Potential trace inequalities via a Calderón-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 $\hat{I}¼={\mathcal{L}}^n$, the Lebesgue measure on ${\mathbb{R}}^n$): There exists a constant $C>0$ such that \[\int_{\mathbb{R}^n} |I_α^μf|^p dν\leq C \|f\|_{L^{p,1}(\mathbb{R}^n,μ)}^p\] for all $f$ in the Lorentz space $L^{p,1}({\mathbb{R}}^n,\hat{I}¼)$, where $\hat{I}¼,\hat{I}½$ are Radon measures such that \[\sup_{Q} \frac{μ(Q)}{l(Q)^{d}} < \infty \quad \text{and} \quad \sup_{μ(Q)>0} \frac{ν(Q)}{\quadμ(Q)^{1-\frac{αp}{d}}} < \infty,\] and $I_{\hat{I}}{\pm }^{\hat{I}}¼$ is the Riesz potential defined with respect to $\hat{I}¼$ of order $\hat{I}\pm \in (0,d)$. 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 Calderón where the target space in one endpoint is a space of bounded functions.

31 pages, to appear in the Journal of the London Mathematical Society