Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, San Francisco, and London, 1975. MR 0450957 (56:9247)
  • [2] S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991 135-141). MR 1059623 (91k:26018)
  • [3] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Univ. Press, Oxford, 1987. MR 929030 (89b:47001)
  • [4] D. E. Edmunds, W. D. Evans, and D. J. Harris, Approximation numbers of certain Volterra integral operators, J. London Math. Soc. (2) 37 (1988), 471-489. MR 939123 (89k:47078)
  • [5] R. K. Juberg, The measure of non-compactness in $ {L^p}$ for a class of integral operators, Indiana Univ. Math. J. 23 (1974), 925-936. MR 0341194 (49:5944)
  • [6] F. J. Martin-Reyes and E. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), 727-733. MR 965246 (90a:26012)
  • [7] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Res. Notes in Math. Ser., vol. 219, Longman Sci. Tech., Harlow, 1990. MR 1069756 (92b:26028)
  • [8] V. D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. London Math. Soc. (2) 45 (1992), 232-242. MR 1171551 (93f:47029)
  • [9] -, Two-weighted estimates for Riemann-Liouville integrals, (Československá akademie věd) Matematický ústav, preprint no. 39, Praha, 1988, pp. 1-28.
  • [10] C. A. Stuart, The measure of non-compactness of some linear integral operators, Proc. Roy. Soc. Edinburgh Sect. A 71 (1973), 167-179. MR 0326501 (48:4845)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 47B07, 47G10

Retrieve articles in all journals with MSC: 47B38, 47B07, 47G10

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society