HEAT KERNELS

HEAT KERNELS

NICK DUNGEY

Received 27 September 2004; Accepted 1 March 2005

This paper is concerned with pointwise estimates for the gradient of the heat kernelKt, t >0, of the Laplace operator on a Riemannian manifoldM. Under standard assumptions onM, we show that∇K_t satisfies Gaussian bounds if and only if it satisfies certain uni- form estimates or estimates inL^pfor some 1≤p≤ ∞. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on∇K_tare stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on covering manifolds and on homogeneous spaces of Lie groups of polynomial growth and boundedness of Riesz transform operators.

Copyright © 2006 Nick Dungey. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetMbe a connected, complete Riemannian manifold. Writed(x,y) for the Riemann- ian distance between x,y∈M and dx for the Riemannian measure. We set B(x,r)= {y∈M:d(x,y)< r}andV(x,r) :=dx(B(x,r)) forr >0. LetH= −div∇be the positive Laplace operator, where div and∇are respectively the Riemannian divergence and gradi- ent onM. We consider the heat semigroupS_t=e^−tH,t≥0, which acts inL^p=L^p(M;dx), 1≤p≤ ∞, and the heat kernelKt(x,y) defined fort >0,x,y∈M. We will assume the volume doubling property, that is, there existsc >0 such that

V(x, 2r)≤cV(x,r) (1.1)

for allx∈Mandr >0, and the Gaussian heat kernel upper bound

Kt(x,y)≤cωVy,t^1/2−1e^{−ωd(x,y)2/t} (1.2)

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 73020, Pages1–10 DOI10.1155/AAA/2006/73020

for allω∈(0, 1/4),t >0, andx,y∈M, wherecωis a constant depending onω. The con- junction of assumptions (1.1) and (1.2) is well understood, and it is known that (1.2) is a consequence of the on-diagonal estimateK_t(x,x)≤cV(x,t^1/2)⁻¹,t >0,x∈M(see, e.g., [7,10] and references therein).

In this paper we study certain gradient estimates forKt, which may or may not hold under the above assumptions. Such estimates have intrinsic interest as regularity proper- ties of the heat kernel, but are also closely connected with the boundedness inL^p(M) of Riesz transform operators (see, e.g., [2]).

Adopt the convention that∇K_t(x,y)= ∇xK_t(x,y) denotes the gradient with respect to the first variable of the two variable kernelKt(·,·), and consider the following conditions.

Condition (I). There existc,b >0 such that

∇Kt(x,y)≤ct^−1/2Vy,t^1/2−1e^{−bd(x,y)2/t} (1.3) for allt >0,x,y∈M(here,| · |denotes the Riemannian length of tangent vectors).

Apparently weaker than Condition (I) is Condition (II). There isc >0 such that

∇Kt(x,y)≤ct^−1/2Vy,t^1/2−1 (1.4) for allt >0,x,y∈M.

We also consider an infinite family of conditions indexed byp∈[1,∞].

Condition (III)_p. There isc >0 such that

∇Kt(x,·)_p≤ct^−1/2Vx,t^{1/2−(1−(1/ p))} (1.5) for allt >0 andx∈M.

Observe that Condition (III)_∞says exactly that

∇K_t(x,y)≤ct^−1/2Vx,t^1/2−1 (1.6) for allt >0,x,y∈M, which is a variation of Condition (II). Also note that Condition (III)1is equivalent to an operator norm estimate

∇St

∞→∞≤ct^−1/2 (1.7)

for allt >0, where in general · p1→p2denotes the norm of a bounded operator fromL^p1 toL^p2.

We next state our main theorem.

Theorem 1.1. Assume that (1.1) and (1.2) hold onM. Then Conditions (I), (II), and (III)p

are equivalent, for each p∈[1,∞]. Moreover, if these conditions hold then estimate (1.3) holds for eachb∈(0, 1/4) with somec=c(b)>0 depending onb.

Some known examples ofMwhere (1.1), (1.2), and Condition (I) hold, are (i) mani- folds of nonnegative Ricci curvature (see Li and Yau [8]) and (ii) Galois covering

manifolds of compact manifolds whose deck transformation group is of polynomial growth (see [4]). Analogous estimates are known for subelliptic sublaplacians on Lie groups of polynomial growth (see, e.g., [13]). There are also simple examples of man- ifolds satisfying (1.1) and (1.2) but not Condition (I): see for example [3, Section 5], whereMis formed by glueing two copies ofRⁿ.

The main tool used in provingTheorem 1.1is the finite speed propagation of the wave equation associated withH, and our proof is influenced by arguments of Sikora [10] and ter Elst, Robinson and Sikora [12, Section 2].

As an application ofTheorem 1.1, we get a result about stability of Condition (I) un- der certain mappings. Letπ:M→M1 be a smooth, surjective mapping of Riemannian manifolds which is a submersion, that is, the differentialπ_∗:TxM→Tπ(x)M1is surjec- tive for allx∈M. Let us say thatπ preserves Laplacians ifH(f ◦π)=(H1f)◦π for all

f ∈C_c^∞(M1) whereH,H1are the Laplace operators onM,M1, respectively.

Theorem 1.2. Letπ:M→M1be a surjective submersion as above, which preserves Lapla- cians. Suppose that (1.1), (1.2), and Condition (I) hold onM. Then (1.1), (1.2), and Con- dition (I) hold onM1.

Important recent work [2] shows that Condition (I) (or Condition (II)), together with (1.1) and (1.2), is sufficient to obtain the boundedness of the Riesz transform operator

∇H^−1/2 inL^p(M) for all 1< p <∞. Applying this result in the context ofTheorem 1.2 gives the following statement.

Corollary 1.3. Adopt the hypotheses ofTheorem 1.2. Then the Riesz transform∇H₁^−1/2is bounded inL^p(M1) for 1< p <∞.

InSection 3we applyTheorem 1.2andCorollary 1.3in some specific cases, namely, co-compact covering manifolds or homogeneous spaces of Lie groups of polynomial growth, to derive new results.

Finally, let us describe a generalization ofTheorem 1.1to a wider context than the above Riemannian setting. Suppose (M,d) is a metric space, endowed with a Borel mea- sureμsuch that the doubling property (1.1) holds withV(x,r) :=μ({y∈M:d(x,y)<

r}). LetH be a nonnegative selfadjoint operator inL²=L²(M;μ), such that the semi- groupSt=e^−tH,t >0, has a continuous kernelKt:M×M→Rsatisfying Gaussian esti- mates (1.2). Assume that the wave equation associated withHhas propagation of finite speed 1, that is, supp(cos(tH^1/2)f)⊆supp_tf for any f ∈L²,t >0. Here, suppf denotes the support of f and supp_t f := {x∈M:d(x, suppf)≤t}.

Next, letEbe a continuous vector bundle overM and let (·,·)xbe a positive-definite inner product on the fibres E_x, x∈M, which varies continuously with x. Using the length|v|x:=(v,v)^1/2_x ,v∈E_x, we may form the spacesL^p(E)=L^p(E;μ) of measurable, p-integrable sections ofE. Let∇denote a linear operator mapping a dense subspace of L²(M) intoL²(E), such that_∇is local, that is, supp(∇f)⊆suppf for any f _∈D(_∇).

ThenTheorem 1.1remains valid under these more general assumptions. Moreover, one could replace the factort^−1/2occurring in Conditions (I), (II), (III)pwitht^−αfor any fixed constantα >0. To prove these assertions requires only trivial modifications in the proof ofTheorem 1.1. Note that the above general setting is essentially that considered in [10].

2. Proofs

For the proof ofTheorem 1.1we first collect a number of preliminary results. In general, c,c,b and so on denote positive constants whose value may change from line to line when convenient. The integral kernel of a linear operatorAacting on functions onM is denoted byKA=KA(x,y) (in particular,Kt=KSt=Ke^−tH fort >0).

A standard consequence of (1.1) is that there is a constantD >0 such that

V(x,γr)≤c(1 +γ)^DV(x,r) (2.1)

for allr >0,γ >0 andx∈M. Observing thatB(y,t^1/2)⊆B(x,t^1/2+d(x,y)) and applying (2.1), we deduce that

Vx,t^1/2≥c1 +d(x,y)t^−1/2−DVy,t^1/2 (2.2) for allt >0 andx,y∈M. This bound allows one, for example, to replaceV(y,t^1/2) with V(x,t^1/2) in (1.2) or in Condition (I). Another straightforward consequence of (1.1) is that for anyb >0 there is ac=c(b)>0 such that

Mdx e^{−bd(x,y)2/t}≤cVy,t^1/2 (2.3)

for allt >0 andy∈M.

Lemma 2.1. Letp∈[1,∞]. Givenm∈Nwithm >2⁻¹D(1−p⁻¹), there existsc=c(m)>

0 such that

K_(I+tH)^−m(·,y)_p=K_(I+tH)^−m(y,·)_p≤cVy,t^{1/2−(1−(1/ p))} (2.4) for allt >0 andy∈M.

Proof. Note that

(I+tH)^−m=Γ(m)^−1∞

0 ds e^−ss^m−1S_st. (2.5) Moreover, (1.2) and (2.3) yield a boundK_u(·,y)p≤V(y,u^1/2)^{−(1−(1/ p))}for all u >0, y∈M. Hence using (2.1), we have

K(I+tH)^−m(·,y)_p≤c _∞

0 e^−ss^m−1Vy, (st)^{1/2−(1−(1/ p))}

≤cVy,t^{1/2−(1−(1/ p))} _∞

0 e^−ss^m−11 +s^{−1(D/2)(1−(1/ p))}

=cVy,t^{1/2−(1−(1/ p))},

(2.6)

where the last integral converges becausem >(D/2)(1−(1/ p)).

Fix a nondecreasing functionψ∈C^∞(R) withψ(x)=0 for allx≤ −1 andψ(x)=1 for allx≥ −2⁻¹, and for eachρ >1 defineFρ∈C^∞(R) by

F_ρ(x)=ψρ|x| −ρ(4π)^−1/2e^−x2/4. (2.7)

Consider the Fourier transformFρofFρ. For eacht >0,ρ >1 andm∈N0= {0, 1, 2,. . .} define the operator

S^(ρ,m)_t =(I+tH)^mF_ρ(tH)^1/2 (2.8)

which is selfadjoint inL²(M), and denote byK_t^(ρ,m)(x,y)=K_t^(ρ,m)(y,x) its integral kernel.

The next lemma, which follows from results of [10], is the place where we use finite speed propagation of the wave equation associated withH.

Lemma 2.2. One has K_t^(ρ,0)(x,y)=Kt(x,y) for all t >0, ρ >1 and x,y∈M such that d(x,y)> ρt^1/2.

For anym∈N0andω∈(0, 1/4), there existsc=c(m,ω)>0 such that

1 +λ^2mFρ(λ)≤ce^−ωρ2 (2.9) for allρ >1 andλ≥0.

Indeed, the first statement ofLemma 2.2 follows in the same way as [10, equation (5.3)], while (2.9) is a cruder version of [10, inequality (5.2)].

We now establish useful pointwise bounds on the kernelsK_t^(ρ,m).

Lemma 2.3. Givenm∈N0,ω∈(0, 1/4), there exist positive constants c=c(m,ω), b= b(m,ω) such that

K_t^(ρ,m)(x,y)≤ce^−ωρ2Vy,t^1/2−1e^{−bd(x,y)2/t} (2.10) for allρ >1,t >0 andx,y∈M.

Proof. In this proof, we fix anω∈(0, 1/4) and denote bycmconstants depending onm.

By the spectral theorem forH, and (2.9), for eachm∈N0one has

S^(ρ,m)_{t 2→2}≤c_me^−ωρ2 (2.11)

for allt >0 andρ >1. Next, writing K_t^(ρ,m)(x,y)=

Mdz K_t^(ρ,m+n)(x,z)K_(I+tH)⁻ⁿ(z,y) (2.12) forn∈N, it follows that

K_t^(ρ,m)(·,y)=S^(ρ,m+n)_t K_(I+tH)⁻ⁿ(·,y). (2.13) Hence by fixingnsufficiently large and applyingLemma 2.1, one finds

K_t^(ρ,m)(·,y)₂≤S^(ρ,m+n)_{t 2→2}K_(I+tH)⁻ⁿ(·,y)₂

≤c_me^−ωρ2Vy,t^1/2−1/2 (2.14)

for all t >0, ρ >1, and y∈M. Replacing mby m+n in this estimate, and applying H¨older’s inequality in (2.12), yields

K_t^(ρ,m)(x,y)≤K_t^(ρ,m+n)(x,·)₂K(I+tH)⁻ⁿ(·,y)₂

≤cme^−ωρ2Vx,t^1/2−1/2Vy,t^1/2−1/2 (2.15) for allt >0,ρ >1 andx,y_∈M. Now, in cased(x,y)_≤ρt^1/2, this estimate yields for any ε∈(0,ω) that

K_t^(ρ,m)(x,y)≤c_me^{−(ω−ε)ρ2}Vx,t^1/2−1/2Vy,t^1/2−1/2e^{−εd(x,y)2/t}. (2.16) In the alternative case whered(x,y)> ρt^1/2,Lemma 2.2implies that

K_t^(ρ,m)(x,y)=

(I+tH)^m_xK_t(x,y)=K_t^(s,m)(x,y) (2.17) for anys∈(ρ,d(x,y)t^−1/2). Then

K_t^(ρ,m)(x,y)≤c_me^−ωs2Vx,t^1/2−1/2Vy,t^1/2−1/2

≤c_me^{−(ω−ε)ρ2}Vx,t^1/2−1/2Vy,t^1/2−1/2e^−εs2 (2.18) for anyε∈(0,ω) ands∈(ρ,d(x,y)t^−1/2), and hence also fors=d(x,y)t^−1/2. Combining the two cases above and using (2.2), for anyε∈(0,ω) we find an estimate

K_t^(ρ,m)(x,y)≤cm,εe^{−(ω−ε)ρ2}Vy,t^1/2−1e^{−2−1εd(x,y)2/t} (2.19) for allt >0,ρ >1 andx,y∈M. The lemma follows.

Remark 2.4. By similar but simpler arguments to those ofLemma 2.3, one may prove the estimates, for eachm∈N0andω∈(0, 1/4),

(I+tH)^m_xKt(x,y)≤c(m,ω)Vy,t^1/2−1e^{−ωd(x,y)2/t} (2.20) for allt >0,x,y∈M. Since (H^m)xK_t(x,y)=(−1)^m(∂/∂t)^mK_t(x,y), the latter estimates amount to estimates on time derivatives ofKt, which are well known consequences of (1.2) (see, e.g., [6] and its references).

Integrating the estimate ofLemma 2.3using (2.3) easily yields the following bounds.

Corollary 2.5. Givenm∈N0,ω∈(0, 1/4) and p∈[1,∞], there is ac >0 such that K_t^(ρ,m)(·,y)_p=K_t^(ρ,m)(y,·)_p≤ce^−ωρ2Vy,t^{1/2−(1−(1/ p))} (2.21) for allt >0,ρ >1 andy∈M.

We are ready to proveTheorem 1.1. Condition (I) implies Condition (II) trivially, and Condition (I) implies Condition (III)pby integration using (2.3).

Next, let us assume Condition (III)pfor a fixedp∈[1,∞], and deduce Condition (I).

First consider the case whered(x,y)> t^1/2. For anyρ∈(1,d(x,y)t^−1/2), byLemma 2.2

∇K_t(x,y)= ∇K_t^(ρ,0)(x,y)=

Mdz∇K_(I+tH)^−m(x,z)K_t^(ρ,m)(z,y). (2.22) From Condition (III)p we may argue as in the proof ofLemma 2.1, to deduce for some fixedm∈Nthat

∇K(I+tH)^−m(x,·)_p≤ct^−1/2Vx,t^{1/2−(1−(1/ p))} (2.23) for allt >0,x∈M. Fixω∈(0, 1/4) and let 1/ p=1−(1/ p). By H¨older’s inequality and Corollary 2.5,

∇Kt(x,y)≤∇K(I+tH)^−m(x,·)_pK_t^(ρ,m)(·,y)_p

≤ct^−1/2Vx,t^{1/2−(1−(1/ p))}Vy,t^{1/2−1/ p}e^−ωρ2 (2.24) wheneverρ∈(1,d(x,y)t^−1/2); this estimate then holds withρ=d(x,y)t^−1/2.

In cased(x,y)≤t^1/2, we write∇Kt(x,y)=

Mdz(∇Kt/2(x,z))Kt/2(z,y) so that ∇K_t(x,y)≤∇K_t/2(x,·)_pK_t/2(·,y)_p

≤ct^−1/2Vx,t^{1/2−(1−(1/ p))}Vy,t^{1/2−1/ p}. (2.25) Combining the two cases above, and using (2.2), we get Condition (I).

Finally, we show that Condition (II) implies Condition (I). From Condition (II), by arguing as inLemma 2.1, we get for some fixedm∈Na bound

∇K_(I+tH)^−m(x,y)_≤ct^−1/2Vy,t^1/2−1 (2.26) for allt >0 andx,y∈M. Whend(x,y)> t^1/2, we insert this estimate in (2.22) and use Lemma 2.3with someω∈(0, 1/4), to get

∇K_t(x,y)≤ct^−1/2

MdzVz,t^1/2−1K_t^(ρ,m)(z,y)

≤ct^−1/2e^−ωρ2

MdzVz,t^1/2−1Vy,t^1/2−1e^{−bd(z,y)2/t}

≤ct^−1/2e^−ωρ2Vy,t^1/2−1

(2.27)

for allρ∈(1,d(x,y)t^−1/2). Here, the last line is a straightforward consequence of (2.2) and (2.3). This establishes the estimate of Condition (I) in cased(x,y)> t^1/2. The case d(x,y)≤t^1/2is immediate from Condition (II). This finishes the proof that Conditions (I), (II), (III)p are equivalent. The last statement ofTheorem 1.1follows easily from the above arguments sinceω∈(0, 1/4) can be chosen arbitrarily. The proof ofTheorem 1.1 is complete.

Let us proveTheorem 1.2. First, work of Saloff-Coste [9] shows that (1.1) and (1.2) hold onM1. Indeed, it is well known that (1.1), (1.2), and Condition (I) onM imply a

global parabolic Harnack inequality onM, and sinceπ:M→M1 preserves Laplacians a Harnack inequality onM1easily follows: see [9, page 36]. But the Harnack inequality implies (1.1) and (1.2) onM1, again according to [9].

To obtain Condition (I) onM1, consider the semigroupsSt=e^−tH onM andS⁽¹⁾_t = e^−tH1onM1. Sinceπpreserves Laplacians, standard semigroup algorithms yield that

St(f◦π)=

S⁽¹⁾_t f◦π (2.28)

for all f _∈L^∞(M1). Moreover, the pointwise lengths of the gradients onM andM1are related by

∇(f◦π)= |∇f| ◦π (2.29)

for all f ∈C^∞(M1). This follows immediately from the standard identity

∇f²= −2⁻¹H f²+fHf (2.30)

for f ∈C^∞(M) and the analogous identity onM1, becauseπpreserves Laplacians. Com- bining these observations, and using the equivalence of Condition (I) with (1.7) given by Theorem 1.1, we obtain

∇S⁽¹⁾_t f_L∞(M1)=∇S⁽¹⁾_t f◦π_L∞(M)=∇St(f ◦π)_L∞(M)

≤ct^−1/2f◦πL^∞(M)=ct^−1/2fL^∞(M1)

(2.31)

for allt >0 and f ∈L^∞(M1). This establishes estimate (1.7) onM1. ThenTheorem 1.2 follows by once again applyingTheorem 1.1.

3. Examples

We give two examples which illustrate our results.

Example (I). LetMbe a Riemannian manifold which is a Galois covering manifold of a compact Riemannian manifoldM0. In other words, there is a discrete, finitely generated groupGof isometries acting properly and freely onM, such that the quotientM0=M/G is compact. Let us also assume that the groupG has polynomial volume growth (see, e.g., [13, Sections VI.2 and X.3] for background). In this situation it is well known that assumptions (1.1) and (1.2) hold onM: see [9, Page 36] or [7, Section 7.6] for instance.

The author [4, Theorem 1.1] proved Condition (I) onM.

LetG1be any (possibly nonnormal) subgroup ofG, and form the quotient manifold M1=M/G1with Riemannian metric induced by the metric ofM. ThenM1 is a cover- ing (though in general not a Galois covering) ofM0. The natural mapπ:M→M1is a covering map and a local isometry, and hence preserves Laplacians. Thus combining the results mentioned above withTheorem 1.2, we obtain the following statement.

Corollary 3.1. LetM1=M/G1be as above. Then Condition (I) holds onM1.

Note that (1.1) and (1.2) and also a parabolic Harnack inequality were already known onM1: see [9, Page 36]. However, Condition (I) seems to be new in this setting. Also, Corollary 1.3applies to give boundedness of the Riesz transform onM1.

Example (II). In this example we consider certain subelliptic operators in place of the Laplace operator, but the principles of Theorems1.1 and1.2 still apply (compare the remarks at the end ofSection 1).

LetGbe a Lie group of polynomial volume growth and consider a sublaplacian H= −

d

i=1

A²_i (3.1)

onG(for background, see [13] and references therein). Here,A1,. . .,Ad is a list of right invariant vector fields on G which generates algebraically the Lie algebrag ofG. The operatorHgenerates a symmetric submarkovian semigroupe^−tHinL^p=L^p(G;dg), 1≤ p≤ ∞, wheredgis a fixed Haar measure forG. Associated withHis a subelliptic gradient

∇f :=(A1f,. . .,A_df), f ∈C^∞(G), and a distancedwith

d(x,y)=supψ(x)−ψ(y) :ψ∈C^∞(G),|∇ψ| ≤1. (3.2) It is well known that estimates (1.1) and (1.2) and Condition (I) hold on G (see [13, Section VIII.2]), and the finite speed propagation property holds forH. In the special case whereA1,. . .,Adform a vector space basis forg, one easily sees thatHis the Laplace operator associated with a right invariant Riemannian metric onG.

Now letG1be any closed subgroup ofG, and form the smooth manifoldM1=G/G1

consisting of all left cosetsgG1,g∈G, ofG1. Consider the projectionπ:G→M1,π(g)= gG1, with differentialπ_∗:TG→TM1. SinceA_iis right invariant, one easily checks that π_∗(Ai|g)=π_∗(Ai|gg1)∈Tπ(g)M1 for allg∈Gand g1∈G1. ThereforeBi:=π_∗(Ai) are well-defined vector fields onM1, and we may form the subelliptic operator

H1= −

d

i=1

B_i² (3.3)

onM1. ThenH(f◦π)=(H1f)◦πfor all f ∈C^∞(M1).

ToH1we associate a gradient∇f =(B1f,. . .,Bdf), f ∈C^∞(M1), and a distancedon M1by a formula analogous to (3.2). Fix a positive Borel measureμonM1which is invari- ant under the standard actiong·(hG1) :=ghG1,g,h∈G, ofGonM1, and consider the spacesL^p(M1)=L^p(M1;μ) (the existence of such a measure follows from the unimod- ularity ofGandG1: see, e.g., [5, Theorem 2.49]). One has (H1f1,f1)≥0, (H1f1,f2)= (f1,H1f2) for all f1,f2∈C_c^∞(M1), and the Friedrichs extension of H1 is a nonnegative selfadjoint operator inL²(M1) which generates a symmetric submarkovian semigroup e^−tH1.

We remark that, according to general theory of partial differential operators on Lie groups (see [11] for instance), one can identifyH1=dU(H) whereUis the unitary rep- resentation ofGinL²(M1) with (U(g)f)(hG1)=f(g⁻¹hG1),g,h∈G.

The proof ofTheorem 1.2goes through in this situation, and therefore (1.1) and (1.2) and Condition (I) hold onM1. To our knowledge, Condition (I) is new in this setting.

Finally, the results of [2] apply to give the boundedness of the Riesz transform∇H₁^−1/2 inL^p(M1), 1< p <∞. This seems to be a new result (the boundedness of the Riesz trans- form on the groupGwas first proved in [1]).

Acknowledgment

This work was carried out with financial support from the Australian Research Council (ARC) Centre of Excellence for Mathematics and Statistics of Complex Systems (MAS- COS).

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Nick Dungey: School of Mathematics, The University of New South Wales, Sydney 2052, Australia E-mail address:[email protected]