Uncertainty principles for the Cherednik transform

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Uncertainty principles for the Cherednik transform

R DAHER¹, S L HAMAD¹, T KAWAZOE²and N SHIMENO³

1Department of Mathematics, Faculty of Sciences of Ain Chock, University of Hassan II, Casablanca, Morocco

2Department of Mathematics, Keio University at Fujisawa, 5322 Endo, Kanagawa 252-8520, Japan

3School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda-City 6691337, Japan

E-mail: ra_daher@yahoo.fr MS received 9 March 2011

Abstract. We shall investigate two uncertainty principles for the Cherednik transform on the Euclidean spacea; Miyachi’s theorem and Beurling’s theorem. We give an analogue of Miyachi’ theorem for the Cherednik transform and under the assumption thatahas a hypergroup structure, an analogue of Beurling’s theorem for the Cherednik transform.

Keywords. Cherednik operators; Cherednik transform; Miyachi’s theorem; Beurling’s theorem.

1. Introduction

Uncertainty principle for the Fourier transform on R was first formulated as Heisen- berg’s inequality and Hardy’s theorem in 1930’s. Then various generalizations have been studied. Many of the variants of uncertainty principles follow from the following three theorems, which we call master theorems for uncertainty principles.

Theorem 1.1 (L^p−L^q Morgan’s theorem). Let a,b > 0, p,q ∈ [1,∞],α ≥ 2 and β > 0 with 1/α+1/β =1 and(aα)¹^/α(bβ)¹^/β > (sin(^π₂(β−1))¹^/β. If a measurable function f onRsatisfies e^a^|^x^|^αf ∈ L^p(R)and e^b^|^y^|^β fˆ ∈ L^q(R), then f = 0 almost everywhere.

Theorem 1.2 (Miyachi’s theorem). Let a,b > 0 and ab = 1/4. If a measu- rable function f on R satisfies the conditions f(x)e^ax² ∈ L^∞(R) + L¹(R) and _∞

−∞log⁺ | ˆf(λ)|e^b^λ²

C(1+ |λ|)^Ndλ <∞for some 0<C,N <∞, then f is a constant multiple of e⁻^ax².

Theorem 1.3 (Generalized Beurling’s theorem). Let f ∈ L²(R)and N ≥ 0. Then _∞

−∞

_∞

−∞

|f(x)|| ˆf(y)|

(1+ |x| + |y|)^Ne²^π|^{x y}^|dxdy<∞if and only if f may be written as f(x)= P(x)e⁻^ax², where a>0 and P is a polynomial of degree< (N−1)/2.

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The L^p−L^qMorgan’s theorem onRwas obtained in [9] for p=q= ∞and generalized onR^din [1] for general p,q ∈ [1,∞]. Miyachi’s theorem onRwas obtained in [8]

and generalized onR^din [5]. Beurling’s theorem, which is the case of N =0 in Theorem 1.3, was found by Beurling and a proof was given in [7]. A generalization of Beurling’s theorem for the Fourier transform onR^d was obtained in [2]. These theorems are also extended to generalized Fourier transformsFon non-Euclidean spaces such as Heisen- berg groups, semisimple Lie groups, Jacobi transforms, Dunkl transforms, etc. We refer to [1,5,12] and references therein. The common key to obtain extensions of uncertainty principles forFis a slice formula, that is, the generalized Fourier transformFis decomposed as a composition of the Euclidean Fourier transform and the so-called Radon transform.

By this formula these extensions resolve into the Euclidean cases. For example, see §3 in [5] for a generalization of Miyachi’s theorem.

In this paper we shall discuss analogous results for the Cherednik transform on the Euclidean spacea, which has no slice formula at present. However, we have a sharp estimate of the heat kernel obtained in [13]. Hence, by using the heat kernel to express the decay of functions, we can reconstruct uncertainty principles for the Cherednik transform. In the previous paper [3], the authors obtained L^p−L^q Morgan’s theorem for the Cherednik transform. Here, we shall obtain Miyachi’s theorem for the Cherednik transform (see §3). As for generalized Beurling’s theorem, a substantial amount of difficulty is anticipated. In §4we shall obtain a partial result on Beurling’s theorem for the Cherednik transform.

2. Harmonic analysis associated with the Cherednik operator

In this section, we review basic results on harmonic analysis associated with the Cherednik operator, which are due to Opdam [10,11].

Letabe a Euclidean space of dimension d equipped with an inner product(., .)and Rbe a root system ina. Let W denote the associated Weyl group, which is generated by orthogonal reflections s_α ∈ O(a)along hyperplanes kerαforα∈ R. Let k =(k_α)_α∈R be a non-negative multiplicity function, that is, k_α 0 and k_α =k_βifα∈Wβ. Choose a setR⁺of positive roots inRand puta₊= {x| ∀α∈R+, (α,x) >0}. We denote bya₊ its closure and put

ρ= 1 2

α∈R⁺

k_αα.

Letξ ∈ aC. The Cherednik operator T_ξ is the differential-difference operator ona defined by

T_ξf(x)=∂_ξf(x)+

α∈R⁺

k_α (α, ξ)

1−e^−(α,^x⁾{f(x)− f(r_αx)} −(ρ, ξ)f(x), where∂_ξis the directional derivative alongξ. For anyξ, η∈a_C,[T_ξ,T_η] =0. Then there exists a neighborhood U of 0 inaand a unique holomorphic function(λ,x)→G_λ(x)on a_C×(a+iU)such that

T_ξG_λ(x)= λ, ξG_λ(x) (∀ξ ∈a) G_λ(0)=1.

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Forλ ∈ a, G_λ is real and strictly positive. Moreover, the following estimate holds on a_C×a:

(a) |G_λ(x)| ≤G_(λ)(x),

(b) |Gλ(x)| ≤G0(x)e^max^w^((wλ),^x⁾,

(c) G0(x)∼

α∈R⁺0,(α,x)≥0

(1+(α,x))e^(−ρ,^x⁺⁾, (1)

whereR⁺₀ is the set of positive indivisible roots and x⁺is the unique conjugate of x ina₊. Letμdenote the measure onadefined by

dμ(x)=

α∈R⁺

2 sinh α,x 2

^2k^αdx. (2)

Let f be a nice function ona, say f belongs to the space C_c^∞(a). The Cherednik transform of f is defined by

Ff(λ)=

a

f(x)G₋iλ(−x)dμ(x).

When f is W -invariant, it can be rewritten as Ff(λ)=

a₊

f(x)F₋iλ(−x)dμ(x), where

F_λ(x)= 1

|W|

w∈W

G_λ(w·x),

which satisfies the same estimates (1)(b), (c) of G_λona_C×a₊.

Let C^∞_R (a), R >0, denote the space of C^∞functions onavanishing outside the ball BR = {x ∈a : ||x|| ≤ R}and let H_R^∞(a)denote the space of holomorphic functions h ona_Csuch that, for every integer N >0,

λ∈asup_C(1+ λ)^Ne⁻^R^(λ)h(λ)<+∞.

The Paley–Wiener theorem and the Plancherel formula hold for the Cherednik transform Fona.

Theorem 2.1. F is an isomorphism of C^∞_R(a)onto H_R^∞(a)for every R > 0 and there exists a measure dνonasuch that

a

f(x)g(−x)dμ(x)=

a

Ff(λ)Fg(λ)dν(λ).

Let p_t^W(x,y), t > 0, denote the W -invariant heat kernel on a ×a and put ht(x) =

|W|⁻¹p^W_2t(0,x). Then htis the W -invariant function onasuch that Fht(λ)=e⁻^t^(λ²^+ρ²⁾.

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Moreover, htis strictly positive and satisfies for all t >0 and x∈a, ht(x)∼t^−γ⁻ⁿ²

α∈R⁺0

(1+ |(α,x)|)(1+t+ |(α,x)|)^k^α⁺^k²^α⁻¹

×e⁻^t^ρ²^−(ρ,^x⁺⁾⁻^x

2

4t , (3)

whereγ =

α∈R+k_α. 3. Miyachi’s theorem

We shall obtain an extension of Miyachi’s theorem for the Cherednik transform. We define the measure dkx onaby

dkx=_α∈R⁺|tanh(α,x)|^2k^α

α∈R⁺0

(1+ |(α,x)|)^k^α⁺^k^2α⁺¹dx. (4)

Here_α∈R+|tanh(α,x)|^2k^αcorresponds to the asymptotic behavior of dμ(x)around x= 0 (see (2)), similarly,

α∈R⁺0(1+ |(α,x)|)and

α∈R⁺0 (1+ |(α,x)|)^k^α⁺^k^2α correspond to the polynomial parts of G0(x)and ht(x)with fixed t respectively (see (1), (3)). We put

L^∞(a)+L¹(a,dkx)= {f1+ f2; f1∈ L^∞(a),f2∈ L¹(a,dkx)}.

Theorem 3.1. Let a,b >0 and ab =1/4. Suppose f is a measurable function ona₊ satisfying

(A) f(x)h⁻₁_/¹_4a(x)∈ L^∞(a)+L¹(a,dkx), (B)

a

log⁺|F(f)(λ)e^b^λ²|

C(1+ λ)^N dλ <∞for some 0<C,N <∞. Then f is a constant multiple of h1/4a.

Proof. The first condition (A) implies that f h⁻₁_/¹_4a = u +v, where u ∈ L^∞(a) and v∈L¹(a,dkx)and hence, f =h1/4au+h1/4av. For the first term, it follows from (1)(a) that for allλ=ξ+iη∈aC,

|F(h1/4au)(λ)| ≤ u_∞

a

h1/4a(x)G_η(x)μ(x)dx

=cFh1/4a(iη)=ce^b^η².

For the second term, first we suppose that max_w∈W(wη,x) = (η,x⁺) > (ρ,x⁺)for x∈a. Then it follows from (3) and (1)(b), (c) that

|F(h1/4av)(λ)| ≤c

a

|v(x)|e^−(ρ,^x⁺⁾⁻^a^x²

α∈R⁺0

(1+ |(α,x)|)^k^α⁺^k²^α

×(1+ |(α,x)|)e^(η−ρ,^x⁺⁾μ(x)dx

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≤c

a

|v(x)|

α∈R⁺0

(tanh|(α,x)|)^2k^α(1+ |(α,x)|)^k^α⁺^k^2α⁺¹

×e⁻^a^x⁺^−η/^2a²dx·e^η²^/^4a

≤cvL¹(R,dkx)e^b^η².

On the other hand, if(η,x⁺)≤(ρ,x⁺)for x∈a, since G_λ(x)is bounded and e⁻^a^x² ≤ ce^−(ρ,^x⁺⁾for x ∈a, it easily follows that

|F(h1/4av)(λ)| ≤cvL¹(R₊,dkx)≤ce^b^η².

Hence,F(f)(λ)is entire and it satisfies |F(f)(λ)| ≤ ce^b^η² for all λ ∈ a_C and (B).

Here we consider F(z)=F(f)(z)e^b^z²/C. Then it is easy to see that F(z)satisfies the assumption of the following lemma.

Lemma 3.2. Suppose F(z)is an entire function onC^dand there exist constants A,B>0 and N ≥0 such that

|F(z)| ≤Ae^B⁽^z⁾² and

Rⁿlog⁺ |F(x)|

(1+ x)^Ndx<∞.

Then F is a constant function.

Proof. When d=1, this lemma was obtained by [8]. Let us suppose that d >1 and put H(z1) = F(z1,z2, . . . ,zd). Then there exists a E ⊂ R^d⁻¹with positive measure such that for each(x2,x3, . . . ,xd)∈ E,

_∞

−∞log⁺ |H(x1)|

C(1+ |x1|)^Ndx1<∞.

Hence H(z1) is constant. Since F is entire, F is also constant. Therefore, F(f)(λ)e^b^λ²/C is constant and F(f)(λ) = ce⁻^b^λ². This implies that f(x) =

ch1/4a(x).

Remark 3.3. The process of the above proof is the same as in [4] for the Jacobi transform on[0,∞). Since the Jacobi functionφ^α,β_λ (x)satisfies the Harish-Chandra integral formula (see §3 of [6]), for example,φ_λ(x)= _Ke⁽ⁱ^λ−ρ)(^H⁽^a^x^k⁾dk in the case of symmetric spaces, it follows that|φ_λ^α,β(x)| ≤ce^(λ−ρ)^x if x >0 andλ−ρ ≥ 0 (see Lemma 11 of [6]).

Hence we can remove the term

α∈R⁺0(1+ |(α,x)|)from (4) (see §3 of [4]). However, in our setting, it is not known that F_λ(x)satisfies the Harish-Chandra integral formula.

Hence the estimate (1)(c) is used even if(λ−ρ,x⁺)≥0.

4. Beurling’s theorem

In this section we shall consider a version of Beurling’s theorem for the Cherednik transform under some assumptions. As in §1, several generalized Beurling’s theorems are known for non-Euclidean Fourier transformsF, and in each case, to prove the generalized

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Beurling’s theorem, one uses the fact thatF is compatible with a convolution structure such thatF(f∗g)=Ff·Fg. Hence, in what follows, we suppose a hypergroup structure ona₊, that is, there exists a positive function K(x,y,z)ona³₊for which

(A) F_λ(x)F_λ(y)=

a₊

F_λ(z)K(x,y,z)dμ(z).

As is well-known, the Jacobi functions on[0,∞)and the spherical functions on semisim- ple Lie groups satisfy this property. However, for the Dunkl kernels, K are not positive in the generic case. By using the kernel K(x,y,z), we can define a translation fy(x)by

fy(x)=

a₊

f(z)K(x,y,z)dμ(z)=

a₊

Ff(λ)F_λ(y)F_λ(x)dν(λ)

and a convolution f ∗g(x)by f ∗g(x)=

a₊

f(y)gx(y)dμ(y)=

a₊

Ff(λ)Fg(λ)F_λ(x)dν(λ) for suitable functions f,g ona₊. Especially it follows that

F(f ∗g)(λ)=Ff(λ)Fg(λ)

forλ∈a. Then the following lemma easily follows.

Lemma 4.1. For t>0 and z,y∈a, e⁻^t^y²

a₊

(ht)z(x)Fy(x)dμ(x)=ctFy(z).

Under the assumption (A) we shall prove the following.

Theorem 4.2. Suppose that f ∈ L²(a)is W -invariant and satisfies

a₊×a+

|f(x)||Ff(y)|Fy(x)dμ(x)dν(y) <∞. (5)

Then f =0.

Proof. We use the same arguments used in the proof of Proposition 2.2 of [2]. As in their first step, we see that f ∈ L¹(a₊,dμ)andFf ∈ L¹(a₊,dν). We put g = f ∗h1

2 and

show that g=0. SinceFf ∈ L¹(a₊,dν), it follows that

a₊

Fg(y)e¹²^y²dν(y) <∞.

and since f ∈ L¹(a₊,dμ), it follows thatFf_∞<∞and thus,

|Fg(y)| ≤ce⁻¹²^y². (6)

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Lemma 4.1 and the fact thatF(ht)(i y)F(ht)(y)=1 yield that

a₊×a+

|g(x)||Fg(y)|Fy(x)dμ(x)dν(y)

≤

a₊×a+

|f| ∗ht(x)|Ff(y)|F(ht)(y)Fy(x)dμ(x)dν(y)

=ct

a₊×a₊

F(|f|)(i y)F(ht)(i y)|Ff(y)|F(ht)(y)dν(y)

=ct

a₊×a₊

F(|f|)(i y)|Ff(y)|dμ(y)

=ct

a₊×a₊

|f(x)||Ff(y)|Fy(x)dμ(x)dν(y) <∞.

Moreover, it follows from (6) and Fy(x)≤ce^x^ythat for each c>2,

x≤R

a₊

|g(x)||Fg(y)|Fy(x)dμ(x)dν(y)

=

x≤R

|g(x)|

y≥c R

+

y<c R

|Fg(y)|Fy(x)dν(y)dμ(x)

≤

x≤R

|g(x)|

y≥c R

ce⁽¹^c⁻¹²⁾^y²dν(y) +

y<c R|Fg(y)|Fy(x)dν(y)

dμ(x)

≤cgL¹(a₊,dμ)+

x≤R

y<c R|g(x)||Fg(y)|Fy(x)dμ(x)dν(y)

<∞.

We note that (6) implies that g admits an holomorphic extension to a_C and|g(z)| ≤ ce¹²^x². Moreover, for all x∈a₊and eⁱ^θ of modulus 1,

|g(eⁱ^θx)| ≤

a₊

|Fg(y)||F₋i eⁱ^θx(y)|dν(y)

≤

a₊

|Fg(y)|Fsinθ·x(y)dν(y)

≤c

a₊

|Fg(y)|Fx(y)dν(y),

where, in the last step we use the asymptotic behavior of Fy(x), x,y ∈ a₊ obtained Remark 3.1 of [13]. Since g andFg are W -invariant, all integrals overa₊that appeared in the previous arguments can be replaced by the ones overa. Then, by defining a function G onaas

G(z)= _z₁

0 · · · _z_d

0

g(u)(g)(i u)dμ(u),

we can deduce that g=0 as in the proof of Proposition 2.2 of [2].

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Remark 4.3 In order to obtain a generalized Beurling’s theorem for the Cherednik transform as in Theorem 1.1 of [2], we need to estimate the following integral:

e⁻^t^y²

a₊

(ht)z(x)

(1+ x)^NFy(x)dμ(x).

At present we have no idea to estimate this integral.

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