Hypoelliptic heat kernel inequalities on the Heisenberg group

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1 Hypoelliptic heat kernel inequalities on the Heisenberg group Bruce K. Driver a,1, Tai Melcher a, a Department of Mathematics, 11, University of California at San Diego, La Jolla, CA, Abstract We study the existence of L p -type gradient estimates for the heat kernel of the natural hypoelliptic Laplacian on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p > 1. The gradient estimate for p = implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p = 1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel. Key words: Heat Kernels, Hypoellipticity, Malliavin Calculus, Heisenberg Group 1991 MSC: E3, 6H7, 58G3 1 Introduction 1.1 Background In the last twenty years or more, a fairly complete and very beautiful theory has been developed applying to elliptic operators on Riemannian manifolds. This theory relates properties of the solutions of elliptic and parabolic equations to properties of the Riemannian geometry. These geometric properties are determined by the principal symbol of the underlying elliptic operator. addresses: driver@math.ucsd.edu Bruce K. Driver), tmelcher@math.ucsd.edu Tai Melcher). 1 The first author was supported in part by NSF Grants and DMS 939. The second author was supported in part by NSF Grants and DMS 939. Preprint submitted to Journal of Functional Analysis 4 June 4

2 The following theorem see for example []) is a typical example of the type of result we have in mind here. Theorem 1.1 Suppose M, g) is a complete Riemannian manifold, and and are the gradient and Laplace-Beltrami operators acting on C M). Let v := g v, v) for all v T M, Ric denote the Ricci curvature tensor, and k denote a constant. Then the following are equivalent: 1) Ric f, f) k f, for all f C c M), ) e t / f e kt e t / f, for all f C c M) and t >, 3) e t / f e kt e t / f, for all f C c M) and t >, 4) there is a function Kt) > such that K) = 1, K) exists, and for all f C c M) and t >. e t / f Kt)e t / f, 1.1) Estimates like 1) 4) are also equivalent to one parameter families of Poincaré and logarithmic Sobolev estimates for the heat kernel. The latter has implications for hypercontractivity of an associated semigroup; see Gross [8]. Also, in [1], Auscher, Coulhon, Duong, and Hofmann study inequalities of the form e t f p Ce ct f, where C and c are positive constants, along with their relation to the Riesz transform on manifolds. As a simple illustration of this theorem, consider the manifold M = R 3 with vector fields x = x, y = y, and z = z. Let and be the standard gradient and Laplacian on R 3 ; = x, y, z ) and = x + y + z. In this case e t / is convolution by the probability density p t x) := 1 πt) 1 e t x R 3/ 3, and e t / f = e t / f, 1.) for all f Cc 1 R 3 ), as follows from basic properties of convolutions; more abstractly, this follows from the commutativity of the Euclidean gradient and Laplacian. Equation 1.) and an application of Hölder s inequality then imply that e t / f p ] [ e t / f ] p e t / f p,

3 for all f Cc 1 R 3 ), where f := x f) + y f) + z f). This paper is a first step toward extending Theorem 1.1 to hypoelliptic operators of the form n L = Xi, 1.3) i=1 where {X i } n i=1 is a collection of smooth vector fields on M satisfying the Hörmander bracket condition. Recall that the Hörmander bracket condition is the assumption T m M = span {Xm) : X L}), m M, where L is the Lie algebra of vector fields generated by the collection {X i } n i=1. By a celebrated theorem of Hörmander, L is hypoelliptic; however, the operator need not be elliptic. The principal symbol of L at ξ T mm is given by σ L ξ) = n i=1 [ξ X i m))]. By definition, the operator L is degenerate at points m M where there exists ξ T mm such that σ L ξ) =. At points of degeneracy of L, the Ricci tensor is not well defined and should be interpreted to take the value in some directions. Hence it is not possible to directly generalize Theorem 1.1 in this setting. Nevertheless it is reasonable to ask if inequalities of the form 1.1) might still hold. More precisely, we let = X 1,..., X n ) and address the following question: do functions K p t) < exist such that e tl/ f p K p t)e tl/ f p, p [1, ), for all f C c M) and t >? In this paper, we give an affirmative answer to this question for p > 1 in the model case of the Heisenberg Lie group; the case p = 1 remains open. Let M = G be R 3 equipped with the Heisenberg group operation given in Eq..1). In this setting, we take L = X + Ỹ, where X and Ỹ are the vector fields X := x 1 y z and Ỹ := y + 1 x z. 1.4) We restrict to this simple case because the basic ideas can already be seen here without the added geometric complications appearing in more general formulations. However, much of the theory generalizes to certain classes of vector fields {X i } n i=1 satisfying the Hörmander bracket condition on more general manifolds. These results will appear in forthcoming papers; see [17]. 3

4 1. Statement of Results Notation 1. Let C p G) denote those functions f C G) such that f and all its partial derivatives have at most polynomial growth. Definition 1.3 The left invariant gradient on G = R 3 is the operator The sublaplacian is = X, Ỹ ). L = X + Ỹ, and we let P t = e tl/ be the semigroup associated to L. Finally, p t g) = P t δ g) = e tl/ δ g) denotes the fundamental solution associated to L, so that for f C p G), P t fg) = p t f g) := G fgh)p t h) dh, where dh denotes right Haar measure and gh is computed relative to the Heisenberg group multiplication in Eq..1) below. Remark 1.4 Since { X, Ỹ } generates the tangent space at all points of G, Hörmander s theorem [9] implies that L is a hypoelliptic operator. Also Malliavin s techniques show p t is a smooth positive function on R 3 ; see Section 3. In this simple setting, an explicit formula for p t g) is p t g) = 1 8π R w sinh wt ) exp 1 4 x w coth where g = x, y, z) G and x = x, y); see for example []. )) wt e iwz dw, 1.5) Notation 1.5 For all p [1, ) and t >, let K p t) be the best function such that P t f p K p t) P t f p, 1.6) for all f C p G). Theorem 1.6 For all p 1, ), K p t) is independent of t, and K p t) = K p <. Moreover, K p > 1 for all p [1, ). Closely related results appear in Kusuoka and Stroock [15]. In particular, Theorem.18 of [15] states that for all p 1, ) there exist finite constants C p such that P t f p C p t p/ P t f p, for all smooth, bounded functions f with bounded derivatives of all orders and t >. 4

5 Section justifies the choice of vector fields made here, a choice which corresponds to left invariant vector fields on R 3 under the Heisenberg group operation. We show that the left invariance of the vector fields leaves the inequality 1.6) translation invariant. Certain scaling arguments imply that the constants K p are also independent of the t parameter. We also show that K p when 1 p and, in general, that K p > 1. Note that at t = the inequality is an empty statement and certainly holds for constant 1. So unlike the elliptic case where K p t) is continuous at t =, there is now a jump discontinuity in K p t) at t =. Independence of the K p with respect to t does not generalize to all Lie groups; however, the discontinuity of K p t) at t = should be a feature which persists in the general hypoelliptic setting. Section 3 briefly reviews some infinite dimensional calculus on Wiener space necessary for the proof of Theorem 1.6. The heat kernel p t g) dg is the distribution in t of the process ξ satisfying Eq. 3.1). Using this representation of p t, we may transform our finite dimensional problem to a problem on Wiener space, where we then may apply Malliavin s probabilistic techniques on proving hypoellipticity. The advantage of the infinite dimensional Wiener space representation of p t g) dg over that in Eq. 1.5) is that it no longer involves an oscillatory integral. Section 4 restates Theorem 1.6 and gives its proof and the proof that this result implies the following Poincaré inequality. Theorem 1.7 Let K be the constant in Theorem 1.6 for p =. Then for all f C p G) and t >. P t f ) P t f) ) K tp t f ), Finally, Section 4. shows that our method can not, without modification, be used to prove K 1 <. Real 3-dimensional Heisenberg Lie group.1 Realization of the Heisenberg Lie group Recall that the real Heisenberg Lie algebra is g = span{x, Y, Z} where Z = [X, Y ] and Z is in the center of g. Thus, g := span{x, Y } is a hypoelliptic subspace of g; that is, the Lie algebra generated by g is g. The Heisenberg group G is the simply connected real Lie group such that LieG) = g. Letting A = ax +by +cz and A = a X +b Y +c Z, we have by the Baker-Campbell- 5

6 Hausdorff formula that e A e A = e A+A + 1 [A,A ]. Thus we may realize G as R 3 with the following group multiplication a, b, c)a, b, c ) = a + a, b + b, c + c + 1 ab a b))..1). Differential operators on G Notation.1 Given an element A g, let à denote the left invariant vector field on G such that Ã) = A.  will denote the right invariant vector field associated to A. Now let X = 1,, ), Y =, 1, ), and Z =,, 1) at the identity G. We extend these to left invariant vector fields on G in the standard way. For g = a, b, c) G, let L g denote left translation by g, and compute as follows: Xa, b, c) = L a,b,c) X = d a, b, c)t,, ) dt = d a + t, b, c 1 dt bt) = 1,, 1 b). So if x, y, z) are the standard coordinates on G = R 3, for f C 1 G), Xf)g) = d fg tx) = f dt x g) 1 y f z g). Performing similar computations for Y and Z, we then have X = x 1 y z, Ỹ = y + 1 x z, and [ X, Ỹ ] = Z = z ; compare with Eq. 1.4). Note then that { X, Ỹ, Z} forms a basis for the tangent space at every point of G. This combined with [ X, Ỹ ] = Z implies that { X, Ỹ } satisfies the Hörmander bracket condition. One may also show that the right invariant vector fields associated to X, Y, and Z are given by ˆX = x + 1 y z, Ŷ = y 1 x z, and [ ˆX, Ŷ ] = Ẑ = z. Remark. The right invariant vector fields associated to X and Y may be expressed as the following linear combinations, ˆX = X + y Z and Ŷ = Ỹ x Z. We will need the following straightforward results. 6

7 Lemma.3 By the left invariance of and P t, the inequality 1.6) holds for all g G, f C p G), and t >, if and only if, for all f C p G) and t >. Proof. If the inequality.) holds, then P t f p ) K p t)p t f p ),.) P t f p g) = P t f) L g p ) = P t f L g ) p ) = P t f L g )) p ) K p t)p t f L g ) p ) = K p t)p t f) L g p ) = K p t)p t f p L g ) = K p t)p t f p g). The converse is trivial. Lemma.4 For A g, ÃP t f) = P t Âf), for all f C p G) and t >. More generally, ÂP t f = P t Âf, from which the previous equation follows, since  = à at. Proof. Heuristically, we know that [Â, B] = for all B g, so that [Â, L] =, and thus [Â, etl/ ] =. Consider ÃP t f) = d P dɛ t fe ɛa ) = d fe ɛa g)p dɛ t g) dg G d = fe ɛa g)p G dɛ t g) dg = Âfg)p t g) dg = P t Âf). To differentiate under the integral, we have used the translation invariance of Haar measure which is Lebesgue measure on R 3 ) and the heat kernel bound G p t g) Ct e ρ g)/ct, where ρg) C x + y + z 1/ ) is the Carnot-Carathéodory distance on G, and C and C are some positive constants; see Theorem in [19] and page 7 of [4]. 7

8 .3 Dilations on G Definition.5 A family of dilations on a Lie algebra g is a family of algebra automorphisms {φ r } r> on g of the form φ r = expw log r), where W is a diagonalizable linear operator on g with positive eigenvalues. So let r > and g = x, y, z), and define φ r : G G by φ r x, y, z) = rx, ry, r z). Notice that φ r a, b, c) x, y, z)) = φ r a + x, b + y, c + z + 1 ay xb)) = φ r ra + rx, rb + ry, r c + r z + r ay xb)) = φ r a, b, c)φ r x, y, z). Thus φ r is in fact an isomorphism of G. The generator W of φ r is given by, W x, y, z) = d φ dr r x, y, z) = x, y, z) x,y,z) r=1 = x x + y y + z z ) 1 = x X + y z + y Using e t X g) = g t,, ) and φ r X φ 1 r ) 1 Ỹ x z + z z = x X + yỹ g) = d t φ dt r e Xφ 1 r g))), + z Z. along with similar formulas involving Ỹ, one shows φ r X φ 1 r The equations in.3) are equivalent to = r X and φ r Ỹ φ 1 r = rỹ..3) Xf φ r ) = r Xf) φ r and Ỹ f φ r) = rỹ f) φ r. Therefore, f φ r ) = r f) φ r,.4) Lf φ r ) = r Lf) φ r, and 8

9 Also, from Eq. 1.5), for g = x, y, z), p r tg) = 1 8π R = 1 8π w sinh wr t w ) exp 1 wr )) t 4 x w coth e iwz dw ) exp 1 wt 4r x w coth )) e iwz/r r dw R r sinh wt = r 4 p t φ r 1)g).5) through the change of variables w r w. Thus, P t f φ r )g) = = that is, f φ r )gh)p t h) dh = fφ r g)φ r h))p t h) dh G fφ r g)h) p t φ r 1h))r 4 dh = fφ r g)h)p r th) dh G G = P r tf φ r )g); P t f φ r ) = e tl/ f φ r ) = e r tl/ f) φ r = P r tf) φ r..6) For a more general exposition on Lie groups which admit dilations, see [6]. The above remarks lead to the following proposition. Proposition.6 If K p is the best constant such that P 1 f p K p P 1 f p, for all f C p G), then K p t) = K p for all t >, where K p t) is the function introduced in Notation 1.5. Proof. By Eqs..4) and.6), P t f φ r 1/) p = [P 1 f) φ r 1/] p = r 1/ P 1 f) φ r 1/ p K p r p/ P 1 f p ) φ r 1/ = K p r p/ P t f p φ r 1/) = K p P t f φ r 1/) p ). Replacing f by f φ r 1/ in the above computation proves the assertion. Moreover, reversing the above argument shows that P t f p K p P t f p implies that P 1 f p K p P 1 f p. G.4 The constant K p > 1 Proposition.7 For p [1, ), let K p be the best constant such that P t f p K p P t f p.7) 9

10 for all f C p G) and t >. Then K p > 1. In particular, K. Proof. First consider the case p = k for some positive integer k, and suppose the constant K k = 1. Then P t f k P t f k, for all t, and f k = P f k = P f k = f k, together would imply that k f k 1) f Lf = d P dt t f k d P dt t f k = 1 L f k..8) We now show that the function fx, y, z) = x + yz violates this inequality. Note that X X Lf = f = f = 1 1y y Ỹ Ỹ z + 1x y = 1 x, Lf = 1 1, f Lf = y y ), and f ) = 1. Hence, k f k 1) f Lf ) ) = k..9) On the other hand, Lφ g) = φ g) Lg + φ g) g, and so setting φ t) = t k and g = f gives L f k = k f k 1) L f + k k 1) f k ) f. From the above, f = yz + 1 xy 1 y z + xy), y + y 3 + xz + 1 x y + 1 x z + xy) and hence f ) =, while L f ) ) =. Therefore 1 ) L f k ) = k..1) 1

11 Inserting the results of Eqs..9) and.1) into Eq..8) would imply that k k, which is absurd. Thus, K k > 1 for any positive integer k. For any p [1, ), there is some integer k such that p k. Thus, P t f k = P t f p ) k/p Kp k/p P t f p ) k/p Kp k/p P t f k..11) Since K k is the optimal constant for which.11) holds and K k > 1, implies that K p > 1. 1 < K k K k/p p We now quantify this estimate this estimate for p =. Since K = P t F sup F Cp G) P t F ) P tf ) =: Ct), P t f where fx, y, z) = x+yz, it follows that K sup t> C t). To finish the proof we compute C t) explicitly. Observe that P t, when acting on polynomials, may be computed using P t = e tl/ = n=1 ) 1 tl n = I + t n! L + 1! t 4 L +. We then have ) P t f = f + t Lf = x + yz) + t x, P 1 + t 1 tf = y y z + 1x y, and P t f = 1 + t ) 1 ) y + z + 1 ) xy = 1 y y4 + z + xyz + 1 ) 4 x y + t ) y + t 8. Also, from before, and so f = 1 1y y z + 1x y, f = 1 y y4 + z + xyz x y, L f = + 3y + x, and L f = = 1. 11

12 Thus, and P t f ) = f ) + t L f ) + t 8 L f ) = 1 t t We can find the maximum value of P t f ) = 1 + t t. Ct) = 1 + t t 1 t t for t by taking derivatives in t to show that C t) takes on maximum value at t = 3. 3 Infinite dimensional calculus Let W R ), F, µ) denote classical two-dimensional Wiener space. That is, W = W R ) is the space of continuous paths ω : [, 1] R such that ω) =, equipped with the supremum norm ω = max t [,1] ωt), µ is standard Wiener measure, and F is the completion of the Borel σ-field on W with respect to µ. W, ) is a Banach space. By definition of µ, the process b t ω) = b 1 t ω), b t ω) ) = ω t is a two-dimensional Brownian motion. For those ω W which are absolutely continuous, let Eω) := 1 ωs) ds denote the energy of ω. The Cameron-Martin Hilbert space is the space of finite energy paths, H 1 = H 1 R ) := {ω W R ) : ω is absolutely continuous and Eω) < }, equipped with the inner product h, k) H 1 := 1 ḣs) ks) ds, h, k H 1. We may identify the Cameron-Martin space with H = L [, 1], R ) in the obvious way h H 1 ḣ H. 1

13 In this way, the spaces are isomorphic, and in the sequel, we make this identification without further comment. To define a notion of differentiation for functions on W, let B = {Bh), h H} be the process given by Bh) = 1 ht) db t. B is an isonormal Gaussian process associated to the Hilbert space H. Denote by S the class of smooth Wiener functionals; that is, random variables F : W R such that F = fbh 1 ),..., Bh n )), for some n 1, h 1,, h n H, and function f C p R n ). Definition 3.1 The derivative of a smooth functional F S is the random process defined by D t F = n i=1 f x i Bh 1 ),..., Bh n ))h i t). Iterations of the derivative for smooth functionals F are given by D k t 1,...,t k F = D t1 D tk F, and are measurable functions defined almost everywhere on [, 1] k W. We will denote the domain of D k in L p [, 1] k W ) by D k,p, which is the completion of the family of smooth Wiener functionals S with respect to the seminorm k,p on S defined by F k,p = E F p ) + 1/p k E D j F p L [,1] j ) ). j=1 Let D = D k,p. p 1 k 1 One may generalize these Sobolev spaces to Hilbert-valued functions, again, given an appropriate notion of differentiation. So let S H be the set of H-valued Wiener functions of the form n F = F j h j, h j H, F j S. j=1 Define D k F = n j=1 D k F j h j for k 1. Then, as in the Euclidean case, we 13

14 may define the seminorm F k,p,h = E F p H) + 1/p k E D j F p L [,1] j,h) ) j=1 on S H for any p 1, and let D k,p H) be the completion of S H in the norm k,p,h, and D H) = D k,p H). p 1 Definition 3. Let D denote the adjoint of the derivative operator D, which has domain in L W [, 1]) consisting of functions G such that k 1 E[DF, G) H ] C F L µ), for all F D 1,, where C is a constant depending on G. For those functions G in the domain of D, D G is the element of L µ) such that E[F D G] = E[DF, G) H ]. It is known that D is a continuous operator from D to D H), and similarly, D is continuous from D H) to D ; see for example Proposition from Nualart [18]. For a more complete exposition of the above definitions, we refer the reader to [5,1 14,16,18,] and references contained therein. 3.1 The Stochastic Differential Equation Let ξ : [, 1] W G denote the solution to the Stratonovich stochastic differential equation dξ t = L ξt X db 1 t + L ξt Y db t = Xξ t ) db 1 t + Ỹ ξ t) db t ξ =. 3.1) Remark 3.3 Since X and Ỹ have smooth coefficients with bounded partial derivatives, Theorem.. in Nualart [18] implies that ξt i D, for i = 1,, 3 and all t [, 1]. 14

15 Because G is a nilpotent Lie group, we may determine an explicit solution of the given SDE. Thus, dξ t = Xξ t 1, ξt, ξt 3 ) db 1 t + Ỹ ξ1 t, ξt, ξt 3 ) db t 1 = db 1 t + 1 db t. 1 ξ t 1 ξ1 t dξ 1 t = db 1 t, dξ t = db t, and dξ 3 t = 1 ξ t db 1 t + 1 ξ1 t db t, and one may verify directly that ξ t = b 1 t, b t, 1 t [ b 1 s db s b s db 1 s ] ) 3.) satisfies the required SDE. Note that the third component of ξ may be recognized as Lévy s stochastic area integral. From Section 3.9 in Gīhman and Skorohod [7] and Theorem 1. in Bell [3], the solution ξ = ξ 1, ξ, ξ 3 ) is a time homogenous Markov process, and P t = e tl/ with L = X + Ỹ is the associated Markov diffusion semigroup to ξ; that is, ν t := ξ t ) µ = p t g) dg is the density of the transition probability of the diffusion process ξ t, and P t f)) = E[fξ t )], 3.3) for any f C p G), where the right hand side is expectation conditioned on ξ =. Proposition 3.4 The Malliavin covariance matrix of ξ t is invertible a.s. for t >, and σ t = Dξ i t, Dξ j t ) H ) 1 i,j 3 det σ) 1 p 1 L p µ) =: L µ). This statement follows from the proof of Theorem.3.3 in Nualart [18] which relies on satisfaction of the Hörmander bracket condition, Lie{X, Y } = g. Remark 3.5 By the general theory, Proposition 3.4 implies ν t = Lawξ t ) is a smooth measure; see for example Theorem.1 and Remark.13 in Bell [3]. 15

16 3. Lifted vector fields and their L -adjoints Given A g, let Ãi be the i th component of the left invariant vector field Ã, hence à = Ã1, Ã, Ã3 ). In particular, we are interested in the vector fields Xx, y, z) = 1,, 1y) and Ỹ x, y, z) =, 1, 1 x). We define the lifted vector field A of à as 3 A = A t := σij 1 à j ξ t )Dξt i H, 3.4) acting on functions F D 1, by i,j=1 AF = DF, A) H. Remark 3.6 Recall that D is a continuous operator from D to D H). Thus, Remark 3.3 implies that à j ξ t ) D and Dξt i D H), for all t [, 1]. So σ ij D for i, j = 1,, 3, and this along with Proposition 3.4 implies that σij 1 D. Hence, A D H). Proposition 3.7 For all f C p G), A[fξ t )] = Ãf)ξ t). Proof. For any function f C p G), fξ t ) D and D[fξ t )] = 3 k=1 f x k ξ t )Dξ k t ; see Proposition 1..3 from Nualart [18]. Then using Eq. 3.4) and the definition of the Malliavin matrix σ, we have as desired. A[fξ t )] = Dfξ t ), A) H ) 3 f = ξ t )Dξt k, σij 1 à j ξ t )Dξt i x k = = i,j,k=1 3 i,j,k=1 3 j,k=1 à j ξ t ) f ξ t ) ) Dξt k, Dξt i x k H σ 1 ij à j ξ t ) f x k ξ t )δ kj = 3 j=1 H à j ξ t ) f x j ξ t ) = Ãf)ξ t) Definition 3.8 For a vector field A acting on functions of W, we will denote the adjoint of A in the L µ) inner product by A, which has domain in L µ) consisting of functions G such that E[AF )G] C F L µ), 16

17 for all F D 1,, for some constant C. For functions G in the domain of A, for all F D 1,. Note that for any F D 1,, E[F A G)] = E[AF )G], E[AF ] = E[DF, A) H ] = E[F D A]. Thus, we must have that A = A 1 = D A a.s. Recall that D is a continuous operator from D H) into D. Thus, for A a vector field on W as defined in Eq. 3.4), Remark 3.6 implies that D A = 3 i,j=1 Thus we have the following proposition. D σij 1 à j ξ t )Dξt) i D. Proposition 3.9 Let à be a left invariant vector field on G with lifted vector field A on W as defined by Eq. 3.4). Then A, the L µ)-adjoint of A, is an element of D. 4 Heat kernel inequalities 4.1 An L p -type gradient estimate p > 1) and a Poincaré inequality Theorem 4.1 For all p > 1, for all f C p G) and t >, where P t f p K p P t f p, 4.1) K p := p/q + p 1 q + 1 ) [ X ξ 1 1 L q µ) + X ξ 1 L q µ)] p/ <, with X the adjoint of the lifted vector field X as in Eq. 3.4) with t = 1, and q = p p 1. Proof. By Proposition.6, we know the constants K p are independent of t. Also, Lemma.3 states that the inequality is translation invariant. Thus, the proof is reduced to verifying the inequality at the identity for t = 1; that is, we must find finite constants K p such that P 1 f p ) K p P 1 f p ), 4.) 17

18 for all f C p G). So applying Remark. and Lemma.4, consider Similarly, Thus, XP 1 f) = P 1 ˆXf) = P 1 X + y Z)f) = P 1 Xf)) + P 1 y Zf)). Ỹ P 1 f) = P 1 Ỹ f)) P 1x Zf)). P 1 f p ) = P 1 f + P y p 1 Zf ) x P 1 f + P y p 1 Zf ) x p/q P 1 f p ) + P y 1 Zf x p ), 4.3) where P y p 1 Zf ) = [ P 1 y Zf) ) + P 1 x Zf) )] p/ x and q = p is the conjugate exponent to p. Let F = F p 1 1, F, F 3 ) := ξ 1 and recall that Z = XỸ Ỹ X. By Eq. 3.3), P 1 y Zf)) = P 1 y XỸ f)) P 1yỸ Xf)) = E[F XỸ f)f )] E[F Ỹ Xf)F )] = E[F XỸ f)f ))] E[F Y Xf)F ))] = E[X F Ỹ f)f )] E[Y F Xf)F )], 4.4) where X and Y are the lifted vector fields of X and Ỹ, as in Eq. 3.4), with t = 1. Hence, P 1 y Zf) ) E[X F Ỹ f)f )] + E[Y F Xf)F )] ) E[X F Ỹ f)f )] + E[Y F Xf)F )] ) [E X F q ) /q P 1 Ỹ f p ) /p ) + E Y F q ) /q P 1 Xf p ) /p )] by Hölder s inequality. Similarly, P 1 x Zf) ) [E X F 1 q ) /q P 1 Ỹ f p ) /p ) + E Y F 1 q ) /q P 1 Xf p ) /p )]. 18

19 Combining this with Eq. 4.3), we have P 1 f p ) P p/q 1 f p ) + [ E X F q ) /q P 1 Ỹ f p ) /p ) + E Y F q ) /q P 1 Xf p ) /p ) + E X F 1 q ) /q P 1 Ỹ f p ) /p ) +E Y F 1 q ) /q P 1 Xf p ) /p ) ] ) p/ P p/q 1 f p ) + [ p/ P 1 Xf p ) /p )[E Y F 1 q ) /q + E Y F q ) /q ] ) +P 1 Ỹ f p ) /p )[E X F 1 q ) /q + E X F q ) /q] p/, where we use Hölder s inequality and that p 1 g) dg is a probability measure to get P 1 f p ) P 1 f p ). So let or equivalently by symmetry, C p := E X F 1 q ) /q + E X F q ) /q, C p = E Y F 1 q ) /q + E Y F q ) /q. Note that C p is a finite constant for all p > 1 by Hölder s inequality, Remark 3.3, and Proposition 3.9, since A F = D F A) for any vector field A on W and F D. Thus, P 1 f p ) p/q P 1 f p ) + C p ) p/ [P 1 Xf p ) /p ) + P 1 Ỹ f p ) /p )] p/ ) p/q + p 1 q + 1 ) Cp p/ P1 f p ), which proves Eq. 4.), and hence, the theorem. Theorem 4. Poincaré Inequality) Let K be the constant in Eq. 4.1) for p = and p t g) dg be the Heisenberg group heat kernel. Then ) f g)p t g) dg fg)p t g) dg K t f g)p t g) dg, R 3 R 3 R 3 for all f C p G) and t >. 19

20 Proof. Let F t g) = P t f)g). Then d ds P t sfs = P t s 1 ) LF s + F s LF s = P t s F s. Integrating this equation on t implies that P t f P t f) = t P t s F s ds = t P t s P s f ds t K P t s P s f ds t = K P t f ds = K tp t f, wherein we have made use of Theorem 4.1. Evaluating the above at gives the desired result. 4. Method fails for the p = 1 case In this section, we show that the argument in the proof of Theorem 4.1 can not be used to prove the inequality 4.1) for p = 1. Proposition 4.3 Let F = F 1, F, F 3 ) := ξ 1. Then X F 1 L µ) + X F L µ) =. 4.5) Proof. Let σ F ) denote the σ algebra generated by F : W G and p t g) dg denote the Heisenberg group heat kernel. Then for f C 1 c R 3 ) E[X F 1 ff )] = E[F 1 Xf)F )] = P 1 x Xf)) = x Xfg)p 1 g) dg G = fg) Xxp 1 g)) dg G = fg)1 + x X ln p 1 g))p 1 g) dg G = E[fF )1 + x X ln p 1 )F )], where in the third line we have applied standard integration by parts. Consequently, we have shown E[X F 1 σf )] = 1 + x X ln p 1 )F ). By a similar computation one also shows E[X F σf )] = y X ln p 1 )F ).

21 Since conditional expectation is L p -contractive and the law of F is absolutely continuous relative to Lebesgue measure, it now follows that X F 1 L µ) + X F L µ) E[X F 1 σf )] L µ) + E[X F σf )] L µ) = 1 + x X ln p 1 L R 3,m) + y X ln p 1 L R 3,m), where m is Lebesgue measure. Hence, it suffices to show that either x X ln p 1 or y X ln p 1 is unbounded. We will show x X ln p 1 is unbounded by making use of the formula for p t g) in Eq. 1.5). Letting t = 1 in Eq. 1.5) and making the change of variables w w, we have p 1 g) = 1 w π R sinh w exp 1 ) x w coth w e iwz dw. Then applying X = x 1 y z yields Xp 1 g) = 1 xw coth w + iyw) π R Setting y = z =, it follows that w sinh w exp 1 ) x w coth w X ln p 1 x,, ) = x w coth wdν x w), R e iwz dw. where dν x w) := 1 w z x sinh w exp 1 ) x w coth w dw 4.6) and z x is the normalizing constant z x := By Lemma 4.4 below, and so lim x R w sinh w exp 1 ) x w coth w dw. lim w coth w dν x w) = 1, x R ) X ln p 1 x,, ) = x lim x w coth w dν x w) =. R Lemma 4.4 Let ψw) = w coth w 1 and ν x be as in Eq. 4.6). Then lim x ψ dν x = ψ) =. 4.7) 1

22 Proof. Since ψ ) = and ψ is continuous, to prove Eq. 4.7) it suffices to show by the usual approximation of δ function arguments that lim ψw) dν x w) = x w ɛ holds for every ɛ >. We begin by rewriting Eq. 4.6) as dν x w) = 1 Z x w sinh w exp 1 ) x ψw) dw where w Z x := R sinh w exp 1 ) x ψw) dw. A glance at the graph of ψ will convince the reader that there are constants α, β > depending on ɛ > ) such that α w ψw) β w for all w ɛ. In fact, one could take β = 1 independent of ɛ). Thus w ɛ w ψw) sinh w exp 1 ) x ψw) dw where in the inequality we have also used that w ɛ = 4β x α βwe αx w/ dw ɛ + x α w 1. sinh w ) e αx ɛ/, Now consider the constant Z x. We know that for w small, there exists a constant γ > such that ψw) γw. So letting ϕw) = Z x w ɛ ɛ ϕw) exp 1 ) x ψw) ɛ ϕw)e γx w / dw = 1 x ɛx ɛx dw w, sinh w ) w ϕ e γw / dw, x where we have made the change of variables w w. So, by the dominated x convergence theorem, ɛx ) w π lim inf xz x) lim inf ϕ e γw / dw = ϕ) e γw / dw = x x ɛx x γ. Thus, Z x 1 π γ 1 x for x sufficiently large, and so 1 lim ψw) dν x w) = lim x w ɛ x as desired. Z x lim x 4β x α w ψw) sinh w exp ) ɛ + x α e αx ɛ/ = w ɛ π γ 1 x 1 ) x ψw) dw

23 References [1] P. Auscher, T. Coulhon, X. T. Duong, S. Hofmann, Riesz transform on manifolds and heat kernel regularity, 3 Preprint. See [] D. Bakry, Ricci curvature and dimension for diffusion semigroups, in: Stochastic processes and their applications in mathematics and physics Bielefeld, 1985), Vol. 61 of Math. Appl., Kluwer Acad. Publ., Dordrecht, 199, pp [3] D. R. Bell, The Malliavin calculus, Vol. 34 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, [4] A. Bellaïche, Tangent space in sub-riemannian geometry, in: Sub-Riemannian Geometry, Vol. 144 of Progress in Mathematics, Birkhäuser, Basel, 1996, pp [5] B. K. Driver, Curved wiener space analysis, Preprint, to appear in Real and Stochastic Analysis: New Perspectives, xxx.lanl.gov/list/math/43. [6] G. B. Folland, E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, N.J., 198. [7] Ĭ. Ī. Gīhman, A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York, 197, translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 7. [8] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 4) 1975) [9] L. Hörmander, Hypoelliptic second order differential equations, Acta Math ) [1] N. Ikeda, S. Watanabe, An introduction to Malliavin s calculus, in: Stochastic analysis Katata/Kyoto, 198), Vol. 3 of North-Holland Math. Library, North- Holland, Amsterdam, 1984, pp [11] N. Ikeda, S. Watanabe, Malliavin calculus of Wiener functionals and its applications, in: From local times to global geometry, control and physics Coventry, 1984/85), Vol. 15 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986, pp [1] N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, nd Edition, Vol. 4 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, [13] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus. I, in: Stochastic analysis Katata/Kyoto, 198), Vol. 3 of North-Holland Math. Library, North- Holland, Amsterdam, 1984, pp

24 [14] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 3 1) 1985) [15] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 ) 1987) [16] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, in: Proceedings of the International Symposium on Stochastic Differential Equations Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Wiley, New York, 1978, pp [17] T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, Ph.D. Thesis, 4. See driver/driver/thesis.html. [18] D. Nualart, The Malliavin calculus and related topics, Probability and its Applications New York), Springer-Verlag, New York, [19] N. T. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and geometry on groups, Vol. 1 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 199. [] S. Watanabe, Analysis of Wiener functionals Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 1) 1987)