Generalization of Titchmarsh’s Theorem for the Jacobi-Dunkl Transform

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Generalization of Titchmarsh’s Theorem for the Jacobi-Dunkl Transform

A. Belkhadir¹ and A. Abouelaz²

1,2Department of Mathematics and Informatics Faculty of Sciences A¨ın Chock

University of Hassan II, Casablanca, Morocco

1E-mail: [email protected]

2E-mail: [email protected] (Received: 23-4-15 / Accepted: 29-5-15)

Abstract

In this paper, using a generalized Jacobi-Dunkl translation operator, we prove a generalization of Titchmarsh’s theorem for functions in the k-Jacobi- Dunkl-Lipschitz class defined by the finite differences of order k ∈ N^∗ and Sobolev spaces associated with the Jacobi-Dunkl operator.

Keywords: Generalized Jacobi-Dunkl translation, Jacobi-Dunkl Lipschitz class, Jacobi-Dunkl transform, Titchmarsh’s theorem.

1 Introduction

Titchmarsh’s theorem characterizes the set of functions satisfying the Cauchy- Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transform, namely we have:

Theorem 1.1. [12] Let α ∈ (0,1) and f ∈ L²(R) . Then the following are equivalents:

1. kf(t+h)−f(t)k=O(h^α) , as h→0 ; 2.

Z

|λ|≥r

|fˆ(λ)|²dλ=O(r^−2α) , as r→+∞ .

where fˆis the Fourier transform of f.

A similar result of theorem 1.1 has been established for the Jacobi transform (see [8], theorem 2.2). Furthermore, a generalization of this result was proved in the Sobolev spaces associated with Jacobi transform (see [1], theorem 2.1 ).

In this paper, we prove a similar result for Jacobi-Dunkl transform, we con- sider functions in Sobolev spacesW_α,β^2,k (associated with Jacobi-Dunkl operator (see [5])) belonging to the k-Jacobi-Dunkl-Lipschitz class defined by the finite difference of order k∈N^∗. For this purpose we use the generalized translation and Jacobi-Dunkl operators.

The paper is organized as follows: in section 2 we recapitulate some results related to the harmonic analysis associated with the Jacobi-Dunkl operator Λα,β (see [2, 3, 4, 5, 7]). Section 3 is devoted to the main result (theorem 3.3). Before, we define the classLip(δ,2, α, β) of functions in W_α,β^2,k satisfying a certain condition correspondent to the generalized Jacobi-Dunkl translation.

Titchmarsh’s theorem for Jacobi-Dunkl transform is given as a corollary of theorem 3.3.

2 Notations and Preliminaries

In the following, α , β and ρ denote 3 reals such that α ≥ β ≥ −¹₂, α6=−¹₂ and ρ=α+β+ 1 .

Notations:

• A_α,β(x) = 2^ρ(sinh|x|)^2α+1(cosh|x|)^2β+1.

• dσ_α,β(λ) = |λ|

8πp

λ² −ρ²|C_α,β(p

λ²−ρ²)|IR\]−ρ,ρ[(λ)dλ where, C_α,β(µ) = 2^ρ−iµΓ(α+ 1)Γ(iµ)

Γ(¹₂(ρ+iµ))Γ(¹₂(α−β+ 1 +iµ)) , µ∈C\(iN) . and IΩ is the characteristic function of Ω.

• L^p(A_α,β) (resp. L^p(σ_α,β), p∈]0,+∞[ , the space of measurable functions g on R such that

||g||_L^p_(Aα,β) = Z

R

|g(t)|^pA_α,β(t)dt 1/p

<+∞.

(resp. ||g||_L^p_(σα,β) = Z

R

|g(λ)|^pdσα,β(λ) 1/p

<+∞).

• D(R) the space of C^∞-functions on R with compact support.

• S(R) the usual Schwartz space ofC^∞-functions onRrapidly decreasing together with their derivatives, equipped with the topology of semi-norms L_m,n , (m, n)∈N², where

L_m,n(f) = sup

x∈R,0≤k≤n

(1 +x²)^m

d^k dx^kf(x)

<+∞.

• S¹(R) = {(cosht)^−2ρf; f ∈ S(R)}.

The topology of this space is given by the semi-normsL¹_m,n , (m, n)∈N², where

L¹_m,n(f) = sup

x∈R,0≤k≤n

(cosht)^−2ρ(1 +x²)^m

d^k dx^kf(x)

<+∞.

• (S¹(R))⁰ the topological dual ofS¹(R) .

Now, we introduce the Jacobi-Dunkl Transform and its basic properties:

The Jacobi-Dunkl function with parameters (α, β) ,α ≥β≥ −¹₂, α6=−¹₂, is defined by :

∀x∈R, ψ^(α,β)_λ (x) = (

ϕ^(α,β)µ (x)− i λ

d

dxϕ^(α,β)µ (x) , if λ∈C\ {0};

1 , if λ= 0.

(1)

with λ² =µ²+ρ², ρ=α+β+ 1 and ϕ^(α,β)µ is the Jacobi function given by:

ϕ^(α,β)_µ (x) = F

ρ+iµ

2 ,ρ−iµ

2 ;α+ 1,−(sinhx)²

, (2)

whereF is the Gaussian hypergeometric function given by F(a, b, c, z) =

∞

X

m=0

(a)_m(b)_m

(c)_mm! z^m ,|z|<1, a, b, z ∈Cand c /∈ −N;

(a)₀ = 1, (a)_m =a(a+ 1)...(a+m−1) . (see [2, 9, 10]).

ψ_λ^(α,β) is the unique C^∞-solution onR of the differentiel-difference equation Λ_α,βu=iλu , λ∈C;

u(0) = 1. (3)

where Λ_α,β is the Jacobi-Dunkl operator given by:

Λ_α,βu(x) = du

dx(x) + A⁰_α,β(x)

A_α,β(x) ×u(x)−u(−x) 2 ; i.e.

Λ_α,βu(x) = du

dx(x) + [(2α+ 1) cothx+ (2β+ 1) tanhx]× u(x)−u(−x)

2 .

The function ψ_λ^(α,β) can be written in the form below (See [3]),

ψ_λ^(α,β)(x) = ϕ^(α,β)_µ (x) +i λ

4(α+ 1)sinh(2x)ϕ^(α+1,β+1)_µ (x) , ∀x∈R , (4) whereλ² =µ²+ρ² , ρ=α+β+ 1.

The Jacobi-Dunkl transform of a function f ∈L¹(A_α,β) is defined by : F_α,β(f)(λ) =

Z

R

f(x)ψ^(α,β)_−λ (x)A_α,β(x)dx, ∀λ∈R ; (5) The inverse Jacobi-Dunkl transform of a function h∈L¹(σ_α,β) is:

F_α,β⁻¹(h)(t) = Z

R

h(λ)ψ_λ^(α,β)(t)dσ_α,β(λ). (6) F_α,β is a topological isomorphism fromS¹(R) ontoS(R) , and extends uniquely to a unitary isomorphism fromL²(A_α,β) ontoL²(σ_α,β) . The Plancherel formula is given by

kfk_L2(A_α,β)=kF_α,β(f)k_L2(σα,β) . (7) Forf ∈ S¹(R) we have the following inversion formula

f(x) = Z

R

F_α,β(f)(λ)ψ_λ^(α,β)(x)dσ_α,β(λ), ∀x∈R, (8) and the relation

F_α,β(Λ_α,βf)(λ) =iλF_α,β(f)(λ). (9) Letf ∈L²(A_α,β) . For allx ∈R the operator of Jacobi-Dunkl translation τ_x is defined by:

τxf(y) = Z

R

f(z)dν_x,y^α,β(z) , ∀y ∈R . (10) whereν_x,y^α,β , x, y ∈R are the signed measures given by

dν_x,y^α,β(z) =







K_α,β(x, y, z)A_α,β(z)dz , if x, y ∈R^∗;

δ_x , if y= 0;

δ_y , if x= 0.

(11)

Here, δ_x is the Dirac measure at x. And

K_α,β(x, y, z) = M_α,β(sinh(|x|) sinh(|y|) sinh(|z|))^−2αI^Ix,y ×Rπ

0 ρ_θ(x, y, z)

×(g_θ(x, y, z))^α−β−1₊ sin^2βθdθ.

I_x,y = [−|x| − |y|,−||x| − |y||]∪[||x|+|y||,|x|+|y|], ρ_θ(x, y, z) = 1−σ_x,y,z^θ +σ_z,x,y^θ +σ^θ_z,y,x

σ_x,y,z^θ =







cosh(x) + cosh(y)−cosh(z) cos(θ)

sinh(x) sinh(y) , if xy6= 0;

0 , if xy= 0.

for all x, y, z ∈R, θ ∈[0, π].

g_θ(x, y, z) = 1−cosh²x−cosh²y−cosh²z+ 2 coshxcoshycoshzcosθ .

t₊ =

t , if t >0;

0 , if t≤0.

and

M_α,β =







2^−2ρΓ(α+ 1)

√πΓ(α−β)Γ(β+ ¹₂) , if α > β;

0 , if α=β.

We have

F_α,β(τ_hf)(λ) =ψ_λ^α,β(h).F_α,β(f)(λ) ; h, λ∈R . (12) Letg ∈L²(σα,β) . Then the distribution Tgσ_α,β defined by

hT_gσα,β, ϕi= Z

R

g(λ)ϕ(λ)dσ_α,β(λ), ϕ∈ D(R), (13) belongs toS⁰(R) .

Letf ∈L²(A_α,β) . Then the distribution T_{f Aα,β} defined by hTf A_α,β, ϕi=

Z

R

f(x)ϕ(x)Aα,β(x)dx , ϕ∈ S¹(R), (14) belongs to (S¹(R))⁰.

Via the correspondance f 7→T_{f Aα,β}, we identify L²(A_α,β) as a subspace of (S¹(R))⁰.

The jacobi-dunkl transform of a distributionT ∈(S¹(R))⁰ is defined by:

hF_α,β(T), ϕi=hT,F_α,β⁻¹( ˇϕ)i, ϕ∈ S(R), (15) where ˇϕ is given by ˇϕ(x) =ϕ(−x) .

It is clear thatF_α,β(T)∈ S⁰(R) .

The jacobi-dunkl transform of a distribution defined by f ∈ L²(Aα,β) is given by the distribution T_{Fα,β(f)σα,β}; i.e.

F_α,β(T_{f Aα,β}) =TF_α,β(f)σα,β. (16) We identify the tempered distribution given byF_α,β(f) and the functionF_α,β(f) . LetT ∈(S¹(R))⁰ and consider the distribution Λ_α,βT defined by

hΛα,β(T), ϕi=−hT,Λα,β(ϕ)i, for all ϕ∈ S¹(R). (17) (Note that S¹(R) is unvariant under Λ_α,β) .

By using (9) it is easy to see that

F_α,β(Λ_α,β(T)) =iλF_α,β(T). (18) For f ∈L²(A_α,β) , we define the finite differences of first and higher order as follows:

∆¹_hf = ∆_hf =τ_hf+τ−hf −2f = (τ_h+τ−h−2E)f;

∆^k_hf = ∆_h(∆^k−1_h )f = (τ_h+τ−h−2E)^kf , k = 2,3, ...;

whereE is the unit operator in L²(A_α,β) .

Lemma 2.1.The following inequalities are valids for Jacobi functionsϕ^α,β_µ (h) 1. |ϕ^(α,β)µ (h)| ≤1 ;

2. |1−ϕ^(α,β)µ (h)| ≤h²λ²; where λ² =µ²+ρ² . Proof. (See [11], Lemmas 3.1-3.2)

Forα ≥ ⁻¹₂ , we introduce the Bessel normalized function of the first kind defined by

j_α(z) = Γ(α+ 1)

∞

X

n=0

(−1)ⁿ(^z₂)²ⁿ

n!Γ(n+α+ 1) , z∈C. We see that lim

z→0

j_α(z)−1

z² 6= 0 , by consequence, there exists c1 >0 and η >0 satisfying

|z| ≤η⇒ |j_α(z)−1| ≥c₁|z|² . (19) Lemma 2.2. Let α ≥ β ≥ ⁻¹₂ , α 6= ⁻¹₂ . Then for |υ| ≤ ρ , there exists a positive constant c₂ such that

|1−ϕ^(α,β)_µ+iυ(t)| ≥c2|1−jα(µt)| . Proof. (See [6], Lemma 9)

3 Main Results

We denote byW_α,β^2,k, k∈N, the Sobolev space constructed by the operator Λ_α,β; i.e.

W_α,β^2,k =

f ∈L²(Aα,β); Λ^j_α,βf ∈L²(Aα,β), j = 0,1,2, ..., k ; (20) where, Λ⁰_α,βf =f, Λ¹_α,βf = Λ_α,βf , Λ^r_α,βf = Λ_α,β(Λ^r−1_α,βf), r= 2,3, ...

Definition 3.1. Let δ∈(0,1) and k ∈N. A function f ∈W_α,β^2,k is said to be in the k-Jacobi-Dunkl-Lipschitz class, denoted by Lip(δ,2, k, r) , if

∆^k+1_h Λ^r_α,βf

L²(Aα,β) =O(h^δ), as h −→0, where r = 0,1, ..., k.

Lemma 3.2. Let f ∈W_α,β^2,k, k ∈N. Then ∆^k+1_h Λ^r_α,βf

2

L²(Aα,β) = 2^2k+2 Z

R

λ^2r|1−ϕ_µ(h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ) , where r= 0,1, ..., k.

Proof. We have

F_α,β(τ_hf+τ_−hf −2f)(λ) = (ψ^(α,β)_λ (h) +ψ_λ^(α,β)(−h)−2).F_α,β(f)(λ).

Since ψ^(α,β)_λ (h) = ϕ^(α,β)µ (h) +i λ

4(α+ 1)sinh(2h)ϕ^(α+1,β+1)µ (h), ψ_λ^(α,β)(−h) =ϕ^(α,β)µ (−h)−i λ

4(α+ 1)sinh(2h)ϕ^(α+1,β+1)µ (−h), and ϕ^(α,β)µ is even [See (2)]; then:

F_α,β(τ_hf +τ−hf −2f)(λ) = 2(ϕ^(α,β)_µ (h)−1).F_α,β(f)(λ).

and

F_α,β(∆^k+1_h f)(λ) = 2^k+1(ϕ^(α,β)_µ (h)−1)^k+1.F_α,β(f)(λ). (21) From formula (18), we obtain

F_α,β(Λ^r_α,βf)(λ) = (iλ)^rF_α,β(f)(λ). (22) Using the formulas (21) and (22) we get

F_α,β(∆^k+1_h Λ^r_α,βf)(λ) = 2^k+1(iλ)^r.(ϕ^(α,β)_µ (h)−1)^k+1.F_α,β(f)(λ).

By the Plancherel formula (7), we have the result.

Theorem 3.3. Let f ∈W_α,β^2,k, k ∈N. Then the following are equivalents:

1. f ∈Lip(δ,2, k, r) ; 2.

Z ∞ s

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ) =O(s^−2δ) , as s→+∞ . Proof. (1) ⇒ (2): Assume that f ∈Lip(δ,2, k, r) ; then

∆^k+1_h Λ^r_α,βf

L²(A_α,β) =O(h^δ) as h −→0.

by lemma 3.2, we have Z

R

λ^2r|1−ϕµ(h)|^2k+2|Fα,β(f)(λ)|²dσα,β(λ) = 1

4^k+1||∆^k+1_h Λ^r_α,βf||²

= O(h^2δ) If|λ| ∈[_2h^η ,^η_h] then |µh| ≤η (recall that λ² =µ²+ρ²).

We get by (19):

|j_α(µh)−1| ≥c₁µ²h². From |λ| ≥ η

2h we have,

µ²h² ≥ η²

4 −ρ²h²;

then we can find an absolute constant c₃ = c₃(η, α, β) such that µ²h² ≥ c₃ (takeh <1) ; thus,

|j_α(µh)−1| ≥c₁c₃. this inequality and lemma 2.2 implys that:

|1−ϕ^(α,β)_µ (h)| ≥c₁c₂c₃ =C Hence,

1≤ 1

C^2k+2|1−ϕ^(α,β)_µ (h)|^2k+2. So,

Z

η 2h≤|λ|≤^η

h

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ) ≤ 1 C^2k+2

Z

η 2h≤|λ|≤^η

h

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2

×|F_α,β(f)(λ)|²dσ_α,β(λ)

≤ 1

C^2k+2 Z

R

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ)

= O(h^2δ).

Then we have, Z

s≤|λ|≤2s

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ) =O(s^−2δ) , as s→+∞.

Or equivalently Z

s≤|λ|≤2s

λ^2r|Fα,β(f)(λ)|²dσα,β(λ)≤K1s^−2δ , as s→+∞, whereK₁ is some absolute constant . It follows that,

Z

|λ|≥s

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ) =

∞

X

i=0

Z

2ⁱs≤|λ|≤2ⁱ⁺¹s

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ)

≤ K1

∞

X

i=0

2ⁱs−2δ

≤ Ks^−2δ. which proves that:

Z

|λ|≥s

λ^2r|Fα,β(f)(λ)|²dσα,β(λ) =O(s^−2δ) , as s→+∞.

(2)⇒(1) : Suppose now that Z

|λ|≥s

λ^2r|Fα,β(f)(λ)|²dσα,β(λ) =O(s^−2δ) , as s→+∞.

we have to show that:

Z

R

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ) =O(h^2δ) , as h→0.

Write:

Z

R

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ) = I₁+I₂, where:

I1 = Z

|λ|≤¹

h

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσα,β(λ) ; I₂ =

Z

|λ|>¹

h

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ).

Estimate I₁ and I₂. From (1) of lemma 2.1 we can write, I₂ ≤ 4^k+1

Z

|λ|>¹

h

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ), (s = 1 h)

= O(h^2δ).

Using the inequalities (1) and (2) of lemma 2.1 we get

I₁ = Z

|λ|≤¹_h

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ)

≤ 2^2k+1 Z

|λ|≤¹_h

λ^2r|1−ϕ^(α,β)_µ (h)|.|F_α,β(f)(λ)|²dσ_α,β(λ)

≤ 2^2k+1h² Z

|λ|≤¹

h

λ^2r.λ²|F_α,β(f)(λ)|²dσ_α,β(λ).

Consider the function ψ(s) =

Z ∞ s

λ^2r|F_α,β(f)(λ)|²dσ_α,β(λ).

An integration by parts gives:

2^2k+1h² Z _h¹

0

λ^2r.λ²|F_α,β(f)(λ)|²dσ_α,β(λ) = 2^2k+1h² Z _h¹

0

−s²ψ⁰(s) ds

= 2^2k+1h² − 1 h²ψ(1

h) + 2 Z _h¹

0

sψ(s)ds

!

≤ 2^2k+2h² Z _h¹

0

sψ(s)ds.

Sinceψ(s) =O(s^−2δ) , we get Z ¹_h

0

sψ(s)ds = O Z ¹_h

0

s^1−2δds

!

= O(h^2δ−2).

Hence,

2^2k+1h² Z _h¹

0

λ^2r.λ²|Fα,β(f)(λ)|²dσα,β(λ) ≤ 2^2k+2h²O(h^2δ−2).

= O(h^2δ)

Finally, Z

R

λ^2r|1−ϕ^(α,β)_µ (h)|^2k+2|F_α,β(f)(λ)|²dσ_α,β(λ) = I₁+I₂

= O(h^2δ) +O(h^2δ)

= O(h^2δ) Which completes the proof of the theorem.

Corollary 3.4. Let f ∈W_α,β^2,k such that f ∈Lip(δ,2, k, r). Then:

Z

|λ|≥s

|F_α,β(f)(λ)|²dσ_α,β(λ) = O s^−2δ−2r

, as s→+∞.

If we take k = 0 in theorem 3.3, we deduce an analog of Titchmarsh’s theorem (theorem 1.1) for the Jacobi-Dunkl transform:

Corollary 3.5. Let δ ∈ (0,1) and f ∈ L²(A_α,β) . Then the following are equivalents:

1. kτ_hf +τ−hf−2fk_L2(Aα,β) =O(h^δ) , as h→0. 2.

Z

|λ|≥s

|F_α,β(f)(λ)|²dσ_α,β(λ) =O(s^−2δ) , as s→+∞.

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