Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

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1 Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces Tim Austin Brown Assaf Naor Courant Institute omain Tessera E.N.S. Lyon Abstract Let H denote the discrete Heisenberg group, equipped with a word metric d W associated to some finite symmetric generating set. We show that if X, is a p-convex Banach space then for any Lipschitz function f : H X there exist x, y H with d W x, y arbitrarily large and fx fy d W x, y log log dw x, y log d W x, y /p. We also show that any embedding into X of a ball of radius 4 in H incurs bi-lipschitz distortion that grows at least as a constant multiple of /p log. log log Both and are sharp up to the iterated logarithm terms. When X is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to and which are sharp up to a universal constant. Introduction Let H def = a, b aba b is central denote the discrete Heisenberg group, with canonical generators a, b H. We let d W, denote the left-invariant word metric on H associated to the symmetric generating set S def = {a, b, a, b }. A Banach space X, X is superreflexive if it admits an equivalent uniformly convex norm, i.e., a norm satisfying α x X x β x X for some α, β > 0 and all x X, such that for all ε 0, there exists δ > 0 for which we have Here we prove the following result: x = y = x y = ε = x + y δ. 3 Theorem.. Let X, X be a superreflexive Banach space. Then there exist c, C > 0 such that for every f : H X which is -Lipschitz with respect to the metric d W, there are x, y X with d W x, y arbitrarily large and fx fy X d W x, y C log d W x, y c. Partially supported by a fellowship from Microsoft Corporation. Supported by NSF grants CCF and CCF , BSF grant , and the Packard Foundation.

2 The fact that H does not admit a bi-lipschitz embedding into any superreflexive Banach space was proved in [8, 7]. These proofs use an argument of Semmes [5], based on a natural extension of Pansu s differentiability theorem []. A natural way to quantify the extent to which H does not admit a bi-lipschitz embedding into X, X is via Gromov s notion [3, Sec. 7.3] of compression rate, defined for a Lipschitz function f : H X as the largest function ω f : 0, [0, such that for all x, y H we have fx fy X ω f d W x, y. The fact that H does not admit a bi-lipschitz embedding into a superreflexive Banach space X means that lim inf t ω f t/t = 0 for all Lipschitz functions f : H X. The differentiability-based proof of this nonembeddability result involves a limiting argument that does not give information on the rate at which ω f t/t vanishes. Theorem. supplies such information, via an approach which is different from the arguments in [8, 7]. Cheeger and Kleiner proved [6] that H does not admit a bi-lipschitz embedding into L. In [8] it was shown that there exists c > 0 such that for any Lipschitz function f : H L we have ω f t/t /log t c for arbitrarily large t. This result covers Theorem. when the superreflexive Banach space X admits a bi-lipschitz embedding into L : such spaces include L p for p, ]. Theorem. is new even for spaces such as L p for p,, which do not admit a bi-lipschitz embedding into L see [3]. Moreover, our method yields sharp results, while the constant c obtained in [8] is far from sharp. In order to state our sharp version of Theorem., we recall the following important theorem of Pisier [3, Thm. 3.]: if X is superreflexive then it admits an equivalent norm for which there exist p and K > 0 satisfying the following improvement of 3: x, y X, x + y p x p + y p K p x y p. 4 A Banach space admitting an equivalent norm satisfying 4 is said to be p-convex. If X, satisfies 4 then the infimum over those K > 0 satisfying 4 is denoted K p X. For concreteness, when p, ] we have K L p / p and for p we have K p L p see []. The following theorem is a refinement of Theorem.. Theorem.. Assume that the Banach space X, satisfies 4. Let f : H X be a -Lipschitz function. Then for every t 3 there exists an integer t n t such that ω f n n log log n /p K p X. 5 log n The estimate 5 is sharp up to the iterated logarithm term. Indeed, L p is p-convex when p [,, and in [6, 7] it was shown that there exists f : H L p satisfying ω f n n log n /p log log n for all n 3 we refer to [7] for a more refined result of this type. Our proof of Theorem. circumvents the difficulties involved with proving quantitative variants of differentiability results by avoiding the need to reason about arbitrary Lipschitz mappings. In 5, and in the rest of this paper, the notation, denotes the corresponding inequalities up to a universal multiplicative factor. The notation A B stands for A B B A.

3 Instead, we start by using a simple result from [] which reduces the problem to equivariant mappings. Specifically, in [, Thm. 9.] it is shown that if X satisfies 4 and f : H X is -Lipschitz, then there exists a Banach space Y that also satisfies 4, with K p Y = K p X in fact, Y is finitely representable in l p X, an action π of H on Y by linear isometric automorphisms, and a -cocycle F : H Y i.e., F xy = πxf y+f x for all x, y H with ω F = ω f. Thus, in proving Theorem. it suffices to assume that f itself is a -cocycle. We note that if X is Hilbert space then Y is also Hilbert space; this is an older result of Gromov see [0]. More generally, when X = L p then it is shown in [] that we can take Y = L p. Having reduced the problem to -cocycles, our starting point is a non-quantitative proof, explained in Section, showing that if X is an ergodic Banach space, then for every -cocycle f : H X we have lim inf t ω f t/t = 0. It turns out that the ideas of this proof, which crucially use the fact that f is a -cocycle, can be nontrivially adapted to yield Theorem.. ecall that X is ergodic if for every linear isometry T : X X and every x X the sequence { n n } T j x converges in norm. eflexive spaces, and hence also superreflexive spaces, are n= ergodic see [, p. 66]. If a Banach space X has the property that all Banach spaces that are finitely representable in X are ergodic, then X must be superreflexive [4]. Thus, when using the reduction to -cocyles based on the result of [], the class of Banach spaces to which it naturally applies is the class of superreflexive spaces.. Bi-Lipschitz distortion of balls For let B = {x H : d W e, x } denote the ball of radius centered at the identity element e H. The bi-lipschitz distortion of B, d W in X,, denoted c X B, is the infimum over those D such that there exists f : B X satisfying x, y B, d W x, y fx fy Dd W x, y. 6 Another way to measure the extent to which H does not admit a bi-lipschitz embedding into X is via the rate at which c X B grows to with. In [8] it was shown that c L B log c 7 for some universal constant c > 0. This result is of importance due to an application to theoretical computer science; see [9, 8, 0] for a detailed discussion. Evaluating the supremum over those c > 0 satisfying 7 remains an important open problem. Theorem. implies the following sharp bound on the bi-lipschitz distortion of B into a p-convex Banach space: Theorem.3. If a Banach space X, satisfies 4 then for every 4 we have c X B log /p. K p X log log Thus in particular for p, ] we have c Lp B p log o and for p we have c Lp B log p o. Theorem.3 is a formal consequence of Theorem.. The simple deduction of Theorem.3 from Theorem. is presented in Section 6. It follows from the results of [, 4] see the explanation in [5, 7] that for every p we have c Lp B log /p. Thus Theorem.3 is sharp up to iterated logarithms. 3

4 . The case of Hilbert space In section 7 we prove the following Poincaré-type inequality for functions on H taking value in Hilbert space: Theorem.4. For every f : H L and every N we have x B k= fxc k fx k x B fxa fx + fxb fx. 8 This result has the following two sharp consequences. First, assume that θ : 0, [0, is nondecreasing, and that θ ω f for some -Lipschitz f : H L. Then since B B and d W c k, e k for all k N, inequality 8 implies that θ t t 3 dt = Combined with [6, Theorem ], we obtain θ s ds s k= k θ k. 9 Corollary.5. A nondecreasing function θ : 0, [0, satisfies θ ω f for some Lipschitz function f : H L if and only if θt t dt t <. 0 A second corollary of Theorem.4 yields a sharp bound up to universal constants on c L B. Indeed, fix and assume that f : B L satisfies d W x, y fx fy Dd W x, y for all x, y B. Let f : H L have Lipschitz constant at most D and coincide with f on B / see equation 4 for an explicit formula defining such an extension f. It follows from 8 applied to f that D k= k log. Thus c L B log. In conjunction with the previously quoted upper bound on c L B, we have Corollary.6. For every we have c L B log. oughly speaking, the proof of Theorem.4 proceeds via a reduction to the case of -cocycles corresponding to the irreducible representations of H see Section 7. But actually, since the representation theory of the continuous Heisenberg group is simpler than the representation theory of the discrete Heisenberg group H, we first apply a discretization argument which reduces Theorem.4 to an inequality on the real Heisenberg group. Then an averaging argument reduces the proof to an inequality on cocycles. Every unitary representation of the continuous Heisenberg group decomposes as a direct integral of irreducibles, and cocycles themselves can be decomposed accordingly. Since the desired inequality involves a sum of squares of norms, it suffices to prove it for cocycles corresponding to irreducible representations that is, for each direct integrand separately. The computation for irreducible representations is carried out in Section 7.. 4

5 Sublinear growth of Heisenberg cocycles in ergodic spaces Write c def = [a, b] = aba b. Thus c lies in the center of H and for every n N we have d W c n, e H = 4n in fact c n = [a n, b n ] = a n b n a n b n. Let X, be a Banach space and π : G AutX be an action of H on X by linear isometric automorphisms. In addition let f Z π be a -cocycle, so f : H X and for all x, y H we have fxy = πxfy + fx. We assume in what follows that f is -Lipschitz, or equivalently that max{ fa, fb } =. In this section we quickly show that if X is an ergodic Banach space then lim inf t ω f t/t = 0, but without obtaining any quantitative bounds. If X is ergodic then the operator on X defined by P x def N = lim πc n x N N is a contraction onto the subspace X 0 X of πc-invariant vectors, and since P x = x for any x X 0 it follows at once that P is idempotent. Also, since c is central in H, the projection P commutes with πg for all g H, and hence the maps g P fg and g I P fg are both still members of Z π. Since P and I P are bounded, these cocycles are both still Lipschitz functions from H to X. We complete the proof by showing that for any ε > 0 we have f n=0 c N εdw e H, c N 8εN for all sufficiently large N. To prove this we consider the two cocycles P f and I P f separately. On the one hand, P f takes values among the πc-invariant vectors, and hence the cocycle identity implies that P f c N = N n=0 πc n P fc = NP fc, and therefore P f c N = N P fc. However, P f c N d W eh, c N N, so these relations are compatible only if P fc = 0. def On the other hand, let f = I P f and for each K, v K def = K K f k= c k. Observe from the cocycle identity and the centrality of c that πgv K + fg = K K k= πg f c k + fg = K = K K f gc k k= K f k= c k g = K K k= π c k fg vk. 5

6 e-arranging gives fg = πgv K v K + K K πc k fg. 3 For any fixed g H the last term of this right-hand side of 3 converges to P I P fg = 0 using again that X is ergodic, and so in particular once K is sufficiently large we obtain that for all g H we have, { max f a ± π a ± v K v K, f b ± π b ± } v K v K ε. Having obtained this approximation to f by a coboundary, let c N = s s s 4N be an expression for c N as a word in S, and observe from another appeal to the cocycle identity that f c N 4N = π s s s i f s i+ = i=0 4N i=0 k= π s s s i π s i+ v K v K + N = πs s s 4N v K v K + N for some remainder N which is a sum of 4N terms all of norm at most ε. Since the action π is isometric and we may let N grow independently of K we obtain f c N vk + N 8εN for all sufficiently large N. Since ε was arbitrary and f above, this completes the proof of. 3 A uniform convexity lemma for ergodic averages c N = f c N by our analysis of P f We prove here a simple lemma on the behavior of ergodic averages in p-convex Banach spaces. Lemma 3.. Assume that X, satisfies 4. Fix z X and an operator T : X X with T. For every integer n 0 denote Then for every l N we have: i=0 l l def s n = n n T j z. p s i+l T jil s il K p z p. 4 Proof. A consequence of 4 is that for every x,..., x n X we have: n n x i p n x j n K p n x i p n p x n i. 5 n i= j= 6 i= i=

7 For the derivation of 5 from 4 see [9, Lem. 3.]. Due to the identity s i+l = l l T jil s il, inequality 5 implies that: l l p T jil s il s i+l K p l p l T jil s il s i+l p The desired inequality 4 now follows by summing 6 over i {0,,...}. 4 Estimates for Heisenberg cocycles K p s il p s i+l p. 6 Let π : H AutX and f Z π be as in Section. For every n N define a linear operator P n : X X by def P n = n n πc j. Thus P n. Lemma 4.. Assume that X, satisfies 4. Then for every l, k, m N there exist integers i [k +, k + m] and j [0, l ] satisfying for all n N, π c jil P i+l f c n P il f c n 6Kn. 7 m/p Proof. Consider the Banach space Y = X X, equipped with the norm x, y Y = x p + y p /p. We also define T : Y Y by T x, y = πcx, πcy. Then T. Since Y, Y satisfies 4 we may apply Lemma 3. to z = fa, fb Y, obtaining the following estimate: 4K p = k+m i=k+ k+m i=k+ l l l m min k+ik+m 0j l l Pi+l f a π c jil P il f a p + P i+l f b π c jil P il f b p π c jil P i+l f a P il f a p + π c jil P i+l f b P il f b p π c jil P i+l f a P il f a p + π c jil P i+l f b P il f b p. It follows that there exist integers i [k +, k + m], j [0, l ] such that { π max c jil P i+l f a P il f a, π c jil } P i+l f b P il f b 7 4K. 8 m/p

8 Consider the operator def Q n = n πa i. n i=0 The cocycle identity implies that fa n = nq n fa. Thus π c jil P i+l fa n P il fa n = n π c jil P i+l P il Q n fa = nq n π c jil P i+l P il fa, 9 where the last equality in 9 holds since c is in the center of H, and therefore Q n commutes with all of {P r } r=0. Since Q n, it follows from 9 and 8 that π c jil P i+l fa n P il fa n 4Kn. 0 m/p Since fa n = πa n fa n, and πa commutes with πc, and hence with all of {P r } r=0, it follows that also π c jil P i+l fa n P il fa n 4Kn. m/p An identical argument implies the analogous bounds with a replaced by b: π c jil P i+l fb n P il fb n 4Kn m /p and π c jil P i+l fb n P il fb n 4Kn. 3 m/p The cocycle identity implies that for all n N, f c n = f [a n, b n ] = πa n b n a n fb n + πa n b n fa n + πa n fb n + fa n. Thus, using 0,, and 3, we conclude the validity of 7. Lemma 4.. For every m, n N we have Proof. Note that for every k N we have Thus, The cocycle identity implies that P m f c n n 5/3. 4 m/3 P m π c k P m = k m πc j m +k m πc j. j= m f P m π c k k P m m. 5 c kn k = π c jn f c n. 6 8

9 Using the fact that f is -Lipschitz, P m and d W e H, c kn 4kn, we deduce from 6 that 4kn Thus, P m f c kn k Pm f c n Pm π c jn f P m c n 5 k Pm f c n k jn m 4n Pm k f c n 4n 3 k 4 m. P m f c n 4n k + 4n3 k m. 7 Choosing k = m /n /3 in 7 roughly the optimal choice of k, we obtain 4. Lemma 4.3. For every m, n N we have: f c n P m f c n m/3 n /3. 8 Proof. In this proof the relation to Section becomes clear. Define f : H X by fh def = f h P m f h = I P m fh. Note that f Z π. Fix an integer k that will be determined later. Consider the vector v X defined by v def = k f c j. k Then v k k j k. 9 Since c is in the center of H, we have the following identity for every h H: πhv + fh = k k Note that k π c j k f h = k k πh f c j + fh = k f hc j = k k = k π c j fh m k f c j h k π c j f h + f c j = k π c j f h v. 30 k m i=0 = m π c j+i fh m i=0 k π c j k k i+k j=i π c j fh. 9

10 Hence, k π c j f h k d W h, e H m m i=0 i k m k d W h, e H. 3 Combining 30 and 3, we see that f is close to a coboundary in the following sense: fh πhv v m k d W h, e H. 3 If we now write c n = h h h 4n for h,..., h 4n { a, a, b, b }, then the cocycle identity for f implies the following bound: f c n 4n = πh h 4n i fh 4n i i=0 3 4n πh h 4n i πh 4n i v v + 4nm k i=0 = π c n v v + 4nm k 9 k + nm k. 33 The optimal choice for k in 33 is k n /3 m/3. For this choice of k, 33 becomes the desired bound 8. 5 Proof of Theorem. As explained in the introduction, using [, Thm. 9.] we may assume without loss of generality that f Z π for some action π of H on X by linear isometric automorphisms. We may also assume that t 8 p. Let m be the largest integer such that t p/3 m m Having defined m, let k be the smallest integer such that and set m 3 p + 3k+ p t, 35 l def = 6 p log m. 36 By Lemma 4. there exist integers i [k +, k + m] and j [0, l ] satisfying for all n N, π c jil P i+l f c n P il f c n 6Kn. 37 m/p Choose n def = m 3 p il,

11 f We may write c n = π c jil P i+l f c n + P il f c n π c jil P i+l f c n + f c n P il f c n. Hence, by Lemma 4., inequality 37, and Lemma 4.3, we obtain the following bound: ω f 4n = ω f d W c n, e H Observe that At the same time, f c n n 5/3 8Kn + i+l/3 m /p + il/3 n / n 38 m 3 p il 3 m p k+l 36 m 3 p + 3k+ p 35 t. Kn. 39 m/p 4n 38 m 3 p il 3 m p k+ml 36 4m 3 p + 3k p m 3m p 35 < 4tm 3m p 34 t. Hence t 4n t. The definition of m implies that m p 3 log n log log n, and therefore 39 becomes: ω f 4n n log log n /p K. log n The proof of Theorem. is complete. 6 Deduction of Theorem.3 from Theorem. Fix 4 and a function f : B X satisfying x, y B, d W x, y fx fy Dd W x, y. 40 Our goal is to bound D from below. Without loss of generality assume that fe = 0. Define f : H X by fx x B /, f x = d W x,e fx x B B /, 4 0 x H B. Then f is D-Lipschitz and coincides with f on B /. Let N H be a maximal 3-separated subset of H. Thus the function f : H X given by f x def = y N f y x only one summand is nonzero for any given x is also D-Lipschitz. Fix a free ultrafilter U on N. Consider the semi-normed space Y = l H, X, Y, where ψ Y def = lim M U B M ψz p z B M Note that since X satisfies 4, so does Y. Y is a semi-normed space rather than a normed space, so we should formally deal below with the quotient Y/{f l H, X : f Y = 0}, but we will /p.

12 ignore this inessential formality in what follows. Complete details are as in the proof of Theorem 9. in []. Alternatively one can note that our proof of Theorem. carries over without change to the class of semi-normed spaces. Define F : H Y by F xz def = f zx f z. This is well defined since the metric d W is left-invariant, and therefore F x Y Dd W x, e H for all x H. Moreover, by leftinvariance, F is D-Lipschitz. Theorem. therefore implies that there exist x, y H such that /4 dw x, y /4 and lim M U B M z B M f zx f zy p d W x, y /p log log /p DK p X. 4 log Fix an integer M > d W x, e + 4 and write m = M d W x, e 4. Since N is a maximal 3-separated subset of H, we have B m x w M wb 3, where M def = {w N : wb 3 B m x }. Hence, since for r we have B r r 4, we can bound the cardinality of M as follows: M dw x, e 4 4 M If w M then there exists z B m and g B 3 such that zx = wg. Hence, for every h B /4 we have d W whx, e = d W zxg hx, e d W z, e + d W x, e + d W g, e + d W h, e < M. Thus the sets {wb /4 x } w M are disjoint and contained in B M. Moreover, if w M and z wb /4 x then d W zx, w /4, and hence also d W zy, w d W zy, zx + d W zx, w /. By the definition of f, this implies that f zx = fw zx and f zy = fw zy. Hence, z B M f zx f zy p d W x, y w M z wb /4 x Theorem.3 now follows from 4 and Embeddings into Hilbert space M B /4 fw zx w zy d W x, y p d W x, e B M. 44 M In this section, we prove Theorem.4. We will deduce it from an inequality on cocycles for the real Heisenberg group. We switch to the real Heisenberg group because its representation theory is simpler. However, this comes at the cost of adding a straightforward discretization step to the proof. The upshot is that we obtain as a byproduct a smooth Poincaré inequality on H of independent interest; see Theorem 7.5. The real Heisenberg group H is defined as the matrix group H def = u w 0 v 0 0 : u, v, w.

13 The discrete Heisenberg group H sits inside H as the cocompact discrete subgroup consisting of unipotent matrices with integer coefficients. We equip the group H with the word metric d S associated with the compact symmetric generating set S = {a u, b v, c w ; u, v, w }, where a u = u , b v = v 0 0, c w = 0 w Let µ denote a Haar measure on H, which coincides with Lebesgue measure under the natural identification of H with 3. Theorem 7.. For every continuous unitary representation π of H, any continuous cocycle γ Z π satisfies the inequality γ c t dt γa u + γb u du. 45 t Section 7. is devoted to the proof of Theorem 7.. It is clearly enough to check inequality 45 when the representation π is irreducible. The proof therefore boils down to a quantitative study of -cocycles with values in an irreducible representation of H. In the next three subsections we deduce Theorem.4 from Theorem 7. by a succession of reductions. Finally, in the last subsection, we state a smooth Poincaré inequality on the real Heisenberg group, whose proof, being very similar to the discrete one, is explained in a few sentences. 7. Proof of Theorem 7. By the Stone-von Neumann theorem see for example [, Ch. ], irreducible representations of H are of two types: those that factor through the center, and, for every λ {0}, the representation π λ on L satisfying h L, π λ a u b v c w hx def = e πiλvx hx + ue πiλw. 46 Note that if a nontrivial irreducible representation π factors through the center then any - cocycle γ Z π must vanish on the center. Indeed, γc w is invariant under πh for every w, which, since the representation is supposed to be irreducible and nontrivial, implies that γc w = 0. Therefore, in proving Theorem 7. we may assume that π = π λ for some λ 0. By [4, Thm. 7] all -cocycles γ Z π λ can be approximated uniformly on compact sets by coboundaries. Hence, it is enough to consider the case where γ is of the form γx = π λ xh h, for some h L. We may assume that h =. By the definition 46, for every w we have γc w = 4 sin πλw, from which we deduce that Also, γc w dw w γa u + γb u du = sin πλw w dw = λ λ. sin πw w dw min{ λ, }. 47 hx + u hx + hx cosπλux dxdu hx + u hx dxdu + hx min{λ x, }dx. 48 3

14 Since = [ u, u ] u + [ u, u ] u + [ u, u ] for every u, we can bound h = from above as follows. hx + u + hx u + hx dx [ u, u ] hx + u hx + hx hx u dx + [ u, u ] hx dx hx + u hx + hx hx u dx + min{λ u hx min{λ x, }dx. 49, } Write k = / λ. By applying 49 with u = kv, and integrating over v [, /] [/, ], we see that γc w dw w 47 min{ λ, } 49 min{ λ, } hx + kv hx dxdv min{ λ, } + min {λ / } hx min{λ x, }dx λ, min{ λ, }k k j= hx + jv hx + j v dxdv + hx min{λ x, }dx = min{ λ, }k hx + u hx dxdu + hx min{λ x, }dx 48 γa u + γb u du. The proof of Theorem 7. is complete. 7. eduction to finitely supported functions Claim 7.. Inequality 8 is a consequence of the following statement. For every finitely supported φ : H L, we have x H k= φxc k φx k x H φxa φx + φxb φx. 50 Proof. Fix N and f : H L. Note that since 8 is not sensitive to adding a constant to the function f, we can assume without loss of generality that the average of f over B 7 is zero. 4

15 Define a cutoff function ξ : H [0, ] by ξx def = x B 5, 6 d W x,e x B 6 B 5, 0 x H B 6, and let φ def = ξf. Then φ is supported on B 6. Since ξ is /-Lipschitz and takes values in [0, ], for all x H and s S, φx φxs ξx ξxs fx + ξxs fx fxs fx + fx fxs. 5 Note that if k {,..., } then d W e, c k 4, and hence for x B we have xc k B 5. Therefore, an application of 50 to φ yields the estimate x B k= fxc k fx k = x B 7 max x H k= s S φxs φx φxc k φx k 5 x H x B 7 fx + max s S φxs φx x B 7 max s S fxs fx. 5 By [7, Thm..] a discrete version of the classical Heisenberg local Poincaré inequality [6], x B 7 fx x B fxa fx + fxb fx, 53 where we used the fact that the average of f on B 7 vanishes. The desired inequality 8 is now a consequence of 5 and eduction to an inequality on the real Heisenberg group Claim 7.3. Inequality 50 is a consequence of the following statement. For every continuous and compactly supported function ψ : H L, we have H ψxc t ψx t dtdµx H sup s S ψxs ψx dµx. 54 Proof. For r > 0 let Br H denote the ball of radius r with respect to the metric d S. Note that H = g H gb. Let σ : H [0, ] be a continuous nonnegative function, which equals on B and 0 outside of B 3. Let σ = g H σ g, where σ g x = σg x. For all x H we have σx C for some C 0,. Writing β = σ/ σ and β g x = βg x, we see that {β g } g H is a continuous partition of unity for H satisfying sup x H g H sup β g xs β g x <. 55 s S 5

16 Throughout the ensuing argument we will use repeatedly the fact that for every x H the number of elements g H for which β g x > 0 is bounded by a constant independent of x, and that g H gb H. Let φ : H L be a finitely supported function on the discrete Heisenberg group H. Define a function on the real Heisenberg group H by ψx = g H φgβ g x. Then ψ is compactly supported. We will eventually apply 54 to ψ, but before doing so we will need some preparatory estimates. The metric d S restricted to H H is bi-lipschitz equivalent to d W [5, Thm ]. It follows that for all g 0 H, if x g 0 B3 then the sum ψx φg 0 = g H β gxφg φg 0 is supported on elements of the form g = g 0 h, with d W h, e K, for some universal constant K N. Thus, using 55 we see that for all x g 0 B, sup ψxs ψx φg 0 h φg s S h B K Integrating 56 over g 0 B gives the following inequality: sup ψxs ψx dµx φg 0 h φg g 0 B s S h B K H By summing 57 over g 0 H we see that sup ψxs ψx dµx max φg s S s S φgs, 58 g H where we used the bound g 0 H φg 0 h φg 0 h B K g H max s S φg φgs, 59 which follows by writing each h B K as a product of at most K elements of S, and using the triangle inequality. In order to deduce from 54 a corresponding bound on φ, we need to bound φ in terms of ψ. To this end, note that for all g 0 H, ψg 0 = φg 0 + βg g 0 φg φg 0 = φg 0 + βh φg 0 h φg 0. g H h B K It follows that for all g 0, g H we have φg 0 φg ψg 0 ψg + h B K φg0 φg 0 h + φg φg h. 60 If x, y H satisfy max{d S x, g 0, d S y, g }, then using 56 we deduce from 60 that φg 0 φg ψx ψy + φg0 φg 0 h + φg φg h. 6 h B K 6

17 Fix g 0 H, x g 0 B, k N and t [k, k+]. Writing g = g 0 c k and y = xc t, we have d S x, g 0 and d S y, g = d S c k g0 xct, e = d S g0 xct k, e d S g0 x, e + d S c t k, e. We may therefore apply 6 and deduce that for all g 0 H, x g 0 B and k N, φg 0 c k φg 0 ψxc t ψx + φg 0 φg 0 h + φg 0c k φg 0 c k h. 6 Integrating 6 over x g 0 B φg 0 c k φg 0 k g 0 B h B K and t [k, k + ], we see that k+ k ψxc t ψx dtdµx + k t h B K φg 0 φg 0 h + φg 0c k φg 0 c k h. 63 Since g 0 B and g 0 B intersect at a set of measure zero if g 0 g 0, by summing 63 over g 0 H and k N, we see that g 0 H k= φg 0 c k φg k H H H ψxc t ψx t dtdµx + g 0 H h B K φg 0 φg 0 h ψxc t ψx t dtdµx + max φg s S φgs g H sup s S ψxs ψx 58 g H max s S φg φgs. dµx + g H max s S φg φgs 7.4 eduction to a -cocycle on H Claim 7.4. Inequality 54 follows from Theorem 7.. Proof. Let ψ : H L be continuous and supported in Br for some r. Take a maximal family { x i B0r } i= of disjoint balls of radius 0r. Define ϕ : H L by ϕg = i= ψx i g. Note that since d S c r, e 4r, for each i N we have H ψxc t ψx r ϕxc t ϕx t dtdµx = x i B5r t dtdµx. 64 Similarly, H sup ψxs ψx dµx = s S x i B r sup s S ϕxs ϕx dµx. 65 7

18 Let X be the space of all finitely supported complex-valued functions on H. We denote by π def the action of H on X given by πxδ y = δ xy, where δ z : H C denotes the function which equals at z H and equals 0 elsewhere. Let U be a free ultrafilter on N. Define a scalar product [, ] on X by: [δ x, δ y ] def = lim n U µbn ϕzx, ϕzy dµz, 66 Bn where, denotes the scalar product on L. Since {B n } n= is a Følner sequence for H, a limit along a free ultrafilter of averages over B n when n is an invariant mean on H. It follows that the scalar product defined in 66 is πh-invariant. Thus π is a unitary representation of H formally we should first pass to the completion of the quotient of X by the subspace consisting of norm-zero elements, but we will ignore this inessential point in what follows. We note that π is also continuous in the strong operator topology. Indeed, since ψ is continuous and compactly supported, ϕ is uniformly continuous. Thus, writing f def X = [f, f] for f X, we have for every y H, lim πxδ y δ y x e X = lim lim x e n U µbn ϕgxy ϕgy dµg Bn lim sup x e g H ϕgxy ϕgy = 0, 67 implying the strong continuity of π. Let γ : H X be given by γx = δ x δ e. Then π Z π and for all x X, γx X = lim n U µbn ϕgx ϕg dµg. 68 B n Arguing as in 67, the uniform continuity of ϕ and 68 imply that γ is a continuous -cocycle. Fix n > 00r large enough so as to ensure that we have µb n B n 00r µb n /. Define I = {i N : x i B5r B n} and write Ω = i I x ib5r and Ω = { i I x ib0r. By the maximality of xi B0r } i= we have Ω B n 00r. Hence µω µb n /. Since H is doubling, µω µω, and therefore I µb5r µω µb n. Note that if g x i B5r for some i N then for every t [, r ] we have gc t x i B 0r. Hence, µbn ϕgc t ϕg dµg Bn µbn i I x i B5r = I µbn ψxc t ψx dµx B5r It follows from 68 and 69 that r µb5r B5r ψxc t ψx r t dµx 68 = sup ϕgc t ϕg dµg µb5r ψxc t ψx dµx. 69 B5r γc t 45 X t dt sup lim s S n U µb n s S γs X B n ϕgs ϕg dµg. 70 8

19 Let J N denote the set of i N such that B n x i B r. Then J µbr µb n+4r. It follows that for all s S, B n ϕgs ϕg dµg ϕgs ϕg dµg i J x i B r = J ψxs ψx dµx µb n+4r B r µbr ψxs ψx dµx. 7 B r Substituting 7 into 70 we conclude that H ψxc t ψx t dtdµx 64 = µb5r µbr sup lim s S n U H µb n+4r µb n B 5r sup s S ψxs ψx r ψxc t ψx t dµx ψxs ψx dµx B r dµx This completes the proof of Claim 7.4, and therefore also the proof of Theorem A smooth Poincaré inequality on H Equip H with the left-invariant iemannian metric given by du + dv + dw udv. In what follows, given a smooth function f : H we let H f denote its gradient with respect to this iemannian structure. Theorem 7.5. For every smooth function f : H, and all > 0, B where C > 0 is a universal constant. fxc t fx dtdµx t B C H fx dµx, The proof of this Poincaré inequality can be obtained from Theorem 7. in a way similar, and actually even shorter than its discrete counterpart. Indeed, the discretization step of Claim 7.3 is not needed here. The other difference lies in the first step, where instead of the discrete Poincaré inequality 53, we use the following smooth version, which is due to [6]. For all > 0, and for all smooth functions f : H whose integral over B is zero, B where c > 0 is a universal constant. fg dµg B c H fg dµg, 9

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