On a Riemannian manifold $(M^n,g)$, let $H_{B,t}^D$ be the heat flow semigroup associated with $\Delta$ and the Dirichlet boundary condition on a ball $B$, and let $h_{B,t}^D$ be its kernel. Let $h_t$ be the heat kernel on the whole manifold $M$. By which properties of heat kernel can we obtain the inequality $$h_{B,t}^D(z,y) \le h_t(z,y) \le h_t^{1/2}(z,z)h_t^{1/2}(y,y)$$?
2026-03-28 14:05:40.1774706740
Properties of heat flow semigroup
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