FOURIER TRANSFORM OF THE HEAT KERNEL ON THE HEISENBERG GROUP

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1 ANALELE ŞTIINŢIFICE ALE UNIVESITĂŢII AL.I. CUZA DIN IAŞI (S.N. MATEMATICĂ, Tomul LIV, 008, f. FOUIE TANSFOM OF THE HEAT KENEL ON THE HEISENBEG GOUP BY MAIAN-DUMITU PANŢIUC Abstract. In tis paper an explicit formula is given for te Fourier transform of te eat kernel on te Heisenberg group at every point of te dual space, as given in Folland. By a result of Siebert we obtain for eac of te above representation a strongly continuous contraction semigroup. For tese semigroups te formula of teir infinitesimal generator is given. Matematics Subject Classification 000: 43A30, 60B5. Key words: Heisenberg group, Fourier transform, convolution semigroups, locally compact groups.. Preliminaries. Te Heiseberg group H n is te simplest noncommutative nilpotent Lie group. Its Laplacian H is not elliptic but a result of Hörmander implies tat it is ypoelliptic. H n is of interest in many domains of analysis and geometry: nilpotent Lie group teory, ypoelliptic second order PDE, probability teory of degenerate diffusion process etc. Te Heisenberg Laplacian generates a convolution semigroup wose measures are absolutely continuous wit respect to te Haar measure. An explicit formula was derived by Hulanicki [5] using representation teory, and by Gaveau [6] using probability teory. It is known tat for a vaguely continuous convolution semigroup on a locally compact Abelian group te Fourier transform of its measures can be Supported in part by CNCSIS-UEFISCSU, contract TD II, code 0 Tis paper was presented at Te 6t Congress of omanian matematicians, July 007 and Conference on Potential Teory and Stocastic, Albac, omania, September 007

2 430 MAIAN-DUMITU PANŢIUC written as te exponential of some continuous negative definite function on te dual group. Performing a Fourier transform in te non-abelian case is more difficult and requires notions of representation teory. For a more extensive introduction in representation teory see Folland [], c. 3. By a result of Siebert [] te Fourier transform of te measures of a vaguely continuous convolution semigroup on a locally compact group at eac representation is a strongly continuous contraction semigroup. An explicit formula for te Fourier transform of te convolution semigroup tat H generates on H n will be given, togeter wit te infinitesimal generator of te contraction semigroup obtained. We recall first some definitions and results from []. Definition.. A (unitary representation of a locally compact group G is a omomorpism π from G to te multiplicative group of unitary operators on a Hilbert space H π continuous in te strong operator topology, tat is a map π : G U(H π suc tat π(xy = π(xπ(y; π(x = π (x = [π(x] ; x π(xu is continuous from G to H π for every u H π. A representation π is called irreducible if te only subspaces of H π tat are invariant for eac π(x, x G are te trivial ones. Two representations π and π of G on H π and H π respectively are (unitary equivalent if tere exists an unitary operator T : H π H π suc tat T π (x = π (xt for all x G Te dual space of a locally compact group G is te space of all classes of equivalence of irreducible representations of G and will be denoted by Ĝ. Several structures were proposed for te dual space. Te one tat yield better results is tat of measurable space wit te so called Mackey-Borel structure. If G is a type I group or equivalently, te Mackey-Borel structure on G is standard ten tere is a measurable field of representations over Ĝ, (ρ π [π] Ĝ suc tat ρ π [π]. Tis way we can identify te points in Ĝ wit representations in tis measurable field. Suppose we fix a measurable field of representations as above. Ten, te Fourier transform of a bounded measure µ M b (G is te measurable

3 3 FOUIE TANSFOM OF THE HEAT KENEL 43 field of operators over Ĝ given by: ˆµ(π = G π(x dµ(x. Consequently te Fourier transform of f L (G is te measurable filed of operators over Ĝ: ˆf(π = f(xπ(x dx. G It is easy to verify te following properties of te Fourier transform: aµ + bν(π = aˆµ(π + bˆν(π, µ ν(π = ˆν(πˆµ(π, f (π = ˆf(π L x f(π = ˆf(ππ(x, x f(π = π(x ˆf(π. Definition.. A family of bounded, positive measures (µ t t>0 on a locally compact group G is called a vaguely continuous convolution semigroup on G if:. µ t (G for all t > 0,. µ t µ s = µ t+s for all s, t > 0, 3. µ t ε 0 vaguely as t 0, were ε 0 is te point-mass at te neutral element of G. If we perform te Fourier transform for te measures of a convolution semigroup on G we will clearly ave a semigroup property for te family of operators ( µ t (π t>0 on H π for eac π: µ t µ s (π = ˆµ t+s (π, for all s, t > 0. By [], Prop. 3. we ave even more: if we denote by µ 0 = I, te identity operator on H π ten ( µ t (π t 0 is a strongly continuous semigroup of operators over H π for every unitary representation π of G. emark.. If G is a locally compact group ten all its irreducible representations are -dimensional and so te representation space can be cosen C for eac suc representation. Moreover, eac irreducible representation can be identified wit a continuous omomorpism γ : G T

4 43 MAIAN-DUMITU PANŢIUC 4 wic allows te dual space to be organized as a group wit pointwise multiplication. Endowing Ĝ wit te topology of compact convergence on G, te dual space becomes a locally compact Abelian group called te dual group of G. In particular if G = n, te dual group n n and te Fourier transform as te classic, well-known formula. For more details on te Abelian case see Berg and Forst [7]. However, if (µ t t>0 is a convolution semigroup on a LCA-group G ten anoter result is available for te Fourier transform of te measures µ t, tat is more precise (see [7]: Teorem.. If (µ t t>0 is a (vaguely continuousconvolution semigroup on a LCA-group G, ten tere exists a uniquely determined continuous, negative definite function ψ on te dual group, Ĝ suc tat ˆµ t (γ = exp( tψ(γ for t > 0 and γ Ĝ. Conversely, given a continuous, negative definite function ψ on Ĝ, ten te formula above determines a convolution semigroup (µ t t>0 on G. As an example, for te brownian semigroup on n : µ t = (4πt n x exp( 4t dx for x n te Fourier transform is ˆµ t (y = exp( t y and te associated negative definite function is y y.. Te Heisenberg group and te eat kernel. Te Heisenberg group H n is n+ = n n wit te composition law: ( (x, ξ, t(x, ξ, t = ( x + x, ξ + ξ, t + t + (x ξ xξ were xξ is te usual scalar product in n. One verifies easily tat H n is a non-abelian group, as te unit =(0,0,0 and te inverse of an element X = (x, ξ, t H n is ( x, ξ, t. It is also obvious tat H n endowed wit te usual topology on n+ is a locally compact group and te Haar measure for tis group is te Lebesgue measure on n+.

5 5 FOUIE TANSFOM OF THE HEAT KENEL 433 H n as a second order differential operator, H wic unlike te usual Laplacian is not elliptic; it is a starting point for many researc in analysis and PDE s. Te Heisenberg Laplacian is were H = n (Xi + Ξ i i= X i = + x i ξ i t and Ξ i = ξ i x i t Te eat kernel for H was first calculated independently by Gaveau in [6] and Hulanicki in [5] as an answer to te requirements: were and K s s = HK s Te Gaveau-Hulanicki result is: K s (x, ξ, t = lim K s(x, ξ, t = δ(x, ξ, t s 0+ (4πs n+ + exp( f(x, ξ, t, τ/sv (τdτ f(x, ξ, t, τ = itτ + ( x + ξ τ 4 ctgτ ( τ/ n V (τ =. In order to compute te Fourier transform of te measures K s (x, ξ, tdxdξdt we need te irreducible representations of H n. Tey can be deduced using te Mackey macine - a way of producing irreducible representations for a group G inducing tem from tose of a normal subgroup (for details see []. All irreducible representations of te Heisenberg group H n are equivalent to one of te following: -dimensional : wit b, β n s τ π b,β : H n T, π b,β (x, ξ, t = e πi(bx+βξ.

6 434 MAIAN-DUMITU PANŢIUC 6 -dimensional: ρ : H n U(L ( n, [ρ (x, ξ, tf](y = e πi(t+ xξ yξ f(y x were is a non-zero real number. Because H n is obviously a type I group, te dual space can be identified wit te measurable field of representations {π b,β ; ρ } b,β,. Tus te Fourier transform of a bounded measure µ M b (H n is a measurable field over Ĥn given by: te complex values ˆµ(π b,β = π b,β ( x, ξ, tdµ(x, ξ, t = H n e πi(bx+βξ dµ(x, ξ, t H n and te operators on L ( n : [ˆµ(ρ φ](y = [ρ ( x, ξ, tφ](ydµ(x, ξ, t H n = xξ+yξ φ(y + xdµ(x, ξ, t. H n e πi( t+ emark.. If f L (H n te complex values ˆf(π b,β : ˆf(π b,β = π b,β ( x, ξ, tf(x, ξ, tdxdξdt H n = e πi(bx+βξ f(x, ξ, tdxdξdt H n are actually te euclidian Fourier transform of f at te point (b, β, 0. Proposition.. Te Fourier transform of K s is given by: K s (π b,β = e 4π ( b + β and ( [ K s (ρ φ](y = s(4πs n φ(xe π (a x y + a x+y dx. n

7 7 FOUIE TANSFOM OF THE HEAT KENEL 435 Proof. For te representations π b,β we ave: K s (π b,β = e πi(bx+βξ K s (x, ξ, tdxdξdt H n were dxdξdt is te Haar measure on te Heisenberg group, and we recall again tat it is actually te usual Lebesgue measure on n+. So, K s (π b,β = ( (4πs n e i tτ τ/ n s n+ s(τ/ n e πi(b jx j +β j ξ j e (x j +ξ j τ 8s ctg( τ dτdxdξdt. j= One can easily see tat we cannot cange te order of integration wit respect to te variable t but tere s no problem to perform it wit te variables τ, x, ξ. We can write ten: K s (π b,β = (4πs n+ n j= e i tτ s ( τ/ n s(τ/ e πi(b jx j +β j ξ j e (x j +ξ j τ 8s ctg( τ dx j dξ j dτdt. Te product under te integral is equal to n ( e πibjx e x τ 8s ctg( τ dx e πiξjx e x τ 8s ctg( τ dx j= ( n π 8π b j s π 8π β j s = j= τ 8s ctg( τ e τctg(τ/ = (8πsn τ n ctg n τ e 8π s( b + β τctg(τ/. τ 8s ctg( τ e τctg(τ/ Tus, K s (π b,β = e i tτ s 4πs c n τ e 8π s( b + β τctg(τ/ dτ dt. Canging te variable τ sτ we obtain: K s (π b,β = π e itτ c n (sτ e 4π (b +β τctg(sτ dτdt = π FFg(0

8 436 MAIAN-DUMITU PANŢIUC 8 were g(τ = 4π ( b + β c n (sτ e τctg(sτ and Fg is te Fourier transform of g. It is obvious tat g(0 = exp( 4π s( b + β and ten K s (π b,β = e 4π s( b + β. To prove te formula for te representations ρ let φ L ( n. We ave: [ K s (ρ φ](y = e πi( t+ xξ+yξ φ(y + xk s (x, ξ, tdxdξdt H n = (4πs n+ e πit e i tτ s V (τ φ(y + x n n e πiξ(x+y e ( x + ξ τ 8s ctg τ dxdξdτdt. Denote by g(τ = V (τ φ(y + xe πiξ(x+y e n n ( x + ξ τ 8s ctg τ dxdξ and cange te variable τ sτ. We obtain: [ K s s (ρ φ](y = (4πs n+ e πit e itτ g(sτdτdt and we recognize now a Fourier transform and its inverse for te function g at te point 4πs. Tus, [ K s (ρ φ](y = = = = (4πs n g(4πs ( n φ(x + ye x π ctg(πs s(πs n e πiξ(x+y e ξ π ctg(πs dξdx ( n n φ(x + ye x π ctg(πs s(πs n n π(x π j +y j πctg(πs e ctg(πs dx j= ( n/ φ(x + y s(πs n e π (ctg(πs x +tg(πs x+y dx.

9 9 FOUIE TANSFOM OF THE HEAT KENEL 437 By an obvious cange of variable and denoting a = ctg(πs we ave: ( [ K n/ s (ρ φ](y = φ(xe π (a x y + a x+y dx. s(πs n emark.. As one can easily see we ave K s (ρ = K s (ρ for eac s, > 0 Te values of te Fourier transform at te π b,β representations reminds of te classical brownian semigroup. For te values at te representations ρ we state some properties in te next two teorems. Te teorems are proved only for te case n = but tey can be verified easily for te general case. Teorem... K s (ρ is a bounded, self-adjoint operator on L ( n ;. K s (ρ is positive, < K s (ρ φ, φ >= K s (ρ φ and K s (ρ <. Proof.. Continuity follows from te obvious relation ˆµ(π µ, µ M b (G and [π] Ĝ and from te fact tat K s L (H n =. Let now φ and ψ L ( n. We ave: < K s (ρ φ, ψ >= K s (ρ φ(yψ(ydy = K s (x, ξ, te πi( t+ xξ+yξ φ(y + xψ(ydxdξdtdy. H n Canging te variable y + x y te above integral is equal to: K s (x, ξ, te πi( t xξ+yξ φ(yψ(y xdxdξdtdy H n = φ(y K s (x, ξ, te πi(t+ xξ yξ ψ(y xdxdξdtdy. H n =< φ, K s (ρ ψ > since K s is a real-valued function. Hence, Ks (ρ is self-adjoint.. Indeed, using. and Teorem. we ave: < K s (ρ φ, φ >=< ˆK s (ρ ˆK s (ρ φ, φ >

10 438 MAIAN-DUMITU PANŢIUC 0 =< K s (ρ φ, K s (ρ φ >= K s (ρ φ 0. Tis implies on te oter and tat and tus ˆK s (ρ = ˆK s (ρ, s > 0 ˆK s (ρ = ˆK s n (ρ n, s > 0, n N. Estimating te norm of ˆK s (ρ we get: ˆK s (ρ φ(y φ(x e π a(x y e π s(4πs ( s(4πs φ L ( Tus, ˆK s (ρ φ L ( a (x+y dx e πa(x y e π a (x+y dx. s(4πs φ L ( e πa(x y e π a (x+y dxdy a s(πs a, wic sows again te continuity of ˆK s (ρ and ˆK s (ρ s(4πs. Letting s tend to we obtain But, for a fixed s > 0 we ave lim ˆK s (ρ = 0. s ˆK n s(ρ = ˆK s (ρ n, n N, so as n te left side of te equality tend to 0 so necessarily ˆK s (ρ <, s > 0.

11 FOUIE TANSFOM OF THE HEAT KENEL 439 Teorem... ( K s (ρ s 0 is a strongly continuous contraction semigroup on L ( n for every 0;. Te infinitesimal generator of te semigroup ( K s (ρ s 0 is A f(y = f(y 4π y f(y, for f C c ( n. Proof.. We can use te general formula µ ν(π = ν(π µ(π to prove te semigroup relation but instead of tat we will give ere a direct proof. It is enoug to consider n =. Let φ L (. Denoting by a = ctg(πs and a = ctg(πs we ave: were ˆK s [ ˆK s (ρ φ](y = e π e π = C a (z+y dz = s(4πs s(4πs s(4πs (a (z y + (z+y a +a (x z + [ ˆK s (ρ φ](ze π (x+z a dxdz φ(x e π (x (a + a +y (a + a A = a + a + a + a, B = y(a a + x(a a, C = Tus, ˆK s [ ˆK s (ρ φ](y = C were f(x, y = s(4πs s(4πs. φ(x a (z y e π (z A zb dzdx φ(x e π (x (a + a +y (a + a f(x, ydx e π (za zb dz = e π (z A B A e π B A dz f(x, y = A e π B A

12 440 MAIAN-DUMITU PANŢIUC But C A = s(4πs s(4πs c(πs s(πs + s(πs c(πs + c(πs s(πs + s(πs c(πs and using basic properties of yperbolic sine and cosine functions we get: C A = s(4π(s + s. It remains to prove tat: x (a + a + y (a + a B A = a(x y + a (x + y, were a = ctg(π(s + s. For tis we will identify te coefficients so we need to sow tat: for x : a + (a a a a + a + a + = a + a a for y : a + (a a a a + a + a + a = a + a and for xy : (a a (a a a + a + a + a = a a, tings tat are easy to verify using again te properties of te yperbolic functions s and c. Tus we ave: ˆK s [ ˆK s (ρ φ](y = φ(xe π a(x y e π a (x+y dx s(4π(s + s = [ ˆK s +s φ](y. Te fact tat ( K s (ρ s 0 is a strongly continuous semigroup follows easily from te vague convergence of measures K s (x, ξ, tdxdξdt to te Dirac measure concentrated at (0, 0, 0 as s tends to 0.

13 3 FOUIE TANSFOM OF THE HEAT KENEL 44. We need to determine ( lim [ s 0+ s K s (ρ φ](y φ(y pointwise for y n and for suitable functions f L ( n. I will give ere a complete calculus only for n =. For te general case it is similar. We terefore ave to find out te following limit: ( lim s 0+ s s(4πs φ(xe π (a(x y + a (x+y dx φ(y were a = ctg(πs. We can rewrite te above limit canging te variable x x + y in te following way: lim s 0+ { s +φ(y s ( ( s(4πs s(4πs (φ(x + y φ(xe π e π (ax + a (x+y dx (ax + a (x+y dx Take te second term of te sum of te limit. Te exponent can be written as: ax + a (x + y = (a + (x a + y + a πy a + a. Canging te variable x x y +a te limit of te second term becomes: ( lim s 0+ s s(4πs e πy ( = lim s 0+ s But a = ctg(πs so lim s 0+ s s(4πs a π a +a } +a π e a x dx a + a π e πy a. +a = tg(4πs so te last quantity is equal to +a ( e πy tg(4πs = 4π y. c(4πs,.

14 44 MAIAN-DUMITU PANŢIUC 4 eturning to te first term in te initial limit one can approximate for appropriate functions, φ(x + y φ(y by xφ (y + x φ (y. Tus we only need to calculate lim s 0+ s + x φ (ye s s(4πs s(4πs e πy tg(4πs +a π a (x+y +a dx. (xφ (y But, clearly, 4π and e πy tg(4πs as s 0+. We can separate again te limit and we need to find: φ (y A = lim s 0 πs xe πctg(4πs(x+y +a dx s and φ (y B = lim s 0 πs x e πctg(4πs(x+y +a dx. s For A, making an obvious cange of variable we get: A = lim s 0+ = φ (y π φ (y πs s + a lim s 0+ π s s s (πs c(4πs π ctg(4πs s(4πs c(4πs = 0. For B we proceed in te same way and aving in mind wat we ve already proved for A, we obtain: φ (y B = lim s 0+ 4 πs x e πctg(4πsx dx s = lim s 0+ φ (y 4 πs s s(4πs πc(4πs s(4πs πc(4πs x e x dx. We ave ten B = φ (y so for n= te infinitesimal generator wic clearly depends on is: A φ(y = φ (y 4π y φ(y. In general, te infinitesimal generator is given by: A φ(y = φ(y 4π y φ(y.

15 5 FOUIE TANSFOM OF THE HEAT KENEL 443 EFEENCES. Folland, G.B. A Course in Abstract Harmonic Analysis, Studies in Advanced Matematics, CC Press, Boca aton, FL, Siebert, E. Fourier analysis and limit teorems for convolution semigroups on a locally compact group, Adv. in Mat. 39 (98, no., Barczy, M.; Pap, G. Fourier transform of a gaussian measure on te Heisenberg group, arxiv:mat/0506v, Beals,.; Gaveau, B.; Greiner, P. Hamilton-Jacobi teory and te eat kernel on Heisenberg groups, J. Mat. Pures Appl. (9 79 (000, no. 7, Hulanicki, A. Te distribution of energy in te Brownian motion in te Gaussian field and analytic-ypoellipticity of certain subelliptic operators on te Heisenberg group, Studia Mat. 56 (976, no., Gaveau, B. Principe de moindre action, propagation de la caleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Mat. 39 (977, no. -, Berg, C.; Forst, G. Potential Teory on Locally Compact Abelian Groups, Springer-Verlag, New York, Siebert, E. A new proof of te generalized continuity teorem of Paul Lévy, Mat. Ann. 33 (978, no. 3, eceived: 7.I.008 Faculty of Matematics, Al. I. Cuza University, Iaşi, OMÂNIA pantirucmarian@yaoo.com

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