ESTIMATES FOR DERIVATIVES OF THE POISSON KERNELS ON HOMOGENEOUS MANIFOLDS OF NEGATIVE CURVATURE

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1 ESTIMATES FOR DERIVATIVES OF THE POISSON KERNELS ON HOMOGENEOUS MANIFOLDS OF NEGATIVE CURVATURE ROMAN URBAN 1 Abstract. We obtain estimates for derivatives of the Poisson kernels for the second order differential operators on homogeneous manifolds of negative curvature both in the coercive and noncoercive case. 1. Introduction and the main result. In this paper we consider a second order differential operator L on a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an abelian group A = R +. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of Ad exp H n, where n is the Lie algebra of N, are all greater than 0. Conversely, every such a group equipped with a suitable left-invariant metric becomes a homogeneous Riemannian manifold with negative curvature, [7]. On S we consider a second order left-invariant operator L = m j=0 Y 2 j + Y. We assume that Y 0, Y 1,..., Y m generate the Lie algebra s of S. We can always make Y 0,..., Y m linearly independent and moreover, we can choose Y 0, Y 1,..., Y m so that Y 1 (e),..., Y m (e) belong to the Lie algebra n of N. Let π : S A = S/N be the canonical homomorphism. Then the image of L under π is a second order left-invariant operator on R +, where γ R. (a a ) 2 γa a, 1991 Mathematics Subject Classification. 22E25, 43A85, 53C30, 31B25. 1 The author was partly supported by KBN grant 2P03A02821, Foundation for Polish Sciences, Subsidy 3/99 and European Commission via TMR Network Harmonic Analysis, contract no. ERB FMRX-CT

2 Finally, the operator we are interested in can be written in the form (1.1) L = L γ = j Φ a (X j ) 2 + Φ a (X) + a 2 2 a + (1 γ)a a, where X, X 1,..., X m are left-invariant vector fields on N, X 1,..., X m linearly independent and generate n, Φ a = Ad exp(log a)y0 = e (log a) ad Y 0 = e (log a)d. D = ad Y0 is a derivation of the Lie algebra n of the Lie group N such that the real parts d j of the eigenvalues λ j of D are positive. By multiplying L γ by a constant i.e. changing Y 0, we can make d j arbitrarily large, [3]. We study derivatives of the Poisson kernels for the operators (1.1). Let µ γ t be the semigroup of measures generated by L γ. It is known, [5], that if γ 0, then there exists a unique (up to a positive multiplicative constant) positive Radon measure ν γ with a smooth density m γ on N such that ˇµ γ t ν γ = ν γ, γ 0. ν γ or its density m γ is called the Poisson kernel for the operator L γ. For γ > 0 the measure ν γ is bounded, while ν 0 is unbounded. These measures have been studied by many authors and in various contexts; see e.g. [3] and literature quoted there. The main goal of this paper is to prove the following estimates for derivatives of the Poisson kernels m γ of L γ for γ 0 (see Theorem 6.1): (1.2) X I m γ (x) C(1 + x ) Q γ I (log(2 + x )) I 0, where stands for a homogeneous norm on N, I is a suitably defined length of the multi-index I and I 0 is a certain number depending on I and the nilpotent part of the derivation D. In particular, I 0 is equal to 0 if the action of A = R + on N, given by Φ a, is diagonal. X 1,..., X n is an appropriately chosen basis of n. For the precise definitions of all the notions that have appeared here see Section 2. Some comments should be made at this moment. First of all, it should be said that the estimate (1.2) for positive γ is known and has been obtained by E. Damek, A. Hulanicki and J. Zienkiewicz in [4] but their approach did not work for γ = 0. The case γ = 0 is essentially different from the case γ > 0 since the operator becomes noncoercive (i.e. there is no ε > 0 such that L 0 + εi admits the Green function) and the Ancona theory [1] is of no use. For the first time the operator L 0 has been studied successfully by E. Damek, A. Hulanicki and the author of this paper in [3], where, among others, the following estimate 2

3 for the Poisson kernel m 0 was obtained C 1 (1 + x) Q m 0 (x) C(1 + x ) Q. However, the question about derivatives in the noncoercive case remained without any answer. In this paper we answer that question. Moreover, our proof works simultaneously for all γ 0 thus we give a new, different than that in [4], proof of the estimate (1.2) for γ > 0. The proof of (1.2) requires both analytic and probabilistic techniques. Some of them have been introduced in [4, 3]. Guide to the paper. The structure of the paper is as follows. In Section 2 we state precisely notation and all necessary definitions. In Section 3 we recall a definition of the Bessel process which appears as the vertical component of the diffusion generated by a 2 L γ on N R +. (cf. [3]). Moreover, we state some lemmas about its properties (without the proofs but we give references where they can be found). In Section 4 we prove the main estimates of the derivatives of the transition probabilities of the evolution on N generated by an appropriate operator which appears as the horizontal component of the diffusion on N R + mentioned above. In Section 5 we prove main probabilistic lemmas which are crucial in the proof of (1.2). This section heavily depends on Sections 3 and 4. Finally, in Section 6 we state precisely the estimate (1.2) (see Theorem 6.1) and we give its proof. Acknowledgements. The author is grateful to Ewa Damek for her helpful remarks concerning the content of this paper. 2. Preliminaries. Let N be a connected and simply connected nilpotent Lie group. Let D be a derivation of the Lie algebra n of N. For every a R + we define an automorphism Φ a of n by the formula Writing x = exp X we put Φ a = e (log a)d. Φ a (x) := exp Φ a (X). Let nc be the complexification of n. Define nc λ = {X nc : k > 0 such that (D λi) k = 0}. 3

4 Then (2.1) n = Imλ 0 where { V λ = V λ (nc n = C λ) n if Imλ 0, nc λ n if Imλ = 0. We assume that the real parts d j of the eigenvalues λ j of the matrix D are strictly greater than 0. We define the number V λ, (2.2) Q = j Re λ j = j d j and we refer to this as a homogeneous dimension of N. In this paper D = ad Y0 (see Introduction). Under the assumption on positivity of d j, (2.1) is a gradation of n. We consider a group S which is a semi-direct product of N and the multiplicative group A = R + = {exp ty 0 : t R} : S = NA = {xa : x N, a A} with multiplication given by the formula (xa)(yb) = (xφ a (y) ab). In N we define a homogeneous norm, (cf. [4, 3]) as follows. Let (, ) be a fixed inner product in n. We define a new inner product 1 ( ) da (2.3) X, Y = Φ a (X), Φ a (Y ) a and the corresponding norm 0 X = X, X 1/2. We put X = (inf{a > 0 : Φ a (X) 1}) 1. One can easily show that for every Y 0 there exists precisely one a > 0 such that Y = Φ a (X) with X = 1. Then we have Y = a. Finally, we define the homogeneous norm on N. For x = exp X we put x = X. Notice that if the action of A = R + on N (given by Φ a ) is diagonal the norm we have just defined is the usual homogeneous norm on N and the number Q in (2.2) is just the homogeneous dimension of N, [6]. Having all that in mind we define appropriate derivatives (see also [4]). We fix an inner product (2.3) in n so that V λj, j = 1,..., k 4

5 are mutually orthogonal and an orthonormal basis X 1,..., X n of n. The enveloping algebra U(n) of n is identified with the polynomials in X 1,..., X n. In U(n) we define X 1... X r, Y 1... Y r = r j=1 X j, Y j. Let Vj r be the symmetric tensor product of r copies of V λj. For I = (i 1,..., i k ) (N {0}) k let Then for X V λj X I = X (i 1) 1... X (i k) k, where X (i j) j V i j j. Φ a (X ) c exp(d j log a + D j log(1 + log a )), where d j = Reλ j and D j = dim V λj 1 and so (2.4) ( k ) k Φ a (X I ) exp i j (d j log a + D j log(1 + log a )) X (i j) j. j=1 Notation. The letter C or c, possibly with a subscript number, occurs in inequalities as a positive constant which is independent of the important parameters of the formula, and may vary from statement to statement, even in the same calculation. 3. Bessel process. Let b t denotes the Bessel process with a parameter α 0 (cf. [9]), i.e. a continuous Markov process with the state space [0, + ) generated by a 2 + 2α+1 a a. The transition function with respect to the measure y 2α+1 dy is given, e.g. in [2, 9], by: ( ) 1 2t (3.1) p t (x, y) = exp x 2 y 2 4t ( 1 exp 2 α (2t) α+1 Γ(α+1) where I α (x) = k=0 ( I xy ) α 2t y 2 4t j=1 1 for x, y > 0, (xy) α ) (x/2) 2k+α k!γ(k + α + 1) for x = 0, y > 0, is the Bessel function (see [8]). Therefore for x 0 and a measurable set B (0, ): P x (b t B) = p t (x, y)y 2α+1 dy. B If b t is the Bessel process with a parameter α starting from x, i.e. b 0 = x, then we will write that b t BESS x (α) or simply b t BESS(α) if the starting point is not important or is clear from the context. 5

6 The following lemmas concerning some properties of the Bessel process are very well known and their proofs are rather standard. Their proofs can be found e.g. in [3], [11], [12]. Lemma 3.1. Let b t BESS(α), α 0. Let D, γ, x 0. There exists a positive constant C such that for every t > s > 0, ( t D E x bsds) γ C(t s) D(1+γ/2). s Lemma 3.2. Let b t BESS(α), α 0. There exist constants c 1, c 2 such that for every x 0, for every λ > 0 and for every t > 0, P x ( sup b s x + λ) c 1 e c 2λ 2 /t. s [0,t] Lemma 3.3. Let b t BESS(α), α 0. There exist constants c 1, c 2 such that for every R > 0 and for every t > 0, P R ( inf b s < R/2) c 1 e c 2R 2 /t. s [0,t] Proof. See Lemma 2.4 in [3] or Lemma 3.6 in [11]. Lemma 3.4. Let b t BESS(α), α 0. Let 0 < η < x be fixed. Then there exist constants c 1, c 2 such that for every t > 0, P x ( inf s [0,t] b s x η) c 1 e c 2/t. Proof. It is enough to rewrite the proof of Lemma 2.4 in [3]. Lemma 3.5. Let b t BESS(α), α 0. There exist positive constants c 1, c 2 such that for every for every x 0, r > 0 and t > 0, P x ( sup b s r) c 1 e c 2 s [0,t] t r. Proof. See Lemma 3.5 in [11] or Lemma 2.3 in [3]. By a straightforward computation, using the definition of the transition function p t (x, y) of the Bessel process (3.1) and asymptotic behavior of the Bessel function (see [8]): { x α 2 I α (x), x 0, α Γ(1+α) exp(x), x, (2πx) 1/2 we get Lemma 3.6. Let b t BESS(α), α 0. There exists a constant C independent of x such that P x (a η b t a + η) Ct (α+1) m([a η, a + η]), 6

7 where m(b) = B y2α+1 dy. 4. Evolutions. Let X, X 1,..., X m be as in (1.1). Let σ : [0, ) [0, ) be a continuous function such that σ(t) > 0 for every t > 0. We study the family of evolutions L σ(t) t, where ( ) (4.1) L σ(t) = σ(t) 2 Φσ(t) (X j ) 2 + Φ σ(t) (X). The main objective of this section is to get pointwise estimates for the derivatives of the transition probabilities p σ (t, s) of the evolution generated by the operator (4.1). For the main result of the paper we are mainly interested in the operator (4.1) with σ(t) beeing a trajectory of an appropriate Bessel process. The estimates we obtain eventually are not by any means optimal. However they allow us to proceed further. In order to get them we make use of the methods developed in [3] and [4]. Since we may assume that X 1,..., X m are linearly independent we select X m+1,..., X n so that X 1,..., X n form a basis of n. For a multiindex I = (i 1,..., i n ), i j algebra n of N we write: X I = X i Xn in k = 0, 1,..., we define Z + and the basis X 1,..., X n of the Lie and I = i i n. For C k = {f : X I f C(N), for I < k + 1} and C k = {f C k : lim X I f(x) exists for I < k + 1}. x For k < the space C k is a Banach space with the norm f C k = X I f C(N). I k Let {U σ (s, t) : 0 s t} be the unique family of bounded operators on C = C 0 which satisfy i) U σ (s, s) = I, ii) U σ (s, r)u σ (r, t) = U σ (s, t), s < r < t, iii) s U σ (s, t)f = L σ(s) U σ (s, t)f for every f C, iv) t U σ (s, t)f = U σ (s, t)l σ(t) f for every f C, v) U σ (s, t) : C 2 C 2. U σ (s, t) is a convolution operator. Namely, U σ (s, t)f = f p σ (t, s), where p σ (t, s) is a smooth density of a probability measure. By ii) we have p σ (t, r) p σ (r, s) = p σ (t, s) for t > r > s. Existence of the family U σ (s, t) follows from [10]. 7

8 For s < t let (4.2) (4.3) ξ σ (s, t) = sup Φ σ(u) n n, u [s,t] η σ (s, t) = sup Φ 1 σ(u) n n. u [s,t] It is not difficult to show (see [4]) that there exist β 1, β 2 > 2 and C > 0 such that for every a > 0, (4.4) Φ a n n C(a β 1 + a β 2 ). Therefore, (4.5) ξ σ (s, t) C(( sup σ(u)) β 1 + ( sup σ(u) β 2 ), u [s,t] u [s,t] η σ (s, t) C(( inf u [s,t] σ(u)) β 1 + ( inf u [s,t] σ(u)) β 2 ). Theorem 3.5 in [4] remains valid in our setting. Therefore we state it verbatim Proposition 4.1. For every multi-index I there are positive constants C, M such that X I p σ (t, s) L (N) C(1 + ξ σ (s, t) + η σ (s, t)) M if 1 t s and X I 1. The next proposition shows what happens in the above Proposition if we allow the difference between t and s to be arbitrarily small Proposition 4.2. For every multi-index I there exist positive constants C, M and ϑ such that X I p σ (t, s) L (N) C(t s) ϑ (1 + ξ σ (s, t) + η σ (s, t)) M if t s 1 and X I 1. Proof. Using a standard homogeneity argument this is simply a corollary from Proposition 4.1. Now we state some properties of the kernels p σ (t, 0) which have been proved in Section 4 of [3] by means of the Nash inequality. Let τ(x) denotes the Riemannian distance between x and the identity element e of N. We define (4.6) A(s, t) = t where β 1 and β 2 are as in (4.4). s σ β 1 2 (u) + σ β 2 2 (u)du, 8

9 The constant D which is appearing in the next theorems is an arbitrary constant between the local dimension and the dimension at infinity of (N, X 1,..., X m ) (see [13]). Theorem 4.3. There exists a positive constant C such that for every r, α > 0, p σ (t, s) L 1 (e ατ ) e αr e C(α+α2 )A(s,t), where A(s, t) is as in (4.6). Theorem 4.4. There exist positive constants C and D Q such that for every t, s, ( t ) D/4 p σ (t, s) L 2 (N) C σ 2(1 Q/D) (u)du s Theorem 4.5. There exist positive constants C and D Q such that for every s < t, ( t D/2 p σ (t, s) L (N) C σ (u)du) 2(1 Q/D). s Theorem 4.6. For every compact set K N which does not contain the identity e of N, there exist positive constants C, ξ, β 1, β 2 and D Q such that for every x K and for every t > 0, ( t D/2 ( p σ (t, 0)(x) C σ (u)du) 2(1 Q/D) exp ξ ), A(0, t) where A(s, t) is as in (4.6) Since f L 2 (e ατ ) f 1/2 L 1 (e ατ ) f 1/2 L 4.3 we get the following 0 by Theorem 4.5 and Theorem Theorem 4.7. There exists a positive constant C such that for every r, α > 0, ( t D/4 p σ (t, s) L 2 (e ατ ) C σ (u)du) 2(1 Q/D) e αr e C(α+α2 )A(s,t), where A(s, t) is as in (4.6). s We will need also the following simple lemma (see e.g. [3]). Lemma 4.8. Let α 0. There is a constant C = C(I, τ, α) such that we have X I f L 2 (e ατ ) C X J f 1/2 L f 1/2 L 1 (e ατ ). J 2 I 9

10 Now we are going to prove the main estimates for the derivatives of the evolution kernels p σ (t, 0). Theorem 4.9. Let 0 < T 1 < T 2 < t. For every multi-index I such that X I 1 and for every compact set K N which does not contain the identity e of N, there exist positive constants ξ, C, M, ϑ such that for every x K and for every t > 0, X I p σ (t, 0)(x) C max{(t 2 T 1 ) ϑ, 1} where A(s, t) is as in (4.6). (1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 )) M e ξ/a(0,t). Proof. The idea of the proof is taken from [4] (see Lemma 4.17 there) and [3]. Let α be a positive number. By subadditivity of the Riemannian distance τ, we have (4.7) X I (p σ (t, 0))(x)e ατ(x) p σ (t, T 2 )(xy 1 )e ατ(xy 1) X I p σ (T 2, 0)(y) e ατ(y) dy N p σ (t, T 2 ) L 1 (e ατ ) X I (p σ (T 2, 0))( )e ατ( ) L. For a multi-index J, let X J be the right-invariant operator corresponding to the left-invariant operator X J. Then, analogously, we obtain (4.8) X J (p σ (T 2, 0))( )e ατ( ) L X J (p σ (T 2, T 1 ))( )e ατ( ) L p σ (T 1, 0) L 1 (e ατ ). Expressing the operator X I in terms of XI and vice versa, i.e. X I = J I ω J X J, where ω J s are polynomials on N (see e.g. [6]), by (4.7) and (4.8) we may estimate (4.9) X I (p σ (t, 0))(x)e ατ(x) C p σ (t, T 2 ) L 1 (e ατ ) p σ (T 1, 0) L 1 (e 2ατ ) X J (p σ (T 2, T 1 ))( )e 2ατ( ) L J I C p σ (t, T 2 ) L 1 (e ατ ) p σ (T 1, 0) L 1 (e 2ατ ) X K (p σ (T 2, T 1 ))( )e 3ατ( ) L. K I Now we estimate more carefully the very last term in (4.9). Taking T 1 < T < T 2 as in (4.7), we have X K (p σ (T 2, T 1 ))(x)e 3ατ(x) p σ (T 2, T ) L 2 (e 3ατ ) X K p σ (T, T 1 ) L 2 (e 3ατ ) 10

11 and in view of Lemma 4.8 (4.10) X K p σ (T 2, T 1 )(x)e 3ατ(x) C p σ (T 2, T ) 1/2 L pσ (T 2, T ) 1/2 L 1 (e 3ατ ) X J p σ (T, T 1 ) 1/2 L pσ (T, T 1 ) 1/2 L 1 (e 3ατ ). J 2 K By (4.9) and (4.10), we finally have (4.11) X I (p σ (t, 0))(x)e ατ(x) C p σ (t, T 2 ) L 1 (e ατ ) p σ (T 1, 0) L 1 (e 2ατ ) p σ (T 2, T ) 1/2 L 1 (e 3ατ ) p σ (T 2, T ) 1/2 L pσ (T, T 1 ) 1/2 X J p σ (T, T 1 ) 1/2 L. L 1 (e 3ατ ) J 2 I Now we take T = (T 1 + T 2 )/2 and we estimate all the terms in (4.11). By Theorem 4.3 p σ (t, T 2 ) L 1 (e ατ ) p σ (T 1, 0) L 1 (e 2ατ ) p σ (T 2, T ) 1/2 L 1 (e 3ατ ) pσ (T, T 1 ) 1/2 L 1 (e 3ατ ) e αr e C(α+α2 )A(0,t) and by Propositions 4.1, 4.2 p σ (T 2, T ) 1/2 L XJ p σ (T, T 1 ) 1/2 L Therefore (4.11) becomes C max{(t 2 T 1 ) ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η ξ (T 1, T 2 )) M. (4.12) X I p σ (t, 0)(x) C max{(t 2 T 1 ) ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 )) M e αr e C(α+α2 )A(0,t) e ατ(x). Now we put α = ετ(x)/a(0, t), where ε > 0 and ε is taken so that ε < 1/2C and r is such that for every x K, r < τ(x)/4 then we obtain X I p σ (t, 0)(x) C max{(t 2 T 1 ) ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 )) M e ετ 2 (x)/4a(0,t) for every x K, and the proof is complete. Theorem Let 0 < T 1 < T 2 < t. For every multi-index I such that X I 1 there exist positive constants C, M, ϑ and D Q such 11

12 that ( t ) D/4 X I p σ (t, 0) L C max{(t 2 T 1 ) ϑ, 1} σ 2(1 Q/D) (u)du T 2 (1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 )) M. Proof. We proceed similarly to the proof of the previous Theorem. Clearly for any positive α we have X I (p σ (t, 0))(x)e ατ(x) p σ (t, T 2 ) L 2 (e 2ατ ) X I p σ (T 2, 0) L 2 (e 2ατ ). By Theorem 4.7 ( t ) D/4 p σ (t, T 2 ) L 2 (e 2ατ ) C σ 2(1 Q/D) (u)du e αr e C(α+α2 )A(T 2,t). T 2 Let, X denotes the right-invariant vector field corresponding to X. As in the previous proof we express the operator X I in terms of XI (see [6]) and we get that X I p σ (T 2, 0) L 2 (e 2ατ ) C X J p σ (T 2, 0) L 2 (e 3ατ ). Therefore, J I ) D/4 ( t (4.13) X I (p σ (t, 0))(x)e ατ(x) C σ 2(1 Q/D) (u)du T 2 e αr e C(α+α2 )A(T 2,t) X J p σ (T 2, 0) L 2 (e 3ατ ). J I Now we want to estimate X J p σ (T 2, 0) L 2 (e 3ατ ). By Lemma 4.8 we can write (4.14) X J p σ (T 2, 0) L 2 (e 3ατ ) C X K p σ (T 2, 0) 1/2 L pσ (T 2, 0) 1/2 L 1 (e 3ατ ) C C K 2 J K 2 J L 2 J X K p σ (T 2, T 1 ) 1/2 L pσ (T 1, 0) 1/2 L 1 p σ (T 2, 0) 1/2 L 1 (e 3ατ ) X L (p σ (T 2, T 1 ))( )e ατ( ) 1/2 L pσ (T 2, 0) 1/2 L 1 (e 3ατ ). 12

13 Hence by (4.13), (4.14) and Theorem 4.3 we have ( t ) D/4 (4.15) X I (p σ (t, 0))(x)e ατ(x) C σ 2(1 Q/D) (u)du T 2 e Cαr e C(α+α2 )A(0,t) X L (p σ (T 2, T 1 ))( )e ατ( ) 1/2 L. L 2 I Proceeding exactly in the same way as in the proof of Theorem 4.9 one shows that (cf. (4.12)) X L (p σ (T 2, T 1 ))(x)e ατ(x) C max{(t 4 T 3 ) ϑ, 1}(1 + ξ σ (T 3, T 4 ) + η σ (T 3, T 4 ) M e Cαr e C(α+α2 )A(0,t), where T 1 < T 3 < T 4 < T 2. Putting T 3 = (3T 1 + T 2 )/4 and T 4 = (T 1 + T 2 )/2 we obtain (4.16) X L (p σ (T 2, T 1 ))(x)e ατ(x) L C max{(t 2 T 1 ) 2ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 ) M e 3αr e C(3α+9α2 )A(0,t). Now by (4.15) and (4.16) we get (4.17) ( t ) D/4 X I (p σ (t, 0))(x) C σ 2(1 Q/D) (u)du e Cαr e C(α+α2 )A(0,t) T 2 max{(t 2 T 1 ) ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 ) M e ατ(x). Since the constant C in (4.17) does not depend on α we can take limit as α tends to 0 and then taking sup over x N we get our result ( t ) D/4 X I (p σ (t, 0)) L C σ 2(1 Q/D) (u)du T 2 max{(t 2 T 1 ) ϑ, 1}(1 + ξ σ (T 1, T 2 ) + η σ (T 1, T 2 ) M. Let (5.1) 5. Two probabilistic lemmas. V ={(x, a) N R + : (x, a) := x + a = 1}, T δ ={(x, a) N R + : x < δ}, δ > 0. 13

14 Lemma 5.1. Let σ t BESS(α), α 0. Let 1/8 < δ < 1/4, and dm(a) = a 2α+1 da. For every multi-index I such that I > 0 there exists a constant C such that for every (x, a) V \ T δ, sup 0<η<1/8 1 0 E 0 X I p σ (t, 0)(x)m([a η, a + η]) 1 1 [a η,a+η] (σ t )dt C, where p σ (t, 0) is the evolution kernel defined in Section 4 corresponding to the operator (4.1). Proof. In order to simplify the notation let I a,η = [a η, a+η]. We divide the set of all trajectories of the Bessel process σ t (with a parameter α) starting from 0 into two subsets: A = {σ : sup σ s 2}, B = {σ : sup σ s > 2}. s [0,t/2] s [0,t/2] Consider the set A. For k 1, let A k = {σ : 1/2 k+1 < sup s [0,t/2] σ s 1/2 k }. Then clearly, A = k= 1 A k. Therefore, where 1 0 E 0 X I p σ (t, 0)(x)m(I a,η ) 1 1 Ia,η (σ t )1 A (σ)dt = I k = 1 Define two stopping times 0 I k, k= 1 E 0 X I p σ (t, 0)(x)m(I a,η ) 1 1 Ia,η (σ t )1 Ak (σ)dt. T 1 = T 1 (k) = inf{s t/2 : σ s = 1/2 k+2 }, T 2 = T 2 (k) = inf{t 1 < s t/2 : σ s = 1/2 k+1 or σ s = 1/2 k+3 }. First we want to estimate I k. By Theorem 4.9 and (4.5) we have that for every σ A k, X I p σ (t, 0)(x) C(T 2 T 1 ) ϑ 2 max{β 1,β 2 }Mk e ξ2k min{β 1,β 2 }/t where ξ is adjusted to the set {x N : δ x 1}. Thus (5.2) I k C2 max{β 1,β 2 }Mk 1 e ξ2k min{β 1,β 2 }/t 0 E 0 (T 2 T 1 ) ϑ m(i a,η ) 1 1 Ia,η (σ t )1 Ak (σ)dt. Now we are going to deal with the expected value above. Note that on the set A k we have T 2 < t/2. Using the Markov property and Lemma 14

15 3.6 we get (5.3) Writing we have E 0 (T 2 T 1 ) ϑ m(i a,η ) 1 1 Ia,η (σ t )1 Ak (σ) E 0 (T 2 T 1 ) ϑ 1 {T2 <t/2}m(i a,η ) 1 E σt2 1 Ia,η (b t T2 ) CE 0 (T 2 T 1 ) ϑ (t T 2 ) (α+1) 1 {T2 <t/2}(σ) C(t/2) (α+1) E 0 (T 2 T 1 ) ϑ 1 {T2 <t/2}(σ) {T 2 < t/2} = E 0 (T 2 T 1 ) ϑ 1 {T2 <t/2} l [log 2 (2/t)] l [log 2 (2/t)] {1/2 l+1 T 2 T 1 < 1/2 l } 2 (l+1)ϑ P 0 (1/2 l+1 T 2 T 1 < 1/2 l ). Now we have to consider two cases. The first one when σ T2 = 1/2 k+1 and the second when σ T2 = 1/2 k+3. In the first case, by Lemma 3.2 P 0 (1/2 l+1 T 2 T 1 < 1/2 l ) P 1/2 k+2( In the second case, by Lemma 3.3 sup s [0,1/2 l ] c 1 e c 22 l /2 2k+4. P 0 (1/2 l+1 T 2 T 1 < 1/2 l ) P 1/2 k+2( In any case (5.4) E 0 (T 2 T 1 ) ϑ 1 {T2 <t/2} C 1 By (5.2), (5.3) and (5.4) we get that inf s [0,1/2 l ] c 1 e c 22 l /2 2k+4. l [log 2 (2/t)] σ s 1/2 k+1 ) σ s 1/2 k+3 ) 2 (l+1)ϑ e C 22 l /2 2k Ce 2kϑ. 1 I k C2 max{β 1,β 2 }Mk e 2kϑ t (α+1) e ξ2k min{β 1,β 2 }/t dt. Thus k= 1 I k C. Now we consider the set B. Define two stopping times T 1 = inf{s t/2 : σ s = 3/4}, 0 T 2 = inf{t 1 < s t/2 : σ s = 3/2 or σ s = 1/2}. By Theorem 4.9 for σ B we have X I p σ (t, 0)(x) C max{(t 2 T 1 ) ϑ, 1}. 15

16 Therefore, we have to estimate (5.5) 1 0 E 0 max{(t 2 T 1 ) ϑ, 1}m(I a,η ) 1 1 Ia,η (σ t )1 B (σ)dt. By the Markov property, Lemma 3.6 and the fact that t T 2 t/2 we can estimate the expected value under the integral sign in (5.5) as follows (5.6) E 0 max{(t 2 T 1 ) ϑ, 1}1 B (σ)m(i a,η ) 1 1 Ia,η E σt2 (b t T2 ) CE 0 max{(t 2 T 1 ) ϑ, 1}1 B (σ)(t T 2 ) (α+1) Ct (α+1) E 0 max{(t 2 T 1 ) ϑ, 1}1 B (σ). Now we are going to estimate E 0 (T 2 T 1 ) ϑ 1 B (σ). For l 0, let C l = {1/2 l+1 < T 2 T 1 1/2 l }. Then E 0 (T 2 T 1 ) ϑ 1 B (σ) = E 0 (T 2 T 1 ) ϑ 1 B (σ)1 Cl (σ) l=0 2 (l+1)ϑ E 0 1 B (σ)1 Cl (σ) l=0 By Lemma (l+1)ϑ P 0 (B) 1/2 P 0 (C l ) 1/2. l=0 (5.7) P 0 (B) c 1 e c 2/t. If σ T2 = 1/2 then by Lemma 3.4 we have (5.8) P 0 (C l ) P 3/4 ( inf σ s 1/2) c 1 e c 22 l s [0,1/2 l ] and if σ T2 = 3/2 then by Lemma 3.2 (5.9) P 0 (C l ) P 3/4 ( sup σ s 3/2) c 1 e c 22 l. s [0,1/2 l ] Using estimates (5.7), (5.8) and (5.9) we get that E 0 (T 2 T 1 ) ϑ 1 B (σ) c 1 e c 2/t. This together with (5.6) complete the proof. Lemma 5.2. Let σ t BESS(α), α 0. Let dm(a) = a 2α+1 da. For every multi-index I such that I > 0 there exists a constant C such that for every (x, a) V, (see (5.1)) (5.10) sup 0<η<1 1 E 0 X I p σ (t, 0)(x)m([a η, a + η]) 1 1 [a η,a+η] (σ t )dt C, 16

17 where p σ (t, 0) is the evolution kernel defined in Section 4 corresponding to the operator (4.1). Proof. For k 0 define the following sets Let A k = { {σ : sup s [0,t/2] σ s 1} for k = 0, {σ : 1/(k + 1) sup s [0,t/2] σ s < 1/k} for k 1. T 1 =T 1 (k) = inf{0 < s < t/2 : σ s = 1/2(k + 1)}, T 2 =T 2 (k) = inf{t 1 < s < t/2 : σ s = 1/(k + 1) or σ s = 1/3(k + 1)}. First we want to estimate the left hand side of (5.10) on the set A k i.e. the absolute value of the following integral (recall that I a,η denotes the interval [a η, a + η]) I k = 1 E 0 X I p σ (t, 0)(x)m(I a,η ) 1 1 Ia,η (σ t )1 Ak (σ)dt. Since on A k we have T 2 t/2 by Theorem 4.10 and (4.5) we get that X I p σ (t, 0)(x) C max{(t 2 T 1 ) ϑ, 1}(k + 1) Mβ ( D/4 3t/4 σu du) 2(1 Q/D), where β = max{β 1, β 2 }. Hence by the Markov property and Lemma 3.6 I k C(k + 1) Mβ E 0 max{(t 2 T 1 ) ϑ, 1}1 Ak (σ) ( 3t/4 t/2 1 t/2 σ 2(1 Q/D) u du) D/4 E σ3t/4 m(i a,η) 1 1 Ia,η (b t/4 )dt C(k + 1) Mβ t (α+1) E 0 max{(t 2 T 1 ) ϑ, 1}1 Ak (σ) ( 3t/4 t/2 1 σ 2(1 Q/D) u du) D/4 dt. 17

18 Now by the Schwarz inequality and Lemma 3.1 we obtain I k C(k + 1) Mβ t (α+1) (E 0 max{(t 2 T 1 ) 4ϑ, 1}) 1/4 ( 3t/4 P 0 (A k ) 1/4 E 0 1 t/2 σu 2(1 Q/D) du ) D/4 C(k + 1) Mβ t (α+1) Q/4 (E 0 max{(t 2 T 1 ) 4ϑ, 1}) 1/4 1 P 0 (A k ) 1/4 dt. But using Lemma 3.4 and Lemma 3.2 E 0 max{(t 2 T 1 ) 4ϑ, 1} 1 + E 0 (T 2 T 1 ) 4ϑ (l+1)4ϑ P 0 (1/2 l+1 T 2 T 1 1/2 l ) 2 + l=0 l=0 P 1/2(k+1) ( 2 + c 1 1/2 2 (l+1)4ϑ max{p 1/2(k+1) ( inf σ s 1/3(k + 1)), s [0,1/2 l ] sup s [0,1/2 l ] l=0 σ s 1/(k + 1))} 2 (l+1)4ϑ e c 22 l /(k+1) 2 C 1 (k + 1) C 2ϑ. On the other hand by Lemma 3.5, P 0 (A k ) c 1 e c 2 tk so we obtain that I k C(k + 1) Mβ+C 2ϑ 1 t (α+1) Q/4 e c 2 tk dt. Since the left hand side of (5.10) is equal to k=0 I k the proof is complete. 6. Estimates of the Poisson kernels In this section we obtain pointwise estimates for derivatives of the Poisson kernels both in the coercive and noncoercive case. Namely we prove the following Theorem 6.1. Let γ 0. For a multi-index I = (i 1,..., i k ) and all X I = X (i 1) 1... X (i k) k, where X (i j) j V i j j, with X I 1, there are constants C such that X I m γ (x) C(1 + x ) Q γ I (log(2 + x )) I 0, 18

19 where I = k j=1 i jd j, d j = Reλ j, and I 0 = k j=1 i jd j, D j = dimv λj 1. For the proof we define appropriate Bessel process and an evolution. Let (6.1) L α = a 2 L γ = a 2 Φ a (X j ) 2 + a 2 Φ a (X) + a 2 + 2α + 1 a, a j where α 0 and α = γ/2. Let G α (x, a; y, b) be the Green function for L α. G α is defined by two conditions: i) L α G α ( ; y, b) = δ (y,b) as distributions (functions are identified with distributions via the right Haar measure), ii) for every yb S, G α ( ; y, b) is a potential for L α. The Green function G α for L α is given by (6.2) G α (x, a; y, b) = 0 p t (x, a; y, b)dt, where T t f(x, a) = f(y, b)p t (x, a; y, b)dyb 2α+1 db is the heat semigroup on L 2 (N R +, dyb 2α+1 db) with the infinitesimal generator L α. In (6.2) we allow (y, b) to be (e, 0) since lim (y,b) (e,0) G(x, a; y, b) exists and is a smooth function of (x, a) (see [3]). On N R + we define dilations : Then and so D t (x, a) = (Φ t (x), ta), t > 0. L α (f D t ) = t 2 L α f D t. (6.3) G α (x, a; y, b) = t Q 2α G α (D t 1(x, a); D t 1(y, b)). It has been proved in [3] that (6.4) m γ (x) = G γ/2 (x 1, 1; e, 0) = G γ/2(e, 0; x 1, 1), γ 0 where G is the Green function for the operator L α = a 2 Φ a (X j ) 2 a 2 Φ a (X) + a 2 + 2α + 1 a, a conjugate to L α with respect to the measure dm(a)dx = a 2α+1 dadx. Moreover, G α(e, 0; x, a) = lim η 0 0 E 0 p σ (t, 0)(x)m([a η, a + η]) 1 1 [a η,a+η] (σ t )dt, 19

20 where the expectation is taken with respect to the distribution of the Bessel process with a parameter α 0 starting from 0 on the space C([0, ), (0, )). Now we are ready to prove our main Theorem. Proof of Theorem 6.1. Let G α(x, a) := G α(e, 0; x, a). By the results of Section 5: Lemma 5.1 and Lemma 5.2 it follows that there exists a constant C such that X I G α(x, a) C for every (x, a) V \ T δ provided that X 1. Since G α is smooth it follows that there is a constant C such that (6.5) X I G α(x, a) C for every (x, a) V, provided that X I 1. Thus by (6.5), estimate (2.4) and the homogeneity of G α (6.3) for (y, b) = D t (x, a), where (x, a) V, (6.6) X I G α(y, b) = (X I G α)(d t (x, a)) = Φ t 1(X I )(G α D t )(x, a) t I (1 + log t 1 ) I 0 t I Q 2α (1 + log t 1 ) I 0 sup Y I (G α D t )(x, a) Y 1 sup Y I G α(x, a) Y 1 C (y, b) I Q 2α (1 + log (y, b) 1 ) I 0. Thus by (6.4) and (6.6) with t = x + 1, X I m γ (x) = X I G γ/2(x 1, 1) C( x + 1) I Q γ (log( x + 2)) I 0. References [1] A. Ancona. Negatively curved manifolds, elliptic operators, and the Martin boundary. Ann. of Math., 125: , [2] A.N. Borodin and P. Salminen. Handbook of Brownian motion - facts and formulae. Birkhäuser Verlag, [3] E. Damek, A. Hulanicki, and R. Urban. Martin boundary for homogeneous Riemannian manifolds of negative curvature at the bottom of the spectrum. To appear in: Rev. Mat. Iberoamericana. [4] E. Damek, A. Hulanicki, and J. Zienkiewicz. Estimates for the Poisson kernels and their derivatives on rank one N A groups. Studia Math., 126(2): , [5] L. Élie. Comportement asymptotique du noyau potentiel sur les groupes de lie. Ann. scient. Éc. Norm. Sup. 4 serie, pages , [6] G.B. Folland and E. Stein. Hardy spaces on homogeneous groups. Princeton Univ. Press, [7] E. Heintze. On homogeneous manifolds of negative curvature. Math. Ann., 211:23 34,

21 [8] N.N. Lebedev. Special functions and their applications. Dover Publications, Inc., [9] D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer- Verlag, [10] H. Tanabe. Equations of evolutions. London, Pitman, [11] R. Urban. Estimates for the Poisson kernels on homogeneous manifold of negative curvature and the boundary Harnack inequality in the noncoercive case. To appear in: Probab. Math. Statist. [12] R. Urban. Estimates for the Poisson kernels on N A groups. A probabilistic method. PhD thesis, Wroclaw University, [13] N.Th. Varopoulos, L. Saloff-Coste, and T. Coulhon. Analysis and geometry on groups, volume 100 of Cambridge Tracts in Math. Cambridge Univ. Press, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA address: urban@math.lsu.edu Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, Wroclaw, Poland address: urban@math.uni.wroc.pl 21