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

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.