Heat kernels on metric measure spaces and an application to semilinear elliptic equations

By Alexander Grigorʼyan, Jiaxin Hu, Ka-Sing Lau. 2003

180 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3ae00f3deda7366f0d882ff5cdf8c462aae5dd6f

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces (2009, background): “Of particular interest is the study of functional inequalities, like Sobolev and Poincaré inequalities, on metric measure spaces; see, for example, [29, 15, 20, 12, 11, 27].”
Estimates of heat kernels for non-local regular Dirichlet forms (2014, unknown): “M (u(x)− u(y))(2) pt(x, y)dμ(y)dμ(x), that is true for any u ∈ F and for any t > 0 (see [17]).”
Eigenvalues of elliptic operators and geometric applications (2004, background): “The nature of the parameter β, which is called the walk dimension, is more complicated (see [26] or [25]).”
Two-sided estimates of heat kernels of jump type Dirichlet forms (2018, background): “This implication was proved in [20], although under the additional assumption that (E ,F) is conservative.”
Heat kernels and function theory on metric measure spaces (2003, background): “Most results surveyed here are taken from [18] (these are Theorems 3.”
Heat kernel upper bounds for jump processes and the first exit time (2009, background): “By a result of [12], (a) implies the lower bound for the volume of balls: there exists c > 0 such that cr ≤ μ(B(x, r)), for all x ∈ M, r > 0.”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “e walk dimension, even though it is, strictly speaking, not a dimension of a geometric object. It is possible to prove that if the metric space (X;d) satises a chain condition, then 2 dW dH+1, see [14,37,39]. When dW = 2, one speaks of Gaussian estimates and when dW >2, one speaks then of sub-Gau”
Characterization of sub‐Gaussian heat kernel estimates on strongly recurrent graphs (2005, methodology): “(see the computations in [28], proof of Theorem 3.”
TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES (2012, unknown): “[2], [33], [26], [29]).”
Calderón reproducing formulas and new Besov spaces associated with operators (2012, result): “Some of more recent results on Besov spaces are [49,55,28,16].”