Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

Abstract

We prove the compactness of the imbedding of the Sobolev space W 1,2 0 (Ω) into L 2 (Ω) for any relatively compact open subset Ω of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Hölder continuous heat kernel. Subject Classification (1991): 53C70, 58J35, 58J50, 31C15, 31C25, 35K05, 53C20, 53C23 Mathematics

Omori-Yau Maximum Principle on Alexandrov Spaces

“…For each x ∈ W p,t , such a point y is unique and we set Φ p,t (x) := y. Definition 2.9 (Infinitesimal Bishop-Gromov condition, [6]). An Alexandrov space X is said to satisfy condition BG (κ, n) if for each p ∈ X the following holds:…”

“…For a BV-Riemannian metric (g ij ) on Reg(X), recall that D k g ij means distributional derivative and |D k g ij | means the total variation measure of D k g ij . Theorem 2.10 (Kuwae-Shioya, [6]). Let X be an Alexandrov space of dimension n ≥ 2.…”

Untitled

“…The measure theoretic conclusions in Theorem 11.1 can be found in [40,Chapter 10], and a proof of the Poincaré inequality in [146] (see also [120]). …”

The Poincaré inequality is an open ended condition

“…Thus by Theorem 1.0.1, each of these conditions is also equivalent to supporting a doubling measure and a (1, 2 − ε)-Poincaré inequality for some ε > 0, quantitatively. Relations between (1, 2)-Poincaré inequalities, heat kernel estimates, and parabolic Harnack inequalities have been established in the setting of Alexandrov spaces by Kuwae, Machigashira, and Shioya ( [37]), and in the setting of complete metric measure spaces that support a doubling Radon measure, by Sturm ([48]). Colding and Minicozzi II [10] proved that on complete noncompact Riemannian manifolds supporting a doubling measure and a (1, 2)-Poincaré inequality, the conjecture of Yau is true: the space of harmonic functions with polynomial growth of fixed rate is finite dimensional.…”