}(1/p(t)) sin(pi/p(t))dt. wherep'(t) = p(t)/(p(t) - 1). The proof hinges on estimates of the norm of the embedding id of L-q(.)(I) in L-r(.) (I), where q, r : l -> (1, infinity) are measurable, bounded away from 1 and infinity, and such that, for some epsilon is an element of (0, 1),r(x) <= q(x) <= r(x) + epsilon for all x is an element of I. It is shown that min(|I|, |I |(epsilon)) <= parallel to id parallel to <= epsilon | I |+ epsilon^(-epsilon), a result that has independent interest.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA201%2F08%2F0383" target="_blank" >GA201/08/0383: Function Spaces, Weighted Inequalities and Interpolation</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2009
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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