The restriction theorem for the Grushin operators

The restriction theorem for the Grushin operators

arXiv:1402.5298v1 [math.FA] 21 Feb 2014

Heping Liu and Manli Song

∗

Abstract Abstract. We study the Grushin operators acting on Rdx1 × Rdt 2 and defined by the formula   d1 d1 d2 X X X L = − ∂x2j −  |xj |2  ∂t2k . j=1

j=1

k=1

We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Muller.

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 [4] on Rn . The result is stated 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 ) holds for all functions f ∈ Lp (Rn ).

A simple duality argument shows that Stein-Thomas theorem is equivalent to the following:

holds for all f ∈ S(Rn ), where radius r.

1 p

+

dr ||p′ ≤ Cr ||f ||p ||f ∗ dσ

1 p′

(1.1)

= 1 and dσr is the surface measure on the sphere with

Moreover, according to the Knapp example [4], the restriction theorem fails if

2n+2 n+3

< p ≤ 2.

From then on, the importance of the restriction theorem has become evident and various new restriction theorems has been proved. On the other hand, the restriction theorem can be generalized to many other spaces, such as Lie groups and compact manifolds (see [2][3][5][7][9]). 2010 Mathematics Subject Classification: 42C, 42C10, 43A90. Key words and phrases: Grushin operators, scaled Hermite operators, 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


The aim of this paper is to study the restriction theorem associated with the Grushin operators, that is, L = −∆x − |x|2 ∆t ,

where (x, t) ∈ Rd1 × Rd2 , while ∆x , ∆t are the corresponding partial Laplacians, and |x| is the Euclidean norm of x. It is obvious that L is self-adjoint. The operator is closely related to the scaled Hermite H(a) = −∆x + a2 |x|2 . Indeed, for a Schwartz function f on Rd1 × Rd2 , R operatorsiλ·t λ let f (x) = Rd2 f (x, t)e dt be the inverse Fourier transform of f in the t variable. Applying R 1 λ (x)e−iλ·t dλ, we see that f the operator L to the Fourier expansion f (x, t) = 2π R d2 Z 1 Lf (x, t) = H(|λ|)f λ (x)e−iλ·t dλ 2π Rd2 Z Z ∞ 1 H(a)f aǫ (x)e−iat·ǫ dσ(ǫ) da ad2 −1 = 2π 0 S d2 −1 Let us recall some results about the special scaled Hermite expansion. For k ∈ N, the Hermite function hk of order k is the function on R defined by √ 2 hk (τ ) = (2k k! π)−1/2 Hk (τ )e−τ /2 . Let ν be a multiindex and x ∈ Rd1 , we define the d1 -dimensional Hermite functions Φν by Φν (x) =

d1 Y

hk (xj ).

j=1

p The eigenfunctions of the scaled Hermite operator H(a) are given by Φaν = |a|1/4 Φν ( |a|x) and H(a)Φaν = (2k +d1 )|a|Φaν . Let Pk (a) stand for the projection of L2 (Rd1 ) onto the k-th eigenspace of H(a). More precisely X Pk (x)ϕ = (ϕ, Φaν )Φaν |ν|=k

Then the spectral decomposition of the opearator H(a) is explicitly known: H(a) =

∞ X (2k + d1 )|a|Pk (a), k=0

Hence the spectral decomposition of the Grushin operator L is given by 1 Lf (x, t) = 2π

Z

∞

0

Z ∞ X d2 (2k + d1 )a k=0

S d2 −1

!

Pk (a)f aǫ (x)e−iat·ǫ dσ(ǫ)

da

Thus the spectrum of L consists of the half line [0, ∞). The Grushin operator is a self-adjoint and positive operator. Furthermore, e−iat·ǫ Φaν (x) is an eigenfunction of L with the eigenvalue (2|ν| + d1 )|a|. Therefore, we have # Z Z ∞ "X ∞ iµt·ǫ µǫ µ µd2 1 − Pk f 2k+d1 (x)e 2k+d1 dσ(ǫ) dµ Lf (x, t) = 2π 0 (2k + d1 )d2 S d2 −1 2k + d1 k=0


=

Z

∞

µPµ f (x, t) dµ

0

where Pµ f (x, t) =

3

1 2π

∞ P

k=0

µd2 −1 (2k+d1 )d2

R

S d2 −1

Pk

µ 2k+d1

iµt·ǫ − 2k+d

µǫ

f 2k+d1 (x)e

1

dσ(ǫ) is an eigenfunction

of L with the eigenvalue µ. Pµ is called the restriction operator. Let Lδk be the Laguerre polynomial of type δ and degree k defined by 1 dk −τ k+δ τ −δ (e x )e x , ∀k ∈ N, δ > −1 k! dτ k We define the normalized Laguerre functions by 1 2 τ δ Γ(k + 1) δ e− 2 τ 2 Lδk (τ ) Lk (τ ) = Γ(k + δ + 1) Lδk (τ ) =

1

2

and set the Laguerre functions ϕk (z) = Ldk1 −1 ( 12 |z|2 )e− 4 |z| , ∀z ∈ Cd1 . Next we introduce the Weyl transform. Let H d1 be the (2d1 + 1)-dimensional Heisenberg group. For each a ∈ R\{0}, (z, s) ∈ H d1 , there is an infinite dimensional representation πa (z, s) which in the Schr¨ odinger realization acts on L2 (Rd1 ) in the following way. For each ϕ ∈ L2 (Rd1 ), z = x + iy, 1 πa (z, s)ϕ(ξ) = eias eia(x·ξ+ 2 x·y) ϕ(ξ + y). For any integrable function g on Cd1 , we define the Weyl transform of g by Z g(z)πa (z, 0) dz Wa (g) = C d1

For each ϕ, ψ ∈

L2 (Rd1 ),

|(Wa (g)ϕ, ψ)| = |

Z

C d1

g(z)(πa (z, 0)ϕ, ψ) dz| ≤ ||g||1 ||ϕ||2 ||ψ||2

This shows that Wa (g) is a bouded operator on L2 (Rd1 ) with ||Wa (g)|| ≤ ||g||1 . Furthermore, from the explicit description of π(z, s) we see that Z 1 eia(x·ξ+ 2 x·y) g(x, y)ϕ(ξ + y) dxdy Wa (g)ϕ(ξ) = C d1

From the above it follows that Wa (g) is an integral operator whose kernel Kga (x, y) is given by Z a g(ξ, y − x)ei 2 ξ·(x+y) dξ Kga (x, y) = R d1

Our main result is the following theorem:

Theorem 1.2 For 1 ≤ p ≤

2(d2 +1) d2 +3

and 1 ≤ q ≤ 2 ≤ r ≤ ∞, the following inequality holds 1

1

||Pµ f ||Lp′ Lr ≤ Cµ2d2 ( p − 2 )+ t

d1 1 ( − 1r )−1 2 q

x

for any Schwartz function f on Rd1 × Rd2 and µ > 0.

||f ||Lpt Lqx

The theorem is stated in terms of the mixed Lebesgue norm q ! q1 Z Z p p dt |f (x, t)| dx ||f ||Lpt Lqx = , 1 ≤ p, q ≤ ∞ R d1

R d2

(with the obvious modifications when p or q are equal to ∞).

4

2


Restriction theorem

To prove the above theorem, first we state some asymptotic properties of the normalized Laguerre functions Lδk (r) (see [10]). We let ν = 4k + 2δ + 2 and assume δ > −1. Lemma 2.1 The Laguerre functions satisfy  (τ ν)δ/2 ,    (τ ν)−1/4 , |Lδk (τ )| ≤ C  ν −1/4 (ν 1/3 + |ν − τ |)−1/4 ,   −γτ e ,

0 ≤ τ ≤ 1/ν 1/ν ≤ τ ≤ ν/2 ν/2 ≤ τ ≤ 3ν/2 τ ≥ 3ν/2

where γ > 0 is a fixed constant.

Using the above estimates for the Laguerre functions, we can get a lower bound for the L1 norm of them. Lemma 2.2

Z

0

d1 −1 − 12 Lk (τ ) τ dτ ≤ C

∞

1 −1 Moreover, there is an interesting result which connects the Laguerre function ϕdk,a (z) = √ d1 −1 ϕk ( az) with the spectral projection Pk (a) (see [11]).

Lemma 2.3 1 −1 Wa (ϕdk,a ) = (2π)d1 |a|−d1 Pk (a)

In order to prove the restriction theorem, we need the estimates of the projections ϕ → Pk (a)ϕ which are given in the following proposition. Proposition 2.1 For ϕ ∈ Lq (Rd1 ), 1 ≤ q ≤ 2, ||Pk (a)ϕ||2 ≤ C|a| Proof.

d1 1 ( − 21 ) 2 q

(2k + d1 )

d1 −1 1 ( q − 21 ) 2

||ϕ||q

As ||Pk (a)ϕ||2 ≤ ||ϕ||2 , it is enough to prove the above estimate when q = 1. Since ||Pk (a)ϕ||22 = (Pk (a)ϕ, Pk (a)ϕ) = (Pk (a)ϕ, ϕ) ≤ ||Pk (a)ϕ||q′ ||ϕ||q ,

it is enough to show that d1

||Pk (a)ϕ||∞ ≤ |a| 2 (2k + d1 )

d1 −1 2

||ϕ||1 .

(2.1)

1 −1 To prove (2.1) we use the fact that Pk (a) = (2π)−d1 |a|d1 Wa (ϕdk,a ). This shows that Pk (a) is an integral operator with the kernel Fk,a (x, y) given by Z a −d1 d1 1 −1 ei 2 ξ·(x+y) ϕdk,a Fk,a (x, y) = (2π) |a| (ξ, x − y) dξ R d1 Z |a| 2 2 2 2 −d1 d1 i a2 ξ·(x+y) d1 −1 |a| Lk (|ξ| + |x − y| ) e− 4 (|ξ| +|x−y| ) dξ = (2π) |a| e 2 R d1


5

Therefore, we have the estimate Z d1 −1 |a| − |a| (|ξ|2 +|x−y|2 ) 2 2 L e 4 |Fk,a (x, y)| ≤ (2π)−d1 |a|d1 (|ξ| + |x − y| ) dξ k 2 R d1 Z ∞ |a| 2 d −1 |a| 2 2 d1 2 1 ≤ C|a| (r + |x − y| ) e− 4 (r +|x−y| ) r d1 −1 dr Lk 2 Z0 ∞ d1 −1 1 2 − 1 r2 d −1 d1 2 ≤ C|a| r e 4 r 1 dr Lk 2 0 Z ∞ d1 d1 −1 d1 −1 − 21 ≤ C|a| 2 (2k + d1 ) 2 Lk (r) r dr 0

Using the estimate of Lemma 2.2, we get d1

|Fk,a (x, y)| ≤ C|a| 2 (2k + d1 )

d1 −1 2

.

This proves (2.1) and hence the proposition.

Now we give a proof of Theorem 1.2. Proof of Theorem 1.2. In order to simplify the notations, we write f as it were the product of two functions, that is f (x, t) = h(t)g(x), with f and g Schwartz functions. However, in the proof we will never use this fact. We take α : Rd1 → C and β : Rd2 → C, α ∈ S(Rd1 ),β ∈ S(Rd2 ). Because the spectral projections associated to the scaled Hermite operator are orthogonal, we have

hPµ f, α ⊗ βi = =

=

= =

Z

Z

Pµ f (x, t)α(x)β(t) dxdt "∞ Z Z Z X µd2 −1 µǫ 1 ˆ h 2π Rd1 Rd2 (2k + d1 )d2 S d2 −1 2k + d1 k=0 iµt·ǫ µ − 2k+d 1 dσ(ǫ) α(x)β(t) dxdt ×Pk g(x)e 2k + d1 Z Z ∞ 1 X µd2 −1 µǫ ˆ h 2π (2k + d1 )d2 Rd1 S d2 −1 2k + d1 k=0 Z µ − iµt·ǫ α(x)β(t)e 2k+d1 dt dxdσ(ǫ) × Pk g(x) 2k + d1 R d2 Z ∞ 1 X µd2 −1 µǫ µǫ µ ˆ ˆ β h dσ(ǫ)hPk g, αi 2π (2k + d1 )d2 S d2 −1 2k + d1 2k + d1 2k + d1 k=0 Z ∞ µǫ µǫ 1 X µd2 −1 ˆ ˆ β h dσ(ǫ) 2π (2k + d1 )d2 S d2 −1 2k + d1 2k + d1 k=0 µ µ g, Pk αi × hPk 2k + d1 2k + d1 R d1

R d2

6


Applying the H¨ older’s inequality to the inner integral we deduce that !1 2 Z ∞ 2 1 X µd2 −1 µǫ ˆ dσ(ǫ) h hPµ f, α ⊗ βi ≤ 2π (2k + d1 )d2 2k + d1 S d2 −1 k=0 ! 1 2 Z 2 µ µǫ µ ˆ × P β P g α dσ(ǫ) k k 2 2 2k + d1 2k + d1 2k + d1 S d2 −1 Lx Lx By Proposition 2.1, we have for any 1 ≤ q ≤ 2 ≤ r ≤ ∞ d1 1 µ ( −1) −1(1−1) Pk g ≤ Cµ 2 q 2 (2k + d1 ) 2 q 2 ||g||Lqx 2k + d1 L2 x d µ ≤ Cµ 21 ( r1′ − 21 ) (2k + d1 )− 12 ( r1′ − 21 ) ||α|| ′ Pk α 2 Lrx 2k + d1

(2.2) (2.3)

Lx

For 1 ≤ p ≤ that

2(d2 +1) d2 +3 ,

Z

Z

S

it follows from the restriction theorem of the Fourier transform on S d2 −1

!1 2 d2 (1− 1 ) 2 p 2k + d µǫ 1 ˆ dσ(ǫ) h ≤C ||h||Lpt d2 −1 2k + d1 µ

S d2 −1

βˆ

(2.4)

!1 1 2 2 2k + d1 d2 (1− p ) µǫ dσ(ǫ) ≤C ||β||Lpt 2k + d1 µ

(2.5)

Therefore, by (2.2), (2.3), (2.4) and (2.5) we have

∞ X d −2d ( 1 − 1 )− 1 ( 1 − 1 ) 2d ( 1 − 1 )+ 1 ( 1 − 1 )−1 hPµ f, α ⊗ βi ≤ C (2k + d1 ) 2 p 2 2 q r µ 2 p 2 2 q r ||f ||Lpt Lqx ||α ⊗ β||Lp Lr′ t

x

k=0

2 +1) If d2 ≥ 2 or 1 ≤ q < 2 ≤ r ≤ ∞ or 1 ≤ q ≤ 2 < r ≤ ∞, because of 1 ≤ p ≤ 2(d d2 +3 , we have 2d2 ( 1p − 21 ) + ( 1q − 21 ) > 1. Hence, the above sum converges and consequently we have

||Pµ f ||Lp′ Lr ≤ Cµ t

x

2d2 ( p1 − 21 )+

d1 1 ( − 1r )−1 2 q

||f ||Lpt Lqx

If d2 = 1 and r = q = 2, we have ∞ iµt iµt µ µ 1 X 1 µ µ − 2k+d − 2k+d 2k+d1 2k+d1 1 1 Pµ f (x, t) = (x)e + Pk (x)e . Pk f f 2π 2k + d1 2k + d1 2k + d1 k=0

Since the operators Pk (a) are orthogonal projections, we have ∞ µ µ µ 1 X 1 µ ||Pk f 2k+d1 ||L2x + ||Pk f 2k+d1 ||L2x ||Pµ f ||L∞ 2 ≤ t Lx 2π 2k + d1 2k + d1 2k + d1 k=0


7

∞ µ µ 1 X µ µ 2k+d1 2k+d1 ≤ f f ||Pk ||L2x + ||Pk ||L2x 2π 2k + d1 2k + d1 k=0 µ 1 2k+d − µ 1 || 2 + ||f 2k+d1 || 2 ≤ ||f Lx Lx 2π 1 ≤ ||f ||L1t L2x π

3

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 sublaplacian is sharp. The example is constructed similarly to the counterexample of M¨ uller [3], which shows that the estimates between Lebesgue spaces for the operators Pµ are necessarily trivial.

Let ϕ ∈ Cc∞ (Rd2 ) be a radial function, such that ϕ(a) = ψ(|a|), where ψ ∈ Cc∞ (R), ψ = 1 on a neighborhood of the point d11 and ψ = 0 near 0. Let h be a Schwartz function on Rd2 and define Z |x|2 −ihλ,ti − |λ| ˆ 2 e |λ|n dλ ϕ(λ)h(λ)e f (x, t) = R d2

R

Denote g(x, t) =

R d2

ϕ(λ)e−

|λ| |x|2 2

|ξ|2

e−ihλ,ti |λ|n dλ =

R

|ξ|2

− 2|λ| −i(hλ,ti+hξ,xi) e

ϕ(λ)e

Rd1 +d2

dξdλ.

\ Hence g(ξ, a) = ϕ(a)e 2|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 second variable. Then, f (x, t) =

Z

0

=

Z

0

=

Z

0

−

+∞

+∞

λ

d1 +d2 −1

− λ2 |x|2

ψ(λ)e

Z

S m−1

d1 −d1 −d2 µd1 +d2 −1 ψ(

−iλhw,ti ˆ h(λw)e dσ(w) dλ

2 µ − µ|x| )e 2d1 d1

Z

µ ˆ µw )e−i d1 hw,ti dσ(w) dµ h( d1

Z

ˆ µw )e−i d1 hw,ti dσ(w) h( d1

S d2 −1

+∞

Pµ f (x, t) dµ

where Pµ f (x, t) = d1

−d1 −d2 d1 +d2 −1

µ

µ − µ|x|2 ψ( )e 2d1 d1

S d2 −1

µ

and it satisfies L(Pµ f ) = µPµ f . Therefore, specially let µ = 1, we have |x|2

P1 f (x, t) = d1

−d1 −d2 − 2d1

e

Z

S d2 −1 |x|2 − 2d 1

= d1 −d1 −d2 e

ˆ w )e−i h( d1

[ h ∗ dσ 1 (t) d1

hw,ti d1

dσ(w)

8


From the restriction theorem associated the Grushin operators, we have the estimate ||P1 f ||Lp′ Lr ≤ t

C||f ||Lpt Lqx . Because of

[ ||P1 f ||Lp′ Lr = C||h ∗ dσ 1 || p′ L

(3.1)

||f ||Lpt Lqx ≤ ||h||Lpt ||g||L1t Lqx . ||h||Lpt

(3.2)

t

x

d1

t

and [ we have ||h ∗ dσ 1 || p′ ≤ C||h||Lp . t L d1

t

From the sharpness of Stein-Tomas theorem which is guaranteed by the Knapp counterexam2 +1) ple, it would imply p ≤ 2(d d2 +3 . Hence the range of p can not be extended. 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 wishes to express her thanks to Professor Detlef M¨ uller for his assistance and generous discussions on restriction theorems.

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x


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[11] S. Thangavelu, An introduction to the uncertainty principle: Hardy’s theorem on Lie groups, Progr. Math. 217(2003). [12] K. Jotsaroop, P. K. Sanjay and S. Thangavelu, Riesz transforms and multipliers for the Grushin opertor, J. Anal. Math. 119(2013), 255-273. 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]