Stability of parabolic Harnack inequalities on metric measure spaces

By Martin T. Barlow, Richard F. Bass, Takashi Kumagai. 2006

160 citations recorded by the source corpus.

Let $(X,d,\mu)$ be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent $\beta\ge 2$ to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d2d78196f422d952a82e2ac912889e402cb6fbc0

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Citations in the corpus

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, background): “ons on metric measure spaces by Sturm [St] and to random walks on graphs by Delmotte [De]. It has been further extended to symmetric anomalous diﬀusions on metric measure spaces including fractals in [BBK1]. In this paper, we consider the stability of parabolic Harnack inequalities for symmetric non”
Stability of elliptic Harnack inequality (2016, background): “r the name rough isometry) for metric spaces; in both cases they just required the conditions (7.1) and (7.2). The condition (7.3) is a natural extension when one treats measure spaces – see [CS] and [BBK]. If two spaces are roughly isometric then they have similar large scale structure. However, as”
Two-sided estimates of heat kernels of jump type Dirichlet forms (2018, background): “The third condition was introduced for the first time by Barlow, Bass and Kumagai in [6].”
Energy inequalities for cutoff functions and some applications (2012, methodology): “x,y) ≤ c 4|x− y|dw−df if |x− y| > 1. (c) The conditions CS(Ψ) and CSA(Ψ) hold for (X,E). Proof. (a) is proved in [BB2, Theorem 6.9], and (b) in [BB2, Corollary 6.10]. That CS(Ψ) holds follows from [BBK]. CSA(Ψ) then follows by Lemma 5.7, or alternatively by Theorem 1.12. Let a(x) > 0, x ∈ X be”
Stability of heat kernel estimates for symmetric non-local Dirichlet forms (2016, background): “d smoothing the connection to make it a manifold. One can naturally construct a Brownian motion on the surfaces of cylinders. Using the stability of heat kernel estimates like (6.3) (see for instance [BBK1] for details), one can show that any divergence operator L = P 2 i,j=1 ∂ ∂xi (aij(x) ∂ j ) in ”
Heat Kernels on Metric Measure Spaces (2014, background): “On the other hand, (PHI) is equivalent to Poincaré inequality + cutoff Sobolev inequality, see [8].”
On the equivalence of parabolic Harnack inequalities and heat kernel estimates (2012, unknown): “Then, for any t ∈ (0, T ), u(t, ·) ≥ PΩt f in Ω. (2.5)”
TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES (2012, background): “4 ) was proved by Barlow and Perkins [12] on SG, by Kumagai [48] on nested fractals, by Fitzsimmons, Hambly and Kumagai [19] on affine nested fractals and by Barlow and Bass on SC [4] and on generalized Sierpinski carpets [5] (see also [2, 7, 44, 47, 50]).”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “7 is stable under rough isometries in the sense of Barlow-Bass-Kumagai [17, 18, 22] and is a part of the general theory of fractional diffusions [10, 12, 13, 15, 16, 19, 20, 38, 50, 51, and references therein].”