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Heat kernels on non-compact Riemannian manifolds: a partial survey. (English) Zbl 0903.58055

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 167-187 (1997).
Let \(M\) be a complete, non-compact Riemannian manifold, \(\Delta\) the Laplace-Beltrami operator, \(e^{t\Delta}\), \(t>0\) the heat semi-group, and \(p_t(x,y)\), \(x,y\in M\), \(t>0\) the heat kernel. The relation between the geometry at infinity of \(M\) and the behaviour of \(\sup_{x\in M}p_t(x,x)\), or \(p_t(x,x)\), for fixed \(x\in M\) as a function of \(t\rightarrow +\infty\) is very important. The paper is a survey of some results particularly related to the volume growth and isoperimetric type properties. Also, the boundedness of Riesz transformations and \(H^1\)-BMO duality is considerd for some manifolds where the heat kernel behaves similarly as in the Euclidean space (§4). In §2, the intermediate role of the heat kernel decay and an isoperimetric inequality is described. It is closely related to the results of Varopoulos. In §3 it is discussed where these three properties coincide. Sections §5 and §6 are dedicated to the characterization of general on-diagonal upper and lower bounds for the heat kernel.
For the entire collection see [Zbl 0882.00016].

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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