Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces   Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in
[1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and
$\supp \mu=X$. We prove that the Li-Yau inequality for the heat flow holds true
on $(X,d,\mu)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack
inequalities for the heat flows are established on $(X,d,\mu)$ for general
$K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.

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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph