Abstract
Let Omega be an open set in a complete, smooth, non-compact, m-dimensional Riemannian manifold M without boundary, where M satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if Omega has infinite measure, and if Omega has finite heat content H_{Omega }(T) for some T>0, then H_{Omega }(t)<infty for all t>0. Comparable two-sided bounds for H_{Omega }(t) are obtained for such Omega .
Highlights
Let (M, g) be a complete, smooth, non-compact, m-dimensional Riemannian manifold without boundary, and let Δ be the Laplace–Beltrami operator acting in L2(M )
For all x, y ∈ M and all t, s > 0, where dz is the Riemannian measure on M
Where B(x; R) = {y ∈ M : d(x, y) < R}, and d(x, y) denotes the geodesic distance between x and y. It was shown independently in [13] and [14] that M satisfying a volume doubling property and a Poincare inequality is equivalent to M satisfying a parabolic Harnack principle, and is equivalent to the Li-Yau bound (1.6) above
Summary
Let (M, g) be a complete, smooth, non-compact, m-dimensional Riemannian manifold without boundary, and let Δ be the Laplace–Beltrami operator acting in L2(M ). It is well known (see [1,2,3,4]) that the heat equation. Has a unique, minimal, positive fundamental solution pM (x, y; t) where x ∈ M , y ∈ M , t > 0. This solution, the heat kernel for M , is symmetric in x, y, strictly positive, jointly smooth in x, y ∈ M and t > 0, and it satisfies the semigroup property pM (x, y; s + t) = dz pM (x, z; s)pM (z, y; t), M (1.2).
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