Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
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DOI：
10.4310/jdg/1214436699
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发表时间：
1982
影响因子：
2.5
通讯作者：
J. Cheeger;M. Gromov;Michael Taylor
中科院分区：
文献类型：
--
作者：
J. Cheeger;M. Gromov;Michael Taylor
deλ是与 /^δ相关的预测值。一个自然的问题是研究代表 /（ /^δ）的明确内核Kf（x）（xx，x2）的行为，以各种几何数量的行为在M上的行为/> 0，e（xλ9jc2，0是a X2的正（对称）x2，固定的t和（说）2> i s * e {xλ，x2，t）的行为与E-P的行为大致相似2（xx，x2）/4欧几里德热核， - （p（xx，x2）表示距离）。
where dEλ is the projection valued measure associated with /^Δ". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of various geometric quantities on M. As a particularly important example we have the heat kernel E(xl9 x2, t) — ke-\2t. By use of the local parametrix and the standard elliptic estimates, one can show that for / > 0, E(xλ9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say) %2> ι s * the domain of all positive powers of Δ as a function of xλ; see e.g. [9]. In works of Garding [19] and Donnelly [16], upper estimates for E(xu x2, t) (and its derivatives) were given under the assumption that M has bounded geometry. They showed that as x2 -> oo, the behavior of E{xλ, x2, t) is roughly similar to that of the e-p 2(xx,x2)/4 Euclidean heat kernel, — (p(xx, x2) denotes distance). Recall that (4ττ/) M is said to have bounded geometry if the injectivity radius i(x) of the