论文标题
$ l^p $ -kato类衡量对称马尔可夫流程下的热内内核估计值
$L^p$-Kato class measures for symmetric Markov processes under heat kernel estimates
论文作者
论文摘要
在本文中，我们建立了两类$ l^p $ -kato类测量的巧合，在对称的马尔可夫工艺的框架中，在轻度条件下，对热核的上和下部估计值进行了。一类$ l^p $ -kato类措施由$ p $ - 正订单分解核的功率定义，另一个是根据与热核估计值相关的一些指数来定义的绿核的$ p $ th幂。我们还证明，如果$ Q $ -th的整合功能在半径$ 1 $的球上具有均匀性相对于中心的统一性，则如果$ l^p $ -kato类别为$ l^p $ q $，则$ q $大于与$ p $相关的常数，并且常数出现在热热kernel的上和下部估计中。这些是Aizenman-Simon的某些结果的完整扩展，以及第二名作者在欧几里得空间上布朗尼动议框架中的最新结果。我们进一步提供了具有AHLFORS规律性的ra量的必要条件，使其属于$ l^p $ -kato类。我们的结果可以适用于许多示例，例如，对称（相对论）稳定过程，$ d $ set上的跳跃过程，riemannian歧管上的布朗动作，分形等上的扩散等等。
In this paper, we establish the coincidence of two classes of $L^p$-Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $L^p$-Kato class measures is defined by the $p$-th power of positive order resolvent kernel, another is defined in terms of the $p$-th power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that $q$-th integrable functions on balls with radius $1$ having uniformity of its norm with respect to centers are of $L^p$-Kato class if $q$ is greater than a constant related to $p$ and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman-Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to $L^p$-Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on $d$-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.
