Spectral Multipliers on Damek Ricci Spaces

Transcription

1 Journal of Lie Theory Volume 7 (27) C 27 Heldermann Verlag Spectral Multipliers on Damek Ricci Spaces Maria Vallarino Communicated by M. Cowling Abstract. Let S be a Damek Ricci space, and be a distinguished Laplacean on S which is left invariant and selfadjoint in L 2 (ρ). We prove that S is a Calderón Zygmund space with respect to the right Haar measure ρ and the left invariant distance. We give sufficient conditions of Hörmander type on a multiplier m so that the operator m( ) is bounded on L p (ρ) when < p <, and of weak type (, ). Mathematics Subject Index 2: 22E3, 42B5, 42B2, 43A8. Keywords and phrases: Multipliers, singular integrals, Calderon Zygmund decomposition, Damek Ricci spaces.. Introduction Let n = v z be an H -type algebra and let N be the connected and simply connected Lie group associated to n (see Section 2 for the details). Let S be a onedimensional harmonic extension of N. Specifically, let S be the one-dimensional extension of N obtained by making A = R + act on N by homogeneous dilations. Let H denote a vector in a acting on n with eigenvalues /2 and (possibly) ; we extend the inner product on n to the algebra s = n a, by requiring n and a to be orthogonal and H to be a unit vector. The Lie algebra s is solvable. Let λ and ρ denote left and right invariant Haar measures on S, and d a left invariant Riemannian metric on S. It is well known that the right Haar measure of geodesic balls is an exponentially growing function of the radius, so that S is a group of exponential growth. Harmonic extensions of H -type groups, now called Damek Ricci spaces, were introduced by E. Damek and F. Ricci [9], [], [], [2], and include all rank one symmetric spaces of the noncompact type. Most of them are nonsymmetric harmonic manifolds, and provide counterexamples to the Lichnerowicz conjecture. The geometry of these extensions was studied by M. Cowling, A. H. Dooley, A. Korányi and Ricci in [5], [6]. Acknowledgement. Work partially supported by the Progetto Cofinanziato MURST Analisi Armonica and the Progetto GNAMPA Calcolo funzionale per generatori di semigruppi e analisi armonica su gruppi ISSN / $2.5 C Heldermann Verlag

2 64 Vallarino Let {E,..., E n } be an orthonormal basis of the algebra s such that E = H, {E,..., E mv } is an orthonormal basis of v and {E mv+,..., E n } is an orthonormal basis of z. Let X, X,..., X n be the left invariant vector fields on S which agree with E, E,..., E n at the identity. Let be the operator defined by n = Xi 2 ; the Laplacean is left invariant and essentially selfadjoint on Cc (S) L 2 (ρ). Therefore there exists a spectral resolution E of the identity for which f = t de (t)f i= f Dom( ). By the spectral theorem, for each bounded measurable function m on R + operator m( ) defined by m( )f = m(t) de (t)f f L 2 (ρ), is bounded on L 2 (ρ); m( ) is called the spectral operator associated to the spectral multiplier m. A classical problem is to find conditions on m that ensure that m( ) extends to a bounded operator from L (ρ) to the Lorentz space L, (ρ) and to a bounded operator on L p (ρ) when < p <. The main result of this paper (see Theorem 4.3 below) is that this holds if m satisfies a Hörmander type condition of suitable order. An analogue r of on the solvable groups coming from the Iwasawa decomposition of noncompact connected semisimple Lie groups of finite centre (any rank) was studied by Cowling, S. Giulini, A. Hulanicki and G. Mauceri [7]. They proved that if a bounded measurable function m on R + belongs locally to a suitable Sobolev space and satisfies Hörmander conditions of suitable order at infinity, then the operator m( r ) is bounded from L (λ) to L, (λ) and on L p (λ) when < p <. A similar result in the case of Damek Ricci spaces was proved by F. Astengo [2]. We warn the reader that actually these authors studied a Laplacean r which is right invariant and selfadjoint on L 2 (λ), whereas is left invariant and selfadjoint on L 2 (ρ). However, it is straightforward to check that where ˇf(x) = f(x ), and hence f = ( r ˇf)ˇ f C c (S), m( )f = (m( r ) ˇf)ˇ f C c (S), for every bounded measurable function m on R +. Since ˇ is an isometry between L p (λ) and L p (ρ) if p, and between L, (λ) and L, (ρ), boundedness results for m( ) on L p (ρ) may be rephrased as boundedness results for m( r ) on L p (λ). Subsequently W. Hebisch and T. Steger [5] sharpened the result in [7] and [2], proving a genuine Hörmander type theorem for spectral multipliers of r in the special case of solvable groups associated to real hyperbolic spaces. the

3 Vallarino 65 In this paper we extend the result in [5] to all Damek Ricci spaces. To be more specific, we prove that if a bounded measurable function m on R + satisfies Hörmander conditions both at infinity and locally, then the operator m( ) is bounded from L (ρ) to L, (ρ) and on L p (ρ) when < p <. The strategy of the proof, which is similar to that of [5, Theorem 2.4], is to show that m( ) may be realized as a singular integral operator, and that such operators are bounded from L (ρ) to L, (ρ) and on L p (ρ) when < p <. The classical theory of higher dimensional singular integrals was developed by A. Calderón and A. Zygmund. One of the basic results of this theory is the so-called Calderón Zygmund decomposition, where each integrable function f is decomposed as g + j= b j, where the good function g is bounded, and each bad function b j may be unbounded but is supported in a ball and its integral vanishes. The Calderón Zygmund theory was extended to spaces of homogeneous type by R. Coifman and G. Weiss [8]. These spaces are measured metric spaces (X, µ, ϱ), where balls satisfy the so-called doubling condition, i.e., there exists a constant C such that µ(b(x, 2r)) C µ(b(x, r)) x X r R +. () The main result of the theory is that an integral operator, which is bounded on L 2 (µ), and whose kernel satisfies the so-called Hörmander condition sup y X sup r> K(x, y) K(x, z) dµ(x) < B(y,κr) c z B(y, r), (2) extends to a bounded operator on L p (µ) when < p < 2 and to a bounded operator from L (µ) to L, (µ). We remark that the doubling condition () plays a fundamental rôle in the theory. Hence, it does not apply to (S, ρ, d), because the doubling condition for balls fails. Recently, Hebisch and Steger [5] realized that a Calderón Zygmund like theory of singular integrals may be developed on some groups of exponential growth. The main idea is that the rôle of balls in the classical theory may be played by other types of sets. First, they defined a Calderón Zygmund space (see Section 3 for the details). Roughly speaking, a metric measured space (X, µ, ϱ) is a Calderón Zygmund space with Calderón Zygmund constant κ, provided that each integrable function f may be decomposed as g + j= b j, where the good function g is bounded, and each bad function b j is unbounded, supported in a set R j and its integral vanishes. The main difference with the classical theory is that R j need not be a ball, but there exist a point x j in X and a positive number r j such that the following two conditions hold: (i) R j B(x j, κ r j ) j N; (ii) µ(r j) κ µ(r j ) where R j = {x X : ϱ(x, R j ) < r j }. Clearly, (ii) is a substitute for the doubling condition for balls.

4 66 Vallarino Then Hebisch and Steger proved that an integral operator on a Calderón Zygmund space that is bounded on L 2 (µ) and whose kernel satisfies the Hörmander condition (2), extends to a bounded operator on L p (µ) when < p < 2, and to a bounded operator from L (µ) to L, (µ). In this paper we prove that all Damek Ricci spaces (S, ρ, d) are Calderón Zygmund spaces. It is worth pointing out that even for complex hyperbolic spaces this result is new. To do so, we shall use a family of suitable sets in S which we call admissible sets and describe in detail in Section 3. It may be worth observing that small sets are balls of small radius, while big sets are rectangles, i.e. products of dyadic sets in N and intervals in A. We apply this result to study the L p (ρ) boundedness of spectral multipliers associated to the Laplacean. Our paper is organized as follows: in Section 2, we recall the definition of an H -type group N and its Damek Ricci extension S ; then we summarize some results of spherical analysis on S. In Section 3, we prove that S, endowed with the right Haar measure ρ and the left invariant metric d, is a Calderón Zygmund space. In Section 4, we prove a Hörmander type theorem for spectral multipliers associated to the Laplacean. The proof of this hinges on an L -estimate of the gradient of the heat kernel associated to. Positive constants are denoted by C ; these may differ from one line to another, and may depend on any quantifiers written, implicitly or explicitly, before the relevant formula. The author would like to thank Stefano Meda for his precious help and encouragement. 2. Damek Ricci spaces In this section we recall the definition of H -type groups, describe their Damek Ricci extensions, and recall the main results of spherical analysis on these spaces. For the details see [], [2], [3], [5], [6], [3]. Let n be a Lie algebra equipped with an inner product, and denote by the corresponding norm. Let v and z be complementary orthogonal subspaces of n such that [n, z] = {} and [n, n] z. According to Kaplan [7], the algebra n is of H -type if for every Z in z of unit length the map J Z : v v, defined by J Z X, Y = Z, [X, Y ] X, Y v, is orthogonal. The connected and simply connected Lie group N associated to n is called an H -type group. We identify N with its Lie algebra n via the exponential map The product law in N is v z N (X, Z) exp(x + Z). (X, Z)(X, Z ) = (X + X, Z + Z + 2 [X, X ]) X, X v Z, Z z.

5 Vallarino 67 The group N is a two-step nilpotent group, hence unimodular, with Haar measure dx dz. We define the following dilations on N : δ a (X, Z) = (a /2 X, az) (X, Z) N a R +. The group N is an homogeneous group with homogeneous norm ( ) X 4 /4 N (X, Z) = 6 + Z 2 (X, Z) N. Note that N (δ a (X, Z)) = a /2 N (X, Z). Given (X, Z ) in N and r >, the homogeneous ball centred at (X, Z ) of radius r is B N ((X, Z ), r) = {(X, Z) N : N ((X, Z ) (X, Z)) < r}. Obviously, B N ( N, r) = δ r 2B N ( N, ). Set Q = (m v + 2m z )/2, where m v and m z denote the dimensions of v and z. The measure of the ball B N ((X, Z ), r) is r 2Q B N ( N, ). Let S and s be as in the Introduction. The map v z R + S (X, Z, a) exp(x + Z) exp(log a H) gives global coordinates on S. The product in S is given by the rule (X, Z, a)(x, Z, a ) = (X + a /2 X, Z + a Z + 2 a/2 [X, X ], a a ) for all (X, Z, a), (X, Z, a ) S. Let e be the identity of the group S. We shall denote by n the dimension m v + m z + of S. The group S is nonunimodular: the right and left Haar measures on S are given by dρ(x, Z, a) = a dx dz da and dλ(x, Z, a) = a (Q+) dx dz da. Then the modular function is δ(x, Z, a) = a Q. We denote by L p (ρ) the space of all measurable functions f such that S f p dρ < and by L, (ρ) the Lorentz space of all measurable functions f such that sup t> t ρ({x S : f(x) > t}) <. We equip S with the left invariant Riemannian metric which agrees with the inner product on s at the identity e. Denote by d the distance induced by this Riemannian structure. From [3, formula (2.8)], for all (X, Z, a) in S, cosh 2 ( d((x, Z, a), e) 2 ) ( a /2 + a /2 = + ) a /2 X a Z 2. (3) We denote by B((X, Z, a ), r) the ball in S with centre (X, Z, a ) and radius r. We write B r for the ball with centre e and radius r; from [3, formula (.8)], there exist positive constants γ, γ 2 such that, for all r in (, ), and for all r in [, ) γ r n ρ(b r ) γ 2 r n, (4) γ e Qr ρ(b r ) γ 2 e Qr. This shows that S, equipped with the right Haar measure ρ, is a group of exponential growth. From (3) we can easily deduce various properties of balls in S.

6 68 Vallarino Proposition 2.. The following hold: (i) there exists a constant c 3 such that B(e, log r) B N ( N, c 3 r) (/r, r) r (, ) ; (ii) if b > and B > /2, then there exists a constant c b,b such that B N ( N, b r B ) (/r, r) B(e, c b,b log r) r [e, + ). A radial function on S is a function that depends only on the distance from the identity. If f is radial, then by [3, formula (.6)], f dλ = f(r) A(r) dr, S where ( ) ( ) r r A(r) = 2 mv+2mz sinh mv+mz cosh mz 2 2 r R +. We shall use repeatedly the fact that ( ) r n A(r) C e Qr + r r R +. (5) A radial function φ is spherical if it is an eigenfunction of the Laplace Beltrami operator L (associated to d) and φ(e) =. For s in C, let φ s be the spherical function with eigenvalue s 2 + Q 2 /4, as in [3, formula (2.6)]. In [2, Lemma ], it is shown that φ (r) C ( + r) e Qr/2 r R +. (6) We shall use the following integration formula on S, whose proof is like that of [7, Lemma.3] and [, Lemma 3]: Lemma 2.2. For every radial function f in Cc (S), δ /2 f dρ = φ (r) f(r) A(r) dr. S The spherical Fourier transform Hf of an integrable radial function f on S is defined by the formula Hf(s) = φ s f dλ. For nice radial functions f on S, an inversion formula and a Plancherel formula hold: f(x) = c S Hf(s) φ s (x) c(s) 2 ds x S, and S S f 2 dλ = c S Hf(s) 2 c(s) 2 ds, where the constant c S depends only on m v and m z, and c denotes the Harish- Chandra function. For later developments, we recall from [2, formula ()] that C s 2 s [, ] c(s) 2 C s n s R [, ].

7 Vallarino 69 Let A denote the Abel transform and let F denote the Fourier transform on the real line, defined by Fg(s) = + g(r) e isr dr, for each integrable function g on R. It is well known that H = F A, hence H = A F. For later use, we recall the inversion formula for the Abel transform [3, formula (2.24)]. We define the differential operators D and D 2 on the real line by D = sinh r r, D 2 = sinh(r/2) r. (7) If m z is even, then A f(r) = a e S D mz/2 D mv/2 2 f(r), (8) where a e S = 2 (3mv+mz)/2 π (mv+mz)/2, while if m z is odd, then A f(r) = a o S r D (mz+)/2 D mv/2 2 f(s) dν(s), (9) where a o S = 2 (3mv+mz)/2 π n/2 and dν(s) = (cosh s cosh r) /2 sinh s ds. Now consider the Laplacean on S, as defined in the introduction. The operator has a special relationship with the Laplace Beltrami operator L. Indeed, denote by L Q the shifted operator L Q 2 /4; then by [, Proposition 2], δ /2 δ /2 f = L Q f () for all smooth compactly supported radial functions f on S. The spectra of L Q on L 2 (λ) and on L 2 (ρ) are both [, + ). Let E LQ and E be the spectral resolution of the identity for which L Q = + t de LQ (t) and = + t de (t). For each bounded measurable function m on R + the operators m(l Q ) and m( ), spectrally defined by m(l Q ) = + m(t) de LQ (t) and m( ) = + m(t) de (t), are bounded on L 2 (λ) and L 2 (ρ) respectively. By () and the spectral theorem, δ /2 m( ) δ /2 f = m(l Q )f, for smooth compactly supported radial functions f on S. Let k m( ) and k m(lq ) denote the convolution kernels of m( ) and m(l Q ) respectively; then m(l Q )f = f k m(lq ) and m( )f = f k m( ) f C c (S), where denotes the convolution on S, defined by f g(x) = f(xy) g(y ) dλ(y) S = f(xy ) g(y) dρ(y), for all functions f, g in C c (S) and x in S. S

8 7 Vallarino Proposition 2.3. Let m be a bounded measurable function on R +. Then k m(lq ) is radial and k m( ) = δ /2 k m(lq ). The spherical transform Hk m(lq ) of k m(lq ) is given by Hk m(lq )(s) = m(s 2 ) s R +. Proof. See [], [3]. 3. Calderón Zygmund decomposition In this section we prove that Damek Ricci spaces are Calderón Zygmund spaces. Recently Hebisch and Steger [5] gave the following axiomatic definition of a Calderón Zygmund space. Definition 3.. Let (X, µ, ϱ) be a metric measured space. Suppose that f is in L (µ), κ is a positive constant and α > κ f L (µ)/µ(x) (if µ(x) =, then the right hand side is taken to be ). A Calderón Zygmund decomposition of f at height α with Calderón Zygmund constant κ is a decomposition f = g + i N b i where there exist sets R i, points x i and positive numbers r i such that: (i) g κ α µ-almost everywhere; (ii) supp(b i ) R i and b i dµ = i N; (iii) R i B(x i, κ r i ) i N; (iv) i µ(r i ) κ f L (µ) α, where R i = {x X : ϱ(x, R i ) < r i }; (v) i b i L (µ) κ f L (µ). A Calderón Zygmund space is a metric measured space (X, µ, ϱ) for which there exists a positive constant κ such that each function f in L (µ) has a Calderón Zygmund decomposition at height α with Calderón Zygmund constant κ whenever α > κ f L (µ)/µ(x). Clearly spaces of homogeneous type are Calderón Zygmund spaces. Note that in this case we may choose R i as balls and Ri as balls of the same centre and dilated radius. It is remarkable that some spaces which are not of homogeneous type are Calderón Zygmund spaces. Indeed, Hebisch and Steger proved that real hyperbolic spaces are Calderón Zygmund spaces. Let S be a Damek Ricci space, as described in Section 2. We shall prove that (S, ρ, d) is a Calderón Zygmund space. The first problem is to define suitable sets R i, as in Definition 3.. As we have already remarked, we cannot use geodesic balls as in the classical case, because their measure increases exponentially: so we define suitable families of big admissible sets and small admissible sets. Big admissible sets. To define big sets we generalize the idea used by Hebisch and Steger in [5]. They defined admissible sets as rectangles which are products of dyadic sets in R n and intervals in R. In the context of H -type groups dyadic sets were introduced by M. Christ [4, Theorem ]. For the readers convenience we recall their properties in the following theorem.

9 Vallarino 7 Theorem 3.2. Let N be an H -type group, endowed with the homogeneous distance and the Haar measure. There exist subsets Q k α of N, where k Z and α is in a countable index set I k, positive constants η, c N, C N and M (η > and M is an integer) such that: (i) N α I k Q k α = k Z; (ii) there are points n k α in N such that B N (n k α, c N η k ) Q k α B N (n k α, C N η k ); (iii) Q k α Q k β = Ø if α β ; (iv) each set Q k α has at most M subsets of type Q k β ; (v) l k and β in I l there is a unique α in I k such that Q l β Q k α ; (vi) if l k, then either Q k α Q l β = Ø or Q l β Q k α. We define big admissible sets as products of dyadic sets in N and intervals in A. Roughly speaking, we may think of these sets as left translates of a family of sets containing the identity (see [8]). For technical reasons we cannot do exactly that, because left translates and dilates of dyadic sets in N may not be dyadic sets. Definition 3.3. A big admissible set is a set of the form Q k α (a /r, a r), where Q k α is a dyadic set in N, a A, r e, a /2 r β η k < a /2 r 4β, () and β is a constant greater than max {3/2, /4 + log η, + log(c 3 /c N )}, where c 3, η, c N are the constants which appear in Proposition 2. and Theorem 3.2. We now investigate some geometric properties of big admissible sets. Proposition 3.4. Denote by R the big admissible set Q k α (a /r, a r), and take c 3, c N, C N and η as in Proposition 2. and Theorem 3.2 (ii). Then the following hold: (i) there exists a constant C N,β such that R B((n k α, a ), C N,β log r); (ii) c 2Q N B N ( N, ) (a /2 r β ) 2Q log r ρ(r) C 2Q N B N ( N, ) (a /2 r 4β ) 2Q log r; (iii) let R be the set {(n, a) S : d((n, a), R) < log r}; then ( ) ρ(r c3 + C 2Q N ) ρ(r). c N Proof. To prove (i), note that by Theorem 3.2 (ii) R B N (n k α, C N η k ) (a /r, a r),

10 72 Vallarino which, in turn, is contained in B N (n k α, C N a /2 r 4β ) (a /r, a r) by the admissibility condition (). By the left invariance of the metric and Proposition 2. (ii), R (n k α, a ) [B N ( N, C N r 4β ) (/r, r)] (n k α, a ) [B(e, C N,β log r)] = B((n k α, a ), C N,β log r), as required. We now prove (ii). Since ρ(r) = Q k α log r, by Theorem 3.2 (ii), c 2Q N B N ( N, ) η 2Qk log r ρ(r) C 2Q N B N ( N, ) η 2Qk log r. Since a /2 r β η k < a /2 r 4β, (ii) follows. To prove (iii), we observe that R = (n,a) R B((n, a), log r). Using the left invariance of the metric and Proposition 2. (i), we deduce that B((n, a), log r) = (n, a) [B(e, log r)] (n, a) [B N ( N, c 3 r) (/r, r)] = B N (n, c 3 a /2 r) (a/r, a r) (n, a) R. Since (n, a) is in R and R is admissible, we see that and (a/r, ar) (a /r 2, a r 2 ) B N (n, c 3 a /2 r) B N (n, c 3 a /2 r 3/2 ) B N (n k α, c 3 a /2 r β + C N η k ) B N (n k α, (c 3 + C N ) η k ). Thus Finally, R B N (n k α, (c 3 + C N ) η k ) (a /r 2, a r 2 ). ρ(r ) ( c 3 + C ) N 2Q BN (n k α, c N η k ) log r c N ( c 3 + C N c N ) 2Q ρ(r), as required. Another useful geometric property is that most big admissible sets may be split up into a finite number of mutually disjoint smaller subsets which are still admissible. More precisely the following lemma holds.

11 Vallarino 73 Lemma 3.5. Let R denote the big admissible set Q k α (a /r, a r ) and let η, M, n k α, c N and C N be as in Theorem 3.2. The following hold: (i) if η k a /2 r β, then there exist J mutually disjoint big admissible sets R,..., R J, where 2 J M, such that R = J i= R i and (c N /(η C N )) 2Q ρ(r) ρ(r i ) ρ(r) i =,..., J ; (ii) if η k < a /2 r β and r e 2, then there exist two disjoint big admissible sets R and R 2 such that R = R R 2 and ρ(r i ) = ρ(r)/2, where i =, 2; (iii) if η k < a /2 r β and r < e 2, then there exists a constant σ N, β such that B((n k α, a ), ) R B((n k α, a ), σ N, β ). (2) Proof. To prove (i), suppose that η k a /2 r β. We split up R in the following way: let Q k i be the subsets of Q k α as in Theorem 3.2, where i J M. Define R i = Q k i (a /r, a r ) i =,..., J. Since η k a /2 r β, the sets R i are admissible. Obviously R = J i= R i and ρ(r i ) ρ(r). By Theorem 3.2 (ii) ρ(r i ) = Q k i log r B N ( N, c N η k ) log r = BN ( N, (c N /(η C N )) C N η k) log r (c N /(η C N )) 2Q B N ( N, C N η k ) log r (c N /(η C N )) 2Q ρ(r), as required. To prove (ii), suppose that η k < a /2 r β and r e 2. Then by the admissibility condition (), Define R and R 2 by a /2 r β η k < η a /2 r β. (3) R = Q k α (a /r, a ) and R 2 = Q k α (a, a r). Clearly the centres of R and R 2 are (n k α, a / r) and (n k α, a r) respectively. Note that r e. To prove that R and R 2 are admissible, we use (3): (a / r) /2 ( r) β a /2 r β η k ; (a / r) /2 ( r) 4β = η r β /4 η a /2 r β > η e β /4 η k > η k.

12 74 Vallarino This proves that R is admissible. The proof of the admissibility of R 2 is similar and is omitted. Obviously R = R R 2 and ρ(r i ) = ρ(r)/2, i =, 2, as required. We now consider (iii). Suppose that η k η a /2 r β and e r < e 2. By the admissibility condition () and the left invariance of the metric, R B N (n k α, C N η k ) (a /r, a r) B N (n k α, C N a /2 r 4β ) (a /r, a r ) = (n k α, a ) [B N ( N, C N r 4β ) (/r, r)]. Since r < e 2, we conclude from Proposition 2. (ii) that R (n k α, a ) [B N ( N, C N e 8β ) (/e 2, e 2 )] B((n k α, a ), σ N, β ), where σ N, β depends only on β and C N. Similarly, () and the left invariance of the metric imply that R B((n k α, a ), ), as required. For later developments it is useful to distinguish big admissible sets that satisfy condition (i) or (ii) in Lemma 3.5, which may be split up into a finite number of smaller big admissible sets, and big admissible sets that satisfy condition (iii) in Lemma 3.5, which cannot be split up in that way. Definition 3.6. A big admissible set Q k α (a /r, a r) is said to be nondivisible if η k < a /2 r β and r < e 2, and to be divisible otherwise. Next we show that there exists a partition of S consisting of big admissible sets whose measure is as large as desired. Lemma 3.7. For all positive σ, there exists a partition P of S which consists of big admissible sets whose measure is greater than σ. Proof. First, we choose r e and k Z such that r β η k < r 4β and r 2βQ log r > σ/c 2Q N B N ( N, ). Set Rα = Q k α (/r, r ), where α I k. The Rα are big admissible sets and ρ(rα) = Q k α log r c 2Q N B( N, ) η 2k Q log r c 2Q N B( N, ) r 2βQ log r > σ α I k. Then the sets R α, where α I k, form a partition of the strip N (/r, r ), consisting of big admissible sets whose measure is greater than σ. Next suppose that a partition of a strip N (a n /r n, a n r n ) which consists of admissible sets whose measure is greater than σ has been chosen. Then we choose r n+ e and k n+ Z such that (a /2 n+ r β n+) 2Q log r n+ > σ c 2Q N B( N, ) and a /2 n+ r β n+ η k < a /2 n+ r 4β n+,

13 Vallarino 75 where a n+ = a n r n r n+. Set Rα n+ = Q k n+ α (a n r n, a n+ r n+ ), where α I kn+. Then the Rα n+ are big admissible sets whose measure is greater than σ, which partition the strip N (a n r n, a n+ r n+ ). By iterating this process we obtain a partition of N (r, ). Similarly, we define a partition of N (, /r ) consisting of big admissible sets with the required property. Small admissible sets. A small admissible set is a ball with radius less than /2. Note that ρ(b((n, a ), r)) = δ ((n, a )) ρ(b r ) = a Q ρ(b r ). By (4), there exist positive constants γ, γ 2 such that γ a Q r n ρ(b((n, a ), r)) γ 2 a Q r n r (, /2). Set γ = + 2(2 e Q γ 2 /γ ) /n. Define B (x, r) = B(x, γ r). Since r < /2, there exists a constant C such that ρ(b (x, r)) C ρ(b(x, r)). (4) The balls of small radius satisfy the following covering lemma. Lemma 3.8. Let B and B 2 be balls of radii less than /2. If B B 2 Ø and ρ(b ) 2 ρ(b 2 ), then B B 2. Proof. Let x i and r i denote the centre and the radius of the ball B i, i =, 2. Since B B 2 Ø, we have that d(x, x 2 ) < r + r 2 <. The condition ρ(b ) 2 ρ(b 2 ) implies that δ(x ) γ r n 2 δ(x 2 ) γ 2 r n 2. Thus r (2 δ(x x 2 ) γ 2 /γ ) /n r 2. Since x x 2 is in B(e, ) we have that δ(x x 2 ) e Q and then It follows that as required. r (2 e Q γ 2 /γ ) /n r 2. B(x, r ) B(x 2, 2r + r 2 ) B ( x 2, ( + 2(2 e Q γ 2 /γ ) /n) r 2 ) = B(x 2, γ r 2 ) = B 2, Let R denote the family of all balls of radius less than /2 and let M R be the noncentred maximal operator M R f(x) = sup f dρ x S, x B ρ(b) B where the supremum is taken over all balls in R. From the covering Lemma 3.8 it follows that M R is bounded from L (ρ) to L, (ρ). We now prove a geometric lemma concerning intersection properties between small balls and big nondivisible sets.

14 76 Vallarino Lemma 3.9. Let B be a ball of radius R, such that /2 R γ/2, where γ = + 2(2 e Q γ 2 /γ ) /n. Let {F l } l be a family of mutually disjoint nondivisible big admissible sets. Then: (i) if B F l Ø, then ρ(b) 2 n (γ /γ 2 ) e Q(2σ N, β+γ/2) ρ(f l ); (ii) the ball B intersects at most (γ 2 /γ ) e Q(+σ N, β+γ/2) sets of the family {F l } l, where σ N, β is the constant which appears in Lemma 3.5. Proof. Let x be the centre of B. Note that B(x, /2) B B(x, γ/2). By (2), there exist points y l such that B(y l, ) F l B(y l, σ N, β ). To prove (i), note that while ρ(b) γ δ (x ) R n γ δ (x ) (/2) n, ρ(f l ) δ (y l ) ρ(b(e, σ N, β )) γ 2 δ (y l ) e Q σ N, β. If B F l Ø, then d(x, y l ) < γ/2 + σ N, β, and so δ(y l x ) e Q(σ N, β+γ/2). Therefore ρ(b)/ρ(f l ) 2 n (γ /γ 2 ) e Q(2σ N, β+γ/2), as required in (i). To prove (ii), note that if l k, then B(y l, ) B(y k, ) = Ø, since F l F k = Ø. Now let I = {l : B F l Ø}. Obviously, if l is in I, then B(y l, ) B(x, γ/2 + + σ N, β ), so that B(y l, ) B(x, γ/2 + + σ N, β ). l I Now consider the left invariant measure of the sets above: Then B intersects at most sets of the family {F l } l. I λ(b(e, )) λ(b(e, γ/2 + + σ N, β )). I λ(b(e, γ/2 + + σ N, β ))/λ(b(e, )) (γ 2 /γ ) e Q(+σ N, β+γ/2) We now prove our first main theorem. Theorem 3.. A Damek Ricci space (S, ρ, d) is a Calderón Zygmund space. Proof. Let f be in L (ρ) and α >. We want to define a Calderón Zygmund decomposition of f at height α. Let P be a partition of S consisting of big admissible sets of measure greater than f L (ρ)/α (it exists by Lemma 3.7). For each R in P, f dρ < α. ρ(r) R

15 Vallarino 77 We split up each divisible set R in P into big admissible disjoint subsets R i, where i J M, as in Lemma 3.5. If f dρ α, ρ(r i ) R i then we stop. Otherwise, if R i is divisible, then we split up R i and stop when we find a subset E such that f dρ α. ρ(e) E By iterating this process, we obtain a family {E i } i of stopping sets. The sets E i have the following properties: (i) E i are mutually disjoint big admissible sets; (ii) f dρ α; ρ(e i ) E i (iii) for each set E i, there exists a set E i such that f dρ < α ρ(e i) E i and Then ρ(e i) max{2, (η C N /c N ) 2Q } ρ(e i ). f dρ max{2, (η C N /c N ) 2Q } f dρ ρ(e i ) E i ρ(e i) E i < max{2, (η C N /c N ) 2Q } α ; (iv) the complement of i E i is the union of mutually disjoint nondivisible big admissible sets {F l } l such that F l f dρ < α ρ(f l ). Define h, g f, and b i f by h = f χ ( i E i ) c, g f = i ( f dρ ) χ Ei and b i f = ( f f dρ ) χ Ei. ρ(e i ) E i ρ(e i ) E i By (iii), g f max{2, (η C N /c N ) 2Q } α. Each function b i f is supported in E i and its integral vanishes. The sum of the L -norms of the functions b i f is b i f L (ρ) 2 f dρ i i E i 2 f L (ρ). By Proposition 3.4 (i), there are x i and r i such that E i B(x i, C N,β log r i ). Moreover, write E i = {x S : d(x, E i ) < log r i }. Then ρ(e i ) ((c 3 + C N )/c N ) 2Q ρ(e i ).

16 78 Vallarino Thus ρ(ei ) ( c 3 + C ) N 2Q ρ(e i ) i c N i ( c 3 + C ) N 2Q f dρ c N α i E i ( c 3 + C ) N 2Q f L (ρ). c N α We now decompose the function h. Let Oα = {x S : M R h(x) > α}. For each point x Oα we choose a ball B x in R such that B x h dρ > α ρ(b x ) and ρ(b x ) > 2 sup{ ρ(b) : B R, x B, h > α }. (5) ρ(b) B Now we select a disjoint subfamily of {B x } x. We choose B x ρ(b x ) > (/2) sup{ρ(b x ) : x Oα}. Next, suppose that B x,..., B xn chosen. Then B xn+ is chosen, disjoint from B x,..., B xn, such that ρ(b xn+ ) > 2 sup{ρ(b x) : x O α, B x B xi = Ø, i =,..., n}. such that have been Then j B xj Oα j Bx j, where Bx j = B(c xj, γ r xj ), γ = + 2(2 e Q γ 2 /γ ) /n and γ, γ 2 are the constants which appear in (4). Indeed, each set B xj is contained in Oα by construction. Moreover, for each point x Oα either B x = B xj Bx j for some index j or B x B xj for all j. In this case there exists an index j such that B x B xj Ø and ρ(b x ) 2 ρ(b xj ). By Lemma 3.8, x B x Bx j. Now define G j = Bx j ( G k ) c ( B xl ) c Oα. k<j l>j It is easy to check that the sets G j are mutually disjoint and that B xj G j Bx j, so that their measures are comparable. Moreover j G j = Oα. Indeed, on the one hand, j G j Oα by construction; on the other hand, if x is in Oα, then there exists an index j such that x is in Bx j. Now either x is in B xl for some index l > j (and then x is in G l ) or x is in G k for some index k < j or x is in G j. We claim that ρ(g j ) G j h dρ C 2 n (γ 2 /γ ) 2 e Q(3 σ N, β+γ+) α, (6) where C is the constant which appears in (4). To see this we first observe that h dρ C h dρ. ρ(g j ) G j ρ(bx j ) Bx j To estimate this average we shall distinguish two cases. First, suppose that B x j is in R. Since ρ(b xj ) 2 ρ(b x j ), by (5) h dρ α. ρ(bx j ) Bx j

17 Vallarino 79 Next, if B x j is not in R, then /2 γ r xj γ/2. Hence we may apply Lemma 3.9 to the ball B x j and the family {F l } l of nondivisible big admissible sets. Let I = {l : B x j F l Ø}. Since h is supported in l F l, by Lemma 3.9, h dρ = h dρ ρ(bx j ) Bx j l I ρ(bx j ) Bx j F l 2 n (γ 2 /γ ) e Q(2σ N, β+γ/2) h dρ l I ρ(f l ) F l I 2 n (γ 2 /γ ) e Q(2σ N, β+γ/2) α 2 n (γ 2 /γ ) 2 e Q(3σ N, β+γ+) α. The claim (6) follows from the last three estimates. We now define the decomposition of h: g h = h χ (O α ) c + j ( h dρ ) χ Gj, b j h ρ(g j ) = ( h h dρ ) χ Gj. G j ρ(g j ) G j By (6), g h C 2 n (γ 2 /γ ) 2 e Q(3 σ N, β+γ+) α on each set G j and g h = h α on (Oα) c. Each function b j h is supported in G j and its integral vanishes. The sum of the L -norms of the functions b j h is b j h L (ρ) 2 h dρ j j G j 2 h dρ Oα 2 h L (ρ) 2 f L (ρ). Now G j Bx j and G j = {x S : d(x, G j ) < r xj } B(c xj, (γ + )r xj ); then there exists a constant C such that ρ(g j) C ρ(g j ). Thus ρ(g j) C j j C ρ(o α) C α C α ρ(g j ) R M L (ρ); L, (ρ) h L (ρ) R M L (ρ); L, (ρ) f L (ρ), since M R is bounded from L (ρ) to L, (ρ). Then f = g f + g h + i b i f + j b j h is a Calderón Zygmund decomposition of the function f at height α. The Calderón Zygmund constant of the space is κ = max { 2, (η C N /c N ) 2Q, C N,β, γ, ( c 3 + C N c N ) 2Q, C 2 n (γ 2 /γ ) 2 e Q(3 σ N, β+γ+), C M R L (ρ); L, (ρ)}.

18 8 Vallarino 4. The multiplier theorem In this section we prove our main result. Let ψ be a function in Cc (R + ), supported in [/4, 4], such that ψ(2 j λ) = λ R +. j Z Let m be a bounded measurable function on R + ; we define m,s and m,s thus: m,s = sup m(t ) ψ( ) H s (R), t< m,s = sup m(t ) ψ( ) H s (R), t where H s (R) denotes the L 2 -Sobolev space of order s on R. Let K m( ) and k m( ) denote the integral kernel and the convolution kernel of the operator m( ) respectively. It is easy to check that K m( ) (x, y) = k m( ) (y x) δ(y) for all x, y S. Our aim is to find sufficient conditions on m that ensure the boundedness of m( ) from L (ρ) to L, (ρ) and on L p (ρ) when < p <. To do it we apply a boundedness theorem for integral operators on Calderón Zygmund spaces proved by Hebisch and Steger [5, Theorem 2.], which we state for the readers convenience. Theorem 4.. Let (X, µ, d) be a Calderón Zygmund space. Let T be a linear operator bounded on L 2 (µ) such that T = j Z T j, where (i) the series converges in the strong topology of L 2 (µ); (ii) every T j is an integral operator with kernel K j ; (iii) there exist positive constants a, A, ε and c > such that X X K j (x, y) ( + c j d(x, y)) ε dµ(x) A y X; (7) K j (x, y) K j (x, z) dµ(x) A (c j d(y, z)) a y, z X. (8) Then T extends from L (µ) L 2 (µ) to an operator of weak type (, ) and to a bounded operator on L p (µ) when < p 2. We shall apply Theorem 4. to the operator m( ). The proof of (7) hinges on a weighted L 2 -estimate (see Lemma 4.2 below) for the kernel of spectral operators associated to with respect to the weight w(x) = δ /2 (x) e Q d(x,e)/2 x S. (9) Lemma 4.2. following hold: There exists a constant C such that for all r in [, ) the

19 Vallarino 8 (i) B r w dρ C r 2 ; (ii) for all compactly supported functions f on R + k f( ) 2 w dρ C r k f( ) 2 dρ. B r B r Proof. From Lemma 2.2, we see that w dρ = B r = B r δ /2 (x) e Qd(x,e)/2 dρ(x) r φ (t) e Qt/2 A(t) dt, which, by (5) and (6), is bounded above by C r ( + t) e Qt/2 e Qt/2( t ) n e Qt dt C r 2 r [, ). + t This proves (i). To prove (ii), let f be compactly supported on R + and let k f(lq ) denote the convolution kernel of the operator f(l Q ). By Proposition 2.3, k f( ) = δ /2 k f(lq ). We split up the ball B r into annuli A R = {x S : R < d(x, e) < R}, R =,..., [r] and C r = {x S : [r] < d(x, e) < r} and estimate the integral of k 2 w on A R and C r separately. We start with the integral on A R. By Lemma 2.2, A R k f( ) 2 w dρ = = A R δ /2 (x) k f(lq )(x) 2 δ /2 (x) e Qd(x,e)/2 dρ(x) R R which, by (6), is bounded above by C R R The proof of the estimate φ (t) k f(lq )(t) 2 e Qt/2 A(t) dt, ( + t) k f(lq )(t) 2 A(t) dt C r k f(lq ) 2 dλ A R = C r k f( ) 2 dρ. A R k f( ) 2 w dρ C r k f( ) 2 dρ C r C r is similar and is omitted. We sum these two estimates to obtain (ii). We now state our main result. Theorem 4.3. Let S be a Damek Ricci space. Suppose that s > 3/2 and that s > max {3/2, n/2}. Let m be a bounded measurable function on R + such that m,s < and m,s <. Then m( ) is bounded from L (ρ) to L, (ρ) and on L p (ρ), for all p in (, ).

20 82 Vallarino Structure of the proof. Take a sufficiently small positive ε that s > 3/2 + ε and s > max {3/2, n/2} + ε, and a bounded measurable function m on R + such that m,s < and m,s <. Since m is bounded, m( ) is bounded on L 2 (ρ) by the spectral theorem. Now define m j (λ) = m(2 j λ) ψ(λ) j Z λ R +. By the spectral theorem the operators m j (2 j ) are bounded on L 2 (ρ); it is straightforward to check that m( ) = j Z m j (2 j ) in the strong topology of L 2 (ρ). To prove the theorem it suffices to show that the integral kernels K mj (2 j ) of the operators m j (2 j ) satisfy estimates (7) and (8) of Theorem 4.. More precisely, we need to show that there exists a constant C such that, for all y in S, C m K mj (2 j )(x, y) ( + 2 j/2 d(x, y)) ε,s j < dρ(x) (2) S C m,s j. The proof of this estimate is very much the same as the proof of [4, Theorem.2], the main difference being that we replace the weight function used by Hebisch by w (see (9)) and [4, Lemma.9] by Lemma 4.2. Furthermore, we need to establish that there exists a constant C such that for all y, z in S K mj (2 j )(x, y) K mj (2 j )(x, z) dρ(x) S C 2 j/2 d(y, z) m,s j < C 2 j/2 d(y, z) m,s j. The proof of this inequality is like the proof of [5, Theorem 2.4] and hinges on an L -estimate of the gradient of the heat kernel associated to which is established in Proposition 4.7. Then the operators m j (2 j ) satisfy the hypothesis of Theorem 4.. It follows that m( ) is bounded from L (ρ) to L, (ρ) and on L p (ρ), for all p in (, 2). The boundedness of m( ) on L p (ρ) for p in (2, ) follows by a duality argument. The proof of the theorem is complete, except for the L -estimate of the gradient of the heat kernel, which is established below. An L -estimate of the gradient of the heat kernel. For all x in S define x = d(x, e) and Ch Q (x) = cosh Q ( x /2). Let h t and q t denote the heat kernels associated to the operators and L Q respectively. By Proposition 2.3, h t = δ /2 q t and Hq t (s) = e ts2 = Fh R t (s) for all s R +, where h R t denotes the heat kernel on R. Then h t (x) = δ /2 (x) (A F ) (Fh R t )( x ) = δ /2 (x) A (h R t )( x ) x S. To prove Proposition 4.7 we shall need various technical results which we prove in Lemmata 4.4 to 4.6. In particular, in Lemma 4.4, we recall various estimates of h R t and its derivatives (see [3, Proposition 5.22]). (2)

21 Vallarino 83 Lemma 4.4. For all r in R + and t in R +, the following hold: (i) for all positive integers j, there exists a constant C, independent of t and r, such that r j h R t (r) C t j/2 h R 2t(r) ; (ii) for all nonnegative integers p and q, D q D p 2 (h R t )(r) = p+q j= t j a j (r) h R t (r), where D and D 2 are the differential operators defined in (7), a j (r) = cosh (p+2q) (r/2) (α j r j + f j (r)), f j, f j are bounded functions on R + and α j are constants. In the next two lemmata we prove various integral estimates. Lemma 4.5. The following hold for all i {,..., n }: (i) X i ( ) ; (ii) t /2 h R 2t( (x) ) X i (δ /2 Ch Q )(x) dρ(x) C t /2 t R + ; (iii) t B c S δ /2 (x) Ch Q (x) h R t ( x ) dρ(x) C t /2 t R +. Proof. For the proof of (i), see [6]. To prove (ii), recall that δ /2 ((X, Z, a)) = a Q/2 and so, by (3), Thus Ch Q ((X, Z, a)) = 2 Q a Q/2 [(a + + X 2 /4) 2 + Z 2 ] Q/2. (δ /2 Ch Q )((X, Z, a)) = 2 Q [(a + + X 2 /4) 2 + Z 2 ] Q/2. By differentiating along the vector field X, we see that X (δ /2 a (a + + X 2 /4) Ch Q )(X, Z, a) C [(a + + X 2 /4) 2 + Z 2 ] Q/2+ a (a + + X 2 /4) Q C [ + (a + + X 2 /4) 2 Z 2 ]. Q/2+ Since h R 2t(r) = C t /2 e r2 /8t and log a < (X, Z, a), t /2 S C t h R 2t( (X, Z, a) ) X (δ /2 Ch Q )(X, Z, a) dρ(x, Z, a) e (log a) 2 8t R + N a (a + + X 2 /4) Q [ + (a + + X 2 /4) 2 Z 2 ] Q/2+ a dx dz da,

22 84 Vallarino which, on changing variables (W = (a + + X 2 /4) Z ), becomes C t a e (log a) 2 8t R + v (a + + X 2 /4) dx dw z Q++mz ( + W 2 ) Q/2+ a da C t e (log a) 2 8t (a + + X 2 /4) mv/2 dx da R + v C t e (log a) 2 8t (a + ) mv/2 ( + (a + ) /2 X 2 /4) mv/2 dx da R + v C t e (log a) 2 8t (a + ) ( + X 2 /4) mv/2 dx da R + v C t 8t a da. e (log a) 2 R + The last integral, on changing variables (s = log a), becomes C t R e s2 /8t ds = C t /2, as required. The proof of (ii) when i =,..., n is similar, and is omitted. To prove (iii), we use Lemma 2.2 and apply (5) and (6): as required. t B c δ /2 (x) Ch Q (x) h R t ( x ) dρ(x) C t 3/2 φ (r) cosh Q (r/2) e r2 /4t A(r) dr C t 3/2 r e Qr/2 e Qr/2 e r2 /4t e Qr dr C t 3/2 r e r2 /4t dr C t /2, Lemma 4.6. Let F be the function defined by F (s) = (α s + f (s)) h R t (s) for all s in R +, where α and f are as in Lemma 4.4. Set I(r, t) = Then the following hold: F (2 arc cosh(cosh(r/2) v) (i) I(r, t) C t /2 h R 2t(r) t R + r [, ); (ii) I (r, t) C h R t (r) t R + r [, ). dv v Q (2v 2 2) /2. Proof. First we observe that by Lemma 4.4 (i) and (ii), F (s) C s h R t (s) C t /2 h R 2t(s),

23 Vallarino 85 and F (s) C ( + s/2t) h R t (s) C h R t (s). We now prove (i): as required. To prove (ii), note that I(r, t) C t /2 h R 2t(r) C t /2 h R 2t(r), dv v Q (v 2 ) /2 I (r, t) C h R t (r) C h R t (r), as required. F (2 arc cosh(cosh(r/2)v)) dv v Q (v 2 ) /2 sinh(r/2)v 2(cosh 2 (r/2)v 2 ) /2 dv v Q (v 2 ) /2 Now we may prove the L -estimate of the gradient of the heat kernel h t. Proposition 4.7. For all t in R +, S h t dρ C t /2. Proof. Denote by q t the heat kernel associated to the operator L Q. From Proposition 2.3, h t = δ /2 q t, so h t C δ /2 ( q t + q t ) t R +. It is well known ([3, Theorem 5.9], [3, Corollary 5.49]) that q t is radial and q t (x) C t ( + x ) ( + + x ) (n 3)/2 e Q x /2 h R t ( x ) t q t (x) C t x ( + + x ) (n )/2 e Q x /2 h R t ( x ), (22) t for every t in R + and x in S. Our purpose is to estimate S h t dρ C = C S δ /2 ( q t + q t ) dρ We study the cases where t < and t separately. φ (r) ( q t (r) + q t (r) ) A(r) dr. (23)

24 86 Vallarino In the case where t <, it suffices to use the pointwise estimates (22) of q t and its gradient in (23). In the case where t, by using (22) in (23), we estimate the integral of h t on the unit ball. The estimate on the complement of the unit ball is more difficult. We have already (2) noted that h t (x) = δ /2 (x) (A F )(Fh R t )( x ) = δ /2 (x) A (h R t )( x ). By the inversion formula for the Abel transform (8) and (9), if m z is even, then h t (x) = C δ /2 (x) D mz/2 D mv/2 2 (h R t )( x ), while if m z is odd, then h t (x) = C δ /2 (x) x D (mz+)/2 D mv/2 2 (h R t )(s) dν(s), (24) for all x in S. We now consider the cases where m z is even and odd separately. In the case where m z is odd, we use Lemma 4.4, taking q and p to be (m z + )/2 and m v /2, and (24), to deduce that n/2 h t (x) = C δ /2 (x) t j a j (s) h R t (s) dν(s) = C δ /2 (x) j= n/2 j= x H j ( x, t). We estimate the gradient of each summand δ /2 H j. When i n and j, we see that X i (δ /2 H j )(x) = C δ /2 (x) H j ( x, t) + C δ /2 (x) H j( x, t) X i ( )(x), and since X i ( ), X i (δ /2 H j )(x) C δ /2 (x) ( H j ( x, t) + H j( x, t) ). (25) and First suppose that j 2. Integration by parts shows that H j (r, t) = 2t j s (a j h R t )(s) (cosh s cosh r) /2 ds, r H j(r, t) = t j s (a j h R sinh r ds t )(s) r [, ). r (cosh s cosh r) /2 By Lemma 4.4 (i) and (ii) s (a j h R t )(s) a j(s) s 2t a j(s) h R t (s) C ( s j 2t + sj ) s e (Q+)s/2 h R t (s) C t (j )/2 s e (Q+)s/2 h R 2t(s).

25 Vallarino 87 By using this in (25), we see that X i (δ /2 H j )(x) C δ /2 (x) t (j+)/2 x s e (Q+)s/2 h R 2t(s) dν(s). Since j 2, clearly t j/2 t and then the integral above is bounded by C t /2 δ /2 (x) t a (s) h R 2t(s) dν(s) C t /2 h 2t (x). We integrate on the complement of the unit ball: B c X i(δ /2 H j )(x) dρ(x) x C t /2 B c h 2t(x) dρ(x) C t /2 j 2 i =,..., n. (26) If j =, then the estimate is more delicate. Note that H (r, t) = t a (s) h R t (s) dν(s) r = t cosh (Q+) (s/2) (α s + f (s)) h R t (s) dν(s) r = t cosh (Q+) (s/2) F (s) dν(s), r where F (s) = (α s+f (s)) h R t (s) and α, f are as in Lemma 4.4 (ii). By changing variables (u = cosh(s/2) cosh (r/2)), the integral for H transforms into Define Thus t cosh Q (r/2) F (2 arc cosh(cosh(r/2)v) I(r, t) = Then, when i =,..., n, X i (δ /2 H )(x) F (2 arc cosh(cosh(r/2)v) H (r, t) = t cosh Q (r/2) I(r, t). dv v Q (2v 2 2) /2. dv v Q (2v 2 2. = t X i (δ /2 Ch Q )(x) I( x, t) + t δ /2 (x) Ch Q (x) I ( x, t) X i ( )(x). From Lemma 4.4 (i) and Lemma 4.6 we deduce that X i (δ /2 H )(x) t X i (δ /2 Ch Q )(x) I( x, t) + t δ /2 Ch Q (x) I ( x, t) C t /2 h R 2t( x ) X i (δ /2 Ch Q )(x) + C t δ /2 Ch Q (x) h R t ( x ). We now integrate the last expression on the complement of the unit ball and apply Lemma 4.5 (ii) and (iii): X i (δ /2 H ) dρ C t /2 h R 2t( (x) ) X i (δ /2 Ch Q )(x) dρ(x) B c S +C t B c C t /2 i =,..., n. δ /2 (x) Ch Q (x) h R t ( x ) dρ(x) (27)

26 88 Vallarino We sum (26) and (27), to conclude that B c h t (x) dρ(x) C n i= n/2 j= B c X i (δ /2 H j )(x) dρ(x) C t /2, as required. In the case where m z is even, the argument is similar but easier, since the inversion formula for the Abel transform is simpler. We omit the details. References [] Astengo F., The maximal ideal space of a heat algebra on solvable extensions of H-type groups, Boll. Un. Mat. Ital. A(7) 9 (995), [2], Multipliers for a distinguished Laplacean on solvable extensions of H- type groups, Monatsh. Math. 2 (995), [3] Anker J. P., E. Damek, and C. Yacoub, Spherical analysis on harmonic AN groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (996), [4] Christ M., A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 6/6 (99), [5] Cowling M., A. H. Dooley, A. Korányi, and F. Ricci, H-type groups and Iwasawa decompositions, Adv. Math. 87 (99), 4. [6], An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. 8 (998), [7] Cowling M., S. Giulini, A. Hulanicki, and G. Mauceri, Spectral multipliers for a distinguished laplacian on certain groups of exponential growth, Studia Math. (994), 3 2. [8] Coifman R. R., and G. Weiss, Analyse harmonique non commutative sur certains espaces homogenes, Lecture Notes in Mathematics 242, Springer (97). [9] Damek E., Curvature of a semidirect extension of a Heisenberg type nilpotent group, Colloq. Math. 53 (987), [], Geometry of a semidirect extension of a Heisenberg type nilpotent group, Colloq. Math. 53 (987), [] Damek E., and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (992), [2], Harmonic analysis on solvable extensions of H -type groups, J. Geom. Anal. 2 (992), [3] Faraut J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, Les Cours du C.I.M.P.A. (983). [4] Hebisch W., Analysis of Laplacians on solvable Lie groups, Notes for ICMS conference, Edinburgh (998). [5] Hebisch W., and T. Steger, Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (23), [6] Hunt G. A., Semigroups of measures on Lie groups, Trans. Amer. Math. Soc. 8 (956),

27 Vallarino 89 [7] Kaplan A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (975), [8] Vallarino M., A maximal function on harmonic extensions of H -type groups, Ann. Math. Blaise Pascal 3 (26), Maria Vallarino Dipartimento di Matematica e Applicazioni Università degli Studi di Milano Bicocca Via R. Cozzi, Milano (Italy) maria.vallarino@unimib.it Received December 22, 24 and in final form October 24, 26