Estimations of the heat kernel on conic manifolds and applications
Let \(N\) be a compact Riemannian manifold with dimension \(n-1\). The cone \(C(N)\) is the space \(\mathbb{R}^+\times N\) endowed with the metric \(dr^2+r^2g_N\). Denote by \(\Delta_N\) (resp. \(\Delta\)) the Laplace-Beltrami operator on \(N\) (resp. \(C(N)\)), and by \(p_t\) the heat kernel of \(\Delta\) on \(C(N)\), subject to the Dirichlet boundary condition if \(\partial N\neq \emptyset\). The author gives a pointwise Gaussian upper estimate for \(p_t\) and \(\left|t\partial_tp_t\right|\) for all \(t>0\) when \(n\geq 3\). The proof relies, in particular, on a theorem of \textit{A. Grigor'yan} [J. Differ. Geom. 45, No. 1, 33-52 (1997; Zbl 0865.58042)]. Under some additional assumptions about the eigenvalues of \(-\Delta_N\), a pointwise upper estimate for the gradient of \(p_t\) is given. As an application of these upper bounds, a theorem about multipliers on conic manifolds is proved, analogous to a result of \textit{G. Alexopoulos} [Tohoku Math. J., II. Ser. 46, No. 4, 457-468 (1994; Zbl 0835.22008)]. As a second application, using the theory of singular integrals, the author proves that the Riesz transform \(\nabla\Delta^{-1/2}\) on \(C(N)\) is of weak type \((1,1)\) and \(L^p\)-bounded for all \(1<p\leq 2\). Under additional assumptions about the first eigenvalue of \(-\Delta_N\), the \(L^p\)-boundedness of the Riesz transform for \(2<p<+\infty\) is also obtained. Finally, the Riesz transform on \(C(C(N))\) is also studied.
scientific article; zbMATH DE number 1540337
scientific article