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

Abstract

We consider generalized Hardy operators \[ T f ( x ) = ∫ a x φ ( x , y ) f ( y ) d y , x ∈ ( a , b ) ⊂ R , Tf(x) = \int _a^x {\varphi (x,y)f(y)\;dy,\quad x \in (a,b) \subset \mathbb {R},} \] acting between two weighted Lebesgue spaces X = L p ( a , b ; v ) X = {L^p}(a,b;v) and Y = L q ( a , b ; w ) , 1 > p ≤ q > ∞ 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 → Y P:X \to Y . The conditions on the kernel φ ( x , y ) \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.