Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 39, pp. 1–11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu RESOLVENT KERNEL ON H-TYPE GROUPS AND A GREEN KERNEL FOR FRACTIONAL POWERS OF ITS SUB-LAPLACIAN ZAKARIYAE MOUHCINE Abstract. In this article, we give an integral representation of the resolvent kernel on H-type groups, then we derive an integral representation of Kaplan’s fundamental solution on this groups. Also we obtain the Green kernel for fractional powers of its sub-Laplacian. 1. Introduction H-type groups form an interesting class of Carnot groups of step two in connec- tion with hypoellipticity questions. Such groups, which were introduced by Kaplan [11] around 1980 in the framework of his research about hypoelliptic partial dif- ferential equations, constitute a direct generalization of Heisenberg groups and are more complicated. This class suggests that this is the largest class of groups for which an elementary expression for the fundamental solution of the sub-Laplacian exists. Many interesting groups are H-type groups, including the two-step nilpo- tent group that appears in the Iwasawa decomposition of a rank-one semisimple Lie group. There has been subsequently a considerable amount of work in the study of such groups [5, 6, 16, 19]. In this article, we are interested in some complex spectral objects associated with the sub-Laplacian L on H-type groups G. Namely, the heat, the resolvent and the green kernels are derived. The first aim is to use the explicit formula for the heat kernel to derive an integral representation of the resolvent kernel. More precisely, one can use the well known formula connecting the resolvent R(ζ,L) = (ζ − L)−1 and the heat T (s) = esL operators [7, p.56] R(ζ,L) = ∫ ∞ 0 e−ζsT (s) ds, to find the resolvent kernel associated with the sub-Laplacian L. We prove that its expression is given in terms of the Whittaker function Wκ,µ(z). As applications of the obtained explicit formula for the resolvent kernel, we derive an integral representation of the Green function on H-type groups G. 2020 Mathematics Subject Classification. 22E25, 22E30, 35K08. Key words and phrases. H-type groups; sub-Laplacian; resolvent kernel; Green kernel; Whittaker function. ©2022. This work is licensed under a CC BY 4.0 license. Submitted December 7, 2021. Published May 19, 2022. 1 2 Z. MOUHCINE EJDE-2022/39 The second aim is to prove that the Kaplan’s fundamental solution obtained in [11] for the sub-Laplacian L on G can also be derived from the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental solution. The third purpose is to use the explicit formula for the resolvent kernel to give the Green kernel of the fractional power of the sub-Laplacian L, i.e. Lα for α ∈]0, 1[, on the on H-type groups G. We prove that its formula is given by a series expansion in terms of the generalized Laguerre polynomials. An interesting relationship with special functions such as Gamma and Bessel functions will appear, showing the underlying harmony of this work. The layout of this article is as follows. The aim of Section 2, is to provide the basic notation and definitions about H-type groups that we shall use throughout the paper. In Sections 3, we establish a new integral representation of the heat kernel obtained in [19]. In Section 4, we obtain an integral representation of the resolvent kernel on G, that plays a major role in the following sections. We ends this section by establishing an integral representation of the Green function on H-type groups G. In Section 5, we prove that the Kaplan’s fundamental solution for the sub-Laplacian L can also be derived from the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental solution. In the section 6, we give a formulas for the Green kernel for fractional powers of the sub-Laplacian L. This article extends the results in [2, 3, 14, 15] from the classical Heisenberg groups C×R, H×R3 and O×R7 to the H-type groups G (Heisenberg groups with multi-dimensional center). 2. Notation and definitions An H-type group G is characterized by being (canonically isomorphic to) R2n × Rm with the group law (x, u) · (y, v) = ( x+ y, u+ v + 1 2 〈x, Uy〉 ) , with x = (x1, . . . , x2n) ∈ R2n, u = (u1, . . . , xm) ∈ Rm and 〈x, Uy〉 = ( 〈x, U (1)y〉, . . . , 〈x, U (m)y〉 ) ∈ Rm, where the U (j)’s have the following properties: (1) U (j) is an m × m skew-symmetric and orthogonal matrix for every j ∈ {1, . . . ,m}, (2) U (i)U (j) + U (j)U (i) = 0, 1 ≤ i 6= j ≤ m. It is clear that the point e = (0, 0) is the identity in G and the inverse operation is (x, u)−1 = (−x,−u). The center of the group G is of dimension m and is given by Z(G) = {(0, u) : u ∈ Rm}. Let U (j) = ( U (j) k,l ) k,l≤2n (1 ≤ j ≤ m). The sub-Laplacian on G is the second- order differential operator L = ∑2n l=1X 2 l , where (Xl)1≤l≤2n are the left-invariant vector fields on G defined by Xl = ∂ ∂xl + 1 2 m∑ j=1 ( 2n∑ k=1 xkU (j) k,l ) ∂ ∂uj . EJDE-2022/39 GREEN KERNELS ON H-TYPE GROUPS 3 Let |x|2 = 2n∑ i=0 x2i , |u|2 = m∑ j=1 u2j , u · v = m∑ j=1 ujvj for v ∈ Rm. We introduce on G the group {δr : 0 < r <∞} of dilations, which is defined by δr(x, u) = (rx, r2u). These dilations satisfy the distributive law δr ((x, u).(y, v)) = (δr(x, u)) . (δr(y, v)) . We also define the norm function on G, which we will call the Kaplan distance, by ρ(x, u) = ( |x|4 + 16|u|2 )1/4 , which satisfies ρ(δr(x, u)) = rρ(x, u). Note that, the Haar measure on G coincides with the Lebesgue measure on R2n×Rm which is denoted by dxdu and the homogeneous dimension of G is Q = 2(n + m). We refer the reader to [5, 11] for further details. 3. Heat kernel on H-type groups The heat kernel of the sub-Laplacian on an H-type group is given in [19]. Theorem 3.1. On an H-type group G ' R2n×Rm, the heat kernel (pt)t>0 has the form pt(x, u) = (2π)−m(4π)−n ∫ Rm ( |λ| sinh(|λ|t) )n e− |λ||x|2 4 coth(|λ|t)−iλ.u dλ, (3.1) for every t > 0 and every (x, u) in G. Using polar coordinates, we establish a new integral representation of the heat kernel (3.1). Proposition 3.2. The heat kernel in (3.1) can be written as pt(x, u) = (2π)− m 2 (4π)−n|u|1−m2 ∫ ∞ 0 e− r|x|2 4 coth(tr) sinhn(tr) Jm 2 −1(|u|r)rn+m 2 dr, (3.2) where Jν is the Bessel functions of the first kind. Proof. We introduce polar coordinates for the λ-variable such that λ = rω, where r = |λ| and ω = (ω1, . . . , ωm) is a point in the unit sphere Sm−1 in Rm with center at the origin. Then dm(λ) = rm−1drdσ(ω), where dσ is the surface measure on Sm−1. By Theorem 3.1, pt(x, u) = (2π)−m(4π)−n ∫ ∞ 0 ∫ Sm−1 ( r sinh(tr) )n e− r|x|2 4 coth(tr)−irω.urm−1 dr dσ(ω) = (2π)−m(4π)−n ∫ ∞ 0 rn+m−1 sinhn(tr) e− r|x|2 4 coth(tr)Iu(r) dr, (3.3) where Iu(r) = ∫ Sm−1 e−irω.u dσ(ω). 4 Z. MOUHCINE EJDE-2022/39 Using the identity [17, p.347]∫ Sm−1 ei〈a,ω〉 dσ(ω) = (2π)ν+1|a|−νJν(|a|), ν = m 2 − 1, for a = −ru, we obtain Iu(r) = (2π) m 2 |u|1−m2 r1−m2 Jm 2 −1(|u|r). (3.4) Substituting (3.4) into the expression of the heat kernel in (3.3), we finally obtain pt(x, u) = (2π)− m 2 (4π)−n|u|1−m2 ∫ ∞ 0 e− r|x|2 4 coth(tr) sinhn(tr) Jm 2 −1(|u|r)rn+m 2 dr, as required. � Remark 3.3. We can proof that, the solution of the Cauchy problem of heat type of L with initial-value is pt ((x, u), (y, v)) := pt ( (x, u).(y, v)−1 ) = pt ( x− y, u− v + 1 2 〈x, Uy〉 ) , (3.5) for all (x, u), (y, v) ∈ G. On the other hand, it is more evident that pt depends only on |x| and |u|. This leads us (throughout this article) to the following notation: ρ := |x− y| and τ := ∣∣u− v + 1 2 〈x, Uy〉 ∣∣. (3.6) Hence, the heat kernel in (3.5) can be written as pt ((x, u), (y, v)) = (2π)− m 2 (4π)−n τ m 2 −1 ∫ ∞ 0 e− rρ2 4 coth(tr) sinhn(tr) Jm 2 −1 (τr) rn+ m 2 dr. (3.7) 4. Resolvent kernel on H-type groups The confluent hypergeometric function [10, p.204] is denoted by 1F1(a, b; z) = Γ(b) Γ(a) ∞∑ j=0 Γ(a+ j) Γ(b+ j) zj j! . (4.1) As in [10, p.264], we define the Kummer’s function of the second kind [1, p.505] U(a, b; z) = Γ(1− b) Γ(a− b+ 1) 1F1(a, b; z) + Γ(b− 1) Γ(a) z1−b 1F1(a− b+ 1, 2− b; z). (4.2) We denote the Whittaker function given by Wκ,µ(z) = e−z/2zµ+ 1 2U ( µ− κ+ 1 2 , 1 + 2µ; z ) . (4.3) Theorem 4.1. Let ζ ∈ C such that <ζ > 0. Then, the resolvent kernel for an H-type group, G, is R (ζ; (x, u), (y, v)) = 2 n−2 2 (2π)−n− m 2 ρnτ m−2 2 ∫ ∞ 0 Γ ( ζ 2r + n 2 ) Jm 2 −1(τr)W− ζ 2r , n−1 2 ( rρ2/2 ) r n+m−2 2 dr, (4.4) where Γ(·) is Euler’s Gamma-function. EJDE-2022/39 GREEN KERNELS ON H-TYPE GROUPS 5 Proof. We use the well known formula connecting the resolvent and the heat kernels R ( ζ; (x, u), (y, v) ) = ∫ ∞ 0 e−ζt pt ( (x, u), (y, v) ) dt; 0, (4.5) as well as the explicit formula for the heat kernel in (3.7), to obtain R (ζ; (x, u), (y, v)) = (2π)− m 2 (4π)−nτ1− m 2 ∫ ∞ 0 Jm 2 −1(τr)Jρ,ζ(r)rn+ m 2 dr, (4.6) where Jρ,ζ(r) = ∫ ∞ 0 e−ζte− rρ2 4 coth(rt) sinh−n(rt) dt. The change of variables s = rt yields Jρ,ζ(r) = 1 r ∫ ∞ 0 e− ζ r se− rρ2 4 coth(s) sinh−n(s) ds. Next, using the integral representation∫ +∞ 0 e−2µse−2β coth(s) ( sinh(s) )2ν ds = 1 4 β 1 2 (ν−1)Γ(µ− ν) [ W−µ+ 1 2 ,ν (4β)− (µ− ν)W−µ− 1 2 ,ν (4β) ] , where 0 and 0, where Q = 2(n+m) is the homogeneous dimension of G and where ρ is the norm function on G given by ρ(x, u) = ( |x|4 + 16|u|2 )1/4 . In other words 〈Lϕ,Φe〉 = ϕ(e), for any function ϕ ∈ C∞0 (G). We prove that the Kaplan’s fundamental solution for the sub-Laplacian L on G can also be derived form the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental solution. Proposition 5.1. Kaplan’s fundamental solution in (5.1) can also be expressed as Φe(x, u) = 2 n−2 2 (2π)− 2n+m+1 2 Γ ( n 2 ) |x|n−1|u|m−2 2 ∫ ∞ 0 Jm 2 −1(|u|r)Kn−1 2 ( r|x|2/4 ) r n+m−1 2 dr. (5.2) Proof. To prove (5.2), we recall first that the resolvent kernel of L has the form R (ζ; (x, u), (y, v)) = 2 n−2 2 (2π)−n− m 2 ρnτ m−2 2 ∫ ∞ 0 Γ ( ζ 2r + n 2 ) Jm 2 −1(τr)W− ζ 2r , n−1 2 ( rρ2/2 ) r n+m−2 2 dr, (5.3) In the limit as ζ → 0 in (5.3), we obtain the Green kernel R0 := R (0; (x, u), (y, v)) of L as pointed out in Remark 4.2. Now, to establish a connection between the EJDE-2022/39 GREEN KERNELS ON H-TYPE GROUPS 7 integral kernel R0 and Kaplan’s fundamental solution, we proceed by computing the integral R0 = 2 n−2 2 (2π)− 2n+m+1 2 Γ ( n 2 ) ρn−1τ m−2 2 ∫ ∞ 0 Jm 2 −1(τr)Kn−1 2 ( rρ2/4 ) r n+m−1 2 dr. (5.4) We use the identity [9, p.684]∫ ∞ 0 r−λKµ(ar)Jν(br) dr = bνΓ ( ν+µ−λ+1 2 ) Γ ( ν−µ−λ+1 2 ) 2λ+1aν−λ+1Γ(1 + ν) 2F1 (ν + µ− λ+ 1 2 , ν − µ− λ+ 1 2 ; ν + 1;− b 2 a2 ) , when <(a± ib) > 0, <(ν − λ+ 1) > |<µ| are fulfilled, and where 2F1(a, b; c; z) = Γ(c) Γ(a)Γ(b) ∞∑ k=0 Γ(a+ k)Γ(b+ k) Γ(c+ k) zk k! , denotes the hypergeometric function [13, p.37]. In our case λ = −n+m−12 , µ = n−1 2 , ν = m 2 − 1, a = ρ2/4, and b = τ , therefore∫ ∞ 0 Jm 2 −1(τr)Kn−1 2 ( rρ2/4 ) r n+m−1 2 dr = 2 3n+5m−5 2 Γ ( n+m−1 2 ) ρn+2m−1τ1− m 2 2F1 (n+m− 1 2 , m 2 ; m 2 ;− (4τ)2 ρ4 ) . Returning to (5.4), we obtain that R0 = 2 4n+5m−7 2 Γ ( n+m−1 2 ) Γ ( n 2 ) (2π) 2n+m+1 2 ρ2(n+m+1) 2F1 (n+m− 1 2 , m 2 ; m 2 ;− (4τ)2 ρ4 ) . The hypergeometric function 2F1 ( n+m−1 2 , m2 ; m2 ;− (4τ)2 ρ4 ) is an elementary function given by Γ (n+m− 1 2 )( 1 + (4τ)2 ρ4 )−n+m−1 2 . It follows that R0 = 2 4n+5m−7 2 Γ ( n 2 ) Γ ( n+m−1 2 ) (2π) 2n+m+1 2 ρ2(n+m+1) ( 1 + (4τ)2 ρ4 )−n+m−1 2 = 2 4n+5m−7 2 Γ ( n+m−1 2 ) Γ ( n 2 ) (2π) 2n+m+1 2 1 (ρ4 + 16τ2) n+m−1 2 . (5.5) In particular, for (y, v) = (0, 0), keeping in mind the expression of ρ and τ given in (3.6), Equation (5.5) reduces to R0 = 2 4n+5m−7 2 Γ ( n+m−1 2 ) Γ ( n 2 ) (2π) 2n+m+1 2 1 (|x|4 + 16|u|2) n+m−1 2 = 2 3Q−6 2 Γ ( n 2 ) Γ ( Q−2 4 ) (4π) Q+1 2 ρ2−Q(x, u), (5.6) 8 Z. MOUHCINE EJDE-2022/39 where Q = 2(n+m) is the homogeneous dimension of G and where ρ is the norm function on G given by ρ(x, u) = ( |x|4 + 16|u|2 )1/4 . By combining (5.1) and (5.6), we obtain R0 = 2 3Q−6 2 Γ ( n 2 ) Γ ( Q−2 4 ) (4π) Q+1 2 c−1Q Φe(x, u), where the constant cQ is as in (5.1) and then can be computed explicitly and it is given by cQ = 2 3Q−6 2 Γ ( n 2 ) Γ ( Q−2 4 ) (4π) Q+1 2 . The asserted formula is established. � Remark 5.2. The constant cQ that appears in Kaplan’s fundamental solution is given by [10] c−1Q = ∫ G |x|2 ( 1 + ρ(x, u)4 )−(Q+6)/4 dx du, which can be also computed explicitly using polar coordinates on H-type groups in [10]. 6. Green kernel for fractional powers of L For 0 < α < 1 one defines the (fractional) power Lα by the usual functional calculus. It is still an unbounded self-adjoint operator. As application of the formula obtained for the resolvent kernel of L, we give the Green kernel of the fractional power operator Lα for α ∈]0, 1[. More precisely, we have the following result. Theorem 6.1. Let α ∈]0, 1[. Then the Green kernel of the fractional power oper- ator Lα is Gα((x, u), (y, v)) = 1 2α(2π) 2n+m 2 τ m−2 2 ∫ ∞ 0 erρ 2/4Wα(r)Jm 2 −1(τr)r 2n+m−2α 2 dr, (6.1) where Wα(r) = ∞∑ k=0 ( k + n 2 )−α L (n+1) k (rρ2/2). Proof. Since L is a self-adjoint operator, its resolvent [12, p.21] satisfies ‖R(s)‖ ≤ 1 s . This estimate enables us to define the fractional powers Lα, α ∈]0, 1[ according to the formula [12, p.127] Lαg = sinπα π ∫ ∞ 0 sα−1R(s)Lg ds, g ∈ D(L). (6.2) EJDE-2022/39 GREEN KERNELS ON H-TYPE GROUPS 9 Thanks to Kato’s formula [12, p.124], the resolvent operator Rα(γ) = (γ − Lα)−1, απ < | arg γ| < π, is given by Rα(γ) = sinπα π ∫ ∞ 0 λαR(λ) λ2α − 2λαγ cosπα+ γ2 dλ. (6.3) The action of Rα(γ) on a function f ∈ L2(G) is Rα(γ)f(x, u) = sinπα π ∫ ∞ 0 λαR(λ)f(x, u) λ2α − 2λαγ cosπα+ γ2 dλ, almost every where. Then the resolvent kernel of Lα is Gα(γ; (x, u), (y, v)) = sinπα π ∫ ∞ 0 λαR(λ; (x, u), (y, v)) λ2α − 2λαγ cosπα+ γ2 dλ. (6.4) The limit value γ = 0 in (6.4) gives a Green kernel of Lα: Gα((x, u), (y, v)) := Gα(0; (x, u), (y, v)) = sinπα π ∫ ∞ 0 λ−αR(λ; (x, u), (y, v)) dλ. (6.5) Using expression in (4.4) and intertwining the integrals, we rewrite (6.5) as Gα((x, u), (y, v)) = 2n/2 sinπα (2π) 2n+m+2 2 ρnτ m−2 2 ∫ ∞ 0 Nα(r) Jm 2 −1(τr) r n+m−2 2 dr, (6.6) where Nα(r) = ∫ ∞ 0 λ−αΓ ( λ 2r + n 2 ) W− λ 2r , n−1 2 ( rρ2/2 ) dλ. (6.7) Next, using the integral representation [4, p.147], Γ(ν)W 1 2− p 2−ν,− p 2 (z) = z1/2−p/2e z 2 ∫ ∞ 0 e−ps(1− e−s)ν−1e−ze s ds; 0. In our case z = rρ2/2, ν = λ 2r + n 2 and p = 1− n, and therefore (6.7) reads Nα(r) = rn/2ρnerρ 2/4 2n/2 ∫ ∞ 0 e(n−1)s(1− e−s)(n−2)/2e−r|x| 2es/2Iα(s) ds, (6.8) where Iα(s) = ∫ ∞ 0 λ−α(1− e−s)λ/2rdλ = ∫ ∞ 0 λ−αe− 1 2r log( es es−1 )λdλ. (6.9) Hence, using [9, p.346],∫ ∞ 0 γν−1e−µγ dγ = Γ(ν) µν ; <µ > 0, <ν > 0, with µ = 1 2r log( es es−1 ) and ν = 1− α, we can write the right hand side in (6.9) as Iα(s) = 21−αr1−αΓ(1− α) log1−α( es es−1 ) . Then the integral in (6.8) reads Nα(r) = Γ(1− α)r n 2 +1−αρnerρ 2/4 2 n 2 +α−1 ∫ ∞ 0 e(n−1)s(1− e−s) n−2 2 e−rρ 2es/2 logα−1 ( es es − 1 ) ds. 10 Z. MOUHCINE EJDE-2022/39 Making the change of variable et = es es−1 , the above equality becomes Nα(r) = Γ(1− α)r n 2 +1−αρnerρ 2/4 2 n 2 +α−1 ∫ ∞ 0 e− n 2 ttα−1(1− e−t)−ne− rρ2e−t 2(1−e−t) dt. (6.10) By using the identity [18, p.101], (1− w)−β−1e− zw 1−w = ∞∑ k=0 L (β) k (z)wk; β, z ∈ C, |w| < 1, (6.11) for β = n+ 1, w = e−t and z = rρ2/2, the integral Nα(r) may therefore be written as Nα(r) = Γ(1− α)r n 2 +1−αρnerρ 2/4 2 n 2 +α−1 ∞∑ k=0 L (n+1) k (rρ2/2) ∫ ∞ 0 tα−1e−( n 2 +k)t dt. (6.12) Making the change variable δ = (n2 + k)t and using the integral representation of the Gamma function Γ(γ) = ∫∞ 0 sγ−1e−s ds, we arrive at Nα(r) = Γ(1− α)r n 2 +1−αρnerρ 2/4 2 n 2 +α−1 ∞∑ k=0 ( k + n 2 )−α L (n+1) k (rρ2/2) ∫ ∞ 0 δα−1e−δ dδ = Γ(α)Γ(1− α)r n 2 +1−αρnerρ 2/4 2 n 2 +α−1 ∞∑ k=0 ( k + n 2 )−α L (n+1) k (rρ2/2) = πr n 2 +1−αρnerρ 2/4 2 n 2 +α−1 sinπα ∞∑ k=0 ( k + n 2 )−α L (n+1) k (rρ2/2). (6.13) The last equality follows using Euler’s reflection formula [9, p.896] Γ(γ)Γ(1− γ) = π sin (πγ) . Substituting (6.13) into the expression of Gα((x, u), (y, v)) in (6.6), we obtain Gα((x, u), (y, v)) = 1 2α(2π) 2n+m 2 τ m−2 2 ∫ ∞ 0 erρ 2/4Wα(r)Jm 2 −1(τr) r 2n+m−2α 2 dr, where Wα(r) = ∞∑ k=0 ( k + n 2 )−α L (n+1) k (rρ2/2). Hence we obtain the formula for the Green function, as asserted. � Remark 6.2. When α approaches 1 in (6.1), we recover the expression of the Green function in Remark 4.2. We hope to return to the case α > 1 in a future work. Acknowledgments. 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Ben Abdellah, Fez, Morocco Email address: zakariyaemouhcine@gmail.com 1. Introduction 2. Notation and definitions 3. Heat kernel on H-type groups 4. Resolvent kernel on H-type groups 5. An integral representation for Kaplan's fundamental solution 6. Green kernel for fractional powers of L Acknowledgments References