Heat kernel and semigroup estimates for sublaplacians with drift on Lie groups

Publ. Mat. 49 (2005), 375–391 HEAT KERNEL AND SEMIGROUP ESTIMATES FOR SUBLAPLACIANS WITH DRIFT ON LIE GROUPS Nick Dungey Abstract Let Gbe a Lie group. The main new result of this paper is an estimate in L2(G) for the Davies perturbation of the semigroup generated by a centered sublaplacian Hon G. When Gis amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth. 1. Introduction In this paper we establish an estimate for the Davies perturbation of the semigroup generated by a centered sublaplacian on a Lie group. This result constitutes a new and useful form of L2off-diagonal estimate for such semigroups. From our result, we derive a new proof of a Gaussian estimate of Varopoulos [18] for the heat kernel of a centered sublaplacian, and also recover a more precise Gaussian estimate of Alexopoulos [2], [1] for the case where the Lie group has polynomial volume growth. The analyses of [18] and [2] are based on probabilistic methods to study the diffusion associated with a sublaplacian. The approach of the present paper, on the other hand, is functional-analytic rather than probabilistic. Note that while [18] and [2] make essential use of the detailed structure theory and geometry for amenable or polynomial growth Lie groups, we largely avoid this and for this reason we feel our approach is often technically simpler. Our methods can be extended to study convolution powers of centered probability densities, but this extension is not trivial and will be described elsewhere [7]. See [3], [18] for centered densities on Lie groups. 2000 Mathematics Subject Classification. 22E30, 35B40, 58J35. Key words. Lie group, heat kernel, Gaussian estimates, sublaplacian. 376 N. Dungey To state our results precisely we fix some notation. Let Gbe a connected Lie group. Denote by dg the left Haar measure on Gand let dbg=d(g−1) = ∆(g−1)dg be the corresponding right Haar measure where ∆: G→(0,∞) is the modular function. Let gbe the Lie algebra of G, consisting of all right invariant vector fields on G. Suppose that A0, A1,...,Ad0are elements of gsuch that A1,...,Ad0algebraically generate the Lie algebra g, and consider the subelliptic sublaplacian H=− d0 X i=1 A2 i+A0 with drift term A0. It is well known (see for example [14, Section IV.4]) that Hgenerates a contraction semigroup St=e−tH in the spaces Lp:= Lp(G;dg), 1 ≤p≤ ∞, and we denote by Kt:G→(0,∞) the corresponding heat kernel which satisfies (1) (Stf)(g)=(Kt∗f)(g)=ZG dh Kt(h)f(h−1g)=ZG db h Kt(gh−1)f(h) for all t > 0, g∈Gand f∈C∞ c(G). The notion of centeredness is defined as follows (cf. [2], [18]). Let G0be the closure of the group [G, G] in G, and consider the canonical homomorphism π0:G→G/G0. Note that G/G0is a connected abelian Lie group and it can therefore be written as a direct product Rn1×Tn2, where T=R/Z. Set G1=π−1 0({0} × Tn2)⊆Gand let g1be the Lie algebra of G1. Then g1is an ideal of gand [g,g]⊆g1. (If Gis simply connected, one just has g1= [g,g].) Observe that G/G1∼ =Rn1. One says that H=−Pd0 i=1 A2 i+A0is centered if A0∈g1. It is not difficult to show that His centered if and only if Hη = 0 for every continuous homomorphism η:G→R(one observes that any such homomorphism vanishes on G1, so that A0η= 0 when A0∈g1). By a well known lemma (see [2, Section 1]), every sublaplacian Hon G is conjugate via a character with a centered sublaplacian. Explicitly, one can find a multiplicative character Φ: G→(0,∞), so Φ(gh) = Φ(g)Φ(h), and a constant β≥0 such that H= Φ−1(Hc+β)Φ where Hcis a centered sublaplacian. In this sense, the study of general sublaplacians essentially reduces to the centered case. Let us state our basic result for the Davies perturbation of the semigroup St. Let R+(G) denote the algebra of all real, right invariant differential operators on Gwithout constant term, so R+(G) is linearly spanned by all monomials X1...Xjwhere j≥1 and X1,...,Xj∈g. Introduce the set Econsisting of all C∞-smooth functions ψ:G→R Sublaplacians with Drift 377 such that Pψ ∈L∞for all P∈ R+(G); note that ψitself may be unbounded. We write k · kp→qfor the norm of a bounded linear operator from Lp=Lp(G;dg) to Lq. Theorem 1.1. Let Hbe centered, let ψ∈ E and set Hλ=eλψHe−λψ. Then there exists k > 0such that Re(Hλf, f)≥ −kλ2kfk2 2 for all λ∈Rand f∈C∞ c(G). Moreover, Hλextends to the generator of a semigroup e−tHλ=eλψSte−λψ in L2satisfying ke−tHλk2→2≤ekλ2t for all t > 0and λ∈R. Davies perturbation estimates are well known for various classes of elliptic or subelliptic operators on manifolds: see, for example, [5], [14] and [19]. In our situation, however, the drift term A0creates new and non-trivial difficulties in the proof. These can be overcome by carefully utilizing the properties of the class Edefined above. When Gis amenable, the estimates of Theorem 1.1 fail for noncentered sublaplacians. We shall give a precise version of this statement in Section 2 below. The main fact about amenability needed there is that Gis amenable if and only µ0= 0, where (2) µ0:= inf 06=f∈C∞ c(G)Pd0 i=1 kAifk2 2 kfk2 2 is the bottom of the L2spectrum of the symmetric sublaplacian − Pd0 i=1 A2 i. If Gis non-amenable, the statements of Theorem 1.1 hold for all sublaplacians, centered or not, as an easy consequence of µ0>0 (see remarks of Section 2 below). See also [18] for discussions of the role of amenability. We define a distance on Gin the following standard way. Fix a compact neighborhood Uof the identity eof Gwhich is symmetric (U=U−1) and define ρ:G→N={1,2,3,...}by ρ(g) = inf{n∈N:g∈Un}, g ∈G, where Un:= {g1g2···gn:gj∈U}. Although ρ /∈ E in general, we shall construct in Section 3 a function ψ0∈ E with the same global growth as ρ, that is, with c−1ρ≤ψ0≤cρ for some c > 1. Together with Theorem 1.1, this leads to a new proof of the following Gaussian estimate of Varopoulos [18]. 378 N. Dungey Theorem 1.2 ([18]).Let Hbe centered. Then there are c, b > 0such that Kt(g)≤c∆(g)−1/2e−bρ(g)2/t for all t≥1,g∈G. Recall (cf. [10]) that Gis said to have polynomial growth of order Dif an estimate c−1nD≤dg(Un)≤cnDholds for some c > 1 and all n∈N; alternatively, if dg(Un)≥aean for some constant a > 0 and all n∈N, then we say Ghas exponential volume growth. We shall give a new proof of the following theorem which was proved by Alexopoulos [2] with a quite difficult proof. (In the special case where Gis nilpotent, an easier proof is given by Melzi [13]. The possibility of using Davies perturbation estimates to prove Alexopoulos’ theorem is mentioned in [18, p. 435] without giving details.) Theorem 1.3 ([2]).Suppose Ghas polynomial growth of order D, and let Hbe centered. Then there are c, b > 0such that Kt(g)≤ct−D/2e−bρ(g)2/t for all t≥1and g∈G. Our proof of Theorem 1.3 is based on Theorem 1.1 and on ideas of [6] and involves certain weighted Nash inequalities for convolution operators. The semigroup St=e−tH generated by a sublaplacian with drift is not bounded analytic in general (it is, however, analytic if A0is in the linear span of the fields {Ai,[Ai, Aj]: i, j ∈ {1,...,d0}}: see [9], [15], [16] for small time estimates in this case). Thus the estimate kHStk2→2=k∂tStk2→2≤ct−1,t > 0, fails in general. Nevertheless, when His centered we have the following interesting large time regularity result for St. Theorem 1.4. Let Hbe centered and let p∈(1,∞). Then there is c=c(p)>0such that kHStkp→p≤ct−1for all t≥1. From Theorem 1.4 and since Pd0 i=1 kAiStfk2 2= Re(HStf, Stf) for all f∈L2, one deduces the estimate of spatial derivatives kAiStk2→2≤ct−1/2 for all i∈ {1,...,d0},t≥1, when His centered. This estimate is apparently new for centered sublaplacians on general Lie groups. In the particular case where Ghas polynomial growth, Theorem 1.4 is contained in results of [2] (which also imply the analogous estimate Sublaplacians with Drift 379 for p= 1 and p=∞). Our proof of Theorem 1.4 for general Gwill be given elsewhere, since the proof seems best approached via a more general study of regularity of the convolution powers K(n)=K∗K∗· · ·∗K of a fixed probability density Kon G, rather than by analyzing directly the sublaplacian H. 2. Proof of Theorem 1.1 In this section, we prove Theorem 1.1, and then give a converse result showing that similar estimates fail for non-centered sublaplacians. In general, c,c0and so on denote positive constants whose value may change from line to line when convenient. Also, sums over the variable i are always taken over the range i∈ {1,...,d0}unless otherwise indicated. A key idea in the proof of Theorem 1.1 is to write the vector field A0∈ g1in terms of second or higher order derivatives (or differences). To do this we require the following algebraic result. Proposition 2.1. There exists a compact, connected, abelian, possibly trivial subgroup Cof Gsuch that g1=c+[g,g], where cis the Lie algebra of Cand +denotes a sum of vector spaces which is not necessarily direct. Some related descriptions of the algebra g1are given in [18, Appendix] in the case where Gis amenable. For completeness, we give a proof of Proposition 2.1 in the Appendix of this paper. We remark that compactness of C, but not the fact that Cis abelian, will be essential in the arguments to follow. To begin the proof of Theorem 1.1, let us fix ψ∈ E. The norm estimate ke−tHλk2→2≤ekλ2tfollows in a routine manner from the first estimate of the theorem. Indeed, using the first estimate one obtains the differential inequality (d/dt)(ke−tHλfk2 2) = −2 Re(Hλe−tHλf, e−tHλf)≤2kλ2ke−tHλfk2 2 which implies the desired norm estimate (compare, for example, [5, Section 4]). Thus we concentrate on proving the first estimate of the theorem. Observing the basic identity eλψX(e−λψf) = Xf −λ(Xψ)ffor λ∈R, X∈g,f∈C∞ c(G), since H=−PiA2 i+A0we find that Hλf=−X i A2 if+A0f+λX i (Aiψ)Aif+λX i Ai((Aiψ)f) −λ2X i (Aiψ)2f−λ(A0ψ)f. 380 N. Dungey Then (Hλf, f) = X i kAifk2 2+ (A0f, f) + λX i (Aif, (Aiψ)f) −λX i ((Aiψ)f, Aif)−λ2X i k(Aiψ)fk2 2−λ((A0ψ)f, f). Because the terms (A0f, f) and (Aif, (Aiψ)f)−((Aiψ)f, Aif) are purely imaginary, then Re(Hλf, f) = X i kAifk2 2−λ2X i k(Aiψ)fk2 2−λ((A0ψ)f, f) ≥X i kAifk2 2−cλ2kfk2 2−λ((A0ψ)f, f), (3) where we used the finiteness of the norms kAiψk∞,i∈ {1,...,d0}. Our main task will be to prove an estimate (4) |((A0ψ)f, f)| ≤ c0X i kAifk2kfk2 for all f∈C∞ c(G). Note the elementary estimate that |λ|kAifk2kfk2≤2−1εkAifk2 2+ 2−1ε−1λ2kfk2 2 for all ε > 0, λ∈Rand i∈ {1,...,d0}. Then by choosing ε > 0 sufficiently small and using (3) and (4), we obtain for some k > 0 an estimate of the desired form Re(Hλf, f)≥ −kλ2kfk2 2,λ∈R. Thus, it remains to establish (4). Apply Proposition 2.1 to decompose A0=A0 0+A00 0with A0 0∈[g,g] and A00 0∈c. We will estimately the terms ((A0 0ψ)f, f) and ((A00 0ψ)f, f) separately. Note that [g,g] is linearly spanned by all commutators of the form [Ai1,...,Aik] with k≥2 and i1,...,ik∈ {1,...,d0}, since A1,...,Ad0 generate the Lie algebra g. Therefore A0 0∈[g,g] is expressible as a sum of terms each of the form AiPfor some i∈ {1,...,d0}and P∈ R+(G). Note that ((AiPψ)f, f) = (Ai((Pψ)f), f)−((Pψ)(Aif), f) =−((Pψ)f, Aif)−(Aif, (Pψ)f) and hence |((AiPψ)f, f)| ≤ 2kP ψk∞kfk2kAifk2. Since kP ψk∞is finite for any P∈ R+(G), this argument shows that |((A0 0ψ)f, f)| ≤ cX i kAifk2kfk2 for all f∈C∞ c(G). Sublaplacians with Drift 381 To handle the term ((A00 0ψ)f, f), define an operator Pacting on C∞ functions ϕ:G→Rby (Pϕ)(g) = ZC ds ϕ(sg) = ZC ds L(s)ϕ(g), g ∈G, where ds denotes Haar measure on the compact group C= exp(c) normalized so that ds(C) = 1. Here, Lis the left regular representation of Gwhich acts by the formula (L(g)ϕ)(h) = ϕ(g−1h), g, h ∈G. Since A00 0∈cit is easy to see that PA00 0ϕ= 0. Defining the difference operators ∂g:= I−L(g) for g∈G, we may therefore write the operator A00 0in the form A00 0= (I− P)A00 0=ZC ds ∂sA00 0. Observing the general identity ∂g(f1f2) = (∂gf1)f2+(L(g)f1)(∂gf2), we have ((∂sA00 0ψ)f, f) = (∂s((A00 0ψ)f), f)−((L(s)A00 0ψ)∂sf, f) = ((A00 0ψ)f, ∂s−1f)−((L(s)A00 0ψ)∂sf, f) for all s∈C, since ∂s−1is adjoint to ∂s. Since kL(s)A00 0ψk∞=kA00 0ψk∞ is finite and k∂sfk2=k∂s−1fk2, we see that |((∂sA00 0ψ)f, f)| ≤ 2kA00 0ψk∞kfk2k∂sfk2 for all s∈C. Recall the standard inequality (cf. [14, pp. 267–268]) (5) k∂gfkp≤cρ(g)X i kAifkp which is valid for all g∈Gand p∈[1,∞]. From the above observations, and the compactness of Cwhich implies that sup{ρ(s): s∈C}is finite, we deduce that |((A00 0ψ)f, f)| ≤ ZC ds |((∂sA00 0ψ)f, f)| ≤ cX i kAifk2kfk2 for all f∈C∞ c(G). This completes the proof of (4) and of Theorem 1.1. Remarks. – In general, the constant kin Theorem 1.1 depends on the choice of ψ∈ E. However, if E0is some subset of Ewhich is uniformly bounded, in the sense that sup ψ∈E0 kPψk∞<∞ 382 N. Dungey for each P∈ R+(G), then it easily follows from the above proof that one can choose the same constant kuniformly for all ψ∈ E0. – The proof of the theorem, and (3), yield the following stronger inequality for (Hλf, f): for each ε∈(0,1) there is a c(ε)>0 such that (6) Re(Hλf, f)≥(1 −ε)X i kAifk2 2−c(ε)λ2kfk2 2 for all f∈C∞ cand λ∈R. This inequality will be needed in the proof of Theorem 1.3. For a symmetric sublaplacian (that is, A0= 0) one can choose ε= 0 in (6), as follows from (3). We do not know if one can take ε= 0 for a general centered sublaplacian. – For an arbitrary sublaplacian (possibly non-centered), inequality (3) implies the crude estimate Re(Hλf, f)≥(µ0−cλ2−c0|λ|)kfk2 2 for λ∈R,f∈C∞ c(G), where µ0is as in (2) and c0=kA0ψk∞. If Gis non-amenable, then since µ0>0 we see that the estimates of Theorem 1.1 hold even for non-centered sublaplacians. We next prove a result converse to Theorem 1.1. Theorem 2.2. Suppose Gis amenable and let Hbe a sublaplacian which is not centered. Then the estimate of Theorem 1.1 fails when λis close to zero. More precisely, there exist a homomorphism Φ: G→R(so Φ(gh) = Φ(g)+Φ(h),g, h ∈G) with Φ∈ E, and constants α, β > 0such that Hλ:= eλΦHe−λΦsatisfies inf 06=f∈C∞ c Re(Hλf, f) kfk2 2 =−βλ2−αλ and (7) ke−tHλk2→2=e(βλ2+αλ)t for all λ∈Rand t > 0. It follows that sup{e−kλ2tke−tHλk2→2:t≥ 1,0< λ ≤t−1/2}=∞for any number k > 0. Proof: Consider the homomorphism π1:G→G/G1∼ =Rn1and identify G/G1with Rn1. The vector fields A0 i:= dπ1(Ai), i∈ {0,1,...,d0}, are constant coefficient fields on Rn1, and A0 06=0 because His not centered. Therefore one can find b∈Rn1such that the function F(x) := hb, xi satisfies A0 0F= 1, where h·,·i is the usual inner product in Rn1. Because A1,...,Ad0generate g, the fields A0 1,...,A0 d0must linearly span Sublaplacians with Drift 383 the tangent space of Rn1, so at least one of the constants λi:= A0 iF, i∈ {1,...,d0}, is non-zero. Then Φ := F◦π1:G→Ris a homomorphism satisfying AiΦ = λi, i∈ {1,...,d0}, and A0Φ = 1. Clearly PΦ is constant for any P∈R+(G), so Φ ∈ E. Note that β:= Pd0 i=1 λ2 i>0, and by calculating as in the proof of Theorem 1.1, Hλ=eλΦHe−λΦ=−X i A2 i+A0+ 2 X i (λλi)Ai−(βλ2+λ). Then Re(Hλf, f)/(kfk2 2) = PikAifk2 2/(kfk2 2)−(βλ2+λ) and taking infimums over f∈C∞ c(G) yields the first statement of the theorem, since amenability means that µ0= 0 in (2). That ke−tHλk2→2≤e(βλ2+λ)tfollows from the first statement of the theorem. To prove the reverse inequality, consider H0 λ:= dπ1(Hλ) which is an elliptic operator on Rn1with constant coefficients and with constant term −βλ2−λ. It is easy to see, via the Fourier theory of L2(Rn1), that ke−tH0 λk2→2=e(βλ2+λ)t. Since Gis amenable, a well known transference theorem [4, Theorem 2.4] gives ke−tH0 λk2→2≤ ke−tHλk2→2, and (7) is proved. The final statement of the theorem follows directly from (7) upon choosing λ∼t−1/2. 3. A smooth distance The aim of this section is to prove the following lemma providing a smooth distance function ψ0∈ E on any connected Lie group. This is required for the applications of Theorem 1.1. Let ρ:G→Nbe defined as in Section 1. Lemma 3.1. There exists a ψ0∈ E satisfying ψ0(g)≥1,ψ0(g) = ψ(g−1),ψ0(gh)≤ψ0(g) + ψ0(h)and c−1ρ(g)≤ψ0(g)≤cρ(g)for all g, h ∈Gand some constant c > 1. As an aside, the function of Lemma 3.1 automatically satisfies Pψ0∈ L∞for left invariant differential operators Pwithout constant term on G. (This follows easily from ψ0(g)=ψ0(g−1) since the inversion map g7→g−1 intertwines left invariant with right invariant operators.) Proof of Lemma 3.1: Consider the modulus ρB:G→[0,∞) associated with a fixed vector space basis B1,...,BNof the Lie algebra g. Then ρB(g) is the distance from gto e, with respect to the right invariant Riemannian metric on Gsuch that B1,...,BNare orthonormal. The functions ρand ρBare equivalent at infinity (cf. [19, p. 41]) in the sense 390 N. Dungey References [1] G. K. Alexopoulos, Sous-laplaciens et densit´es centr´es sur les groupes de Lie `a croissance polynomiale du volume, C. R. Acad. Sci. Paris S´er. I Math. 326(5) (1998), 539–542. [2] G. K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc. 155(739) (2002), 101 pp. [3] G. K. Alexopoulos, Centered densities on Lie groups of polynomial volume growth, Probab. Theory Related Fields 124(1) (2002), 112–150. [4] R. R. Coifman and G. Weiss,“Transference methods in analysis”, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 31, American Mathematical Society, Providence, R.I., 1976. [5] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132(1) (1995), 141–169. [6] N. Dungey, On Gaussian kernel estimates on groups, Colloq. Math. 100(1) (2004), 77–90. [7] N. Dungey, Properties of random walks on discrete groups: time regularity and off-diagonal estimates, Preprint (2005). [8] N. Dungey, A. F. M. ter Elst and D. W. Robinson,“Analysis on Lie groups with polynomial growth”, Progress in Mathematics 214, Birkh¨auser Boston, Inc., Boston, MA, 2003. [9] A. F. M. ter Elst and D. W. Robinson, Weighted subcoercive operators on Lie groups, J. Funct. Anal. 157(1) (1998), 88–163. [10] Y. Guivarc’h, Croissance polynomiale et p´eriodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379. [11] W. Hebisch, Estimates on the semigroups generated by left invariant operators on Lie groups, J. Reine Angew. Math. 423 (1992), 1–45. [12] G. Hochschild,“The structure of Lie groups”, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. [13] C. Melzi, Large time estimates for heat kernels in nilpotent Lie groups, Bull. Sci. Math. 126(1) (2002), 71–86. [14] D. W. Robinson,“Elliptic operators and Lie groups”, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. [15] N. Th. Varopoulos, Small time Gaussian estimates of heat diffusion kernels. I. The semigroup technique, Bull. Sci. Math. 113(3) (1989), 253–277. Sublaplacians with Drift 391 [16] N. Th. Varopoulos, Small time Gaussian estimates of heat diffusion kernels. II. The theory of large deviations, J. Funct. Anal. 93(1) (1990), 1–33. [17] N. Th. Varopoulos, Analysis on Lie groups, Rev. Mat. Iberoamericana 12(3) (1996), 791–917. [18] N. Th. Varopoulos, Geometric and potential theoretic results on Lie groups, Canad. J. Math. 52(2) (2000), 412–437. [19] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis and geometry on groups”, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992. School of Mathematics The University of New South Wales Sydney 2052 Australia E-mail address:[email protected] Primera versi´o rebuda el 15 de desembre de 2004, darrera versi´o rebuda el 22 d’abril de 2005.