Lower estimates of the heat kernel on conic manifolds and Riesz transform
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 \textit{J. Cheeger} [J. Differ. Geom. 18, 575--657 (1983; Zbl 0529.58034)], and does not go through any Poincaré inequality.
scientific article; zbMATH DE number 2113842