Second-order subelliptic operators on Lie groups II: real measurable principal coefficients

A.F.M. ter Elst^{1} and Derek W. Robinson^{2}

March 1998

AMS Subject Classification: 35J15, 35K05, 22E30.

Home institutions:

Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands Centre for Mathematics and its Applications School of Mathematical Sciences Australian National University Canberra, ACT 0200 Australia

Abstract

Let G be a connected Lie group with Lie algebra 𝔤 and a_{1}, …, a_{d′} an algebraic basis of 𝔤. Further let A_{i} denote the generators of left translations, acting on the L_{p}- spaces L_{p}(G; dg) formed with left Haar measure dg, in the directions a_{i}. We consider second-order operators

H = − ∑_{i,j=1}^{d′} A_{i} c_{ij} A_{j} + ∑_{i=1}^{d′}(c_{i} A_{i} + A_{i} c′_{i}) + c_{0} I

corresponding to a quadratic form with real measurable coefficients c_{ij} and complex c_{i}, c′_{i}, c_{0} ∈ L_{∞}. The matrix C = (c_{ij}) of principal coefficients, which is not necessarily symmetric, is assumed to satisfy the subellipticity condition

ℜC = 2^{−1}(C + C^{∗}) ≥ μI > 0

uniformly over G.

We prove that H generates a strongly continuous holomorphic semigroup S on L_{2} with a kernel K which satisfies Gaussian bounds

|K_{z}(g; h)| ≤ a |z|^{−D′/2}e^{ω|z|}e^{−b(|gh^{−1}|′)^{2}|z|^{−1}}

for g, h ∈ G and z in a subsector Λ(θ) of the sector of holomorphy. Moreover, the kernel is Hölder continuous and there is a ν ∈ ⟨0, 1⟩ such that for all κ > 0 one has estimates

|K_{z}(k^{−1}g; l^{−1}h) − K_{z}(g; h)|
≤ a |z|^{−D′/2}e^{ω|z|}(\frac{|k|′ + |l|′}{|z|^{1/2} + |gh^{−1}|′} )^{ν}e^{−b(|gh^{−1}|′)^{2}|z|^{−1}}

for g, h, k, l ∈ G and z in the subsector with |k|′ + |l|′ ≤ κ |z|^{1/2} + 2^{−1}|gh^{−1}|′.

In addition, if all the coefficients of H are real-valued then

K_{t}(g; h) ≥ a′ t^{−D′/2}e^{−ω′t}e^{−b′(|gh^{−1}|′)^{2}t^{−1}}

for some a′, b′ > 0 and ω′ ≥ 0 uniformly for g, h ∈ G and t > 0.

1 Introduction

We continue the analysis of the semigroup kernels associated with second-order subelliptic operators with variable coefficients acting on the L_{p}-spaces over a d-dimensional Lie group G begun in [ElR3]. In the latter paper we considered operators

H = − ∑_{i,j=1}^{d′} A_{i}c_{ij}A_{j} + ∑_{i=1}^{d′}(c_{i}A_{i} + A_{i}c′_{i}) + c_{0}  (1)

with complex coefficients c_{ij}, c_{i}, c′_{i}, c_{0} ∈ L_{∞} where A_{i} = dL(a_{i}) denotes the generator of left translations L, acting on one of the classical function spaces, in the direction a_{i} of the Lie algebra 𝔤 of G, and a_{1}, …, a_{d′} is an algebraic basis of 𝔤. Subellipticity corresponds to the condition

ℜC = 2^{−1}(C + C^{∗}) ≥ μI > 0 ,

in the sense of d′ × d′-matrices, uniformly over G. In this paper we analyze operators for which the principal coefficients c_{ij} are real-valued but the lower-order coefficients can still be complex. Moreover, we do not assume that the matrix C = (c_{ij}) of principal coefficients is symmetric.

Our main result establishes Hölder continuity estimates comparable to the classic results of Morrey [Mor], Nash [Nas] and De Giorgi [Gio] for strongly elliptic operators with real measurable coefficients on R^{d} and Gaussian upper bounds of the type first obtained by Aronson [Aro]. In addition we derive Gaussian lower bounds on the kernels associated with operators for which all coefficients are real. The proofs are again a combination of parabolic and elliptic techniques based on a mixture of the methods introduced by Nash [Nas], as refined by Fabes, Saloff-Coste and Stroock [FaS] [SaS], together with the method of De Giorgi [Gio] as influenced by the exposition of Giaquinta [Gia] and recent work of Auscher [Aus]. In particular De Giorgi estimates are combined with arguments involving Morrey–Campanato spaces.

Throughout the paper we adopt the notation and definitions of [ElR3]. In particular the operator H formally given by (1) is defined as a closed sectorial operator on the L_{2}-space, L_{2}(G; dg), where dg denotes left Haar measure. Then H is the generator of a strongly continuous semigroup S on L_{2} with a holomorphic extension to a sector Λ(θ) = {z ∈ C\{0} : | arg z| < θ} in the complex plane with θ ≥ θ_{C} = arccot(γ_{C}/μ_{C}) where

μ_{C} = sup{ μ : 2^{−1}(C + C^{∗}) ≥ μI } , γ_{C} = inf{ γ : γI ≥ (2i)^{−1}(C − C^{∗}) ≥ −γI } .

(The value of μ_{C} corresponds to the ellipticity constant.) Moreover, S has a distribution kernel K_{t} ∈ D′(G × G) such that

(ψ, S_{t}φ) = ∫_{G} dg \overline{ψ(g)} ∫_{G} dĥ K_{t}(g; h) φ(h)

for all φ, ψ ∈ C_{c}^{∞}(G) and t > 0 where dĥ denotes right Haar measure. We prove that the kernel has a holomorphic extension K_{z} to a sector Λ(θ) ⊇ Λ(θ_{C}) satisfying

K_{z_{1}+z_{2}}(g; h) = ∫_{G} dk̂ K_{z_{1}}(g; k) K_{z_{2}}(k; h)  (2)

for all z_{1}, z_{2} ∈ Λ(θ_{C}). In addition, we derive bounds on K in terms of the right invariant distance d′( ·; ·), the control distance, canonically associated with the algebraic basis a_{1}, …, a_{d′} (see, for example, [Rob] Sections IV.2 and IV.4c). This distance has the characterization

d′(g; h) = sup{ |ψ(g) − ψ(h)| : ψ ∈ C_{c}^{∞}(G) , ∑_{i=1}^{d′} |(A_{i}ψ)|^{2} ≤ 1, ψ real }

(see, for example, [Rob], Lemma IV.2.3, or [ElR2], Lemma 4.2). Other parameters which enter the estimates are the subelliptic modulus g ↦ |g|′ = d′(g; e), where e is the identity of G, and the local dimension D′, i.e., the integer for which the left Haar measure |B′(g; r)| of the ball B′(g; r) = { h ∈ G : d′(g; h) < r } satisfies bounds

c^{−1}r^{D′} ≤ |B′(e; r)| ≤ c r^{D′}  (3)

for some c > 0 and all r ≤ 1.

Theorem 1.1 Let H be a subelliptic operator of the form (1) with real measurable principal coefficients c_{ij} and complex measurable lower order coefficients c_{i}, c′_{i} and c_{0}. Then, for each θ ∈ ⟨0, θ_{C}⟩, there exist a, b > 0 and ω ≥ 0 such that the kernel K satisfies

|K_{z}(g; h)| ≤ a |z|^{−D′/2}e^{ω|z|}e^{−b(|gh^{−1}|′)^{2}|z|^{−1}}  (4)

uniformly for g, h ∈ G and z ∈ Λ(θ). The kernel has the semigroup property (2) and is Hölder continuous. In particular there is a ν ∈ ⟨0, 1⟩ such that for each θ ∈ ⟨0, θ_{C}⟩ and κ > 0 there exist a, b > 0 and ω ≥ 0 such that

|K_{z}(k^{−1}g; l^{−1}h) − K_{z}(g; h)| ≤ a |z|^{−D′/2}e^{ω|z|}(\frac{|k|′ + |l|′}{|z|^{1/2} + |gh^{−1}|′} )^{ν}e^{−b(|gh^{−1}|′)^{2}|z|^{−1}} (5)

for all g, h, k, l ∈ G and z ∈ Λ(θ) with |k|′ + |l|′ ≤ κ |z|^{1/2} + 2^{−1}|gh^{−1}|′.

Moreover, if all the coefficients of H are real-valued then

K_{t}(g; h) ≥ a′ t^{−D′/2}e^{−ω′t}e^{−b′(|gh^{−1}|′)^{2}t^{−1}}  (6)

for some a′, b′ > 0 and ω′ ≥ 0 uniformly for g, h ∈ G and t > 0.

The theorem gives a satisfactory description of the structure of the kernel of secondorder subelliptic operators on Lie groups for small times, i.e., small t. It improves on earlier results [BrR], [SaS], insofar it removes all unnecessary regularity, symmetry and reality restrictions on the coefficients and is valid for all groups, modular or unimodular, with polynomial or exponential growth. It does, however, leave open questions of optimization of the Gaussian factor in the bounds and the large time behaviour. These properties will be discussed in the concluding remarks, Section 4.

Bounds such as (4) and (5) are well known for subelliptic operators of all orders if the coefficients are smooth (see, for example, [ElR1]). It is also known that positivity and bounds such as (6) are only possible for second-order operators with real coefficients (see [ABR] and [Rob]). Note that the theorem establishes the bounds (4) and (5) throughout any subsector of the sector Λ(θ_{C}) and consequently if the matrix of principal coefficients C = (c_{ij}) is symmetric the results are valid in any subsector of the open right half-plane. In fact the bounds (4) follow for complex z from their real z counterparts together with the L_{2}-holomorphy of the semigroup S (see [Dav], Theorem 3.4.8) and similar reasoning could be applied to the continuity estimates (5). These bounds allow one to extend the semigroup S to a holomorphic semigroup on all the L_{p}-spaces with a holomorphy sector containing the p-independent subsector Λ(θ_{C}).

The proof of the upper bounds and the Hölder continuity uses the strategy developed in [ElR3] following ideas of [Aus]. It suffices to prove De Giorgi estimates for the principal part of the operator. These estimates are established by a refinement of the Nash–De Giorgi techniques based on the general reasoning given in Chapter 5 of Giaquinta’s book [Gia].

For the proof of the lower bounds we use the strategy of [BrR] (see also [Str1]). First one replaces H by a sequence of subelliptic operators H_{n} obtained by regularizing the coefficients of H. Secondly, one establishes kernel bounds and continuity estimates uniformly for the regularizing sequence. Thirdly, one deduces the required result by a limit over the regularizing sequence. The regularization and limiting processes are described in detail in [ElR1] and [ElR3]. Hence we concentrate on deriving the uniform bounds.

2 De Giorgi estimates

The key to bounds on the semigroup kernel is the derivation of De Giorgi estimates for weak solutions of the elliptic equation Hφ = 0 on small balls B′(g; r) = { h ∈ G : d′(g; h) < r }. In [ElR3] these estimates were derived for operators with complex uniformly continuous coefficients by a series of L_{2}-estimates based on the Poincaré and Caccioppoli inequalities. In the present context we need more refined L_{p}-estimates with p ∈ [1, 2⟩. A principal new ingredient is the Poincaré–Sobolev inequality.

First, for each bounded open subset Ω ⊆ G, introduce the spaces

H′_{p;1}(Ω) = {φ ∈ L_{p}(Ω ; dg) : A_{i}φ ∈ L_{p}(Ω ; dg) for all i ∈ {1, …, d′}} ,

where A_{i}φ denotes the distributional derivative in D′(Ω). We use the notation ∇′φ = (A_{1}φ, …, A_{d′}φ) and equip the spaces with the norms φ ↦ (∥φ∥_{p,Ω}^{p} + ∥∇′φ∥_{p,Ω}^{p})^{1/p} where

∥φ∥_{p,Ω} = ( ∫_{Ω} dh |φ(h)|^{p})^{1/p}

and

∥∇′φ∥_{p,Ω} = ( ∫_{Ω} dh ( ∑_{i=1}^{d′} |(A_{i}φ)(h)|^{2})^{p/2})^{1/p} .

Moreover, for φ ∈ L_{1,loc} we denote by ⟨φ⟩_{Ω} the average of φ over Ω. Finally, if Ω = B′(g; r) we simplify notation by setting ∥ · ∥_{p,g,r} = ∥ · ∥_{p,Ω} and for g = e we drop the e, e.g., B′(r) = B′(e; r), ∥∇′φ∥_{2,r} = ∥∇′φ∥_{2,e,r} etc..

Proposition 2.1 Let p ∈ [1, D′⟩ and 1/q = 1/p − 1/D′. Then there exist c_{p} > 0 and R_{p} ∈ ⟨0, 1] such that

∥φ − ⟨φ⟩_{g,r}∥_{q,g,r} ≤ c_{p} ∥∇′φ∥_{p,g,r}  (7)

for all g ∈ G, r ∈ ⟨0, R_{p}], ε > 0 and φ ∈ H′_{q;1}(B′(g; r(1 + ε))).

Proof The proof of the estimates (7) for g = e and φ ∈ C^{∞}(\overline{B′(r)}) follows from [FLW], Theorem 1. Their extension to φ ∈ H′_{q;1}(B′(g; r(1 + ε))) then follows the proof of Proposition 2.4 in [ElR3]. □

Our aim is to establish that the principal part H_{P} of the subelliptic operator H, and the principal part H_{P}^{∗} of its adjoint H^{∗}, satisfy De Giorgi estimates. These estimates then yield the upper bounds and continuity estimates for the semigroup kernel stated in Theorem 1.1. Since these results were already established for dimensions d = 1 and d = 2 in [ElR4], even if the principal coefficients are complex, it suffices to assume d ≥ 3. In particular D′ ≥ 2.

The De Giorgi estimates for G = R^{d} are established in Section 5.1 of [Gia]. The proof contains two technical ingredients. The first is the classical Sobolev–Poincaré inequality on balls of radius R and Proposition 2.1 provides the Lie group replacement. The second is the existence of a sequence of cut-off functions η with η = 1 on the ball of radius r, with supp η contained in the ball of radius R and ∥∇′η∥_{∞} ≤ c (R − r)^{−1} for all small r < R, where c is independent of r and R. It is particularly important that these cut-off functions exist for all r arbitrarily close to R. These functions exist by the following lemma.

Lemma 2.2 There exists a b_{1} > 0 and, for all r, R ∈ ⟨0, 1] with r < R, an η_{r,R} ∈ C_{c}^{∞}(B′_{R}) with 0 ≤ η_{r,R} ≤ 1 and η_{r,R} = 1 on B′_{r} such that ∥A_{i}η_{r,R}∥_{∞} ≤ b_{1}(R − r)^{−1} for all i ∈ {1, …, d′}.

Proof By Lemma 2.6 of [ElR3] there exist σ ∈ ⟨0, 1⟩, c_{1} > 0 and for all r ∈ ⟨0, 1] there exists an η_{r} ∈ C_{c}^{∞}(B′_{r}) such that 0 ≤ η_{r} ≤ 1, η_{r} = 0 on B′_{σr} and ∥A_{i}η_{r}∥_{∞} ≤ c_{1} r^{−1} for all i ∈ {1, …, d′} and r ∈ ⟨0, 1]. Then there is a c_{2} > 0 such that ∥η_{r}∥_{1}^{−1}|B′_{r}| ≤ c_{2} uniformly for all r ∈ ⟨0, 1]. For all s, t ∈ ⟨0, 1] with s < t define φ_{s,t} = ∥η_{s}∥_{1}^{−1}η_{s} ∗1_{B′_{t}}. Then φ_{s,t} ∈ C_{c}^{∞}(G), supp φ_{s,t} ⊂ B′_{s+t} and φ_{s,t}(g) = 1 for all g ∈ B′_{t−s}. Moreover,

|(A_{i}φ_{s,t})(g)| = ∥η_{s}∥_{1}^{−1}|((A_{i}η_{s}) ∗ 1_{B′_{t}})(g)| ≤ |(∥η_{s}∥_{1}^{−1}∥A_{i}η_{s}∥_{∞}1_{B′_{s}}) ∗ 1_{B′_{t}})(g)|
≤ ∥∥η_{s}∥_{1}^{−1}∥A_{i}η_{s}∥_{∞}1_{B′_{s}} ∥_{1}∥1_{B′_{t}}∥_{∞} ≤ c_{1} c_{2} s^{−1}

for all i ∈ {1, …, d′} and g ∈ G. Finally, for all 0 < r < R ≤ 1 choose s = 2^{−1}(R − r) and t = 2^{−1}(R + r). □

For M, μ > 0 let E_{r,P}^{div}(μ, M) be the set of all pure second-order subelliptic operators

H = − ∑_{i,j=1}^{d′} A_{i} c_{ij} A_{j}

with real-valued coefficients c_{ij} such that μ_{C} ≥ μ and ∥C∥_{∞} = sup_{g∈G} ∥C(g)∥ ≤ M, where ∥C(g)∥ denotes the l_{2}-norm of the matrix C(g) = (c_{ij}(g)). We aim to prove estimates uniform for all operators in E_{r,P}^{div}(μ, M). We first prove a version of the Caccioppoli inequalities, as in [ElR3], Lemma 2.7.

Lemma 2.3 For all M, μ > 0 there exists a b_{2} > 0 such that

∫_{B′_{r}} |∇′((φ − k)^{+})|^{2} ≤ b_{2}(R − r)^{−2} ∫_{B′_{R}} |(φ − k)^{+}|^{2}

uniformly for all H ∈ E_{r,P}^{div}(μ, M), all r, R ∈ R with 0 < r < R ≤ 1, all k ∈ R and all real-valued φ ∈ H′_{2;1}(B′_{R}) satisfying Hφ = 0 weakly on B′_{R}.

Proof We may assume that k = 0, since one also has H(φ − k) = 0 weakly on B′_{R}. Next, let η_{r,R} and b_{1} be as in Lemma 2.2. Note that φ^{+} ∈ H′_{2;1}(B′_{R}) and A_{i}(φ^{+}) = 1_{[φ>0]}A_{i}φ by an argument similar to the proof of Lemma 7.6 in [GiT] but using the estimates underlying the proof of Lemma 2.4 in [ElR1]. Therefore A_{i}(φ^{+}) A_{j}(φ^{+}) = 1_{[φ>0]}A_{i}φ A_{j}φ = A_{i}(φ^{+}) A_{j}φ and φ^{+}A_{i}φ = φ^{+}A_{i}(φ^{+}). Then

μ_{C} ∥η_{r,R}∇′(φ^{+})∥_{2}^{2} ≤ (η_{r,R}∇′(φ^{+}), Cη_{r,R}∇′(φ^{+}))
= (η_{r,R}^{2}∇′(φ^{+}), C∇′φ) = ∑_{i,j=1}^{d′}([η_{r,R}^{2}, A_{i}](φ^{+}), c_{ij}A_{j}φ)

where the last step uses Hφ = 0 weakly on B′_{R}. But [η_{r,R}^{2}, A_{i}] = −2η_{r,R}(A_{i}η_{r,R}) and hence

μ_{C} ∥η_{r,R}∇′(φ^{+})∥_{2}^{2} ≤ −2((∇′η_{r,R})(φ^{+}), Cη_{r,R}∇′φ)
= −2((∇′η_{r,R})φ^{+}, Cη_{r,R}∇′(φ^{+}))
≤ 2 ∥C∥_{∞} ∥(∇′η_{r,R})φ^{+}∥_{2} ∥η_{r,R}∇′(φ^{+})∥_{2} .

Therefore

∫_{B′_{r}} |∇′((φ)^{+})|^{2} ≤ ∥η_{r,R}∇′(φ^{+})∥_{2}^{2} ≤ 4 μ_{C}^{−2}∥C∥_{∞}^{2} ∥(∇′η_{r,R})φ^{+}∥_{2}^{2}
≤ 4 b_{1}^{2} μ_{C}^{−2}∥C∥_{∞}^{2} (R − r)^{−2} ∫_{B′_{R}} |φ^{+}|^{2} ,

by the estimates of Lemma 2.2. □

Set θ = 2^{−1} + (4^{−1} + 2(D′)^{−1})^{1/2} > 1. Then θ^{2} − θ − 2(D′)^{−1} = 0. Moreover, if k ∈ R, 0 < r ≤ R < ∞ and φ ∈ H′_{2;1}(B′_{R}) set

A(k, r) = [φ > k] ∩ B′_{r} .

It will be clear from the context which φ is used.

Proposition 2.4 For all M, μ > 0 there exists a b_{3} > 0 such that

ess sup_{g∈B′_{R/2}} φ(g) ≤ k + b_{3}(R^{−D′} ∫_{A(k,R)} |φ − k|^{2})^{1/2}(R^{−D′}|A(k, R)|)^{(θ−1)/2}

uniformly for all H ∈ E_{r,P}^{div}(μ, M), R ∈ ⟨0, 1], k ∈ R and real-valued φ ∈ H′_{2;1}(B′_{R}) satisfying Hφ = 0 weakly on B′_{R}.

Proof The proof is almost the same as the proof of Theorem 5.1 in [Gia]. The only problem arises with the application of the Sobolev inequality.

Let η_{r,R} and b_{1} be as in Lemma 2.2 and b_{2} be as in Lemma 2.3. Let H ∈ E_{r,P}^{div}(μ, M), S ∈ ⟨0, 1] and φ ∈ H′_{2;1}(B′_{S}) be a real-valued function satisfying Hφ = 0 weakly on B′_{S}. Then for all 0 < r < R ≤ S and k ∈ R one has

∫_{B′_{(r+R)/2}} |∇′(η_{r,(r+R)/2}(φ − k)^{+})|^{2} ≤ 2 ∫_{B′_{(r+R)/2}} |∇′η_{r,(r+R)/2}|^{2} |(φ − k)^{+})|^{2}
+ 2 ∫_{B′_{(r+R)/2}} |η_{r,(r+R)/2}|^{2} |∇′((φ − k)^{+})|^{2}
≤ (8d′ b_{1}^{2} + 8b_{2})(R − r)^{−2} ∫_{B′_{R}} |(φ − k)^{+}|^{2} .

Next, by the subelliptic Sobolev inequalities for Lie groups, [Rob], Theorem IV.5.6, there exists a b > 0 such that

( ∫_{G} |ψ|^{2D′/(D′−2)})^{(D′−2)/D′} ≤ b ∫_{G} |∇′ψ|^{2} + b ∫_{G} |ψ|^{2}

uniformly for all ψ ∈ L′_{2;1}(G). Therefore

( ∫_{B′_{r}} |(φ − k)^{+}|^{2D′/(D′−2)})^{(D′−2)/D′}

≤ ( ∫_{G} |η_{r,(r+R)/2}(φ − k)^{+}|^{2D′/(D′−2)})^{(D′−2)/D′}
≤ b ∫_{G} |∇′(η_{r,(r+R)/2}(φ − k)^{+})|^{2} + b ∫_{G} |η_{r,(r+R)/2}(φ − k)^{+}|^{2}
= b ∫_{B′_{(r+R)/2}} |∇′(η_{r,(r+R)/2}(φ − k)^{+})|^{2} + b ∫_{B′_{(r+R)/2}} |η_{r,(r+R)/2}(φ − k)^{+}|^{2}
≤ b(8d′ b_{1}^{2} + 8b_{2} + 1)(R − r)^{−2} ∫_{B′_{R}} |(φ − k)^{+}|^{2} .

The rest of the proof is as in [Gia]. The uniformity of the constant follows from the proof. Actually, one can take b_{3} = 2^{(θ−1)^{−1}2^{D′θ/2}θ(1+D′θ/2)}(b(8d′b_{1}^{2} + 8b_{2} + 1))^{D′θ/4}c^{(θ−1)/2}, where the constant c is as in (3). □

Corollary 2.5 For all M, μ > 0 there exists a b_{4} > 0 such that

ess sup_{g∈B′(R/2)} |φ(g)| ≤ b_{4}(R^{−D′} ∫_{B′(R)} |φ|^{2})^{1/2}

uniformly for all H ∈ E_{r,P}^{div}(μ, M), R ∈ ⟨0, 1], k ∈ R and all real-valued φ ∈ H′_{2;1}(B′(R)) satisfying Hφ = 0 weakly on B′(R).

Proof Let b_{3} be as in Proposition 2.4. Then applying Proposition 2.4 with k = 0 one deduces that

ess sup_{g∈B′(R/2)} φ(g) ≤ b_{3}(R^{−D′} ∫_{B′(R)} |φ|^{2})^{1/2}(R^{−D′}|B′(R)|)^{(θ−1)/2} .

But these estimates are also valid for φ replaced by −φ. Now the corollary follows from the volume estimates (3). □

Before we can prove the De Giorgi estimates for subelliptic operators with real measurable coefficients we need one more technical lemma.

Lemma 2.6 For all M, μ > 0 there exists a b_{5} > 0 such that

R^{−D′}|A(k_{n}, R)| ≤ b_{5} n^{−β}

uniformly for all H ∈ E_{r,P}^{div}(μ, M), R ∈ ⟨0, R_{1}/4], n ∈ N and real-valued φ ∈ H′_{2;1}(B′(4R)) satisfying Hφ = 0 weakly on B′(4R) and |A(k_{0}, R)| ≤ 2^{−1}|B′(R)| where

k_{n} = ess sup_{g∈B′(2R)} φ(g) − 2^{−(n+1)}( ess sup_{g∈B′(2R)} φ(g) − ess inf_{g∈B′(2R)} φ(g)) ,

R_{1} is as in Proposition 2.1 and β = (2D′ − 2)^{−1}D′.

Proof Note that the essential suprema and infima are finite by Corollary 2.5. It follows from the proof of [Gia], Proposition 5.2, that

n(R^{−D′}|A(k_{n}, R)|)^{1/β} ≤ b R^{−D′}|A(k_{0}, R)|

for all n ∈ N, where b = 8(1 − 2^{−1/D′})^{−4}b_{2}^{2}c^{2}c_{1}^{4}σ^{−2D′} and the constants b_{2}, c and c_{1} are as in Lemma 2.3, (3) and Proposition 2.1. But |A(k_{0}, R)| ≤ |B′(R)| ≤ c R^{D′}. Therefore R^{−D′}|A(k_{n}, R)| ≤ (b c)^{β} n^{−β} and the lemma follows. □

At this point we can state and prove the uniform De Giorgi estimates which we require.

Proposition 2.7 For all M, μ > 0 there exist ν > 0 and c_{DG} > 0 such that

∫_{B′(g;r)} |∇′φ|^{2} ≤ c_{DG} (r/R)^{D′−2+2ν} ∫_{B′(g;R)} |∇′φ|^{2}

uniformly for all H ∈ E_{r,P}^{div}(μ, M), r, R ∈ R with 0 < r ≤ R ≤ 1, g ∈ G and all (complexvalued) φ ∈ H′_{2;1}(B′(R)) satisfying Hφ = 0 weakly on B′(g; R).

Proof As in the proof of Proposition 3.3 in [ElR3] it suffices to prove the estimates for g = e, R ∈ ⟨0, R_{1}∧R_{N}], r ≤ R/4 and φ ∈ H′_{2;1}(B′(2R)) satisfying Hφ = 0 weakly on B′(2R) and ⟨φ⟩_{R} = 0, where R_{N} is the constant in the Poincaré inequality (11) of Proposition 2.4 in [ElR3] and R_{1} is as in Proposition 2.1. Moreover, since H is a real operator, it suffices to prove the inequality for real-valued φ.

Let b_{2}, b_{3}, b_{4}, b_{5} and β be as in Lemma 2.3, Proposition 2.4, Corollary 2.5 and Lemma 2.6. Moreover, let c be as in (3) and c_{N} be as in the Neumann-type Poincaré inequality of Proposition 2.4 in [ElR3]. Let R and φ be as above. We follow the ideas of [Gia] Section 5.1. For all r ∈ ⟨0, R/2] the function φ is essentially bounded on B′(r) by Corollary 2.5 and we set

m(r) = ess inf_{g∈B′(r)} φ(g) , M(r) = ess sup_{g∈B′(r)} φ(g) .

Now suppose r ∈ ⟨0, R/4]. Set k_{0} = 2^{−1}(M(2r) + m(2r)). We may assume that |A(k_{0}, r)| ≤ 2^{−1}|B′(r)|, otherwise replace φ by −φ. Next, for all n ∈ N we set k_{n} = M(2r) − 2^{−(n+1)}(M(2r) − m(2r)). Then it follows from Proposition 2.4 that

M(r/2) ≤ k_{n} + b_{3}(r^{−D′} ∫_{A(k_{n},r)} |M(r) − k_{n}|^{2})^{1/2}(r^{−D′}|A(k_{n}, r)|)^{(θ−1)/2}
≤ k_{n} + b_{3} c^{1/2}(M(2r) − k_{n})(b_{5} n^{−β})^{(θ−1)/2}

uniformly for all n ∈ N, where Lemma 2.6 is used in the last inequality. Next fix N ∈ N such that

b_{3} c^{1/2}(b_{5} N^{−β})^{(θ−1)/2} ≤ 2^{−1} .

Note that N depends only on M and μ. Then

M(r/2) ≤ M(2r) − 2^{−(N+1)}(M(2r) − m(2r)) + 2^{−1}2^{−(N+1)}(M(2r) − m(2r))
= M(2r) − 2^{−(N+2)}(M(2r) − m(2r))

and hence

M(r/2) − m(r/2) ≤ M(2r) − m(2r) − 2^{−(N+2)}(M(2r) − m(2r))
= (1 − 2^{−(N+2)})(M(2r) − m(2r)) .

This is valid for all r ∈ ⟨0, R/4]. Therefore one deduces by induction that

M(2^{−(2n+1)}r) − m(2^{−(2n+1)}r) ≤ (1 − 2^{−(N+2)})^{n}(M(R/2) − m(R/2))

for all n ∈ N_{0} and

M(r) − m(r) ≤ 4^{ν} (r/R)^{ν}(M(R/2) − m(R/2))

for all r ∈ ⟨0, R/2], where ν = −(2 log 2)^{−1} log(1 − 2^{−(N+2)}) > 0. Finally, applying Lemma 2.3 to φ with k = ⟨φ⟩_{2r} and to −φ with k = −⟨φ⟩_{2r} one deduces that for all r ∈ ⟨0, R/4]

∫_{B′(r)} |∇′φ|^{2} ≤ 4b_{2} (2r)^{−2} ∫_{B′(2r)} |φ − ⟨φ⟩_{2r}|^{2}
≤ 4b_{2} (2r)^{−2} ∫_{B′(2r)} |M(2r) − m(2r)|^{2}
≤ 4a^{2} b_{2} c (2r)^{D′−2}(2r/R)^{2ν}(M(R/2) − m(R/2))^{2}
≤ 4a^{2} b_{2} c (2r)^{D′+2ν−2}R^{−2ν}(2 ess sup_{g∈B′(R/2)} |φ(g)|)^{2}
≤ 16a^{2} b_{2} b_{4}^{2} c (2r)^{D′+2ν−2}R^{−(D′+2ν)} ∫_{B′(R)} |φ|^{2}
≤ 16a^{2} b_{2} b_{4}^{2} c c_{N} 2^{D′+2ν−2}(r/R)^{D′+2ν−2} ∫_{B′(R)} |∇′φ|^{2} ,

where we have used ⟨φ⟩_{R} = 0 and the Neumann-type Poincaré inequality in the last step. This completes the proof of the De Giorgi estimates for operators with real measurable coefficients. □

3 Kernel estimates

In this section we prove the estimates for the semigroup kernel stated in Theorem 1.1.

The upper bounds and continuity estimates were already established for dimensions one and two in [ElR4]. In fact these low-dimensional results do not require the principal coefficients of H to be real-valued. If, however, D′ ≥ 2 then the bounds and continuity are a direct corollary of Theorem 4.1 of [ElR3] and the De Giorgi estimates of Proposition 2.7. Note that this latter proposition can be applied both to H and its adjoint H^{∗}.

It remains to prove the Gaussian lower bounds in the last statement of Theorem 1.1. There is a large literature on lower bounds of semigroup kernels associated with real secondorder elliptic operators. (References directly relevant to the current paper can be found in the books [Dav], [Rob], [VSC].) Most of this work is based on the arguments of Nash [Nas] for pure second-order operators on R^{d} and the subsequent discussion is largely composed of arguments already contained in the literature. The only essential new feature arises from the lack of symmetry of the principal coefficients. Previous Lie group results also require some smoothness of the coefficients or place restrictions on the growth properties of the group. Therefore we have to rearrange the reasoning to take care of these difficulties.

First we reduce to the case of smooth coefficients by regularization as in [BrR], [Str1], [ElR3]. If τ_{n} ∈ C_{c}^{∞}(G), τ_{n} ≥ 0, ∥τ_{n}∥_{1} = 1 is an approximation to the identity and c ∈ L_{∞} the regularization c^{(n)} is defined by

c^{(n)}(g) = (R(τ_{n})c)(g) = ∫_{G} dh τ_{n}(h) c(gh) ,

where R is the right regular representation. Then let H_{n} denote the operators constructed from H by regularization of the coefficients as in [ElR3], Section 2.1, i.e., one replaces the c_{ij} in H by c_{ij}^{(n)}, etc.. Since τ_{n} ≥ 0 and ∥τ_{n}∥_{1} = 1 it follows that μ_{C^{(n)}} ≥ μ_{C}, ∥C^{(n)}∥_{∞} ≤ ∥C∥_{∞}, ∥c^{(n)}∥_{∞} ≤ ∥c∥_{∞} and ∥c_{0}^{(n)}∥_{∞} ≤ ∥c_{0}∥_{∞}. Moreover, if K^{(n)} denotes the kernel of the semigroup generated by H_{n} it follows from the uniform bounds of Proposition 2.7 together with Propositions 4.5 and 2.3 of [ElR3] that lim_{n→∞} K_{t}^{(n)} = K_{t} uniformly on compacta of G × G. Thus we can effectively assume the coefficients of H are smooth as long as the final estimates are independent of the smoothness. For operators with smooth coefficients it follows from [ElR1], Corollary 3.5, that the kernel belongs to C_{b;∞}(G × G).

Next we observe that the kernel K_{t} associated with an operator H with all real coefficients is pointwise positive, i.e., K_{t}(g; h) ≥ 0 for all g, h ∈ G. If the principal coefficients are symmetric this is a standard result but it also follows for non-symmetric c_{ij} by the dispersivity of H (see [ABR], Proposition 2.7). It is not a priori evident, however, that K_{t} is strictly positive and this complicates the ensuing arguments. To avoid this difficulty we replace K_{t} by K_{t}^{(δ)} = K_{t} + δ with δ ∈ ⟨0, 1] and eventually take the limit δ → 0. Now we turn to Nash’s arguments. The first important step is to deduce local lower bounds on the kernel.

The basic ingredient in Nash’s method is an L_{1}-lower bound

inf_{t∈⟨0,1]} inf_{g∈G} ∫_{G} dĥ K_{t}(g; h) ≥ c > 0 .  (8)

This is straightforward for pure second-order operators since

∫_{G} dĥ K_{t}(g; h) = 1 .

The lower-order terms present a difficulty, however, which can be circumvented by an argument of Stroock [Str1] which we apply to the modified kernels K^{(δ)}.

Choose a real χ ∈ C^{∞}(G) such that ∫_{G} dĥ e^{−χ(h)} = 1 and A_{i}χ ∈ L_{∞} for all i ∈ {1, …, d′}. If G is not compact a function with these properties can be constructed by regularization of the modulus g ↦ |g|′ with a C^{∞}-function. (See, for example, [Rob], pages 201–202.) Next fix g ∈ G and define ψ by setting ψ(h) = χ(hg^{−1}). Then ψ satisfies the properties required of χ and in addition ∥A_{i}ψ∥_{∞} = ∥A_{i}χ∥_{∞}. Further define k_{t} by setting k_{t}(h) = K_{t}^{(δ)}(g; h) = K_{t}(g; h) + δ. Next for each γ ∈ ⟨0, 1] introduce H_{γ}: ⟨0, ∞⟩ → ⟨0, ∞⟩ by

H_{γ}(t) = ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ}(h) ,

where k_{t}^{γ}(h) = (k_{t}(h))^{γ}. Note that the factor e^{−ψ} is necessary since k_{t}^{γ} ≥ δ^{γ}. Then

H′_{γ}(t) = −γ ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ−1}(h) (H̃_{h}K_{t})(g; h)

where H̃_{h} denotes the adjoint of H with respect to right Haar measure acting on the h variable. (Note that k_{t}^{γ−1} is bounded for γ ∈ ⟨0, 1⟩ because k_{t} ≥ δ. This together with the boundedness of h ↦ (H̃_{h}K_{t})(g; h) ensures that the integral exists.) Using the explicit form of H̃ we can now integrate by parts, i.e., evaluate H̃ in terms of the associated form. This operation causes no difficulty because we are assuming the coefficients are smooth and hence the kernel is a C^{∞}-function over G × G. One finds

H′_{γ}(t) = γ(1 − γ) ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ−2}(h) (A_{i}k_{t})(h) c_{ji}(h) (A_{j}k_{t})(h)

+ γ ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ−1}(h) (A_{i}ψ)(h) c_{ji}(h) (A_{j}k_{t})(h) + R

where the first terms on the right hand side denote the contribution to the differential equation of the terms in H̃ with two derivatives and R indicates the contribution of the lower order terms. Note that we have identified A_{j}k_{t} with the derivative in the j-th direction, with respect to the second variable, of K_{t}. Simple rearrangement then gives

H′_{γ}(t) = 4γ^{−1}(1 − γ) ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}(A_{i}k_{t}^{γ/2})(h) c_{ji}(h) (A_{j}k_{t}^{γ/2})(h)

+ 2 ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ/2}(h) (A_{i}ψ)(h) c_{ji}(h) (A_{j}k_{t}^{γ/2})(h) + R .

The first term on the right hand side, which only depends on the hermitian part of the matrix C = (c_{ij}) of principal coefficients, can now be bounded below using the subellipticity condition,

∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}(A_{i}k_{t}^{γ/2})(h) c_{ji}(h) (A_{j}k_{t}^{γ/2})(h) ≥ μ_{C} ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}|(A_{i}k_{t}^{γ/2})(h)|^{2} ,

and the second term can be bounded by an ε, ε^{−1} argument, e.g.,

| ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}k_{t}^{γ/2}(h) (A_{i}ψ)(h) c_{ji}(h) (A_{j}k_{t}^{γ/2})(h)|

≤ ∥C∥_{∞} ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}(ε |(A_{i}k_{t}^{γ/2})(h)|^{2} + (4ε)^{−1}k_{t}^{γ}(h)∥A_{i}χ∥_{∞}^{2})

for all ε > 0.

The lower order terms contained in the remainder R can be bounded in a similar manner with an ε, ε^{−1} argument. (Care has to be taken with the terms with no derivatives on the right hand side. These contain a factor K_{t}(g; h) but this can be replaced by k_{t}(h) at the cost of introducing an extra factor K_{t}(g; h)/K_{t}^{(δ)}(g; h). But this factor satisfies 0 ≤ K_{t}(g; h)/K_{t}^{(δ)}(g; h) ≤ 1 and consequently plays no role in the estimate.) Therefore one obtains a differential inequality

H′_{γ}(t) ≥ 2γ^{−1}(1 − γ)μ_{C} ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}|(A_{i}k_{t}^{γ/2})(h)|^{2} − ν_{γ}H_{γ}(t)  (9)

where ν_{γ} is a constant which depends only on γ, μ_{C}, ∥C∥_{∞}, ∥c∥_{∞}, ∥c_{0}∥_{∞} and ∥A_{i}χ∥_{∞}. Similarly,

H′_{1}(t) = ∑_{i,j=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}(A_{i}ψ)(h) c_{ji}(h) (A_{j}k_{t})(h) + R .

So there exists a c > 0 such that

H′_{1}(t) ≥ −c ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}|(A_{i}k_{t})(h)| − c ∫_{G} dĥ e^{−ψ(h)}k_{t}(h)

where the value of c depends only on μ_{C}, ∥C∥_{∞}, ∥c∥_{∞}, ∥c_{0}∥_{∞} and ∥A_{i}χ∥_{∞}. But

|(A_{i}k_{t})(h)| ≤ ε k_{t}^{γ−2}(h) |(A_{i}k_{t})(h)|^{2} + (4ε)^{−1}k_{t}^{2−γ}(h)

for any ε > 0. Therefore

H′_{1}(t) ≥ −4c γ^{−2}ε ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}|(A_{i}k_{t}^{γ/2})(h)|^{2}
− (4ε)^{−1}c d′ ∫_{G} dĥ e^{−ψ(h)}k_{t}^{2−γ}(h) − c ∫_{G} dĥ e^{−ψ(h)}k_{t}(h) .

Next using the upper bounds for K_{t} of Theorem 1.1, which have already been established, and making the choice ε = (2c)^{−1}γ (1 − γ) μ_{C} one deduces that there exists τ_{γ} > 0, whose value depends on γ, μ_{C}, ∥C∥_{∞}, ∥c∥_{∞}, ∥c_{0}∥_{∞} and ∥A_{i}χ∥_{∞} such that

H′_{1}(t) ≥ −2γ^{−1}(1 − γ) μ_{C} ∑_{i=1}^{d′} ∫_{G} dĥ e^{−ψ(h)}|(A_{i}k_{t}^{γ/2})(h)|^{2} − τ_{γ}t^{−(1−γ)D′/2}H_{1}(t)  (10)

uniformly for all t ∈ ⟨0, 1]. Hence

(H_{γ} + H_{1})′(t) ≥ −(ν_{γ} + τ_{γ}t^{−(1−γ)D′/2})(H_{γ} + H_{1})(t)

by (9) and (10). Thus with γ = 1 − (D′)^{−1} and ω_{γ} = ν_{γ} + τ_{γ} one has

(H_{γ} + H_{1})′(t) ≥ −ω_{γ}t^{−1/2}(H_{γ} + H_{1})(t)

and since this inequality is valid for all t ∈ ⟨0, 1] one may integrate between t_{0} and t and conclude that

log(H_{γ}(t) + H_{1}(t)) ≥ −2ω_{γ} + log(H_{γ}(t_{0}) + H_{1}(t_{0})) ≥ −2ω_{γ} + log H_{1}(t_{0}) .

But lim_{t_{0}→0} H_{1}(t_{0}) = e^{−ψ(g)} + δ ≥ e^{−χ(e)}. Therefore

H_{γ}(t) + H_{1}(t) ≥ c_{γ}

for some c_{γ} > 0, whose value depends on the coefficients only through μ_{C}, ∥C∥_{∞}, ∥c∥_{∞} and ∥c_{0}∥_{∞}, uniformly for g ∈ G and t ∈ ⟨0, 1]. Note that c_{γ} is independent of the parameter δ used to modify the kernel. Since H_{γ}(t) ≤ (H_{1}(t))^{γ}, it follows that H_{1}(t) ≥ c for a suitable c > 0, which is again independent of δ, uniformly for all t ∈ ⟨0, 1] and g ∈ G. Therefore

∫_{G} dĥ K_{t}(g; h) ≥ ∫_{G} dĥ e^{−ψ(h)}K_{t}(g; h) = H_{1}(t) − δ ≥ c − δ .

Finally taking the limit δ → 0 one concludes that (8) is valid.

Combination of the Gaussian upper bounds of of Theorem 1.1 and the L_{1}-lower bounds of (8) now give bounds that indicate that K_{t} is localized near the diagonal g = h uniformly for all small t.

Lemma 3.1 There exist c′, λ > 0 such that

∫_{B′(g;λt^{1/2})} dĥ K_{t}(g; h) ≥ c′

uniformly for all t ∈ ⟨0, 1] and g ∈ G.

Proof It follows from the Gaussian upper bounds on the kernel and a quadrature estimate (see, for example, [Rob] pages 223–224) that one has bounds

∫_{G\B′(g;λt^{1/2})} dĥ K_{t}(g; h) ≤ a e^{ωt}λ^{−1}

for all λ, t > 0, uniform in g ∈ G. Therefore using (8) one deduces that

∫_{B′(g;λt^{1/2})} dĥ K_{t}(g; h) ≥ c − a e^{ω}λ^{−1}

for all t ∈ ⟨0, 1] and g ∈ G. Consequently the statement of the lemma holds, with c′ = 2^{−1}c, for all λ ≥ 2 a c^{−1}e^{ω}. □

The next lemma is the key to pointwise local lower bounds. It is a version of an estimate first given by Nash [Nas] in his fundamental analysis of strongly elliptic operators on R^{d}. Nash’s idea was to use the relative entropy as a measure of localization. Nash considered the entropy of the semigroup kernel relative to a Gaussian measure and used its properties to obtain bounds

K_{1}(x; y) ≥ c > 0  (11)

for all x, y such that |x − y| ≤ κ. Then a scaling argument gives the crucial local bounds

K_{t}(x; y) ≥ c t^{−d/2}  (12)

for all t ∈ ⟨0, 1] and all x, y such that |x − y| ≤ κ t^{1/2}.

One can apply Nash’s arguments directly in the current context and deduce the Lie group equivalent of (11) but these bounds are difficult to exploit since one does not have scaling arguments. Therefore there is no obvious way of deducing the analogue of the bounds (12). But Saloff-Coste and Stroock [SaS] realized that this difficulty can be circumvented by considering the relative entropy with respect to a suitably chosen family of measures with compact support. We will closely follow their reasoning.

For r > 0 define ρ_{r}, σ_{r}: G → [0, ∞⟩ by ρ_{r} = (σ_{r})^{2} and σ_{r}(g) = 1 − |g|′r^{−1} if |g|′ ≤ r and σ_{r}(g) = 0 if |g|′ > r. Both ρ_{r} and σ_{r} are weakly^{∗} differentiable and it follows from the triangle inequality that ∑_{i=1}^{d′} ∥A_{i}σ_{r}∥_{∞}^{2} ≤ a r^{−2} for a suitable a > 0. Moreover, there exists a local weighted Poincaré inequality with density function ρ_{r}. Define the weighted average of φ by ⟨φ⟩_{r,ρ} = ( ∫_{G} dĝ ρ_{r}(g))^{−1} ∫_{G} dĝ ρ_{r}(g) φ(g).

Proposition 3.2 There exist R_{0} ∈ ⟨0, 1] and c > 0 such that

∫_{G} dĝ ρ_{r}(g)|φ(g) − ⟨φ⟩_{r,ρ}|^{2} ≤ c r^{2} ∫_{G} dĝ ρ_{r}(g) ∑_{i=1}^{d′} |(A_{i}φ)(g)|^{2}

uniformly for all r ∈ ⟨0, R_{0}] and φ ∈ L′_{2;1}(B′(2)).

Proof This has been proved essentially in the appendix of [SaS]. In [SaS] the group has polynomial growth, but for a general group all the estimates are valid locally, just as in Jerison [Jer]. □

This Poincaré inequality will be used in the derivation of the following result.

Lemma 3.3 There exist c, λ > 0 and t_{0} ∈ ⟨0, 1] such that

∫_{G} dĥ ρ_{λt^{1/2}}(h) log (t^{D′/2}K_{t}(g; h)) ≥ −c ∫_{G} dĥ ρ_{λt^{1/2}}(h)

uniformly for all t ∈ ⟨0, t_{0}] and g ∈ B′(4^{−1}λt^{1/2}).

Remark 3.4 It is not evident that the relative entropy defined by the integral on the left hand side of the above inequality is finite. But this will be established in the following proof.

Proof Let λ ≥ 1 and R_{0} ∈ ⟨0, 1] be as in Lemma 3.1 and Proposition 3.2. Set t_{0} = λ^{−2}R_{0}^{2}. Next fix t ∈ ⟨0, t_{0}] and g ∈ B′(4^{−1}λt^{1/2}) and for s, δ ∈ ⟨0, 1] set l_{s}^{(δ)}(h) = t^{D′/2}K_{st}^{(δ)}(g; h) where K_{t}^{(δ)} = K_{t} + δ as above. We simplify notation by setting ρ(h) = ρ_{λt^{1/2}}(h) and σ(h) = σ_{λt^{1/2}}(h). Then l_{s}^{(δ)} ≥ t^{D′/2}δ > 0 and we can introduce the functions Φ_{δ}: ⟨0, 1] → R by

Φ_{δ}(s) = ( ∫_{G} dĥ ρ(h))^{−1} ∫_{G} dĥ ρ(h) log l_{s}^{(δ)}(h) .

One must prove that for suitable c, λ > 0 one has Φ_{δ}(1) ≥ −c uniformly for all δ ∈ ⟨0, 1], t ∈ ⟨0, t_{0}] and g ∈ B′(4^{−1}λt^{1/2}). Once this is established it follows from the monotone convergence theorem, by setting δ = n^{−1} and taking the limit n → ∞, that the relative entropy integral exists. Moreover, in the limit δ → 0, the required bounds are valid.

Differentiating Φ_{δ} and then integrating by parts one finds

∫_{G} dĥ ρ(h) Φ′_{δ}(s) = −t ∫_{G} dĥ ρ(h) l_{s}^{(δ)}(h)^{−1}(H̃l_{s}^{(0)})(h)

= t{ ∫_{G} dĥ ρ(h) ∑_{i,j=1}^{d′}(A_{i} log l_{s}^{(δ)})(h) c_{ji}(h) (A_{j} log l_{s}^{(δ)})(h)

− 2 ∫_{G} dĥ ρ(h) ∑_{i,j=1}^{d′}(A_{i}σ)(h) c_{ji}(h) σ(h)(A_{j} log l_{s}^{(δ)})(h)} + R

where we have again explicitly exhibited the contribution to the differential equation of the part of H̃ containing two derivatives and used R to indicate the contribution of the lower order terms. Note that we have also used the identity A_{j}l_{s}^{(0)} = A_{j}l_{s}^{(δ)}.

Therefore by using subellipticity on the first term on the right and an ε, ε^{−1} estimate on the second term one obtains bounds

∫_{G} dĥ ρ(h) Φ′_{δ}(s) ≥ (1 − ε)μ_{C} t ∫_{G} dĥ ρ(h) ∑_{i=1}^{d′} |(A_{i} log l_{s}^{(δ)})(h)|^{2}

− ε^{−1}μ_{C}^{−1} ∥C∥_{∞}^{2} t ∫_{G} dĥ ρ(h) ∑_{i=1}^{d′} |(A_{i}σ)(h)|^{2} + R

for all ε ∈ ⟨0, 1]. Since ∑_{i=1}^{d′} ∥A_{i}σ∥_{∞}^{2} ≤ aλ^{−2}t^{−1} one then finds

∫_{G} dĥ ρ(h) Φ′_{δ}(s) ≥ (1 − ε) μ_{C} t ∫_{G} dĥ ρ(h) ∑_{i=1}^{d′} |(A_{i} log l_{s}^{(δ)})(h)|^{2}
− ε^{−1}μ_{C}^{−1} ∥C∥_{∞}^{2} a λ^{−2}|B′_{λt^{1/2}}| + R .

The remainder R can, however, be dealt with by similar ε, ε^{−1} estimates. (Care has to be taken again with the terms in H̃ with no derivatives on the right hand side. These contain a factor l_{s}^{(0)} but this can be replaced by l_{s}^{(δ)} at the cost of introducing an extra factor l_{s}^{(0)}/l_{s}^{(δ)}. But this factor takes values in [0, 1] and plays no essential role in the estimate which is expressed totally in terms of l^{(δ)} with all constants independent of δ.) Then, choosing ε appropriately, one obtains bounds

∫_{G} dĥ ρ(h) Φ′_{δ}(s) ≥ 2^{−1}μ_{C} t ∫_{G} dĥ ρ(h) ∑_{i=1}^{d′} |(A_{i} log l_{s}^{(δ)})(h)|^{2} − ν |B′_{λt^{1/2}}|

with the value of ν independent of δ, t and g. Next we use Proposition 3.2 to deduce that

∫_{G} dĥ ρ(h) ∑_{i=1}^{d′} |(A_{i} log l_{s}^{(δ)})(h)|^{2} ≥ a λ^{−2} t^{−1} ∫_{G} dĥ ρ(h)| log l_{s}^{(δ)}(h) − Φ_{δ}(s)|^{2}

where a is a constant independent of δ, t, g and the coefficients of H. Therefore combining these bounds and using a straightforward estimate on the integral of ρ one concludes that

Φ′_{δ}(s) ≥ μ( ∫_{G} dĥ ρ(h))^{−1} ∫_{G} dĥ ρ(h)| log l_{s}^{(δ)}(h) − Φ_{δ}(s)|^{2} − ν′

with the values of ν′ and μ dependent on μ_{C}, ∥C∥_{∞}, ∥c∥_{∞} and ∥c_{0}∥_{∞} but independent of the choice of δ, t and g. This is the Nash differential inequality, [Nas], Part II, which is ‘solved’ by repetition of his original arguments (see, for example, [Dav] p. 95). One needs bounds ∫_{B′(g;4^{−1}λt^{1/2})} dĥ l_{s}(h) ≥ c t^{D′/2} uniformly for all s ∈ ⟨0, 1], t ∈ ⟨0, t_{0}] and g ∈ B′(4^{−1}λt), but these have been proved under weaker restrictions in Lemma 3.1. The conclusion is that Φ_{δ}(1) ≥ −c uniformly for all δ ∈ ⟨0, 1], t ∈ ⟨0, t_{0}] and g ∈ B′(4^{−1}λt). Finally taking the limit δ → 0 gives the statement of the lemma. □

The required local lower bounds on the kernel follow straightforwardly from the entropy estimate.

Lemma 3.5 There exist a, λ > 0 and t_{0} ∈ ⟨0, 1] such that

K_{t}(g; h) ≥ a t^{−D′/2}

uniformly for all t ∈ ⟨0, t_{0}] and g, h ∈ G with |gh^{−1}|′ ≤ λ t^{1/2}.

Proof First note that as K_{t}(h; g) = K̃_{t}(g; h) where K̃_{t} is the kernel associated with the adjoint semigroup on the Lp̂-spaces it follows from Lemma 3.3 that one has estimates

∫_{G} dĥ ρ_{λt^{1/2}}(h) log (t^{D′/2}K_{t}(g; h)) ≥ −c ∫_{G} dĥ ρ_{λt^{1/2}}(h)
∫_{G} dĥ ρ_{λt^{1/2}}(h) log (t^{D′/2}K_{t}(h; g)) ≥ −c ∫_{G} dĥ ρ_{λt^{1/2}}(h)

uniformly for all t ∈ ⟨0, t_{0}] and g ∈ B′(λt^{1/2}), for suitable c, λ > 0 and t_{0} ∈ ⟨0, 1]. Now using the convolution semigroup property one has

t^{D′}K_{2t}(g; k) = ∫_{G} dĥ t^{D′/2}K_{t}(g; h)t^{D′/2} K_{t}(h; k)
≥ ∫_{G} dĥ ρ_{λt^{1/2}}(h) t^{D′/2}K_{t}(g; h) t^{D′/2}K_{t}(h; k) .

Therefore using concavity of the logarithm one deduces that

log (( ∫_{G} dĥ ρ_{λt^{1/2}}(h))^{−1}t^{D′}K_{2t}(g; k))

≥ ( ∫_{G} dĥ ρ_{λt^{1/2}}(h))^{−1} ∫_{G} dĥ ρ_{λt^{1/2}}(h)( log (t^{D′/2}K_{t}(g; h)) + log (t^{D′/2}K_{t}(h; k)))
≥ −2 c

uniformly for all t ∈ ⟨0, t_{0}] and g, k ∈ B′(λt^{1/2}). Hence

K_{2t}(g; k) ≥ e^{−2c}t^{−D′} ∫_{G} dĥ ρ_{λt^{1/2}}(h) ≥ a t^{−D′/2}

under the same restrictions on g, k and t. Now the restrictions on g and k can be weakened by noting that (g, k) ↦ K_{t}(gh; kh) is the kernel associated with the subelliptic operator H conjugated with right translations by h. Since the coefficients of the conjugated operator are the right translates of the original coefficients the ellipticity constant μ_{C} and the parameters ∥C∥_{∞} etc. are unchanged. Hence one has similar estimates

K_{2t}(gh; kh) ≥ a t^{−D′/2}

valid under the previous restrictions but for all h ∈ G. The statement of the lemma follows immediately. □

The final step in the proof of the Gaussian lower bounds on the kernel is to convert the local lower bounds into the Gaussian bounds by the use of the convolution semigroup property. This is achieved by a now standard procedure (see, for example, [Rob], Proposition III.5.2).

Finally we note that if one can establish the local lower bounds for all t > 0 then the Gaussian lower bounds of Theorem 1.1 are valid with ω′ = 0.

4 Concluding remarks

There are several direction in which the bounds of Theorem 1.1 can be improved under more restrictive circumstances and this leaves several questions unanswered. First if the matrix of principal coefficients c_{ij} of H is symmetric and one introduces the distance

d_{c}(g; h) = sup{ |ψ(g) − ψ(h)| : ψ ∈ C_{c}^{∞}(G) , ∑_{i=1}^{d′}(A_{i}ψ)c_{ij}(A_{j}ψ) ≤ 1, ψ real }

then there exists an ω ≥ 0 and for each ε > 0 an a_{ε} > 0 such that

|K_{t}(g; h)| ≤ a_{ε} t^{−D′/2}e^{ωt}e^{−d_{c}(g;h)^{2}(4(1+ε)t)^{−1}}

uniformly for g, h ∈ G and t > 0. These upper bounds have been established by a variety of arguments (see, for example, [Dav] [FaS]) and the argument of the Gaussian is near optimal. If, however, the matrix of principal coefficients is non-symmetric none of the reasonings appear to allow optimization of the argument.

Secondly, assume that a_{1}, …, a_{d′} is a vector space basis of 𝔤 and all coefficients of H are real. Then if the matrix of principal coefficients is symmetric and the c_{ij} are (left) uniformly continuous one can establish lower bounds

K_{t}(g; h) ≥ a_{ε} t^{−D′/2}e^{−ωt}e^{−d_{c}(g;h)^{2}(4(1−ε)t)^{−1}}

for each ε ∈ ⟨0, 1⟩ uniformly for g, h ∈ G and t > 0. These are derived by a variation of Nash’s original reasoning as developed by Stroock [Str2]. The arguments do not seem to extend to the subelliptic situation nor to the case that c_{ij} is non-symmetric.

Thirdly, it is possible to derive good large time bounds for groups of polynomial growth. If the operator H is pure second-order and c_{ij} is symmetric one has upper and lower bounds

a′ V (t)^{−1/2}e^{−b′(|gh^{−1}|′)^{2}t^{−1}} ≤ K_{t}(g; h) ≤ a V (t)^{−1/2}e^{−d_{c}(g;h)^{2}(4(1+ε)t)^{−1}}

uniformly for g, h ∈ G and t > 0 where V (t) = |B′(e; t)|. These follow from [SaS] together with a regularization argument (see, for example, [ElR3], Section 2.1). The distance d_{c} is, however, not necessarily optimal for large time bounds. The situation of non-symmetric principal coefficients is again unclear.

Acknowledgements

Parts of this work were carried out whilst the first author was visiting the Centre of Mathematics and its Applications at the ANU and the second author was visiting the Technical University in Eindhoven. Both authors wish to thank the respective institutes for supporting these visits.

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