A functional calculus and restriction theorem on H-type groups

arXiv:1402.4340v1 [math.FA] 18 Feb 2014

A functional calculus and restriction theorem on H-type groups Heping Liu and Manli Song

∗

Abstract Let L be the sublaplacian and T the partial Laplacian with respect to central variables on H-type groups. We investigate a class of invariant differential operators by the joint functional calculus of L and T . We establish Stein-Tomas type restriction theorems for these operators. In particular, the asymptotic behaviors of restriction estimates are given.

1

Introduction

The restriction theorem for the Fourier transform plays an important role in harmonic analysis as well as in the theory of partial differential equations. The initial work on restriction theorem was given by E. M. Stein [8] that the Fourier transform of an Lp -function on Rn has a welldefined restriction to the unit sphere S n−1 which is square integral on S n−1 . The result is listed as follows: Theorem 1.1 (Stein-Tomas)

Let 1 ≤ p ≤

2n+2 n+3 .

Then the estimate

||fˆ||L2 (S n−1 ) ≤ C||f ||Lp (Rn )

(1.1)

holds for all functions f ∈ Lp (Rn ). A simple duality argument shows that the estimate (1.1) is equivalent to the following estimate: dr || ′ ≤ Cr ||f ||p (1.2) ||f ∗ dσ p

for all Schwartz functions f on Rn , where sphere with radius r.

1 p

+

1 p′

= 1 and dσr is the surface measure on the

Moreover, according to the Knapp example [8], the above estimates (1.1) and (1.2) fail if < p ≤ 2.

2n+2 n+3

2010 Mathematics Subject Classification: 42B10, 43A65, 47A60. Key words and phrases: H-type group, scaled special Hermite expansion, joint functional calculus, restriction operator. The first author is supported by National Natural Science Foundation of China under Grant #11371036 and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant #2012000110059. The second author is supported by the China Scholarship Council under Grant #201206010098. ∗ Corresponding author.

1

2

Heping Liu and Manli Song

Later, many authors have worked on the topic and various new restriction theorems have been proved. The study of restriction theorems has recently obtained more and more attention. Recent progress on the restriction theorems can be found in [10]. To generalize the restriction theorem on Heisenberg group, D. M¨ uller [6] established the boundedness of the restriction opp erator with respect to the mixed L −norm and also give a counterexample to show that the estimate between Lebesgue spaces for the restriction operator is necessarily trivial, due to the fact that the center of the Heisenberg group is of dimension 1. On an H-type group, let T be the Laplacian on the centre and L the sublaplacian. It is R +∞ well known that L is positive and essentially self-adjoint. Let L = 0 λdE(λ) be the spectral decomposition of L. Then the restriction operator can be formally written Pλ f = δλ (L)f = lim χ(λ−ǫ,λ+ǫ)(L)f which is well defined for a Schwartz function f , where χ(λ−ǫ,λ+ǫ) is the

ǫ→+∞

characteristic function of the interval (λ − ǫ, λ + ǫ). Liu and Wang [5] investigated the restriction theorem for the sublaplacian L on H-type groups with the center whose dimension is more than 1. They give the following result: Theorem 1.2 Let G be an H-type group with the underlying manlifold R2n+m , where m > 1 is the dimension of the center. Suppose 1 ≤ p ≤ 2m+2 m+3 . Then the following estimate ||Pλ f ||p′ ≤ Cλ

2(n+m)( 1p − 21 )−1

||f ||p , λ > 0

holds for all Schwartz functions f on G. In a recent paper, V. Casarino and P. Ciatti [2] extend the work of M¨ uller, Liu and Wang on M´ etivier groups. They prove the following result. Theorem 1.3 Let G be a M´ etivier group, with Lie algebra g. Let z and v denote, respectively, the centre of g and its orthogonal complement. If dimz = d and dimv = 2n, and if 1 ≤ r ≤ 2d+2 d+3 , then for all p, q satisfying 1 ≤ p ≤ 2 ≤ q ≤ ∞ and for all Schwartz functions f , we have 2

1

1

||Pλ f ||Lr′ (z)Lq (v) ≤ Cλd( r −1)+n( p − q )−1 ||f ||Lr (z)Lp (v) , λ > 0 Although V. Casarino and P. Ciatti [2] investigate the joint functional calculus of L and T on the Heisenberg group, they only prove a restriction theorem for the sublaplacian L on M´ etivier groups. The invariant differential operators related to the joint functional calculus of L and T on H-type groups do not have the homogeneous properties in general. Thus the asymptotic behaviors of restriction estimates for these operators are also interested. In this article we will show restriction theorems for these operators on H-type groups. The range of p depends on the dimension of the center. In particular, the asymptotic behaviors of restriction estimates are given. The outline of the paper is as follows. In the second section, we provide the necessary background for the H-type group. In the next section, by introducing the joint functional calculus of L and T , the restriction operator can be computed explicitly. In the fourth section, we prove the restriction theorem on H-type groups. Finally, in the last section, we show that the range of p in the restriction theorem is sharp.

A functional calculus and restriction theorem on H-type groups

2

3

Preliminaries

Definition 2.1. (H-type Group) Let g be a two step nilpotent Lie algebra endowed with an inner product h·, ·i. Its center is denoted by z. g is said to be of H-type if [z⊥ , z⊥ ] = z and for every t ∈ z, the map Jt : z⊥ → z⊥ defined by hJt u, wi := ht, [u, w]i, ∀u, w ∈ z is an orthogonal map whenever |t| = 1. An H-type group is a connected and simply connected Lie group G whose Lie algebra is of H-type. For a given 0 6= a ∈ z∗ , the dual of z, we can define a skew-symmetric mapping B(a) on z⊥ by hB(a)u, wi = a([u, w]), ∀u, w ∈ z⊥ We denote by za be the element of z determined by hB(a)u, wi = a([u, w]) = hJza u, wi Since B(a) is skew symmetric and non-degenerate, the dimension of z⊥ is even, i.e. dimz⊥ = 2n. For a given 0 6= a ∈ z∗ , we can choose an orthonormal basis {E1 (a), E2 (a), · · · , En (a), E 1 (a), E 2 (a), · · · , E n (a)} of z⊥ such that B(a)Ei (a) = |za |J |zza | Ei (a) = |a|E i (a) a

and B(a)E i (a) = −|a|Ei (a) We set m = dimz. Throughout this paper we assume that m > 1. We can choose an orthonormal basis {ǫ1 , ǫ2 , · · · , ǫm } of z such that a(ǫ1 ) = |a|, a(ǫj ) = 0, j = 2, 3, · · · , m. Then we can denote the element of g by n m X X (xi Ei + yi E i ) + tj ǫj (z, t) = (x, y, t) = i=1

j=1

We identify G with its Lie algebra g by exponential map. The group law on H-type group G has the form 1 (z, t)(z ′ , t′ ) = (z + z ′ , t + t′ + [z, z ′ ]) (2.1) 2 where [z, z ′ ]j = hz, U j z ′ i for a suitable skew symmetric matrix U j , j = 1, 2, · · · , m. Theorem 2.1 G is an H-type group with underlying manifold R2n+m , with the group law (2.1) and the matrix U j , j = 1, 2, · · · , m satisfies the following conditions: (i) U j is a 2n × 2n skew symmetric and orthogonal matrix, j = 1, 2, · · · , m. (ii) U i U j + U j U i = 0, i, j = 1, 2, · · · , m with i 6= j. Proof. See [1].

4


Remark 2.1 In particular, hz, U 1 z ′ i =

n P

(x′j yj − yj′ xj ).

j=1

Remark 2.2 All the above expressions depend on a given 0 6= a ∈ z∗ , but we will suppress a from them for simplification. Remark 2.3 It is well know that H-type algebras are closely related to Clifford modules [7]. 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 m + 1 ≤ 2n (see [4]). The left invariant vector fields which agree respectively with

∂ ∂ ∂xj , ∂yj

at the origin are given

by m 2n ∂ 1X X k Xj = zl Ul,j + ∂xj 2

Yj =

∂ 1 + ∂yj 2

k=1 m X k=1

l=1 2n X

!

k zl Ul,j+n

l=1

∂ ∂tk !

∂ ∂tk

where zl = xl , zl+n = yl , l = 1, 2, · · · , n. The vector fields Tk = ∂t∂k , k = 1, 2, · · · , m correspond to the center of G. In terms of these vector fields we introduce the sublaplacian L and full Laplacian ∆ respectively L=−

n m X X 1 (Xj2 + Yj2 ) = −∆z + |z|2 T − hz, U k ∇z iTk 4 j=1

(2.2)

k=1

∆=L+T

(2.3) where ∆z =

m 2n X X ∂ ∂ t ∂ ∂2 ∂2 , ,··· , ). , ∇z = ( , T = − 2 2 ∂z1 ∂z2 ∂z2n ∂zj ∂tk j=1 k=1

3

Restriction operator

First we recall some results about the scaled special Hermite expansion. We refer the reader to [12] and [13] for details. Let λ > 0. The twisted Laplacian (or the scaled special Hermite expansion) Lλ is defined by Lλ = −∆z +

n X ∂ λ2 |z|2 ∂ − iλ − yj xj . 4 ∂yj ∂xj j=1

where we identify z = x + iy ∈ Cn with z = (x, y) ∈ R2n .


5

For f, g ∈ L1 (Cn ),we define the λ-twisted convolution by Z iλ f (z − w)g(w)e 2 Im z·w dw f ×λ g = Cn

1

2

Set Laguerre function ϕλk (z) = Lkn−1 ( 12 λ|z|2 )e− 4 λ|z| , k = 0, 1, 2, · · · ,where Lkn−1 is the Laguerre polynomial of type (n − 1) and degree k. For f ∈ S (Cn ),we have the scaled special Hermite expansion f (z) =

λ 2π

n X +∞

f ×λ ϕλk (z)

(3.1)

k=0

which is an orthogonal form. We also have n X +∞ λ 2 ||f || = ||f ×λ ϕλk ||2 2π

(3.2)

k=0

Moreover, f ×λ ϕλk ia an eigenfunction of Lλ with the eigenvalue (2k + n)λ and n( p1 − 21 )− 12

||f ×λ ϕλk ||2 ≤ (2k + n)

λ

n( p1 − 23 )

||f ||p , f or 1 ≤ p
1 is the dimension of the center. Let h(ξ, η) = ξ α + η β , α, β > 0. Then for 1 ≤ p ≤ 2m+2 m+3 , we have if α < 2β, 2 α (n+ 2β m)( 1p − 12 )−1 ||Pµh f ||p′ ≤ Cµ α ||f ||p , µ > 1 and

2

||Pµh f ||p′ ≤ Cµ α if α > 2β,

(n+m)( 1p − 21 )−1

1

2

||f ||p , 0 < µ ≤ 1

1

||Pµh f ||p′ ≤ Cµ α (n+m)( p − 2 )−1 ||f ||p , µ > 1 and

2

||Pµh f ||p′ ≤ Cµ α if α = 2β,

2

||Pµh f ||p′ ≤ Cµ α

α (n+ 2β m)( 1p − 21 )−1

(n+m)( p1 − 12 )−1

||f ||p , 0 < µ ≤ 1

||f ||p , 0 < µ < +∞

hold for all Schwartz functions f . First we prove the following abstract statement. Proposition 4.1 h((2k + n)λ, λ2 ) is a strictly monotonic differentiable positive function of λ on R+ , with the domain (A, B) where 0 ≤ A < B ≤ +∞. Then for 1 ≤ p ≤ 2m+2 m+3 , the estimate holds ||Pµh f ||p′ ≤ Cµ ||f ||p where Cµ ≤ C

∞ X 1 1 2(n+m)( 1p − 12 )−1 (2k + n)2n( p − 2 )−1 λk (µ)|λ′k (µ)| k=0

for all Schwartz functions f and all positive µ ∈ (A, B).

Because the scaled special Hermite expansion is orthogonal, we have D E Z Pµh f, g = Pµh f (z, t)g(z, t) dzdt Proof.

G

=

Z G

∞

X 1 λkn+m−1 (µ)|λ′k (µ)| (2π)n+m k=0

Z

λ (µ)˜ a

S m−1

f ∗ ek k

(z, t) dσ(˜ a) g(z.t) dzdt

(4.1)

8

Heping Liu and Manli Song ∞

X 1 = λn+m−1 (µ)|λ′k (µ)| k n+m (2π) k=0

=

1 (2π)2n+m

∞ X

Z

S m−1

λ2n+m−1 (µ)|λ′k (µ)| k

k=0

Z

Z

λ (µ)

f λk (µ)ã ×λk (µ) ϕk k

(z)gλk (µ)ã (z) dzdσ(˜ a)

R2n

S m−1

Z

λ (µ)

f λk (µ)ã ×λk (µ) ϕk k

(z)

R2n

λ (µ)

a) × g λk (µ)ã ×λk (µ) ϕk k (z) dzdσ(˜ Z ∞ X 1 λ (µ) ||f λk (µ)ã ×λk (µ) ϕk k ||2 λ2n+m−1 (µ)|λ′k (µ)| ≤ k (2π)2n+m S m−1 k=0

λ (µ)

× ||g λk (µ)ã ×λk (µ) ϕk k

||2 dσ(˜ a)

Because of (3.4), we have λ (µ)

||f λk (µ)ã ×λk (µ) ϕk k

n( p1 − 21 )− 21

||2 ≤ C(2k + n)

n( p1 − 23 )

λk

(µ)||f λk (µ)ã ||p , 1 ≤ p
0 such that for A > 0 and n ∈ Z+ , we have X (2m + n)ν ≤ Cν Aν+1 , ν < −1;

(4.3)

m∈N 2m+n≥A

X

(2m + n)ν ≤ Cν Aν+1 ,

ν > −1.

(4.4)

m∈N 2m+n≤A

Now theorem 4.1 follows from the result in the following lemma. Lemma 4.2 Let h(ξ, η) = ξ α + η β , α, β > 0. The series in (4.1) has the estimate if α < 2β, 2 α (n+ 2β m)( 1p − 21 )−1 Cµ ≤ Cµ α ,µ>1 and

2

Cµ ≤ Cµ α if α > 2β,

(n+m)( p1 − 12 )−1

2

Cµ ≤ Cµ α and

2

,01

1

Cµ ≤ Cµ α (n+ 2β m)( p − 2 )−1 , 0 < µ ≤ 1

if α = 2β,

2

1

1

Cµ ≤ Cµ α (n+m)( p − 2 )−1 , 0 < µ < +∞ Proof.

h(ξ, η) = ξ α + η β , α, β > 0, thus we have µ = (2k + n)α λαk (µ) + λ2β k (µ), which yields λ′k (µ) =

1 α(2k +

(µ) + n)α λα−1 k

2βλk2β−1 (µ)

To study the convergence of this series, we need to distinguish three cases according to the relative of α and 2β:α < 2β,α > 2β and α = 2β. In order not to burden the exposition, we only prove the case α < 2β, and the other cases are analogous.

10


If α < 2β:

1

when µ ≤ 1, it is easy to see that λk (µ) ∼ Cµ ≤ C

µα 2k+n

and λ′k (µ) ∼

1

µ α −1 2k+n ,

so that the series

∞ 1 1 X 2n( 1 − 1 )−1 2(n+m)( p − 2 )−1 (2k + n) p 2 λk (µ)|λ′k (µ)| k=0

2(n+m)( − )−1 1 −1 1 ∞ X p 2 µα µα 2n( p1 − 12 )−1 (2k + n) ≤C 2k + n 2k + n 1

1

k=0 2

1

1

≤ Cµ α (n+m)( p − 2 )−1

∞ X

1

k=0 (2k

1

2

1

≤ Cµ α (n+m)( p − 2 )−1

(4.5)

2m( 1p − 21 )+1

+ n)

converges. When µ > 1, we split the sum into two parts, the sum over those k such that (2k +n)α λαk (µ) ≥ λ2β (µ) and those such that (2k +n)α λαk (µ) < λ2β (µ). They are denoted by I and II respectively. For the first part, (2k + n)α λαk (µ) ≥ λ2β (µ) implies 1

1

λk (µ) ∼

µ α −1 µα , λ′k (µ) ∼ , and 2k + n 2k + n

2k + n ≥ µ

2β−α 2αβ

Then we control the first part I by X

I≤C

2k+n≥µ

1

2n( 1p − 21 )−1

(2k + n)

2β−α 2k+n≥µ 2αβ 2

≤ Cµ α

(µ)|λ′k (µ)|

2β−α 2αβ

X

≤C

2(n+m)( 1p − 21 )−1

1

(2k + n)2n( p − 2 )−1 λk

(n+m)( 1p − 12 )−1

X

2k+n≥µ

1

µα 2k + n

2(n+m)( 1 − 1 )−1 p

2

1

µ α −1 2k + n

1

2β−α 2αβ

2m( p1 − 21 )+1

(2k + n)

Then by (4.3), we have 1

2

1

1

I ≤ Cµ α (n+m)( p − 2 )−1 (µ

2β−α 2αβ

2

2m( 1p − 12 )

1

α

1

≤ Cµ α (n+ 2β m)( p − 2 )−1

)

For the second part, (2k + n)α λαk (µ) < λ2β (µ) implies 1

1

λk (µ) ∼ µ 2β , λ′k (µ) ∼ µ 2β −1 , and 2k + n < µ

2β−α 2αβ

Then we control the second part II by II ≤ C

X

2k+n 2β, ||Pµh f ||p′ ≤ Cµ and

1

α

2m+2 m+3 ,

α 2 (n+ 2β m)( p1 − 21 )−1 −α

2

1

||f ||p , µ > 1

1

||Pµh f ||p′ ≤ Cµ− α (n+m)( p − 2 )−1 ||f ||p , 0 < µ ≤ 1

if α = 2β, ||Pµh f ||p′ ≤ Cµ hold for all Schwartz functions f .

2 −α (n+m)( 1p − 21 )−1

||f ||p , 0 < µ < +∞

we have

12


Theorem 4.3 Let h(ξ, η) = (1 + ξ)−1 . For 1 ≤ p ≤ 1

2m+2 m+3 ,

the estimates

1

||Pµh f ||p′ ≤ Cµ−2(n+m)( p − 2 )−1 ||f ||p , when µ → 0+ and

2(n+m)( 1p − 21 )−1

||Pµh f ||p′ ≤ C(1 − µ)

||f ||p , when µ → 1−

hold for all Schwartz functions f . More generally, we have Theorem 4.4 Let h(ξ, η) = (ξ α + η β )γ , α, β, γ > 0. Then for 1 ≤ p ≤ if α < 2β, 2

||Pµh f ||p′ ≤ Cµ αγ and

2

||Pµh f ||p′ ≤ Cµ αγ

α (n+ 2β m)( p1 − 21 )−1

(n+m)( p1 − 12 )−1

2m+2 m+3 ,

we have

||f ||p , µ > 1

||f ||p , 0 < µ ≤ 1

if α > 2β, 2

||Pµh f ||p′ ≤ Cµ αγ and

2

||Pµh f ||p′ ≤ Cµ αγ

(n+m)( 1p − 12 )−1

α (n+ 2β m)( p1 − 12 )−1

||f ||p , µ > 1

||f ||p , 0 < µ ≤ 1

if α = 2β, 1

2

1

||Pµh f ||p′ ≤ Cµ αγ (n+m)( p − 2 )−1 ||f ||p , 0 < µ < +∞ hold for all Schwartz functions f .

Theorem 4.5 Let h(ξ, η) = (ξ α + η β )−γ , α, β, γ > 0. Then for 1 ≤ p ≤ if α < 2β, 1

2

1

||Pµh f ||p′ ≤ Cµ− αγ (n+m)( p − 2 )−1 ||f ||p , µ > 1 and

2

α

1

1

||Pµh f ||p′ ≤ Cµ− αγ (n+ 2β m)( p − 2 )−1 ||f ||p , 0 < µ ≤ 1 if α > 2β, ||Pµh f ||p′ ≤ Cµ and ||Pµh f ||p′ ≤ Cµ

α 2 (n+ 2β m)( 1p − 12 )−1 − αγ

2 − αγ (n+m)( 1p − 21 )−1

||f ||p , µ > 1

||f ||p , 0 < µ ≤ 1

if α = 2β, ||Pµh f ||p′ ≤ Cµ hold for all Schwartz functions f .

2 − αγ (n+m)( 1p − 21 )−1

||f ||p , 0 < µ < +∞

2m+2 m+3 ,

we have


13

Theorem 4.6 Let h(ξ, η) = (1 + ξ α + η β )−γ , α, β, γ > 0. For 1 ≤ p ≤ 2m+2 m+3 , the estimates if α ≤ 2β, α − 2 (n+ 2β m)( 1p − 12 )−1 ||Pµh f ||p′ ≤ Cµ αγ ||f ||p , when µ → 0+ and

1

2

1

1

||Pµh f ||p′ ≤ C(1 − µ γ ) α (n+m)( p − 2 )−1 ||f ||p , when µ → 1−

if α > 2β,

1

2

1

||Pµh f ||p′ ≤ Cµ− αγ (n+m)( p − 2 )−1 ||f ||p , when µ → 0+ and

1

2

||Pµh f ||p′ ≤ C(1 − µ γ ) α

α (n+ 2β m)( 1p − 21 )−1

||f ||p , when µ → 1−

hold for all Schwartz functions f .

5

Sharpness of the range p

In this section we only give an example to show that the range of p in the restriction theorem associated with the full Laplacian ∆ is sharp. The example is constructed similarly to the counterexample of M¨ uller [6], which shows that the estimates between Lebesgue spaces for the operators Pµ∆ are necessarily trivial. Let ϕ ∈ Cc∞ (Rm ) be a radial function, such that ϕ(a) = ψ(|a|), where ψ ∈ Cc∞ (R), ψ = 1 on a neighborhood of the point n and ψ = 0 near 0. Let h be a Schwartz function on Rm and define Z |z|2 −iha,ti − |a| ˆ 4 e |a|n da ϕ(a)h(a)e f (z, t) = Rm

Denote g(z, t) =

R

Rm

ϕ(a)e− − |ξ|

|a| |z|2 4

e−iha,ti |a|n da =

2

R

2

Rm+2n

− |ξ| |a|

ϕ(a)e

e−i(ha,ti+hξ,zi) dξda.

\ Hence g(ξ, a) = ϕ(a)e |a| , which shows that gˆ and consequently g are Schwartz functions. On the other hand, we have f = h ∗t g, where ” ∗t ” denotes the involution about the central |a| |a| 2 2 variable. By Lemma 3.1, we have ∆ e−iha,ti e− 4 |z| = (nλ + λ2 )e−iha,ti e− 4 |z| . Therefore, we write f by the integration with polar coordinates as Z Z +∞ −iλhw,ti n+m−1 − λ4 |z|2 ˆ h(λw)e dσ(w) dλ f (z, t) = λ ψ(λ)e 0

=

=

Z

Z

0

+∞

+∞ 0

S m−1

λ∆ (µ)n+m−1 λ′∆ (µ)ψ

λ∆ (µ) 2 λ∆ (µ) e− 4 |z|

Z

ˆ λ∆ (µ)w e h

S m−1

−iλ∆ (µ)hw,ti

dσ(w) dµ

Pµ∆ f (z, t) dµ

where λ∆ (µ) 2 Pµ∆ f (z, t) = λ∆ (µ)n+m−1 λ′∆ (µ)ψ λ∆ (µ) e− 4 |z|

Z

S m−1

ˆh λ∆ (µ)w e−iλ∆ (µ)hw,ti dσ(w)

14


λ∆ (µ) =

p

n2 + 4µ − n 2

Therefore, specially let µ = 2n2 , we have λ∆ (2n2 ) = n, λ′∆ (2n2 ) = ∆ P2n 2 f (z, t)

n|z|2 1 = nn+m−2 e− 4 3

Z

1 3n

and

−inhw,ti ˆ h(nw)e dσ(w)

S m−1

1 = nn−1 e 3

n|z|2 − 4

dn (t) h ∗ dσ

From the restriction theorem associated the full Laplacian on H-type groups, we have the ∆ f || ′ estimate ||P2n 2 Lp (G) ≤ C||f ||Lp (G) . Because of and

∆ d ||P2n 2 f ||Lp′ (G) = C||h ∗ dσn || p′ L (Rm )

(5.1)

||f ||Lp (G) ≤ ||h||Lp (Rm ) ||g||L1t Lpz . ||h||Lp (Rm ) ,

(5.2)

where the mixed Lebesgue norm is defined by ||g||

L1t Lpz

=

Z

R2 n

Z

|f (z, t)| dt Rm

p

1

p

dz

,

dn || p′ m ≤ C||h||Lp (Rm ) . we have ||h ∗ dσ L (R )

From the sharpness of Stein-Tomas theorem which is guaranteed by the Knapp counterexample, it would imply p ≤ 2m+2 m+3 . Hence the range of p can not be extended. With the same tricks we can also prove the range of p for the restriction theorem associated with the functional calculus is also sharp. Acknowledgements: The work is performed while the second author studies as a joint Ph.D. student in the mathematics department of Christian-Albrechts-Universit¨ at zu Kiel. She is deeply grateful to Professor Detlef M¨ uller for generous discussions and his continuous encouragement.

References [1] A. Bonfiglioli and F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type groups, J. Funct. Anal. 207(2004), 161-215. [2] V. Casarino and P. Ciatti, A restriction theorem for M´ etivier groups,Adv. Math. 245(2013), 52-77. [3] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. [4] A. Kaplan and F. Ricci, Harmonic analysis on groups of Heisenberg type, Harmonic analysis, Lecture Nothes in Math. 992, Springer, Berlin (1983), 416-435.


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[5] H. Liu and Y. Wang, A restriction theorem for the H-type groups, Proc. Amer. Math. Soc.139(2011), 2713-2720. [6] D. M¨ uller, A restriction theorem for the Heisenberg group, Ann. of. Math. 131(1990), 567587. [7] H. M. Reimann, H-type groups and Clifford modules, Adv. Appl. Clifford Algebras 11(2001), 277-287. [8] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, N.J.(1993). [9] R. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87(1989), 51-148. [10] T. Tao, Some recent progeress on the restriction conjecture, Fourier analysis and convexity, App. Numer. Harmon. Anal., 2004,217-243. [11] S. Thangavelu, Some restriction theorems for the Heisenberg group, Studia Math. 99(1991), 11-21. [12] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Math. Notes, Princeton Univ. Press, No.42(1993). [13] S. Thangavelu, An introduction to the uncertainty principle: Hardy’s theorem on Lie groups, Progr. Math. 217(2003). [14] Q. Yang and F. Zhu, The heat kernel on H-type groups, Proc. Amer. Math. Soc 136(2008), 1457-1464. Heping Liu School of Mathematical Sciences Peking University Beijing 100871 People’s Republic of China E-mail address: [email protected] Manli Song School of Mathematical Sciences Peking University Beijing 100871 People’s Republic of China E-mail address: [email protected]