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

Opic, Bohumír

doi:10.1090/S0002-9939-1994-1200178-3

On the distance of the Riemann-Liouville operator from compact operators
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Proc. Amer. Math. Soc. 122 (1994), 495-501 Request permission

Abstract:

We consider generalized Hardy operators \[ Tf(x) = \int _a^x {\varphi (x,y)f(y)\;dy,\quad x \in (a,b) \subset \mathbb {R},} \] 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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