On one-sided potentials with multiple kernels

Abstract

Necessary and sufficient conditions for weight function v governing the boundedness of the Volterra, truncated and ball fractional integral operators with multiple kernels from L p to are established, where 1 < p ≤ q < ∞. The result for the operator with multiple positive kernels enables us to derive a trace inequality for the Riemann–Liouville transform with α, β > 1/p. An example of weight function v presented in the work shows that the condition which arises applying twice one-dimensional boundedness result, for example, to the operator is different from the criterion guaranteeing the inequality To prove the main statements, we use higher dimensional Hardy's inequalities which are also studied in this article.