Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds
The main results of the paper are some precise estimates for the gradient of the heat kernel on certain non-compact manifolds and boundedness of the Riesz transforms in spaces \(L^p\). Here is a brief summary of the paper. Let \(M\) be a Riemannian manifold, with cocompact, smooth, proper and free action of a discrete, finitely generated nilpotent (or, more generally, of polynomial growth) group \(G\); assume also that \(G\) acts on \(M\) by isometries. Let \(H\) be the positive Laplacian on \(M\), \(K(t,x,y)\) -- the corresponding heat kernel. Let \(\nabla\) be the gradient, \(d(x, y)\) -- the Riemannian distance, and \(V(x, r)\) -- the volume of the ball on \(M\). The main result of the paper is the following estimate: \[ |\nabla_x K(t,x,y)|\leq ct^{-1/2}V(x, t^{1/2})^{-1}\exp(- bd(x,y)^2/t), \] for some constants \(c\) and \(b\). The author also proves that \(\|\nabla f\|_p\leq c_p\| H^{1/2}f\|_p\) for some constant \(c\) and \(1< p<\infty\), thus establishing boundedness of the Riesz transform \(\nabla H^{-1/2}\) in spaces \(L^p\). The main idea of the proof is to consider separately two cases: the case of ``small'' time \((0< t< 1)\) heat propagation, which is not related to the group action and can be established using estimates in a (relatively compact) fundamental domain \(X\subseteq M\), and the case of ``large'' time \((t\geq 1)\) heat propagation, which can be related to the group action, and treated as the case of operators acting in the space \(L^2(G;L^2(X))\) using some elementary relations between ball volumes in \(M\) and the order of polynomial growth of the group \(G\), and also some earlier estimate obtained by the author for the length function on nilpotent groups. For this second case the author establishes some lovely estimates for the discrete oscillation of the gradient of the heat kernel. The proof of the main theorems is obtained by combining estimates for these two cases.
scientific article; zbMATH DE number 2133506