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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the distance of the Riemann-Liouville operator from compact operators


Author: Bohumír Opic
Journal: Proc. Amer. Math. Soc. 122 (1994), 495-501
MSC: Primary 47B38; Secondary 47B07, 47G10
MathSciNet review: 1200178
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Abstract: We consider generalized Hardy operators

$\displaystyle Tf(x) = \int_a^x {\varphi (x,y)f(y)\;dy,\quad x \in (a,b) \subset \mathbb{R},} $

acting between two weighted Lebesgue spaces $ X = {L^p}(a,b;v)$ and $ Y = {L^q}(a,b;w), 1 < p \leq q < \infty $, and present lower and upper bounds on the distance of T from the space of all compact linear operators P, $ P:X \to Y$. The conditions on the kernel $ \varphi (x,y)$ are patterned in such a way that the above mentioned class of operators T contains the Riemann-Liouville fractional operators of orders equal to or greater than one.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1200178-3
PII: S 0002-9939(1994)1200178-3
Article copyright: © Copyright 1994 American Mathematical Society