It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [CKS] for Nash-type inequalities with power order considerably and also extend that of [G1] for second order differential operators on a complete non-compact manifold.

中文翻译：

---

对称马尔可夫半群的热核上限

众所周知，对称Dirichlet形式的Nash型不等式等于相关对称Markov半群的对角热核上限。在本文中，我们表明在某些温和的假设下，这两者都隐含（并因此等价于）非对角热核上限。我们的方法基于一种新的广义Davies方法。我们的结果极大地扩展了具有幂阶的Nash型不等式的[CKS]的结果，并且还扩展了完整非紧凑流形上二阶微分算子的[G1]的结果。