In this paper, we introduce the notion of \(L^p\) -Green-tight measures of \(L^p\) -Kato class in the framework of symmetric Markov processes. The class of \(L^p\) -Green-tight measures of \(L^p\) -Kato class is defined by the p-th power of resolvent kernels. We first prove that under the \(L^p\) -Green tightness of the measure \(\mu \) , the embedding of extended Dirichlet space into \(L^{2p}(E;\mu )\) is compact under the absolute continuity condition for transient Markov processes, which is an extension of recent seminal work by Takeda. Secondly, we prove the coincidence between two classes of \(L^p\) -Green-tightness, one is originally introduced by Zhao, and another one is invented by Chen. Finally, we prove that our class of \(L^p\) -Green-tight measures of \(L^p\) -Kato class coincides with the class of \(L^p\) -Green tight measures of Kato class in terms of Green kernel under the global heat kernel estimates. We apply our results to d-dimensional Brownian motion and rotationally symmetric relativistic \(\alpha \) -stable processes on \(\mathbb {R}^d\) .
