Song Journal of Inequalities and Applications (2016) 2016:246 DOI 10.1186/s13660-016-1187-9

RESEARCH

Open Access

Decay estimates for fractional wave equations on H-type groups Manli Song* *

Correspondence: [email protected] School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, China

Abstract The aim of this paper is to establish the decay estimate for the fractional wave α equation semigroup on H-type groups given by eit , 0 < α < 1. Combining the dispersive estimate and a standard duality argument, we also derive the corresponding Strichartz inequalities. MSC: 22E25; 33C45; 35H20; 35B40 Keywords: fractional wave equation; decay estimate; H-type groups

1 Introduction In this paper, we study the decay estimate for a class of dispersive equations: i∂t u + α u = f ,

u() = u ,

()

where  is the sub-Laplacian on H-type groups G, α > . The partial differential equation in () is significantly interesting in mathematics. When α =  , it is reduced to the wave equation; when α = , it is reduced to the Schrödinger equation. The two equations are most important fundamental types of partial differential equations. In , Bahouri et al. [] derived the Strichartz inequalities for the wave equation on the Heisenberg group via a sharp dispersive estimate and a standard duality argument (see [] and []). The dispersive estimate  itα  e ϕ  ∞ ≤ C|t|–θ L

()

plays a crucial role, where ϕ is the kernel function on the Heisenberg group related to a Littlewood-Paley decomposition introduced in Section  and θ > . Such an estimate does not exist for the Schrödinger equation (see []). The sharp dispersive estimate is also generalized to H-type groups for the wave equation and the Schrödinger equation (see [– ]). Motivated by Guo et al. [] on the Euclidean space, we consider the fractional wave equation () on H-type groups and will prove a sharp dispersive estimate. © 2016 Song. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Theorem . Let N be the homogeneous dimension of the H-type group G, and p the dimension of its center. For  < α < , we have  itα  e u  ≤ Cα |t|–p/ u  N–p/ , ∞ B˙ ,

and the result is sharp in time. Here, the constant Cα >  does not depend on u , t, and B˙ ρq,r is the homogeneous Besov space associated to the sublaplacian  introduced in the next section. Following the work by Keel and Tao [] or by Ginibre and Velo [], we also get a useful estimate on the solution of the fractional wave equation. Corollary . If  < α <  and u is the solution of the fractional wave equation (), then for q ∈ [(N – p)/p, +∞) and r such that /q + N/r = N/ – , we have the estimate   uLq ((,T),Lr ) ≤ Cα u H˙  + f L ((,T),H˙  ) , where the constant Cα >  does not depend on u , f or T. Remark . In this article, we assume  < α < . For α = , the decay estimate has been proved (see []). For other cases, we could investigate the problem in a similar way to  < α ≤ .

2 Preliminaries 2.1 H-Type groups Let g be a two step nilpotent Lie algebra endowed with an inner product ·, ·. Its center is denoted by z. g is said to be of H-type if [z⊥ , z⊥ ] = z, and, for every s ∈ z, the map Js : z⊥ → z⊥ defined by   Js u, w := s, [u, w] ,

∀u, w ∈ z⊥ ,

is an orthogonal map whenever |s| = . An H-type group is a connected and simply connected Lie group G whose Lie algebra is of H-type. Given  = a ∈ z∗ , the dual of z, we can define a skew-symmetric mapping B(a) on z⊥ by     B(a)u, w = a [u, w] ,

∀u, w ∈ z⊥ .

We denote by za the element of z determined by     B(a)u, w = a [u, w] = Jza u, w.

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Since B(a) is skew symmetric and non-degenerate, the dimension of z⊥ is even, i.e. dim z⊥ = d. We can choose an orthonormal basis   E (a), E (a), . . . , Ed (a), E (a), E (a), . . . , Ed (a) of z⊥ such that B(a)Ei (a) = |za |J za Ei (a) = |a|Ei (a) |za |

and B(a)Ei (a) = –|a|Ei (a). We set p = dim z. We can choose an orthonormal basis { ,  , . . . , p } of z such that a( ) = |a|, a(j ) = , j = , , . . . , p. Then we can denote the element of g by p d (z, t) = (x, y, t) = (xi Ei + yi Ei ) + sj j . i=

j=

We identify G with its Lie algebra g by the exponential map. The group law on H-type group G has the form

   (z, s) z , s = z + z , s + s + z, z , 

()

where [z, z ]j = z, U j z  for a suitable skew-symmetric matrix U j , j = , , . . . , p. Theorem . G is an H-type group with underlying manifold Rd+p , with the group law () and the matrix U j , j = , , . . . , p, satisfies the following conditions: (i) U j is a d × d skew-symmetric and orthogonal matrix, j = , , . . . , p. (ii) U i U j + U j U i = , i, j = , , . . . , p with i = j. Proof See [].



Remark . It is well known that H-type algebras are closely related to Clifford modules (see []). H-type algebras can be classified by the standard theory of Clifford algebras. Specially, on H-type group G, there is a relation between the dimension of the center and its orthogonal complement space. That is p +  ≤ d (see []). Remark . We identify G with Rd × Rp and denote by n = d + p its topological dimension. Following Folland and Stein (see []), we will exploit the canonical homogeneous structure, given by the family of dilations {δr }r> ,   δr (z, s) = rz, r s . We then define the homogeneous dimension of G by N = d + p.

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The left invariant vector fields which agree, respectively, with given by

∂ , ∂ ∂xj ∂yj

at the origin are

 d  p ∂  ∂ k + zl Ul,j , Xj = ∂xj  ∂sk k= l=  d  p ∂  ∂ k + zl Ul,j+d , Yj = ∂yj  ∂sk k=

l=

where zl = xl , zl+d = yl , l = , , . . . , d. In terms of these vector fields we introduce the sublaplacian  by

=–

d 

 Xj + Yj .

j=

2.2 Spherical Fourier transform Korányi [], Damek, and Ricci [] have computed the spherical functions associated to the Gelfand pair (G, O(d)) (we identify O(d) with O(d) ⊗ Idp ). They involve, as on the Heisenberg group, the Laguerre functions ) (γ ) –τ / L(γ , m (τ ) = Lm (τ )e

τ ∈ R, m, γ ∈ N,

(γ )

where Lm is the Laguerre polynomial of type γ and degree m. We say a function f on G is radial if the value of f (z, s) depends only on |z| and s. We q denote, respectively, by Srad (G) and Lrad (G),  ≤ q ≤ ∞, the spaces of radial functions in S (G) and Lp (G). In particular, the set of Lrad (G) endowed with the convolution product  f ∗ f (g) =

    f gg – f g dg ,

g ∈ G,

G

is a commutative algebra. Let f ∈ Lrad (G). We define the spherical Fourier transform, m ∈ N, λ ∈ Rp ,  – 

|λ|  ˆf (λ, m) = m + d –  |z| dz ds. eiλs f (z, s)L(d–) m  m Rd+p ˆ ˆ By a direct computation, we have f  ∗ f = f · f . Thanks to a partial integration on the sphere Sp– , we deduce from the Plancherel theorem on the Heisenberg group its analog for the H-type groups. Proposition . For all f ∈ Srad (G) such that   m+d–    ˆf (λ, m)|λ|d dλ < ∞, m Rp m∈N

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we have

f (z, s) =

 π

d+p  p m∈N R

e–iλ·s fˆ (λ, m)L(d–) m



|λ|  |z| |λ|d dλ, 

()

the sum being convergent in L∞ norm. Moreover, if f ∈ Srad (G), the functions f is also in Srad (G) and its spherical Fourier transform is given by  (λ, m) = (m + d)|λ|fˆ (λ, m). f The sublaplacian  is a positive self-adjoint operator densely defined on L (G). So by the spectral theorem, for any bounded Borel function h on R, we have    (λ, m) = h (m + d)|λ| fˆ (λ, m). h()f

2.3 Homogeneous Besov spaces We shall recall the homogeneous Besov spaces given in []. Let R be a non-negative, even function in Cc∞ (R) such that supp R ⊆ {τ ∈ R :  ≤ |τ | ≤ } and   R –j τ = ,

∀τ = .

j∈Z

For j ∈ Z, we denote by ϕ and ϕj , respectively, the kernel of the operator R() and R(–j ). As R ∈ Cc∞ (R), Hulanicki [] proved that ϕ ∈ Srad (G) and obviously ϕj (z, s) = Nj ϕ(δj (z, s)). For any f ∈ S (G), we set j f = f ∗ ϕj . By the spectral theorem, for any f ∈ L (G), the following homogeneous Littlewood-Paley decomposition holds:

f=

j f

in L (G).

j∈Z

So f L∞ (G) ≤



j f L∞ (G) ,

f ∈ L (G),

()

j∈Z

where both sides of () are allowed to be infinite. Let  ≤ q, r ≤ ∞, ρ < N/q, we define the homogeneous Besov space B˙ ρq,r as the set of distributions f ∈ S (G) such that f 

ρ B˙ q,r

=





jρr

j f rq

r

< ∞,

j∈Z

 and f = j∈Z j f in S (G). ˙ ρ is Let ρ < N/q. The homogeneous Sobolev space H ˙ ρ = B˙ , , H

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which is equivalent to ˙ρ u∈H

⇔

ρ/ u ∈ L .

Analogous to Proposition  of [] on the Heisenberg group, we list some properties of ρ the spaces B˙ q,r in the following proposition. Proposition . Let q, r ∈ [, ∞] and ρ < N/q. (i) The space B˙ ρq,r is a Banach space with the norm  · B˙ ρq,r ; ρ (ii) the definition of B˙ q,r does not depend on the choice of the function R in the Littlewood-Paley decomposition; ρ –ρ < ρ < Nq the dual space of B˙ q,r is B˙ q ,r ; (iii) for – N q ρ ρ–σ (iv) for any u ∈ S (G) and σ > , then u ∈ B˙ q,r if and only if Lσ / u ∈ B˙ q,r ; (v) for any q , q ∈ [, ∞], the continuous inclusion holds: B˙ ρq,r ⊆ B˙ ρq,r ,

 ρ  ρ = – – , q N q N

ρ ≥ ρ ;

(vi) for all q ∈ [, ∞] we have the continuous inclusion B˙ q, ⊆ Lq ; (vii) B˙ , = L .

3 Technical lemmas α By the inversion Fourier formula (), we may write eit ϕ explicitly into a sum of a list of oscillatory integrals. In order to estimate the oscillatory integrals, we recall the stationary phase lemma. Lemma . (see []) Let g ∈ C ∞ ([a, b]) be real-valued such that 

 g (x) ≥ δ for any x ∈ [a, b] with δ > . Then for any function h ∈ C ∞ ([a, b]), there exists a constant C which does not depend on δ, a, b, g or h, such that    

b a

     eig(x) h(x) dx ≤ Cδ –/ h∞ + h  .

In order to prove the sharpness of the time decay in Theorem ., we describe the asymptotic expansion of oscillating integrals. Lemma . (see []) Suppose φ is a smooth function on Rp and has a non-degenerate critical point at λ¯ . If ψ is supported in a sufficiently small neighborhood of λ¯ , then    

Rp

  eitφ(λ) ψ(λ) dλ ∼ |t|–p/ ,

as t → ∞.

Besides, it will involve the Laguerre functions when we estimate the oscillatory integrals. We need the following estimates.

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Lemma . (see [])  

γ   (d–)  ≤ Cγ ,d (m + d)d–/  τ d L (τ ) m   dτ for all  ≤ γ ≤ d. Finally, we introduce the following properties of the Fourier transform of surface-carried measures. Theorem . (see []) Let S be a smooth hypersurface in Rp with non-vanishing Gaussian curvature and dμ a C∞ measure on S. Suppose that  ⊂ Rp \ {} is the cone consisting of all ξ which are normal of some point x ∈ S belonging to a fixed relatively compact neighborhood N of supp dμ. Then

∂ ∂ξ

ν

    ) = O  + |ξ | –M , dμ(ξ

 )= dμ(ξ



e–i(xj ,ξ ) aj (ξ ),

∀M ∈ N, if ξ ∈/ ,

if ξ ∈ ,

where the (finite) sum is taken over all points x ∈ N having ξ as a normal and  ν   ∂      ≤ Cν  + |ξ | –(p–)/–|ν| . a (ξ ) j  ∂ξ  Here, we need the following properties of the Fourier transform of the measure dσ on  is radial. By Theorem ., we have the radical decay propthe sphere Sp– . Obviously, dσ erties of the Fourier transform of the spherical measure. Lemma . For any ξ ∈ Rp , the estimate holds      (ξ ) = ei|ξ | φ+ |ξ | + e–i|ξ | φ– |ξ | , dσ where  (k)  φ± (r) ≤ ck ( + r)–(p–)/–k ,

for all r > , k ∈ N.

4 Dispersive estimates Lemma . Let  < α < . The kernel of ϕ of R() introduced in Section  satisfies the estimate  α  supeit ϕ(z, s) ≤ Cα |t|–/ |s|(–p)/ . z

Proof By the inversion Fourier formula () and polar coordinate changes, we have itα

e

ϕ(z, s) =

 π

d+p  m∈N

Rp

e–iλ·s+it(m+d)

α |λ|α



  |λ|  |z| |λ|d dλ × R (m + d)|λ| L(d–) m 

Song Journal of Inequalities and Applications (2016) 2016:246

=

 π

d+p 

Page 8 of 12



+∞

e–iλε·s+it(m+d)

p– m∈N S

α λα





  (d–) λ  d+p– |z| λ × R (m + d)λ Lm dλ dσ (ε). 

()

The expression after the Sp– integral sign in () is very similar to an integral computed in [] or [] (see the proof of Lemma .). Integrating the result over Sp– gives us  α    supeit ϕ(z, s) ≤ Cα min , |t|–/ ,

()

z

and Lemma . will come out only if we prove the case for p ≥  and |s| > . By switching the order of the integration in (), it follows from Lemma . that

α

eit ϕ(z, s) =

 π

d+p 

+∞

   (λs)eit(m+d)α λα R (m + d)λ dσ

m∈N 

λ  d+p– |z| λ dλ 

d+p  +∞  iλ|s|   –iλ|s|   it(m+d)α λα  e φ+ λ|s| + e φ– λ|s| e = π m∈N 

  (d–) λ  d+p– |z| λ × R (m + d)λ Lm dλ 

d+p +   – Im + Im . := π m∈N × L(d–) m

Then it suffices to study ± Im



+∞

=

i(±λ|s|+t(m+d)α λα )

e

φ±







   (d–) λ  d+p– λ|s| R (m + d)λ Lm dλ. |z| λ 

Performing the change of variables, μ = (m + d)λ, recall that R ∈ Cc∞ (R), ± Im =





±

eitgm,s,t (μ) hm,s,z (μ) dλ, /

where μ|s| + μα , (m + d)t



μ|s| μ|z| μd+p– (d–) hm,s,z (μ) = φ± R(μ)Lm . m + d (m + d) (m + d)d+p

± gm,s,t (μ) = ±

By Lemma . and Lemma ., we get   hm,s,z ∞ + h m,s,z  ≤ C(m + d)–(p+)/ |s|–(p–)/ . ± ± )

| ≥ α|α – |–α– , applying Lemma . on Im gives us Since |(gm,s,t

 ± I  ≤ Cα (m + d)–(p+)/ |t|–/ |s|–(p–)/ . m

()

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To conclude it suffices to sum these estimates since

(m + d)–(p+)/ < +∞.

m∈N

The decay estimate of time is sharp in the joint space-time cone   (s, t) ∈ Rp × R : s = Ct .



We will prove the sharp dispersive estimate. Lemma . Let  < α < . The kernel of ϕ of R() introduced in Section  satisfies the estimate   α supeit ϕ(z, s) ≤ Cα |t|–p/ . z,s

Proof From (), it suffices to show the inequality |t| > . Recall from () that

α

eit ϕ(z, s) =

 π

d+p  p– m∈N S

Im,ε dσ (ε),

where 

+∞

e–iλε·s+it(m+d)

Im,ε = 



α λα



  λ  d+p– λ R (m + d)λ L(d–) dλ |z| m 



eitGm,ε,s,t (μ) Hm,z (μ) dμ

= /

with μ ε · s, (m + d)t

μd+p– μ|z| (d–) . Hm,z (μ) = R(μ)Lm (m + d) (m + d)d+p

Gm,ε,s,t (μ) = μα –

We will try to apply Q times a non-critical phase estimate to the oscillatory integral Im,ε . Case : |s| ≥ α–α– (m + d)|t|. By (),    

Sp–

   + – ≤ Cα (m + d)–p–/ |t|–p/ . Im,ε dσ (ε) = Im + Im

Case : |s| ≤ α–α– (m + d)|t|. We get

Gm,ε,s,t (μ) = αμα– –

ε·s |s| ≥ α–α– – ≥ α–α– . (m + d)t (m + d)|t|

Here the phase function Gm,ε,s,t has no critical point on [/, ]. By Q-fold ( ≤ Q ≤ d) integration by parts, we have 



Im,ε = (it)–Q /

  eitGm,ε,s,t (μ) DQ Hm,z (μ) dμ,

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where the differential operator D is defined by

Hm,z (μ) d DHm,z = .

dμ Gm,ε,s,t (μ) By a direct induction,

DQ Hm,z =

Q

(Q+) βQ+

)β · · · (Gm,ε,s,t ) Hm,z (Gm,ε,s,t ,

k (Gm,ε,s,t ) (β )

C(β, k, Q)

k=Q β=k

where β = (β , . . . , βQ+ ) ∈ {, . . . , Q} × NQ and β = A direct calculation shows that

Q+ j=

jβj .

l– l–     (l) Gm,ε,s,t (μ) = α (j – α)μ–l+α ≤ l+α α (j – α) ≤ C(α, Q), j=

l ≥ .

j=

Using Lemma .,  (β )  H  (μ) ≤ C(β )(m + d)–p–/ . m,z Hence, we have  (β )    ≤ C(α, Q)|t|–Q (m + d)–p–/ . |Im,ε | ≤ C(α, Q)|t|–Q sup Hm,z ∞ ≤β ≤Q

Taking Q = d, since |t| >  and p ≤ d – , which implies p/ < d, it follows that |Im,ε | ≤ Cα |t|–p/ (m + d)–p–/ . It immediately leads to    

Sp–

  Im,ε dσ (ε) ≤ Cα |t|–p/ (m + d)–p–/ .

Combining the two cases, by a straightforward summation  itα  e ϕ(z, s) ≤ Cα |t|–p/ (m + d)–p–/ ≤ Cα |t|–p/ . m∈N

The lemma is proved.



Proof of Theorem . The dispersive inequality in Theorem . is a direct consequence of Lemma . (see []). It suffices to show the sharpness of the estimate. Let Q ∈ Cc∞ ([/, ]) with Q() = . Choose u such that ⎧ ⎨Q(|λ|), m = , uˆ  (λ, m) = ⎩, m ≥ .

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By the inversion Fourier formula (), then we have

α

eit u (z, s) =

 π

d+p  Rp

e–iλ·s+itd

α |λ|α

   Q |λ| e–|λ||z| / |λ|d dλ.

α

Consider eit u (, t¯s) for a fixed s¯ = αdα (, . . . , , ). The above oscillatory integral has a phase (λ) = –λ · s¯ + dα |λ|α with a unique non-degenerate critical point λ¯ = α – d–α s¯ = (, . . . , , ). Indeed, the Hessian is equal to ⎛  ⎜ ⎜   H(λ¯ ) = αdα |λ¯ |α– (α – )λ¯ k λ¯ l + |λ¯ | δk,l ≤k,l≤p = αdα ⎜ ⎜ ⎝

⎞ ..

⎟ ⎟ ⎟. ⎟ ⎠

.  α–

So by Lemma ., it yields α

eit u (, t¯s) ∼ C|t|–p/ .



5 Strichartz inequalities In this section, we shall prove the Strichartz inequalities by the decay estimate in Lemma .. We obtain the intermediate results as follows. We omit the proof and refer to [, ]. Theorem . Let  < α < . For i = , , let qi , ri ∈ [, ∞] and ρi ∈ R such that () /qi = p(/ – /ri ); () ρi = –(N – p/)(/ – /ri ), except for (qi , ri , p) = (, ∞, ). Let qi , ri denote the conjugate exponent of qi , ri for i = , . Then the following estimates are satisfied:  itα  e u  q ˙ ρ ≤ Cu L , L  (R,B ) r ,

  t    ei(t–τ )α f (τ ) dτ   

ρ 

Lq ((,T),B˙ r , )



≤ Cf 

–ρ q L  ((,T),B˙  )

,

r ,

where the constant C >  does not depend on u , f or T. Consider the non-homogeneous fractional wave equation (). The general solution is given by itα

u(t) = e



t

α

ei(t–τ ) f (τ ) dτ .

u – i 

Song Journal of Inequalities and Applications (2016) 2016:246

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Theorem . Under the same hypotheses as in Theorem ., the solution of the fractional wave equation () satisfies the following estimate:  uLq ((,T),B˙ ρ ) ≤ C u L + f  r ,

q

L

–ρ ((,T),B˙  )

 ,

r ,

where the constant C >  does not depend on u , f or T. Applying Proposition ., by direct Besov space injections, we immediately obtain the Strichartz inequalities on Lebesgue spaces in Corollary ..

Competing interests The author declares to have no competing interests. Acknowledgements The work is supported by the National Natural Science Foundation of China (Grant No. 11371036) and the Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY068). Received: 1 September 2016 Accepted: 26 September 2016 References 1. Bahouri, H, Gérard, P, Xu, C-J: Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg. J. Anal. Math. 82, 93-118 (2000) 2. Ginibre, J, Velo, G: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50-68 (1995) 3. Keel, M, Tao, T: Endpoints Strichartz estimates. Am. J. Math. 120, 955-980 (1998) 4. Del Hierro, M: Dispersive and Strichartz estimates on H-type groups. Stud. Math. 169, 1-20 (2005) 5. Furioli, G, Melzi, C, Veneruso, A: Strichartz inequalities for the wave equation with the full Laplacian on the Heisenberg group. Can. J. Math. 59(6), 1301-1322 (2007) 6. Furioli, G, Veneruso, A: Strichartz inequalities for the Schrödinger equation with the full Laplacian on the Heisenberg group. Stud. Math. 160, 157-178 (2004) 7. Liu, H, Song, M: Strichartz inequalities for the wave equation with the full Laplacian on H-type groups. Abstr. Appl. Anal. 2014, 3 (2014) 8. Song, N, Zhao, J: Strichartz estimates on the quaternion Heisenberg group. Bull. Sci. Math. 138(2), 293-315 (2014) 9. Guo, Z, Peng, L, Wang, B: Decay estimates for a class of wave equations. J. Funct. Anal. 254(6), 1642-1660 (2008) 10. Bonfiglioli, A, Uguzzoni, F: Nonlinear Liouville theorems for some critical problems on H-type groups. J. Funct. Anal. 207, 161-215 (2004) 11. Reimann, HM: H-Type groups and Clifford modules. Adv. Appl. Clifford Algebras 11, 277-287 (2001) 12. Kaplan, A, Ricci, F: Harmonic analysis on groups of Heisenberg type. In: Harmonic Analysis. Lecture Notes in Math., vol. 992, pp. 416-435 (1983) 13. Folland, GB, Stein, EM: Hardy Spaces on Homogeneous Groups. Math. Notes. Princeton University Press, Princeton (1992) 14. Korányi, A: Some applications of Gelfand pairs in classical analysis. In: Harmonic Analysis and Group Representations, pp. 333-348 (1982) 15. Damek, E, Ricci, F: Harmonic analysis on solvable extensions of H-type groups. J. Geom. Anal. 2, 213-248 (1992) 16. Hulanicki, A: A functional calculus for Rockland operators on nilpotent Lie groups. Stud. Math. 78, 253-266 (1984) 17. Stein, EM: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993) 18. Sogge, CD: Fourier Integrals in Classical Analysis. Cambridge University Press, Cambridge (1993)