Potential trace inequalities via a Calderón-type theorem

In this paper, we develop a general theoretical tool for the establishment of the boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous properties of operators that are easier to handle (such as fractional maximal operators). A principal example of the new results one obtains by our analysis is the following inequality, which generalizes a result of Korobkov and Kristensen (who had treated the case (Formula presented.), the Lebesgue measure on (Formula presented.)): There exists a constant (Formula presented.) such that (Formula presented.) for all (Formula presented.) in the Lorentz space (Formula presented.), where (Formula presented.) are Radon measures such that (Formula presented.) and (Formula presented.) is the Riesz potential defined with respect to (Formula presented.) of order (Formula presented.). More broadly, we obtain inequalities in this spirit in the context of rearrangement-invariant spaces through a result of independent interest, an extension of an interpolation theorem of Calderón where the target space in one endpoint is a space of bounded functions.