Potential trace inequalities via a Calder\'on-type theorem

Abstract

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 $\mu=\mathcal{L}^n$, the Lebesgue measure on $\mathbb{R}^n$): There exists a constant $C>0$ such that \[\int_{\mathbb{R}^n} |I_\alpha^\mu f|^p d\nu \leq C \|f\|_{L^{p,1}(\mathbb{R}^n,\mu)}^p\] for all $f$ in the Lorentz space $L^{p,1}(\mathbb{R}^n,\mu)$, where $\mu, \nu$ are Radon measures such that \[\sup_{Q} \frac{\mu(Q)}{l(Q)^{d}} < \infty \quad \text{and} \quad \sup_{\mu(Q)>0} \frac{\nu(Q)}{\quad\mu(Q)^{1-\frac{\alpha p}{d}}} < \infty,\] and $I_\alpha^\mu$ is the Riesz potential defined with respect to $\mu$ of order $\alpha \in (0,d)$. 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 Calder\'on where the target space in one endpoint is a space of bounded functions.