# Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0$ such that $$\frac{c}{t^{n/2}} e^{-\frac{1}{4t}d(x, y)^2} \leq p_t(x, y) \leq \frac{C}{t^{n/2}} e^{-\frac{1}{4t}d(x, y)^2}$$ uniformly for all $t \in (0, T]$ and $d(x, y)$ small.

Now, consider a ball $B:= B(p, r)$ on a compact Riemannian manifold $M$, and let $p_B(t, x, y)$ be the Dirichlet heat kernel corresponding to $B$. My question is:

Are there known two-sided bounds (like the above Gaussian bounds) on $p_B(t, x, y)$?

• Could you please give me a reference where the above double inequality is stated? This is exactly what I was looking for. Aug 25, 2017 at 17:32
• As far as I know, the two-sided estimate of the heat kernel given in this question are generally believed to be false. In the case of the sphere, a lovely article Sharp estimates of the spherical heat kernel by A. Nowak, P. Sjögren and T.Z. Szarek, DOI:10.1016/j.matpur.2018.10.002 gives an exact bound of the heat kernel which is different from what is written in the question. Jul 6, 2020 at 21:36

The answer to the question is yes for the upper bound (albeit with different constants in the gaussian factor), and no for the lower bound.

For the upper bound, and for any domain $$\Omega \subset M$$, by domain monotonicity of the heat kernel (e.g. cf. [1, Exercise 7.40]), you have

$$p_\Omega(t,x,y) \leq p(t,x,y), \qquad \forall x,y \in B$$

and at this point just use the gaussian upper bound for the heat kernel $$p(t,x,y)$$ of the whole manifold $$M$$, e.g. the famous Li-Yau estimate [3, Thm. 3.2]. Here one should be careful though as it seems that you are asking for a precise gaussian factor $$\exp(-d^2(x,y)/4t)$$ which is in general not possible to achieve globally, as these estimate will fail at the cut locus, but of course one can prove global estimates with a gaussian factor, such as

$$p(t,x,y) \leq \frac{C}{t^{n/2}} e^{-\frac{d^2(x,y)}{(4+\epsilon)t}}$$

for some $$\epsilon >0$$, cf. the aforementioned Li-Yau reference.

For the lower bound, if e.g. $$x$$ is a point at the boundary of your ball, then $$p_B(t,x,\cdot)=0$$, so you cannot have a lower bound as you ask. However, if we fix the central point of the ball, that there is a classical results by Debiard-Gaveau-Mazet and Cheeger-Yau which might be what you need. It says that if $$o\in M$$, $$B=B_R(o)$$ and $$k$$ is a lower bound for the Ricci curvature on $$B$$, that is $$\mathrm{Ric}(B)\geq k(n-1)$$, then

$$p_B(t,o,y) \geq Q(t,d(o,y))$$

where $$Q$$ denotes the heat heat kernel on a ball of radius R of the simply connected n-dimensional manifold with constant curvature equal to k. You can find a statement and references in [2]

[1] Grigor’yan, Alexander, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-4935-4/hbk). xvii, 482 p. (2009). ZBL1206.58008.

[2] Chavel, Isaac, Eigenvalues in Riemannian geometry. With a chapter by Burton Randol. With an appendix by Jozef Dodziuk, Pure and Applied Mathematics, 115. Orlando etc.: Academic Press, Inc. XIV, 362 p. {$} 62.00; \textsterling 46.50 (1984). ZBL0551.53001. [3] Li, Peter; Yau, Shing Tung, On the parabolic kernel of the Schrödinger operator, Acta Math. 156, 154-201 (1986). ZBL0611.58045. I think you will find what you need here: http://ac.els-cdn.com/002212369090106U/1-s2.0-002212369090106U-main.pdf?_tid=c2567084-37e6-11e6-a2ed-00000aab0f6b&acdnat=1466537672_d9cacfc968cbe3fc89394cbabb17f2c8 • The article that you are pointing to is about the heat kernel on open subsets of$\Bbb R^n\$, while the OP asks about the heat kernel of balls in Riemannian manifolds. The presence of the curvature changes a lot of things, such that your reference is not usable in this setting. Aug 25, 2017 at 17:30

This article contains upper and lower bounds which, if I understand correctly, apply to this case; however the bounds are different from the ones you wrote when $$x$$ or $$y$$ is within a distance of order $$\sqrt{t}$$ from the boundary of $$B$$.