  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.
