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

TL;DR

This paper introduces a new theoretical framework for establishing the boundedness of complex operators like potentials on specific function spaces, generalizing previous results and extending interpolation theorems.

Contribution

It develops a Calderón-type theorem to derive trace inequalities for potentials on rearrangement-invariant spaces, broadening the scope of operator boundedness results.

Findings

Established a generalized inequality for Riesz potentials with Radon measures.

Extended Calderón's interpolation theorem to include spaces of bounded functions.

Provided conditions on measures for boundedness of potential operators.

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.