$L^p$-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)${}_{p^{\prime }}$ implies (Sob)${}_p$ for $1\le p\le p^{\prime }<\infty$, and prove (Sob)${}_{p^{\prime }}$ implies (Dyn)${}_p$ for $1\le p<p^{\prime }<\infty$. 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.

22 pages; title of paper changed, to appear in Journal of Functional Analysis

Cited by in corpus (2)

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