×

zbMATH — the first resource for mathematics

Lower estimates of the heat kernel on conic manifolds and Riesz transform. (Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz.) (French) Zbl 1076.58017
In [Bull. Sci. Math. 124, 365–384 (2000; Zbl 0977.58024)], the second author established Gaussian upper estimates for the heat kernel \(p_{t}\) associated to the Laplace-Beltrami operator \(\Delta\) on \(C(N)\), where \(N\) is a compact connected Riemannian manifold without boundary and with dimension \(n\). Namely, \(p_{t}(x,y)\leq C_{1}t^{-n/2}e^{-d^{2}(x,y)/C_{2}t}\).
In the present paper, the authors prove the corresponding lower bound, namely \(p_{t}(x,y)\geq c_{1}t^{-n/2}e^{-d^{2}(x,y)/c_{2}t}\).
Thus, on such manifolds, Gaussian upper and lower bounds for the heat kernel are valid, and the volume of balls has polynomial growth with exponent \(n\), although the so-called Riesz transforms (namely, the operator \(\nabla \Delta^{-1/2}\)) are not \(L^p\)-bounded for all \(p>2\) (this was proven by the second author in [J. Funct. Anal. 168, 145–238 (1999; Zbl 0937.43004)]).
The proof of the lower bound for \(p_{t}\) relies on a formula for \(p_{t}\) due to J. Cheeger [J. Differ. Geom. 18, 575–657 (1983; Zbl 0529.58034)], and does not go through any Poincaré inequality.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
PDF BibTeX XML Cite
Full Text: DOI