THE HEAT KERNEL ON H-TYPE GROUPS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 4, April 2008, Pages S ( X Article electronically published on December 21, 2007 THE HEAT KERNEL ON H-TYPE GROUPS QIAOHUA YANG AND FULIU ZHU (Communicated by David S. Tartaoff Abstract. In this paper we present an explicit calculation of the heat ernel for the sub-laplacian on an H-type group G by using irreducible unitary representations of G and the heat ernel for the associated Hermite operator. 1. Introduction As is well nown, the heat ernel plays an important role in many problems in harmonic analysis. An explicit expression for the heat ernel on the Heisenberg group H n = C R was obtained by Hulanici [9], Gaveau [7] and Staubach [11]. Gaveau [7] also obtained the heat ernel for free nilpotent Lie groups of step two. Cygan [6] obtained the heat ernel for all nilpotent Lie groups of step two. Zhu [13] used the method of the stochastic integral due to Gaveau to construct the heat ernel for the quaternionic Heisenberg groups. But neither Gaveau s expression for free nilpotent Lie groups nor Cygan s expression for arbitrary nilpotent Lie groups of step two were as explicit as those in the cases of Heisenberg groups and quaternionic Heisenberg groups. The Hulanici-Gaveau formula for the heat ernel on the Heisenberg groups has many interesting applications (see [13]. Although very impressive, these applications depend heavily on explicit expressions for the heat ernel. A natural and interesting question is : Are there other nilpotent Lie groups for which the expressions for the heat ernel are as explicit as those in the cases of the Heisenberg groups and quaternionic Heisenberg groups? The aim of this paper is to loo for such a formula for the heat ernel on H-type groups, a remarable class of stratified groups of step two introduced by Kaplan [10]. On these groups there is a natural sub-laplacian with an associated heat ernel. We use the method of Thangavelu [12], the irreducible unitary representation and the same argument as in [11], to calculate the heat ernel on the H-type groups and obtain a closed form of expression which closely resembles those of the heat ernel on Heisenberg groups and quaternionic Heisenberg groups. As we now, the H-type groups are the only nilpotent Lie groups on which an explicit formula for the heat ernel has been obtained up to now. Received by the editors March 30, Mathematics Subject Classification. Primary 22E25; 35A08. Key words and phrases. H-type groups, sub-laplacian, heat ernel, Hermite operator. The first author was supported by the National Science Foundation of China under grant number c 2007 American Mathematical Society Reverts to public domain 28 years from publication

2 1458 QIAOHUA YANG AND FULIU ZHU 2. Irreducible unitary representations We begin by describing the Lie groups and Lie algebras under consideration. An H-type group G is a Carnot group of step two with the following properties: the Lie algebra g of G is endowed with an inner product, such that, if z is the center of g, then[z, z ]=z and, moreover, for every fixed z z, themapj z : z z defined by (2.1 J z (v,ω = z, [v, ω], ω z is an orthogonal map whenever z, z = 1 (see [3], [10]. For reasons that will soon be clear we put 2n =dimz and m =dimz. Fix λ z, the dual of z, and define the sew-symmetric linear mapping B(λ onz by B(λX, Y = λ([x, Y ] X, Y z. We denote by z λ the element of λ z determined by B(λX, Y = λ([x, Y ] = J zλ (X,Y. Denote the ernel of B(λ byr λ,andletm λ be the orthogonal complement of r λ in z.sinceb(λ is sew-symmetric, m λ is B(λ-invariant and its dimension dim m λ is even. Denote by Λ the Zarisi-open subset of z of the vectors λ for which dim m λ is a maximum. From (2.1 we now that dim m λ =dimz =2n and r λ = {0} for all λ ΛandΛ=z \{0}. Forλ, µ Λ, we put λ, µ = z λ,z µ and λ = λ, λ. We denote by Sym m B(λ the symmetric function of degree 2n in the roots of B(λ. Fix λ in Λ. There are orthogonal vectors E 1 (λ,,e n (λ, E 1 (λ,, E n (λ in m λ such that B(λE i (λ = λ E i (λ and B(λE i (λ = λ E i (λ, and Sym 2n B(λ = λ 2n. We may further decompose z. Denote by x λ and y λ the subspaces span{e 1 (λ,,e n (λ} and span{e 1 (λ,, E n (λ}, respectively. Then we may write V in z as X + Y with X x λ and Y y λ. With respect to this basis, the group law on an H-type group G has the form (z, t (z,t =(x, y, t (x,y,t (2.2 ( = z i + z i, i =1, 2,, 2n t j + t j z, U(j z, j =1, 2,,m where z =(x, y R 2n and the matrices U (1,U (2,,U (2n satisfy the following conditions: (1 U j is an m m sew-symmetric and orthogonal matrix, for every j = 1, 2,, 2n; (2 U (i U (j + U (j U (i =0foreveryi, j {1, 2,, 2n} with i j. The irreducible unitary representations parameterized by λ Λmaybede- scribed as follows (see [1], [5]: (π λ (x, y, tφ(ξ =e i m j=1 λ jt j +i n j=1 (x jξ j x jy j λ φ(ξ + y, for all φ L 2 (y λ. For f L 1 (G andλ Λ, its Fourier transform f(λ isthe operator-valued function defined by f(λ = f(x, y, tπ λ (x, y, tdxdydt. z x λ y λ

3 THE HEAT KERNEL ON H-TYPE GROUPS 1459 This means that for each ϕ, ψ L 2 (y λ, ( f(λϕ, ψ = f(x, y, t(π λ (x, y, tϕ, ψdxdydt. z y λ x λ We now describe the sub-laplacian on an H-type group G (see also [3]. The vector fields in the algebra g of G =(R 2n+m, that agrees at the origin with x j, y j (j =1, 2,,naregivenby ( X j = + 1 z l U ( l,j, x j 2 t l=1 ( Y j = + 1 z l U ( l,j+n y j 2 t with (U ( l,j 2n 2n = U (. The Kohn s sub-laplacian on the H-type group G is given by L = (Xj 2 + Yj 2 = z 1 4 z 2 t z, U ( z, t j=1 where z = z 2 j=1 j = ( 2 x 2 j=1 j l=1 + 2 yj 2, t = t 2 j and z =( x, y T =(,,,,,,, T. x 1 x n y 1 y n We recall that this essentially selfadjoint positive operator does not depend on the choice of an orthonormal basis of z. Moreover, on functions ũ(z, t =u( z,t, we obtain (see [3] (2.3 z, U ( z ũ(z, t =0, =1, 2,,m, and hence L has the form Lũ(z, t = z ũ(z, t 1 4 z 2 u ũ(z, t. The representation π λ of G determines a representation πλ of its Lie algebra g on the space of C vectors. Recall that f is said to be a C vector for the representation π λ if (x, y, t π λ f is a C function from G into the Hilbert space. The representation πλ is defined by ( d πλ(xf = dt π λ(exp txf t=0 for every X in the Lie algebra g. We can then extend πλ to the universal enveloping algebra of left-invariant differential operators on G. IfA is any such operator, then a simple computation shows that A(π λ (x, y, tf,g =(π λ (x, y, tπλ(af,g. Therefore, one way of obtaining entry functions that are eigenfunctions of A is to tae f to be an eigenfunction of the operator πλ (A. An easy calculation yields that πλ(x j =i λ ξ j, πλ(y j = ξ j,

4 1460 QIAOHUA YANG AND FULIU ZHU for j =1, 2,,n, so that H(λ =π λ(l = ξ 2 j=1 j + λ 2 ξ 2 is the Hermite operator. The eigenfunctions of H(λ aregivenby Φ λ α(ξ = λ n 4 Φα ( λ ξ, α =(α 1,,α n, where Φ α (ξ is the product ψ α1 (ξ 1 ψ αn (ξ n andψ αj (ξ j (j =1,,nisthe eigenfunction of 2 + ξ 2 ξj 2 j with eigenvalue 2α i + 1 (see [4], [12]. Note that H(λΦ λ α =(2 α + n λ Φ λ α, α = α α n ; therefore, L(π λ (x, y, tφ λ α, Φ λ β=(2 α + n λ (π λ (x, y, tφ λ α, Φ λ β. Thus the entry functions (π λ (x, y, tφ λ α, Φ λ β asα, β range over Nn give a family of eigenfunctions for the sub-laplacian. Let L λ be the operator defined by the relation L(e i m j=1 λ jt j f(x, y = e i m j=1 λ jt j L λ f(x, y. Explicitly, L λ is given by the expression L λ = z λ 2 z 2 N, where N = i m λ z, U ( z. Put π λ (x, y =π λ (x, y, 0. Since (π λ (x, y, tφ λ α, Φ λ β=e i m j=1 λ jt j (π λ (x, yφ λ α, Φ λ β, the functions Φ λ αβ(z =(2π n n 2 λ 2 (πλ (x, yφ λ α, Φ λ β are eigenfunctions of the operator L λ : (2.4 L λ Φ λ αβ(z =(2 α + n λ Φ λ αβ(z. They are also eigenfunctions of the operators z λ 2 z 2 (see [12], page 54: (2.5 ( z λ 2 z 2 Φ λ αβ(z =( α + β + n λ Φ λ αβ(z. From (2.4 and (2.5 we obtain (2.6 N Φ λ αβ(z =( α β λ Φ λ αβ(z. We now define the Laguerre functions ϕ n 1 (z onr 2n by ϕ n 1 (z =L n 1 ( 1 2 z 2 e 1 4 z 2, where L n 1 (s =e s t s 1 d n 1! ds n 1 (e s s n 1+ for s>0. We also define ϕ n 1, λ (z =ϕn 1 ( λz for λ 0. Thenϕ n 1, λ satisfies (see [12], page 58 (2.7 ϕ n 1, λ (z =(2π n n 2 λ 2 Φ λ αα(z. α =

5 THE HEAT KERNEL ON H-TYPE GROUPS The heat ernel In this section, we calculate the heat ernel of the sub-laplacian L, givenby L = (Xj 2 + Yj 2 = z 1 4 z 2 t z, U ( z t = j=1 ( 2 x 2 j=1 j + 2 y 2 j 1 4 ( x 2 + y 2 t 2 j z, U ( z. t We observe that {X j,y j } n j=1 and their first-order commutators span the whole Lie algebra g. Thus the sub-laplacian is an example of a step-two hypoelliptic operator, studied in Hörmander [8]. We start by looing at the heat equation: p s (x, y, t = Lp s (x, y, t, s > 0, x,y R n, t R m, s p 0 (x, y, t =δ 0, where δ 0 is the Dirac mass at the origin. The calculation of the heat ernel proceeds along the same lines as [11]. Taing the Fourier transform in the t variable yields p λ s s = n ( 2 x 2 j=1 j + 2 y 2 j p λ s + 1 m 4 ( x 2 + y 2 λ 2 p λ s i λ z, U ( z p λ s, p λ 0(x, y, λ =δ 1 λ, where p λ s = e iλ t p s (x, y, tdt R m and δ = δ(x, y isthediracmassinr 2n. We rewrite the equation as p λ s s =( z λ 2 z 2 p λ s Np λ s, where N = i m λ z, U ( z, introduced earlier. First we shall show that the Lie bracet [ z λ 2 z 2, N ] is zero. Lemma 3.1. [ z λ 2 z 2, N ]=0. Proof. Since N = i m λ z, U ( z, a simple computation shows that ( N = i λ U ( j,l + N z j z l z j l=1 and zj 2 Therefore we obtain (3.1 z N = N =2i z 2 j=1 j λ ( l=1 N =2i λ U ( j,l z l j=1 l=1 + N 2 z j zj 2. U ( j,l + N z. z l z j

6 1462 QIAOHUA YANG AND FULIU ZHU Recalling U ( =1, 2,,m is sew-symmetric, we have, from (3.1, Thus U ( j,l j=1 l=1 z l z j =0, =1, 2,,m, z N = N z. [ z λ 2 z 2, N ]= z N λ 2 z 2 N + N z N 1 4 λ 2 z 2 = 1 4 λ 2 z 2 N 1 4 λ 2 N ( z λ 2 z 2 N =0. Here we use the fact N z 2 = 0 (see (2.3. Lemma 3.1 is quite useful and enables us to write (3.2 e s( z+ 1 4 λ 2 z 2 N = e sn e s( z+ 1 4 λ 2 z 2. So we are led to study the heat semigroup e s( z+ 1 4 λ 2 z 2. Let h s (x, y be the heat ernel of the Hermite operator z λ 2 z 2,thatis, h s (x, y =( z + 1 s 4 λ 2 z 2 h s (x, y with lim h s (x, yf(x, y =f(0, 0. s 0 R m An explicit calculation of h s (x, y is given by (see e.g. [4], Theorem 2.1 ( n λ h s (x, y =(4π n exp{ λ z 2 coth λ s} sinh λ s 4 and hence (see [12], page 85, (3.3 h s (x, y =(2π n λ n =0 e (2+n λ s ϕ n 1, λ, where ϕ n 1, λ was introduced earlier. From (2.6, (2.7 and (3.3, the explicit spectral decomposition of N is given by (3.4 N h s (x, y =(2π n λ n =(2π n λ n =(2π n λ n =0. =0 =0 =0 e (2+n λ s N ϕ n 1, λ e (2+n λ s (2π n 2 λ n 2 N e (2+n λ s (2π n 2 λ n 2 0 α = Φ λ αα(z

7 THE HEAT KERNEL ON H-TYPE GROUPS 1463 Therefore, from (3.2, (3.3 and (3.4, p λ s is given by p λ s = e sn e s( z+ 1 4 λ 2 z 2 (δ 1 λ = e sn h s (x, y = h s (x, y. This gives us, after taing the inverse Fourier transform, the heat ernel for the sub-laplacian: ( n λ p s (x, y, t =(2π m (4π n exp{ λ z 2 coth λ s iλ t}dλ. R sinh λ s 4 m We summarize the computations and the formula as a theorem: Theorem 3.2. The heat ernel of the sub-laplacian on an H-type group is given by ( n λ p s (x, y, t =(2π m (4π n exp{ λ z 2 coth λ s iλ t}dλ. sinh λ s 4 R m Acnowledgments The authors thans the referee for his/her careful reading and very useful comments which improved the final version of this paper. References 1. F. Astengo, M. Cowling, B. Di Blasio, M. Sundari, Hardy s uncertainty principle on certain Lie groups, J. London Math. Soc., 62 (2000, MR (2002b: R. Beals, B. Gaveau, P. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, Bull. Sci. Math. 121 (1997, MR (98b:35032a 3. A. Bonfiglioli, F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type groups, J.FunctionalAnalysis,207 (2004, MR (2004: D.C. Chang, J.Z. Tie, A note on Hermite and subelliptic operators, Acta Math. Sin., 21 (2005, MR (2006e: L.J. Corwin, F.P. Greenleaf, Representations of nilpotent Lie groups and their applications. 1: Basic theory and examples, Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, MR (92b: J. Cygan, Heat ernels for class 2 nilpotent groups, Studia Math., 64 (1979, MR (82b: B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139 (1977, MR (57: L. Hörmander, Hypoelliptic second order differential equations, Acta Math., Uppsala, 119 (1967, MR (36: A. Hulanici, The distribution of energy of the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math., 56 (1976, MR (54: A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans.Amer.Math.Soc.,258 (1980, MR (81c: W. Staubach, Wiener path integrals and the fundamental solution for the Heisenberg Laplacian, J. d Analyse Math., 91 (2003, MR (2005m: S. Thangavelu, An introduction to the uncertainty principle: Hardy s theorem on Lie groups, Progress in Mathematics, 217, Birhäuser, Boston, MR (2004j: Fuliu Zhu, The heat ernel and the Riesz transforms on the quaternionic Heisenberg groups, Pacific J. Math. 209 (2003, MR (2004e:43013

8 1464 QIAOHUA YANG AND FULIU ZHU School of Mathematics and Statistics, Wuhan University, Wuhan, , People s Republic of China Current address: Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P.O. Box 71010, Wuhan , People s Republic of China address: qaohyang2465@yahoo.com.cn School of Mathematics and Statistics, Wuhan University, Wuhan, , People s Republic of China address: flzhu@whu.edu.cn