Heat Kernel Bounds on Metric Measure Spaces and Some Applications
By Renjin Jiang, Huaiqian Li, Hui-Chun Zhang. 2014
108 citations recorded by the source corpus.
Let (X,d,μ) be a RCD∗(K,N) space with K∈ℝ$K\in \mathbb {R}$ and N∈[1,∞). We derive the upper and lower bounds of the heat kernel on (X,d,μ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the Lp boundedness of (local) Riesz transforms.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/1787128229502add0c0d987171965047e6489929
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In Applied Mathematics
- Elia Brué
- Luigi Ambrosio
- Daniele Semola
- Luca Tamanini
- Nicola Gigli
- Shouhei Honda
- David Tewodrose
- Filippo Santambrogio
Citations in the corpus
- Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “see [36], which adapts the approach of [53], [54] to the RCD setting.”
- Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “In the last few years, research in this area has increased considerably, with contributions by many authors (see, for instance, [6, 8, 17, 20, 33, 39, 51, 65]), which have often given new insights into more classical questions, both of analytical and geometric nature.”
- Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “2) of the heat kernel on RCD∗(K , N )-spaces established in [33], we prove the pointwise convergence of heat kernels defined on a converging sequence of RCD∗(K , N ) spaces, and we deduce from this fact the short-time estimate of the heat kernel on regular sets of RCD∗(K , N ) spaces.”