GAUSSIAN UPPER BOUNDS FOR THE HEAT KERNEL

ON ARBITRARY MANIFOLDS

Alexander Grigor’yan^{†}

1. Introduction

In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word ”Gaussian” we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in IR^{n} :

p(x, y, t) = \frac{1}{(4πt)^{n/2}} exp\{−\frac{|x − y|^{2}}{4t} \}

The history of the heat kernel Gaussian estimates started with the works of Nash [25] and Aronson [2] where the double-sided Gaussian estimates were obtained for the heat kernel of a uniformly parabolic equation in IR^{n} in a divergence form (see also [15] for improvement of the original Nash’s argument and [26] for a consistent account of the Aronson’s results and related topics). In particular, the Aronson’s upper bound for the case of time-independent coefficients which is of interest for us reads as follows:

p(x, y, t) ≤ \frac{const}{t^{n/2}} exp\{−\frac{|x − y|^{2}}{Ct} \}

where C is a large enough constant.

In a series of works of Gushin [21], [22] he extended the Gaussian upper bounds to parabolic equations in unbounded domains in IR^{n} with the Neumann boundary condition.

As far as Riemannian manifolds are concerned, the heat kernel Gaussian upper bound first appeared in the work of Cheng, Li and Yau [7] for the case of complete manifolds of a bounded sectional curvature, which was extended soon by a different method in [6] to manifolds with bounded geometry. The most advanced and sharp results under the curvature assumptions were obtained by Li and Yau [24] by using their famous gradient estimates.

Given a Riemannian manifold M, one considers the associated Laplace operator Δ, its (minimal) heat kernel p(x, y, t), and expects to have a Gaussian upper bound as follows

p(x, y, t) ≤ \frac{const}{f(t)} exp\{−\frac{r^{2}}{Ct}\}  (1.1)

where r = dist(x, y) is a geodesic distance between x, y, and f(t) is some increasing function. In the works, cited above, such estimate was shown to be true on certain manifolds

† The author is supported by the EPSRC Fellowship B/94/AF/1782

Grigor’yan A., Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. , 45 (1997) 33-52. subject to curvature restrictions, with the constant C arbitrarily close to the ideal value 4.

The next crucial step was done by B.Davies who developed in a series of his works [9], [10], [11], [12] a powerful abstract method which enabled him to deduce the heat kernel Gaussian upper bounds from the log-Sobolev inequality. This method is robust in contrast to those based on Riemannian curvature. For example, it is invariant under a quasi-isometric transformation of the metric.

An alternative robust method based on a Faber-Krahn type inequalities was introduced by the author in [17] (see also [18], [19]). In particular, it was shown in [17] that any complete manifolds admits the estimate

p(x, y, t) ≤ φ(x, t)φ(y, t) exp\{−\frac{r^{2}}{Ct}\}

(where the function φ(x, t) is expressed in geometric terms) which suggests that the Gaussian exponential factor has a non-geometric nature.

The common achievement of the the upper bound works is understanding that the Gaussian upper bound (1.1) is virtually equivalent to a (logically) weaker on-diagonal bound

p(x, y, t) ≤ \frac{const}{f(t)}  (1.2)

(which does not take into account the distance between x, y). Indeed, let first f(t) = t^{n/2}, where n = dim M. Then the fact that (1.2) is true for all x, y ∈ M and for all t > 0 is equivalent to each of the following functional inequalities 1^{∘} a proper Sobolev inequality as proved by Varopoulos [28]; 2^{∘} a Nash type inequality by Carlen, Kusuoka, Stroock [3]; 3^{∘} a log-Sobolev inequality by Davies [9]; 4^{∘} a Faber-Krahn type inequality by [17] and by Carron [4]. On the other hand, each of these functional inequalities implies also the Gaussian upper bound (1.1) with the same function f(t). See [5] for a more geometric approach based on modified isoperimetric constants. See [23], [13] for the setting of graphs, [8], [29], [30] for Lie groups, and [1] for symmetric spaces.

The fact that an on-diagonal upper bound implies a Gaussian one was extended to more general class of functions f(t) (including those of superpolynomial growth) by Davies [12] and by the author [17], again by using a bridging functional inequality.

At the same time, there is a direct way of deducing a Gaussian upper bound from an ondiagonal one which appeared first in the work of Ushakov [27] for the case of a polynomial function f(t) and for the parabolic equation in the Euclidean space. This method was adapted later for manifolds in [16] but still within a polynomial setting.

The main purpose of the present paper is to extend this method to a wider class of the functions f(t) including sub- and superpolynomially growing functions. The main result is the following theorem.

Theorem 1.1 Let x, y be two points on an arbitrary smooth connected Riemannian manifold M, and let us have for all t ∈ (0, T) (where T may be equal to ∞ or be a positive number)

p(x, x, t) ≤ \frac{1}{f(t)}  (1.3)

and

p(y, y, t) ≤ \frac{1}{g(t)}  (1.4)

where f, g are regular in some sense functions (seeSection 2 below for the definition). Then for any C > 4, for some δ = δ(C) > 0, and all t ∈ (0, T)

p(x, y, t) ≤ \frac{const}{\sqrt{f(δt)g(δt)}} exp\{−\frac{r^{2}}{Ct}\}  (1.5)

where r = dist(x, y). Let us emphasize the fact that unlike the functional-theoretic methods cited above, this theorem assumes the on-diagonal upper bounds only at two points x, y rather than at any point. The regularity condition is wide enough to include such functions as log^{a} t, t^{b}, exp t^{c} and their combinations.

Needless to say that Theorem 1.1 recovers all Gaussian upper bounds obtained previously, and provides a simple way to produce such bounds automatically whenever one has proved a (much simpler) on-diagonal estimate.

2. Integral Estimates of Solutions

Let M denote any smooth connected Riemannian manifold (not necessarily complete), and let Ω be a pre-compact region on M with a smooth boundary. We allow M to have a boundary. If this is the case, then part of the boundary of Ω may be located on ∂M. In fact, \overline{Ω} will be treated as a compact manifold with a boundary.

We consider a function u(x, t) defined on \overline{Ω} × (0, +∞) which is smooth enough and satisfies the following conditions:

\{ \begin{matrix}u(u_{t} − Δu) ≤ 0 \\ u\frac{∂u}{∂ν}|_{x∈∂Ω,t>0} ≤ 0 \\ u|_{x∉K,t=0} = 0 \end{matrix}  (2.1)

where ν is the outward normal vector field on the boundary ∂Ω, K is a compact in Ω (the initial condition is understood in the sense that u(x, t) → 0 as t → 0+ locally uniformly in x ∈ Ω \ K).

For example, u may be a solution to the Dirichlet or Neumann problem for the heat equation in Ω × (0, +∞) (with an initial condition having a support on K) or a positive subsolution, or a negative supersolution.

We will consider two integrals of u :

I(t) = ∫_{Ω} u^{2}(x, t)dx
E_{D}(t) = ∫_{Ω} u^{2}(x, t) exp\{\frac{dist^{2}(x, K)}{Dt}\} dx

where D is a positive number. Of course, we have always

I(t) ≤ E(t).

The main result to be proved here is that to some extend there is a reverse inequality. But before we are able to state that, we have to introduce a technical regularity hypothesis on a function of a single variable.

Definition We say that a function f(t) defined for t ∈ (0, ∞) is regular if

1^{∘} it is positive; 2^{∘} it is monotonically increasing; 3^{∘} there are numbers A ≥ 1 and γ > 1 such that for all 0 < t_{1} < t_{2} the following

inequality holds:

\frac{f(γt_{1})}{f(t_{1})} ≤ A\frac{f(γt_{2})}{f(t_{2})}.  (2.2)

Examples. 1. Let the function f(t) be of at most polynomial growth in the sense that for all t > 0 and some γ > 1

f(γt) ≤ Af(t).  (2.3)

Then (2.2) is obviously true. Indeed, we have

\frac{f(γt_{1})}{f(t_{1})} ≤ A ≤ A\frac{f(γt_{2})}{f(t_{2})}  (2.4)

because f(γt_{2}) ≥ f(t_{2}). Examples of the functions satisfying (2.3) are: f(t) = t^{n}, f(t) = log^{n}(1 + t) (where n > 0) etc.

2. Let f(t) be of at least polynomial growth in the sense that for some γ > 1 the quotient

\frac{f(γt)}{f(t)}  (2.5)

is increasing in t. Then (2.2) holds again at this time with A = 1 because by monotonicity of (2.5)

\frac{f(γt_{1})}{f(t_{1})} ≤ \frac{f(γt_{2})}{f(t_{2})}  (2.6)

Examples of such functions are: f(t) = t^{n}, f(t) = e^{t^{n}} etc.

3. Let us combine the two situations above: suppose that there is some T > 0 such that for all t < T the inequality (2.3) holds while for t > T the ratio (2.5) is increasing. Then f(t) is regular again. Indeed, in order to check (2.2) , let us consider two cases: t_{1} < T and t_{1} ≥ T. In the first case, we have again (2.4) , while in the second case we repeat (2.6) .

An example of a function which fits this case is:

f(t) = \{ \begin{matrix}c_{1}t^{n} & , t < T \\ c_{2}e^{t^{m}} & , t > T \end{matrix}

where n > 0, m > 0, and the constants c_{1,2} are chosen to ensure continuity (and, therefore, also monotonicity) of f(t).

Now we can state our main technical result.

Theorem 2.1 Let us suppose that u(x, t) satisfies (2.1) , and for any t > 0 we have

I(t) ≤ \frac{1}{f(t)}  (2.7)

where the function f(t) is regular as above. Then for any D > 2 and for all t > 0

E_{D}(t) ≤ \frac{4A}{f(δt)}

where δ = δ(D, γ) > 0.

Proof of the theorem. The proof will consist of three steps. In the first step, we will estimate the integral

I_{R}(t) = ∫_{Ω\B(K,R)} u^{2}(x, t)dx

where B(K, R) is the open R−neighbourhood of the set K. In the second step, we will estimate E_{D}(t) for large D applying the upper bounds for I_{R}(t), and, finally, in the third step, we will finish the proof for all D > 2.

STEP 1. Let us prove the following key lemma. The statement of this kind seems to have appeared for the first time in the paper of Ushakov [27] for the case of a polynomial function f(t) and in the Euclidean space. We modified his approach and made it work for a more general function f(t).

Lemma 2.2 Under the hypotheses of Theorem 2.1, there exists D_{0} = D_{0}(γ) > 2 such that

I_{R}(t) ≤ \frac{2A}{f(t/γ)} exp\{−\frac{R^{2}}{D_{0}t}\}

for all R > 0 and t > 0 (where the constants A, γ are those from the regularity hypothesis (2.2) ).

Proof of lemma 2.2. The idea of the proof is to compare the quantities I_{R}(t) and I_{ρ}(τ) for ρ < R, τ < t in the following way:

I_{R}(t) ≤ I_{ρ}(τ) + \frac{1}{f(τ)} exp\{−\frac{(R − ρ)^{2}}{2(t − τ)} \} .  (2.8)

After we have shown (2.8) ,we will arrange sequences {R_{k}}, {t_{k}} (where k = 0, 1, 2, ...) which start with R and t respectively and are decreasing so that R_{k} → R/2 and t_{k} → 0 as k → ∞. Applying (2.8) to the consecutive pairs (R_{k}, t_{k}) and (R_{k+1}, t_{k+1}) and summing up all those inequalities, we obtain an upper bound for I_{R}(t) in terms of a series which can be dealt with by taking specific sequences R_{k}, t_{k}.

We will finish this argument later but first we turn to the proof of (2.8) . We apply the integral maximum principle, which states the following.

Proposition 2.3 If u(x, t) is smooth enough in \overline{Ω} × (0, T) and satisfies the conditions

\{ \begin{matrix}u(u_{t} − Δu) ≤ 0 \\ u\frac{∂u}{∂ν}|_{x∈∂Ω,t∈(0,T)} ≤ 0 \end{matrix}  (2.9)

then the integral

∫_{Ω} u^{2}(x, t)e^{ξ(x,t)}dx  (2.10)

is a decreasing function of t ∈ (0, T) provided the function ξ(x, t) is Lipschitz in \overline{Ω} × (0, T) and satisfies the inequality

ξ_{t} + \frac{1}{2} |∇ξ|^{2} ≤ 0.  (2.11)

This property of solutions to the heat equation is well known and goes back to the famous Aronson’s paper [2]. Thereafter, it was proved for different settings in various works (see, for example, [7], [26], [18]). The actual proof is very simple and consists of taking the time derivative of the integral (2.10) and of applying integration by parts:

\frac{d}{dt} ∫_{Ω} u^{2}e^{ξ} = ∫_{Ω} ξ_{t}u^{2}e^{ξ} + ∫_{Ω} 2uu_{t}e^{ξ}

≤ ∫_{Ω} ξ_{t}u^{2}e^{ξ} + ∫_{Ω} 2uΔue^{ξ}

≤ ∫_{Ω} ξ_{t}u^{2}e^{ξ} + ∫_{∂Ω} 2u\frac{∂u}{∂ν}e^{ξ} − 2 ∫_{Ω}(∇(ue^{ξ}), ∇u)

≤ −\frac{1}{2} ∫_{Ω} |∇ξ|^{2} u^{2}e^{ξ} − 2 ∫_{Ω} u(∇ξ, ∇u)e^{ξ} − 2 ∫_{Ω} |∇u|^{2} e^{ξ}

= −\frac{1}{2} ∫_{Ω}(u∇ξ + 2∇u)^{2}e^{ξ} ≤ 0.

⊓ ⊔

Let us note that we have used here at full strength the hypotheses (2.9) and (2.11) on the functions u and ξ respectively. Moreover, this is the only place where we need (2.9) and (2.11) . In the proof of Theorem 2.1 which follows, we will apply the two first conditions from (2.1) (”the equation” and ”the boundary condition”) only via Proposition 2.3. On the contrary, we will use the initial condition of (2.1) explicitly.

We will be applying (2.11) with different functions ξ. Let us note that any function of the form

ξ(x, t) = \frac{d(x)}{D(t − s)}

fits (2.11) provided d(x) is a distance function to some subset of M, D ≥ 2, and the point s does not belong to (0, T) (to ensure no singularities).

In order to prove (2.8) , we choose some s, T such that s > T > t, and put d(x) to be the distance to the exterior of the set B(K, R), i.e.

d(x) = \{ \begin{matrix}R − dist(x, K) & , x ∈ B(K, R) \\ 0 & , x ∉ B(K, R) \end{matrix} .

By the integral maximum principle, we have

∫_{Ω} u^{2}(x, t) exp\{−\frac{d(x)^{2}}{2(s − t)}\} ≤ ∫_{Ω} u^{2}(x, τ) exp\{−\frac{d(x)^{2}}{2(s − τ)}\}

Now we replace the left hand side integral by a less value I_{R}(t) because d(x) vanishes off the set B(K, R). Then, we split the right hand side integral into two parts: over interior of B(K, ρ) and over its exterior. In the first part, the exponential weight is bounded from above by

exp\{−\frac{(R − ρ)^{2}}{2(s − τ)} \}

since the distance from any point of B(K, ρ) to the exterior of B(K, R) is at least R − ρ. In the second part, we replace the exponential weight simply by the larger 1, obtaining, thus, I_{ρ}(τ). Therefore, we have

I_{R}(t) ≤ exp\{−\frac{(R − ρ)^{2}}{2(s − τ)} \} ∫_{B(K,ρ)} u^{2}(x, τ)dx + I_{ρ}(τ).

Finally, we apply the hypothesis (2.7) in the form

∫_{B(K,ρ)} u^{2}(x, τ)dx ≤ \frac{1}{f(τ)}

and let s → t+ whence (2.8) follows.

Given R and t, we consider the sequences {R_{k}} and {t_{k}}, k = 0, 1, 2, ... such that 1^{∘} {R_{k}} and {t_{k}} are decreasing in k; 2^{∘} R_{0} = R and R_{k} → \frac{1}{2}R as k → ∞; 3^{∘} t_{0} = t and t_{k} → 0 as k → ∞. Applying (2.8) for the consecutive pairs (R_{k}, t_{k}) and (R_{k+1}, t_{k+1}) we obtain

I_{R_{k}}(t_{k}) ≤ I_{R_{k+1}}(t_{k+1}) + \frac{1}{f(t_{k+1})} exp\{−\frac{(R_{k} − R_{k+1})^{2}}{2(t_{k} − t_{k+1})} \}  (2.12)

Let us note that I_{R_{k}}(t_{k}) → 0 as k → ∞. Indeed,

lim_{k→∞} I_{R_{k}}(t_{k}) = ∫_{Ω\B(K, \frac{1}{2} R)} u^{2}(x, 0)dx = 0

because of the initial condition (2.1) for the function u.

Hence, we can sum up the inequalities (2.12) over all k = 0, 1, 2, ... and obtain

I_{R}(t) ≤ ∑_{k=0}^{∞} \frac{1}{f(t_{k+1})} exp\{−\frac{(R_{k} − R_{k+1})^{2}}{2(t_{k} − t_{k+1})} \} .  (2.13)

Let us specify the sequences R_{k}, t_{k} in the following way: we take

R_{k} = (\frac{1}{2} + \frac{1}{k + 2})R

and

t_{k} = t/γ^{k}.

Since

R_{k} − R_{k+1} ≥ \frac{R}{(k + 3)^{2}}

and

t_{k} − t_{k+1} = \frac{(γ − 1)t}{γ^{k+1}}

we can expand (2.13) as follows

I_{R}(t) ≤ ∑_{k=0}^{∞} \frac{1}{f(t_{k+1})} exp\{−\frac{γ^{k+1}}{(k + 3)^{4}(γ − 1)} \frac{R^{2}}{2t} \} .

Let us estimate f(t_{k+1}) as follows. According to the property of the function f(·) to be regular, we have the following sequence of inequalities:

\frac{f(t_{k})}{f(t_{k+1})} ≤ A\frac{f(t_{0})}{f(t_{1})}
\frac{f(t_{k−1})}{f(t_{k})} ≤ A\frac{f(t_{0})}{f(t_{1})}
...
\frac{f(t_{0})}{f(t_{1})} ≤ A\frac{f(t_{0})}{f(t_{1})}

Multiplying all them we derive

\frac{f(t_{0})}{f(t_{k+1})} ≤\{A\frac{f(t_{0})}{f(t_{1})}\}^{k+1}

whence we get

I_{R}(t) ≤ \frac{1}{f(t)} ∑_{k=0}^{∞} exp\{(k + 1) log A\frac{f(t_{0})}{f(t_{1})} − \frac{γ^{k+1}}{(k + 3)^{4}(γ − 1)} \frac{R^{2}}{2t} \} .  (2.14)

The main idea of the proof is that the numerator γ^{k+1} grows in k much faster than the denominator (k + 3)^{4} whenever γ > 1. In particular, there exists a positive number m = m(γ) such that

\frac{γ^{k+1}}{(k + 3)^{4}(γ − 1)} ≥ m(k + 2)

for any k ≥ 0. We can just take

m = inf_{k≥0} \frac{γ^{k+1}}{(k + 3)^{4}(k + 2)(γ − 1)}.

Let us denote for simplicity L = log A\frac{f(t_{0})}{f(t_{1})} and rewrite the inequality (2.14) as follows

I_{R}(t) ≤ \frac{1}{f(t)} ∑_{k=0}^{∞} exp\{(k + 1)L − m(k + 2)\frac{R^{2}}{2t} \}

= \frac{1}{f(t)} exp\{−m\frac{R^{2}}{2t} \} ∑_{k=0}^{∞} exp\{−(k + 1)(m\frac{R^{2}}{2t} − L)\}

We have either

m\frac{R^{2}}{2t} − L ≥ log 2

or

m\frac{R^{2}}{2t} − L < log 2.

In the former case, we obtain obviously

I_{R}(t) ≤ \frac{1}{f(t)} exp\{−m\frac{R^{2}}{2t} \} ∑_{k=0}^{∞} 2^{−(k+1)} = \frac{1}{f(t)} exp\{−m\frac{R^{2}}{2t} \}  (2.15)

while in the latter case we estimate I_{R}(t) in a different way:

I_{R}(t) ≤ I(t) ≤ \frac{1}{f(t)}

≤ \frac{1}{f(t)} exp\{L + log 2 − m\frac{R^{2}}{2t} \}

= \frac{2}{f(t)}A\frac{f(t)}{f(t/γ)} exp\{−m\frac{R^{2}}{2t} \}

= \frac{2A}{f(t/γ)} exp\{−m\frac{R^{2}}{2t} \}

Combining this together with (2.15) , we obtain finally for both cases:

I_{R}(t) ≤ \frac{2A}{f(t/γ)} exp\{−m\frac{R^{2}}{2t} \}

⊓ ⊔

STEP 2. The purpose of this part of the proof is to show that for D > D_{1} ≡ 5D_{0} and for all t > 0 we have

E_{D}(t) ≤ \frac{4A}{f(t/γ)}.  (2.16)

To that end, we split the integral

E_{D}(t) = ∫_{Ω} u^{2}(x, t) exp\{\frac{r^{2}(x)}{Dt}\} dx

(where r(x) ≡ dist(x, K)) into a series

\begin{matrix}E_{D}(t) = & ∫_{\{r(x)≤R\}} u^{2}(x, t) exp\{\frac{r^{2}(x)}{Dt}\} dx \\ + & ∑_{k=0}^{∞} ∫_{\{2^{k}R≤r(x)≤2^{k+1}R\}} u^{2}(x, t) exp\{\frac{r^{2}(x)}{Dt}\} dx \end{matrix}  (2.17)

where R > 0 is an arbitrary number.

The first integral on the right hand side (2.17) is bounded from above by

exp\{\frac{R^{2}}{Dt}\} ∫_{Ω} u^{2}(x, t)dx ≤ \frac{1}{f(t)} exp\{\frac{R^{2}}{Dt}\} .  (2.18)

The k-th term in the sum (2.17) is estimated from above by using Lemma 2.2 as

≤ exp\{\frac{4^{k+1}R^{2}}{Dt} \} ∫_{Ω\B(K,2^{k}R)} u^{2}(x, t)dx

≤ \frac{2A}{f(t/γ)} exp\{\frac{4^{k+1}R^{2}}{Dt} − \frac{4^{k}R^{2}}{D_{0}t} \}

≤ \frac{2A}{f(t/γ)} exp\{−\frac{4^{k}R^{2}}{Dt} \}

(2.19)

where we have used D_{0} ≤ D/5.

Combining (2.18) and (2.19) , we obtain

E_{D}(t) ≤ \frac{1}{f(t)} exp\{\frac{R^{2}}{Dt}\} + \frac{2A}{f(t/γ)} ∑_{k=0}^{∞} exp\{−\frac{4^{k}R^{2}}{Dt} \}  (2.20)

We can choose R here arbitrarily. Let us define R to satisfy the identity R^{2}/Dt = log 2 and deduce from (2.20) :

E_{D}(t) ≤ \frac{2}{f(t)} + \frac{2A}{f(t/γ)} ∑_{k=0}^{∞} 2^{−4^{k}} ≤ \frac{2 + 2A}{f(t/γ)}

whence (2.16) follows since A ≥ 1.

Remark. By taking another (more optimal) value for R, namely,

R^{2} = Dt log(1 + \sqrt{2A})

we could replace the coefficient 2 + 2A in the formula above by a better value 1 + 2\sqrt{2A}.

STEP 3. Now we will finish the proof of Theorem 2.1. In view of the previous step, it suffices to consider the case 2 < D < D_{1}. The integral maximum principle (Proposition 2.3) implies that for any s > 0 the integral

∫_{Ω} u^{2}(x, t) exp\{\frac{r^{2}(x)}{2(t + s)}\} dx

is decreasing in t ∈ (0, ∞) (where r(x) = dist(x, K)). Therefore, for any τ ∈ (0, t)

∫_{Ω} u^{2}(x, t) exp\{\frac{r^{2}(x)}{2(t + s)}\} dx ≤ ∫_{Ω} u^{2}(x, τ) exp\{\frac{r^{2}(x)}{2(τ + s)}\} dx.  (2.21)

Given t > 0 and D, 2 < D < D_{1}, let us find suitable values of s, τ so that the left hand side of (2.21) is equal to E_{D}(t) while the right hand side is to be equal to E_{D_{1}}(τ). To that end, we must solve simultaneously the equations

\{ \begin{matrix}2(t + s) = Dt \\ 2(τ + s) = D_{1}τ \end{matrix}

which yields s = \frac{D−2}{2}t and τ = \frac{D−2}{D_{1}−2}t < t. Therefore, for this value of τ, we have

E_{D}(t) ≤ E_{D_{1}}(τ),

and applying the inequality

E_{D_{1}}(τ) ≤ \frac{4A}{f(τ/γ)}

known from the previous step of the proof, we get finally

E_{D}(t) ≤ \frac{4A}{f(\frac{D−2}{D_{1}−2}t/γ)}.

Thus, we have proved Theorem 2.1 with

δ = δ(D, γ) = γ^{−1} min\{1, \frac{D − 2}{D_{1} − 2}\} .

⊓ ⊔

3. Pointwise Estimates of the Heat Kernel

On any smooth connected Riemannian manifold M, we define the heat kernel p(x, y, t) as the smallest positive fundamental solution to the heat equation. It exists, it is unique, and it can be constructed as follows. Let us take an increasing sequence of pre-compact regions Ω_{k} ⊂ M, k = 1, 2, 3, ... which exhausts M, and in each Ω_{k} construct the Green function p_{k}(x, y, t) to the Dirichlet problem for the heat equation. Then, on one hand, we have by the maximum principle

0 ≤ p_{k} ≤ p_{k+1}

while on the other hand

∫_{Ω_{k}} p_{k}(x, y, t)dx ≤ 1.

These two properties ensure that there is a limit

p(x, y, t) ≡ lim_{k→∞} p_{k}(x, y, t)

which is, by definition, the (minimal) heat kernel (see [14] for detailed justification of this construction).

If the manifold M has a boundary, then the exhausting regions Ω_{k} will necessarily have for large k a part of their boundary on ∂M, so one can put a boundary condition (normally Dirichlet or Neumann one) on ∂M to be satisfied by all p_{k} and, therefore, by p(x, y, t).

Our main result is the following theorem.

Theorem 3.1 Let x, y be two points on an arbitrary manifold M, and let us have for all t > 0

p(x, x, t) ≤ \frac{1}{f(t)}  (3.1)

and

p(y, y, t) ≤ \frac{1}{g(t)}  (3.2)

where f, g are regular functions in the sense of the previous section. Then for any C > 4 and all t > 0

p(x, y, t) ≤ \frac{4A}{\sqrt{f(δt)g(δt)}} exp\{−\frac{r^{2}(x, y)}{Ct}\}  (3.3)

where r(x, y) = dist(x, y), δ = δ(C, γ), and A, γ are the constants from (2.2) .

Remark. The theorem is applicable also if the inequalities (3.1) and (3.2) hold only on a bounded time interval (0, T) as stated in Theorem 1.1 in the Introduction (with an obvious modification of the notion of regularity for a bounded interval). Indeed, the on-diagonal heat kernel p(x, x, t) is known to a decreasing function of t. Therefore, if we extend the functions f(t) and g(t) beyond the point T as constants then (3.1) and (3.2) will be valid for all t > 0. Moreover, it is evident that the extended functions will preserve regularity, so that we can apply Theorem 3.1 and obtain that (3.3) holds, in particular, for all t ∈ (0, T). Proof of Theorem 3.1. Let us apply the following universal inequality which is true always:

p(x, y, t) ≤ \sqrt{E_{D}(x, t/2), E_{D}(y, t/2)} exp\{−\frac{r^{2}(x, y)}{2Dt}\}  (3.4)

where D is a positive constant and

E_{D}(z, t) ≡ ∫_{M} p^{2}(z, ζ, t) exp\{\frac{r^{2}(z, ζ)}{Dt}\} dζ.

This inequality was proved in [17] but the proof is very short so that we can reproduce it here for the sake of completeness. Indeed, we have by the semigroup property of the heat kernel and by the triangle inequality r^{2}(x, y) ≤ 2(r^{2}(x, z) + r^{2}(y, z)) :

p(x, y, t) = ∫_{M} p(x, z, t/2)p(z, y, t/2)dz

≤ ∫_{M} p(x, z, t/2) exp\{\frac{r^{2}(x, z)}{Dt}\} p(z, y, t/2) exp\{\frac{r^{2}(y, z)}{Dt}\} exp\{−\frac{r^{2}(x, y)}{2Dt}\} dz

≤ exp\{−\frac{r^{2}(x, y)}{2Dt}\}

×\{∫_{M} p^{2}(x, z, t/2) exp\{\frac{r^{2}(x, z)}{Dt/2}\} dz\}^{\frac{1}{2}}\{∫_{M} p^{2}(y, z, t/2) exp\{\frac{r^{2}(y, z)}{Dt/2}\} dz\}^{\frac{1}{2}}

= exp\{−\frac{r^{2}(x, y)}{2Dt}\}\{E_{D}(x, t/2)E_{D}(y, t/2)\}^{\frac{1}{2}}

what was to be proved.

The rest of the proof of Theorem 3.1 follows from the next theorem.

Theorem 3.2 Let x be a point on an arbitrary manifold M, and let us have for all t > 0

p(x, x, t) ≤ \frac{1}{f(t)}  (3.5)

where f(t) is a regular function. Then for any D > 2 and all t > 0

E_{D}(x, t) ≤ \frac{4A}{f(2δt)}  (3.6)

where δ = δ(D, γ), and A, γ are the constants from (2.2) .

To finish the proof of Theorem 3.1 we are left to notice that (3.1) , (3.2) imply (3.6) and a similar inequality for the point y, which together with (3.4) imply (3.3) (we have to replace in the final result 2D by C). Proof of Theorem 3.2. Let us consider one of the sets Ω_{k} containing the point x. Since p_{k} ≤ p then the inequality (3.5) is valid for p_{k}, too. It is sufficient to show that (3.6) holds for the integral

E_{D,k}(x, t) ≡ ∫_{Ω_{k}} p_{k}^{2}(x, z, t) exp\{\frac{r^{2}(x, z)}{Dt}\} dz

since thereafter we could pass to the limit as k → ∞ and establish the same upper bound for E_{D}(x, t).

Let us apply Theorem 2.1 to estimate E_{D,k}(x, t). Indeed, the function u(z, t) ≡ p_{k}(x, z, t) satisfies the conditions (2.1) with the single-point compact K = {x}, and we have for this function

I(t) = ∫_{Ω} u^{2}(z, t)dz = ∫_{Ω} p_{k}(x, z, t)p_{k}(z, x, t)dz = p_{k}(x, x, 2t) ≤ \frac{1}{f(2t).}

Therefore, by Theorem 2.1 we have for any D > 2

E_{D,k}(x, t) = ∫_{Ω} u^{2}(z, t) exp\{\frac{r^{2}(x, z)}{Dt}\} dz ≤ \frac{4A}{f(2δt)}

what was to be proved. ⊓ ⊔

Theorem 3.2 may have other applications. For example, in conjunction with the result of [20] it can give upper bounds of derivatives of the heat kernel. Indeed, as proved in [20], any upper bound for E_{D}(x, t)

E_{D}(x, t) ≤ \frac{1}{h(t)}

(which is supposed to be true for some x and all t > 0) implies

E_{D}^{(1)}(x, t) ≡ ∫_{M} |∇_{z}p|^{2} (x, z, t) exp\{\frac{r^{2}(x, z)}{Dt}\} dz ≤ \frac{const_{D}}{h^{(1)}(t)}

and

E_{D}^{(2)}(x, t) ≡ ∫_{M} |Δ_{z}p|^{2} (x, z, t) exp\{\frac{r^{2}(x, z)}{Dt}\} dz ≤ \frac{const_{D}}{h^{(2)}(t)}  (3.7)

where

\begin{aligned}h^{(1)}(t) & = ∫_{0}^{t} h(s)ds \\ h^{(2)}(t) & = ∫_{0}^{t} h^{(1)}(s)ds \end{aligned}

and D > 2.

Another result of [20] is an inequality similar to (3.4)

|\frac{∂p}{∂t}| (x, y, t) ≤ \sqrt{E_{D}^{(2)}(x, t/2), E_{D}(y, t/2)} exp\{−\frac{r^{2}(x, y)}{2Dt}\} .  (3.8)

Combinig together the inequalities (3.8) , (3.7) and (3.6) , we obtain the following statement.

Corollary 3.3 Under hypotheses of Theorem 3.1 we have in addition to (3.3) also

|\frac{∂p}{∂t}| (x, y, t) ≤ \begin{matrix}const & A,C,γ \\ √ & \overline{f^{(2)}(δt)g(δt)} \end{matrix} exp\{−\frac{r^{2}(x, y)}{Ct}\} .

provided C > 4. Similar estimates can be proved also for the higher order time derivatives of the heat kernel.

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