论文标题
$ l^p $ -kato类测量及其与Sobolev嵌入dirichlet空间定理的关系
$L^p$-Kato class measures and their relations with Sobolev embedding theorems for Dirichlet spaces
论文作者
论文摘要
在本文中，我们讨论了dirichlet空间的连续嵌入$（\ MATHCAL {f}，\ MATHCAL {E} _1）$与Lebesgue空间中的关系与关联的分辨率内核$r_α（x，y）$的集成性。对于阳性度量$μ$，我们考虑以下两个属性；第一个是将dirichlet空间$（\ Mathcal {f}，\ Mathcal {e} _1）$连续嵌入到$ l^{2p}（e;μ）$（我们写为（SOB）$ _ p $）中，第二个是1阶的$ _1 resolvent $ _1（ y）\} _ {x \ in E} $相对于$ y $的$ y $均匀的$ p $ - 相对于$ y $ $ y $（我们将其写为（dyn）$ _ p $）。在某些假设下，对于满足（dyn）$ _ 1 $的度量，我们证明了（dyn）$ _ {p'} $ insect（sob）$ _ p $ for $ 1 \ leq p \ leq p \ leq p \ leq p \ leq p \ leq p \ leq p'<\ infty $，prove（sob）$ _ {p'} $ _ infim $ p'为了证明这些结果，我们介绍了$ l^p $ -kato类，$ l^p $ - 对Kato Class措施的介绍，并讨论其属性。我们还提供了与Gagliardo-Nirenberg型插值不平等相对应的这种关系的变体。作为应用程序，我们讨论了及时的交叉点测量的连续性。
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 $r_α(x, y)$. For a positive measure $μ$, we consider the following two properties; the first one is that the Dirichlet space $(\mathcal{F}, \mathcal{E}_1)$ is continuously embedded into $L^{2p}(E;μ)$ (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 $μ$ (which we write as (Dyn)$_p$). Under some assumptions, for a measure $μ$ 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.
