Subelliptic Li-Yau estimates on three dimensional model spaces

arXiv:0806.2547v1 [math.AP] 16 Jun 2008 Subelliptic Li-Yau estimates on three dimensional model spaces Dominique Bakry ∗, Fabrice Baudoin †, Michel Bonnefont ‡, Bin Qian§ Institut de Mathématiques de Toulouse Université de Toulouse CNRS 5219 June 16, 2008 Abstract We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU (2), Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the Γ2 techniques that we adapt to those elementary model spaces. The important feature developed here is that although the usual notion of Ricci curvature is meaningless (or more precisely leads to bounds of the form −∞ for the Ricci curvature), we describe a parameter ρ which plays the same rôle as the lower bound on the Ricci curvature, and from which one deduces the same kind of results as one does in Riemannian geometry, like heat kernel upper bounds, Sobolev inequalities and diameter estimates. 1 Framework and Introduction The estimation of heat kernel measures is a topic which had been under thorough investigation for the last thirty years at least, see [12, 8]. Among the many techniques developed for that, the famous Li-Yau parabolic inequality [12] is a very powerful tool, which relies in Riemannian geometry bounds on the gradient on heat kernels to lower bounds on the Ricci curvature. More precisely, in the simplest form, it asserts that, if E is a smooth Riemannian manifold with dimension n and non negative Ricci curvature, then if f is any positive solution of the heat equation ∂t f = ∆f, where ∆ is the Laplace Beltrami operator of E, then, if u = log f ∂t u ≥ |∇u|2 − ∗ n t. 2 [email protected] [email protected] ‡ [email protected] § [email protected], This author would like to express sincere thanks to China Scholarship council for financial support † 1 This is a very precise and powerful estimate. For the model case, which is here the Euclidean space E = Rn and when f is the heat kernel (that is the solution of the heat equation starting at time t = 0 from a Dirac mass), then this inequality is in fact an equality. From this inequality, one may easily deduce Harnack inequalities and hence precise bounds on the heat kernel. Many generalizations of this inequality have been developed, all of them including lower bounds on the Ricci tensor. In particular, it works for a general elliptic operator L under the assumption that it satisfies a curvature-dimension inequality CD(ρ, n), which is the furthermost generalization on the notion of lower bound on the Ricci curvature, see [6, 4]. In the non elliptic case, things appear to be infinitely more complicated. In particular, most of the hypoelliptic systems do not satisfy any CD(ρ, n) inequality (any reasonable notion of lower bound on the Ricci tensor leads to the value −∞). Nevertheless, some Li-Yau inequalities may be obtained [9]. In what follows, we shall use the Γ2 techniques developed in [4] to produce these Li-Yau bounds. The method developed here works quite well on the simple models developed here (Heisenberg groups, SU (2), SL(2)), but could be easily generalized to a larger class of hypoelliptic operators. We shall not try to present here the most general results, but concentrate for simplicity on the three model cases mentioned above. In fact, they should be thought of as the analogous of the model spaces of Riemmanian geometry (Euclidean spaces, Spheres and Hyperbolic spaces). In all what follows, given an elliptic second order operator L on a smooth manifold, with no constant term, one defines 1 Γ(f, g) = (L(f g) − f Lg − gLf ) 2 which stands for ∇f · ∇g in the Riemannian case, and the curvature dimension inequality is defined from the operator Γ2 Γ2 (f, f ) = 1 (Γ(f, f ) − 2Γ(f, Lf ). 2 Then, L is said to satisfy a CD(ρ, n) inequality if, for any smooth function f , one has Γ2 (f, f ) ≥ ρΓ(f, f ) + 1 (Lf )2 . n The parabolic Li-Yau inequality is then described in terms of the quantity |∇f |2 = Γ(f, f ) and the parameters ρ and n. For the Laplace Beltrami operator L = ∆ on a smooth Riemannian manifold, this amounts to say that the dimension is at most n and that the Ricci curvature is bounded below by ρ. In the hypoelliptic models that we describe below, however, no such inequality holds (the best possible constant ρ is −∞), but we shall produce some analogous of the Li-Yau inequality through a parameter ρ which therefore plays the rôle of a substitute for the Ricci curvature. In what follows we consider a three-dimensional Lie group G with Lie algebra g and we assume that there is a basis {X, Y, Z} of g such that [X, Y ] = Z [X, Z] = −ρY 2 [Y, Z] = ρX where ρ ∈ R. Example 1.1 (SU(2), ρ = 1) The Lie group SU(2) is the group of 2 × 2, complex, unitary matrices of determinant 1. Its Lie algebra su(2) consists of 2×2, complex, skew-adjoint matrices of trace 0. A basis of su(2) is formed by the Pauli matrices:       1 1 1 0 1 0 i i 0 X= , Y = ,Z = , 0 i 0 0 −i 2 −1 2 2 for which the following relationships hold [X, Y ] = Z, [X, Z] = −Y, [Y, Z] = X. (1.1) Example 1.2 (Heisenberg group, ρ = 0) The Heisenberg group H is the group of 3 × 3 matrices:   1 x z  0 1 y  , x, y, z ∈ R. 0 0 1 The Lie algebra of H is spanned by the matrices    0 0 0 1 0    0 0 0 0 0 , Y = X= 0 0 0 0 0 for which the following equalities hold   0 0   0 1 and Z = 0 0 0 0 0  1 0 , 0 [X, Y ] = Z, [X, Z] = [Y, Z] = 0. Example 1.3 (SL(2), ρ = −1) The Lie group SL(2) is the group of 2 × 2, real matrices of determinant 1. Its Lie algebra sl(2) consists of 2 × 2 matrices of trace 0. A basis of sl(2) is formed by the matrices:       1 1 1 1 0 0 1 0 1 , Y = , Z= , X= 0 −1 1 0 0 2 2 2 −1 for which the following relationships hold [X, Y ] = Z, [X, Z] = Y, [Y, Z] = −X. (1.2) We consider on the Lie group G the subelliptic, left-invariant, second order differential operator L = X 2 + Y 2, as well as the heat semigroup Pt = etL . We also set 1 Γ(f, f ) = (Lf 2 − 2f Lf ) = (Xf )2 + (Y f )2 , 2 3 and Γ2 = 1 (LΓ(f, f ) − 2Γ(f, Lf )). 2 In the present setting, Γ2 (f, f ) = (X 2 f )2 + (Y 2 f )2 + 1 1 ((XY + Y X)f )2 + (Zf )2 + ρΓ(f, f ) − 2(Xf )(Y Zf ) + 2(Y f )(XZf ). 2 2 (1.3) The mixt terms −2(Xf )(Y Zf )+2(Y f )(XZf ) prevents to find any lower bound on this quantity involving Γ(f, f ) and (Lf )2 only, whence the absence of any CD(ρ, n) inequality. 2 Li-Yau type estimates for the heat semigroup The classical method of Li and Yau [12] consists in applying the maximum principle to a carefully chosen expression. The method developed in [4] is quite different. Considering a positive solution of the heat equation ∂t f = Lf , and denoting f 7→ Pt f the associated heat kernel, one writes u = log f and look at the expression Φ(s) = Ps (f (t − s)Γ(u(t − s), u(t − s))), defined for 0 < s < t. Then, one obtains through the CD(ρ, n) inequality a differential inequality Φ′ (s) ≥ (AΦ(s) + B)2 + C, where A, B, C are expressions which are constant in t but may depend on the function f . Then, the parabolic Li-Yau inequality is obtained as a consequence of this differential inequality. Here, we shall develop this method a bit further, looking at more complicated quantities like Ps (f (t − s)(a(s)Γ(u(t − s), u(t − s)) + b(s)(Zu(t − s))2 )), and try to get some differential inequality on it. The computations developed here are not restricted to Lie group, since we only use an generalized CD(ρ, n) inequality. There are many hypoelliptic systems that may be treated under the same lines. The reason why we restrict ourselves to those model cases described previously are mainly for pedagogical reasons. We have the following inequality, which is our technical starting point: Proposition 2.1 Let f : G → R be positive. Let t > 0, for all x ∈ G and s ∈ [0, t], consider the expressions Φ1 (s) = Ps ((Pt−s f )Γ(ln Pt−s f ))(x) and Φ2 (s) = Ps ((Pt−s f )(Z ln Pt−s f )2 )(x). Then, for every differentiable, non-negative and decreasing function b : [0, t] → R,
′ −b′ Φ1 + bΦ2 (s) ≥ −b′ (s)  !  2  b′ (s) 1 b′′ (s) b′ (s) b′′ (s) +2 + 2ρ LPt f (x) − +2 + 2ρ Pt f (x) . b′ (s) b(s) 4 b′ (s) b(s) 4 Proof. We fix a positive function f , t > 0 and we perform all the following computations at a given point x. With the same notations as Proposition 2.1, straightforward (but quite tedious) computations show that Φ′1 (s) = 2Ps ((Pt−s f )Γ2 (ln Pt−s f )) and Φ′2 (s) = 2Ps ((Pt−s f )Γ(Z ln Pt−s f )). For the last equality we use the crucial facts that [L, Z] = 0 and X(f )Z(f )[X, Z](f ) + Y (f )Z(f )[Y, Z](f ) = 0. Now, thanks to the Cauchy-Schwarz inequality, the expression (1.3), shows that for every λ > 0, and every smooth function g,   1 1 1 2 2 Γ2 (g) ≥ (Lg) + (Zg) + ρ − Γ(g) − λΓ(Zg). 2 2 λ We therefore obtain the following differential inequality Φ′1 (s)  2 ≥ Ps ((Pt−s f )(L ln Pt−s f ) ) + Φ2 (s) + 2ρ − λ 2  Φ1 (s) − λΦ′2 (s). We now have that for every γ ∈ R, (L ln Pt−s f )2 ≥ 2γL ln Pt−s f − γ 2 , and L ln Pt−s f = Γ(Pt−s f ) LPt−s f − . Pt−s f (Pt−s f )2 Thus, for every λ > 0 and every γ ∈ R,   2 ′ Φ1 (s) ≥ 2ρ − − 2γ Φ1 (s) + Φ2 (s) − 2λΦ′2 (s) + 2γLPt f − γ 2 Pt f. λ Now for two functions a and b defined on the time interval [0, t) with a positive, we have   2 ′ ′ (aΦ1 + bΦ2 ) ≥ a + (2ρ − − 2γ)a Φ1 + (a + b′ )Φ2 + (−aλ + b)Φ′2 + 2aγLPt f − aγ 2 Pt f. λ So, if b is a positive decreasing function on the time interval [0, t), by choosing in the previous inequality a = −b′ , b λ = − ′, b 5 and 1 γ= 2 we get the desired result.   b′′ b′ + 2 + 2ρ , b′ b  As a first corollary, by using the function b(s) = (t − s)α , α>2 and integrating from 0 to t, we deduce Corollary 2.2 For all α > 2, for every positive function f and t > 0,   3α − 1 2ρt LPt f ρ2 t ρ(3α − 1) (3α − 1)2 1 t − + − + . Γ(ln Pt f ) + (Z ln Pt f )2 ≤ α α−1 α Pt f α α−1 α−2 t Observe that this takes a simpler form when ρ ≥ 0, since then one can use proposition 2.1 with ρ = 0 and get Corollary 2.3 When ρ ≥ 0, there exist constants A, B and C such that, with u = ln(Pt f ) ∂t u ≥ AΓ(u) + Bt(Zu)2 − C . t In particular, one gets ∂t u ≥ −C/t, which gives Pt f ≤ t−C P1 f. On the Heisenberg group, one sees that the behavior of Pt f when t goes to 0 is of order t−2 (a simple dilation argument shows that). Therefore, one sees that the optimal constant C in the previous inequality is C = 2. Unfortunately, it can be shown by some elementary considerations similar to those developed in the proof of corollary 2.5 that the best constant one may obtain from the previous proposition shall always produce a constant C > 2. This is a strong difference with the classical parabolic Li-Yau inequality where the inequality ∂t u ≥ − n 2t gives the right order of magnitude of the heat kernel near t = 0. Now when ρ > 0, we easily get an exponential decay by using the function:  2ρs  2ρt α b(s) = e− 3α − e− 3α , α > 2. This writes: Corollary 2.4 For every α > 2, for every positive function f , x ∈ G and t > 0,
2ρt 4ρt 1 2 2ρt LPt f (x) 1 − 3 1 − e− 3α e− 3α 3α − 1 3 2 − 3α 3α Γ(ln Pt f )(x) + (Z ln Pt f ) (x) ≤ e + ρ 2ρt . 2 ρ α−1 Pt f (x) 2 1 − α2 1 − e− 3α Moreover for ρ > 0 and t large, with more work we actually can do better. 6 Corollary 2.5 Let us assume ρ > 0. There exist t0 > 0 and C > 0, such that for any positive function f ,   ρt |∂t ln Pt f (x)| ≤ C exp − , t ≥ t0 , x ∈ G. 3 Proof. To make this proof we have to be more precise in the study of the differential inequality of Theorem 2.1. Start with this inequality and set V (b) = −b2 b′ for b a positive decreasing function such that b(t) = b′ (t) = 0. The constraints that the non negative function V on [0, b0 ] must satisfy are Z b0 x2 dx t= V (x) 0 and  We then get with ut = ln Pt f and a0 = V (x) x2  = 0. x=0 V (b0 ) b20 a0 Γ(ut ) + b0 (Zut )2 ≤ A∂t ut + B, where for any choice of such a function V , one has  2 Z  Z b0  ′ 1 b0 V ′ V − 2ρt dx, B = − 2ρt dx. A= x2 4 0 x2 0 In this system, we see that changing V (s) into V λ(λs) and b0 into bλ0 leaves t unchanged and 3 multiply every constant a0 , A and B by λ1 . Therefore, we may assume that b0 = 1 without any loss. Also, changing V (s) into cV (s) allows us to reduce to the case t = 1. So finally we have rephrased the problem as follows. For any non negative function V on [0, 1] such that   Z 1 2 x V (x) dx = 1, = 0, x2 x=0 0 V and for any u = log Pt f with f ≥ 0 one has V (1)Γ(u) + t(Zu)2 ≤ (α(V ) − 2ρt)∂t u + where α(V ) = Z 0 1
1 β(V ) − α2 (V ) + (α(V ) − 2ρt)2 , 4t V′ dx, β(V ) = x2 Z 0 1 V′ x2 2 dx. The preceding calculus is valid for any ρ. An easy integration by parts shows us the term α(V ) is non negative whatever V is. But now for ρ > 0, observe that this time the term α(V ) − 2ρt can be made negative, and therefore we may get as in the elliptic case with strictly positive Ricci bound a universal upper bound on |∂t u|. One has the obvious inequalities   V (x) + 8, β(V ) > α(V )2 , α(V ) > V (1) − x2 x=0 7 and in the previous, no equality may occur (in the first one because then β = ∞ and in the second one because of the constraint on V .) The first inequality comes from Z 1 ′ Z 1 V V dx = V (1) + 2 dx, 2 3 0 x 0 x and Z 1 0 V dx x3 Z 1 0 x2 dx ≥ V Z 1 0 dx √ x 2 = 4. To make the term β(V ) − α2 (V ) small we are lead to choose V = λx3 on [ǫ, 1] and V = λǫ3−γ xγ on [0, ǫ], for some fixed γ ∈ (5/2, 3). The constraint on V implies λ = − log ǫ + Meanwhile, we have and 1 . 3−γ  3−γ α = λ 3 + 2ǫ γ−2 β=λ so that 2  β − α2 = λ2 ǫ By taking  , (15 − γ)(3 − γ) 9+ǫ 2γ − 5 (3 − γ)2 γ−2   , γ + 10 4 +ǫ 2γ − 5 γ−2  2ρ 1 ǫ = exp − t + +R 3 3−γ for t large enough to ensure ε < 1, one obtains  .  α − 2ρt ≃ −3R and  2ρt β − α ≃ Ct ε ≃ Ct exp − 3 2 2 2  . With R = ct exp(− ρ3 ) the terms (α − 2ρt)2 and β − α2 are of the same order and playing now with the sign of c, one gets   t . |∂t u| ≤ C exp − 3  Interestingly, only from these estimates, we can deduce that for ρ > 0 the Lie group G has to be compact. (This is of course not new since the Lie algebra is that of a compact semi-simple Lie group). But we also get an upper bound on the diameter similar to the classical upper bound of the Myers’s theorem, together with some precise information on the Sobolev constants and the spectral gap. Those considerations in fact show that this parameter ρ may serve as a substitute of the Ricci lower bound for a Riemannian manifold. We proceed first by showing that in that case there is a spectral gap. 8 Proposition 2.6 Let us assume ρ > 0. The spectrum of −L lies in {0} ∪ [ ρ3 , +∞]. Proof. We fix x ∈ G and denote by pt (x, ·) the heat kernel starting from x. We have for t ≥ t0 ,   ρt | ∂t ln pt (x, y) |≤ C exp − . (2.4) 3 This shows us that ln pt converges when t → ∞. Let us call ln p∞ this limit. Moreover, from Corollary 2.4, Γ(ln pt ) is bounded above by a constant C(t) which goes to 0 when p t goes to ∞. Since the oscillation between ln pt (x, y1 ) and ln pt (x, y2 ) is bounded above by C(t)d(y1 , y2 ), for the associated Carnot-Carathéodory distance, which may be defined (see [2]) as d(x, y) = sup {f,Γ(f,f )≤1} f (x) − f (y), (2.5) √ such that if Γ(f, f ) ≤ C, then f (x) − f (y) ≤ Cd(x, y). In the limit, ln p∞ (x, ·) is a constant. We deduce from this that the invariant measure µ is finite. We may then as well suppose that this measure is a probability, in which case p∞ = 1. By integrating the inequality (2.4) from t to ∞ we therefore obtain for t ≥ t0 :   ρt | ln pt (x, y) |≤ C2 exp − 3 and thus    ρt ≤ pt (x, y) ≤ exp C2 exp − . 3 R This implies by the Cauchy-Schwarz inequality that for f ∈ L2 (µ) such that f dµ = 0,  Z 2ρt 2 (Pt f ) ≤ C3 exp − f 2 dµ. 3   ρt exp −C2 exp − 3  For a symmetric Markov semigroup Pt , this is a standard fact (see [2] for example) that this is equivalent to say that the spectrum of −L lies in {0} ∪ [ρ/3, ∞), or equivalently that we have a spectral gap inequality: for any function f in L2 such that ∇f is in L2 , one has Z 2 f dµ ≤ Z f dµ 2 3 + ρ Z |∇f |2 dµ. (2.6)  Remark 2.7 It can be shown that the spectral gap is actually ρ 2 and not ρ3 . We can now conclude with a substitute of the Myers’s theorem: Proposition 2.8 Assume that ρ > 0, then the diameter of L for the Carnot-Caratheodory distance is finite. 9 Proof. We are now going to prove a Sobolev inequality for the invariant measure µ. Indeed, for 0 < t ≤ t0 we have ∂t ln pt ≥ −C/t, from which we get ln pt0 − ln pt ≥ −C log(t0 /t), and therefore ln pt ≤ A − C log t where A is a constant. This gives the ultracontractivity of the semigroup Pt with a polynomial bound t−C when t → 0. Now it is a well known fact (see [13, 2]) that this last property is equivalent to a Sobolev inequality Z  C−1 Z Z C 2C ≤ A f 2 dµ + B k∇f k2 dµ. (2.7) f C−1 dµ When we have both Sobolev inequality (2.7) and spectral gap inequality (2.6) then (see [2]) we have a tight Sobolev inequality, that is the Sobolev inequality (2.7) with A = 1. In this situation, the diameter of E with respect to the distance defined in 2.5 is finite (see [5]), which concludes the proof.  References [1] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. Taniguchi symposium. New trends in stochastic analysis (Charingworth, 1994), 43–75, World Sci. Publ., River Edge, NJ, 1997. [2] D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), 1–114, Lecture Notes in Math., 1581, Springer, Berlin, 1994. [3] D. Bakry, F. Baudoin, M. Bonnefont, D. Chafai: On gradient bounds for the heat kernel on the Heisenberg group, Arxiv preprint 0710.3139, (2007). [4] D. Bakry, M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Revista Mat. Iberoamericana, 22 (2006), 683–702. [5] , D. Bakry, M. Ledoux, Myer’s theorem and Sobolev inequalities, Duke Math. J., 85, 1 (1996), 253–270, [6] D. Bakry, Z. Qian, Harnack inequalities on a manifold with positive or negative Ricci curvature, Rev. Mat. Iberoamericana, 15 (1999), 143–179. [7] F. Baudoin, M. Bonnefont: The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds, Arxiv preprint, (2007). 10 [8] E.B. Davies :Heat kernels and spectral theory, Cambridge tracts in Math. Cambridge Univ. Press 92, 1989 [9] Huai-Dong Cao, Shing-Tung Yau, Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields, Math. Z. 121 (1992), 485–504. [10] M. Ledoux, The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 2, 305–366. [11] H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J. Funct. Anal. 236 (2006), no. 2, 369–394. [12] P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math., 56 (1986), 153–201. [13] N. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal., 52 (1985), 240– 260. 11