Abstract
We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.
Highlights
There are a lot of works devoted to the mapping properties of the Riemann-Liouville operator Iα
In this work we investigate the problems of boundedness and compactness of the operator Tα,β defined by ( . ) from Lp,w to Lq,v when < α
These two theorems can be reformulated as the following new information in the theory of Hardy type inequalities
Summary
Let v and u be almost everywhere positive functions, which are locally integrable on the interval I. =. Denote by Lp,v ≡ Lp(v, I) the set of all functions f measurable on I such that f p,v. Let W be a non-negative, strictly increasing and locally absolutely continuous function on. When u ≡ and β = the operator Tα,β is called the fractional integral operator of a function f with respect to a function W ). When u ≡ and W (x) = x the operator ) becomes the Riemann-Liouville operator Iα defined by x sβ f (s) ds.
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