On the boundedness of generalized Cesàro operators on Sobolev spaces

For β>0 and p≥1, the generalized Cesàro operatorCβf(t):=βtβ∫0t(t−s)β−1f(s)ds and its companion operator Cβ⁎ defined on Sobolev spaces Tp(α)(tα) and Tp(α)(|t|α) (where α≥0 is the fractional order of derivation and are embedded in Lp(R+) and Lp(R) respectively) are studied. We prove that if p>1,...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2014-11, Vol.419 (1), p.373-394
Main Authors: Lizama, Carlos, Miana, Pedro J., Ponce, Rodrigo, Sánchez-Lajusticia, Luis
Format: Article
Language:English
Subjects:
Boundedness
Cesàro operators
Sobolev spaces
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Summary:For β>0 and p≥1, the generalized Cesàro operatorCβf(t):=βtβ∫0t(t−s)β−1f(s)ds and its companion operator Cβ⁎ defined on Sobolev spaces Tp(α)(tα) and Tp(α)(|t|α) (where α≥0 is the fractional order of derivation and are embedded in Lp(R+) and Lp(R) respectively) are studied. We prove that if p>1, then Cβ and Cβ⁎ are bounded operators and commute on Tp(α)(tα) and Tp(α)(|t|α). We calculate explicitly their spectra σ(Cβ) and σ(Cβ⁎) and their operator norms (which depend on p). For 1
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2014.04.047