$L^p$-Kato class measures and their relations with Sobolev embedding theorems for Dirichlet spaces

TL;DR

This paper explores the relationship between Sobolev embedding theorems and resolvent kernel integrability for Dirichlet spaces, introducing $L^p$-Kato class measures and their properties, with applications to intersection measures.

Contribution

It introduces the $L^p$-Kato class measures and establishes new relations between Sobolev embeddings and resolvent kernel integrability for Dirichlet spaces.

Findings

Proves implications between (Dyn) and (Sob) properties under certain conditions.

Introduces and analyzes the properties of $L^p$-Kato class measures.

Provides variants related to Gagliardo-Nirenberg inequalities and applications to intersection measures.

Abstract

In this paper, we discuss relationships between the continuous embeddings of Dirichlet spaces $(\mathcal{F}, \mathcal{E}_1)$ into Lebesgue spaces and the integrability of the associated resolvent kernel $r_\alpha(x, y)$. For a positive measure $\mu$, we consider the following two properties; the first one is that the Dirichlet space $(\mathcal{F}, \mathcal{E}_1)$ is continuously embedded into $L^{2p}(E;\mu)$ (which we write as (Sob)$_p$), and the second one is that the family of 1-order resolvent kernels $\{r_1(x, y)\}_{x\in E}$ is uniformly $p$-th integrable in $y$ with respect to the measure $\mu$ (which we write as (Dyn)$_p$). Under some assumptions, for a measure $\mu$ satisfying (Dyn)$_1$, we prove (Dyn)$_{p'}$ implies (Sob)$_p$ for $1\leq p \leq p'<\infty$, and prove (Sob)$_{p'}$ implies (Dyn)$_p$ for $1\leq p < p'<\infty$. To prove these results we introduce $L^p$-Kato class, an $L^p$-version of the set of Kato class measures, and discuss its properties. We also give variants of such relations corresponding to the Gagliardo-Nirenberg type interpolation inequalities. As an application, we discuss the continuity of intersection measures in time.