Potential trace inequalities via a Calderón-type theorem
Zdeněk Mihula, Luboš Pick, Daniel Spector
TL;DR
This work develops a Calderón-type framework to obtain trace-type bounds for potential operators $I_\alpha^\mu$ on rearrangement-invariant spaces by leveraging boundedness results for smoother operators such as fractional maximal operators. Central to the approach are the notions of $(p,q)$-sawyerable operators and the single-variable operator $R_{p,q}$, which together enable a reduction of boundedness for the harder operator $I_\alpha^\mu$ to boundedness of a dyadic-maximal-type object on one-dimensional rearrangement spaces. The authors prove a pair of abstract theorems: a Calderón-type equivalence for sawyerable operators and a general principle that transfers endpoint bounds to the endpoint space $\Lambda_X(\mathcal{R},\mu)$, yielding a generalized Korobkov–Kristensen inequality in the two-measure setting. Specializing to Lorentz spaces, the results provide explicit boundedness criteria and extend trace inequalities to measures with nontrivial growth, unifying Lebesgue, Lorentz, and other rearrangement-invariant spaces within a single framework. Overall, the paper offers a versatile, measure-theoretic, and function-space toolkit for analyzing potential-type operators in non-Euclidean settings with broad implications for trace inequalities and Sobolev-type embeddings.
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 $μ=\mathcal{L}^n$, 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 $L^{p,1}(\mathbb{R}^n,μ)$, where $μ, ν$ are Radon measures such that \[ \sup_{x\in\mathbb{R}^n, r>0} \frac{μ(B(x,r))}{r^{d}} < \infty \quad \text{and} \quad \sup_{μ(Q)>0} \frac{ν(Q)}{\quadμ(Q)^{1-\frac{αp}{d}}} < \infty, \] and $I_α^μ$ is the Riesz potential defined with respect to $μ$ of order $α\in (0,d)$. 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.