Some new estimates for the Cherednik–Opdam transform in the space L2α,β(R)

DOI 10.1007/s11868-015-0138-4

Some new estimates for the Cherednik–Opdam transform in the space L²_α,β(R)

S. El Ouadih¹ · R. Daher¹

Received: 3 October 2015 / Accepted: 11 October 2015

© Springer Basel 2015

Abstract New estimates are proved for the Cherednik–Opdam transform in the space L²_α,β(R)on certain classes of functions characterized by the generalized continuity modulus.

Keywords Cherednik–Opdam operator·Cherednik–Opdam transform· Generalized translation

Mathematics Subject Classification 42B37

1 Introduction

Abilov et al. [1], proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator.

In this paper, we prove new estimates in certain classes of functions characterized by a generalized continuity modulus and connected with the Cherednik–Opdam transform associated to T^(α,β) in L²_α,β(R) analogs of the statements proved in [1]. For this purpose, we use a generalized translation operator.

In Sect.2, we give some definitions and preliminaries concerning the Cherednik–

Opdam transform. The new estimates are proved in Sect.3.

B S. El Ouadih

salahwadih@gmail.com R. Daher

rjdaher024@gmail.com

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

2 Preliminaries

In this section, we develop some results from harmonic analysis related to the differential-difference operatorT^(α,β). Further details can be found in [2,3]. In the following we fix parametersα,βsubject to the constraintsα≥β ≥ −¹₂andα > ⁻₂¹. Letρ = α+β +1 and λ ∈ C. The Opdam hypergeometric functions G^(α,β)_λ onRare eigenfunctionsT^(α,β)G^(α,β)_λ (x)=iλG^(α,β)_λ (x)of the differential-difference operator

T^(α,β)f(x)=f(x)+ [(2α+1)cothx+(2β+1)tanhx]f(x)− f(−x)

2 −ρf(−x)

that are normalized such thatG^(α,β)_λ (0)=1. In the notation of Cherednik one would writeT^(α,β)as

T(k1+k2)f(x)= f(x)+

2k1

1+e^−2x + 4k2

1−e^−4x

(f(x)− f(−x))

−(k1+2k2)f(x),

withα=k1+k2−¹₂andβ=k2−¹₂. Herek1is the multiplicity of a simply positive root andk2the (possibly vanishing) multiplicity of a multiple of this root. By [2] or [3], the eigenfunctionG^(α,β)_λ is given by

G^(α,β)_λ (x)=ϕ^α,β_λ (x)− 1 ρ−iλ

∂

∂xϕ_λ^α,β(x)=ϕ_λ^α,β(x)+ ρ

4(α+1)sinh(2x)ϕ_λ^α+1,β+1(x), whereϕ^α,β_λ (x)=2 F1(^ρ+₂^iλ;^ρ−₂^iλ;α+1; −sinh²x)is the classical Jacobi function.

Lemma 2.1 [4]The following inequalities are valids for Jacobi functionsϕ^α,β_λ (x) (i) |ϕ_λ^α,β(x)| ≤1.

(ii) 1−ϕ^α,β_λ (x)≤x²(λ²+ρ²).

DenoteL²_α,β(R), the space of measurable functions f onRsuch that

f2,α,β =

R|f(x)|²A_α,β(x)d x 1/2

<+∞,

where

A_α,β(x)=(sinh|x|)^2α+1(cosh|x|)^2β+1. The Cherednik–Opdam transform of f ∈Cc(R)is defined by

Hf(λ)=

R f(x)G^(α,β)_λ (−x)A_α,β(x)d x for all λ∈C.

The inverse transform is given as H⁻¹g(x)=

Rg(λ)G^(α,β)_λ (x) 1− ρ

iλ

dλ 8π|cα,β(λ)|² here

c_α,β(λ)= 2^ρ−iλ (α+1) (iλ)

1

2(ρ+iλ) ₁

2(α−β+1+iλ).

The corresponding Plancherel formula was established in [2], to the effect that

R|f(x)|²A_α,β(x)d x = _+∞

0

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ),

where fˇ(x):= f(−x)anddσ is the measure given by dσ(λ)= dλ

16π|c_α,β(λ)|².

According to [3] there exists a family of signed measures μ^(α,β)x,y such that the product formula

G^(α,β)_λ (x)G^(α,β)_λ (y)=

RG^(α,β)_λ (z)dμ^(α,β)x,y (z), holds for allx,y∈Randλ∈C, where

dμ^(α,β)_x,y (z)=

⎧⎨

⎩

K_α,β(x,y,z)A_α,β(z)d z if x y=0

dδx(z) if y=0

dδy(z) if x=0

and

K_α,β(x,y,z)=M_α,β|sinhx.sinhy.sinhz|^−2α _π

0

g(x,y,z, χ)^α−β−₊ ¹

×

1−σx^χ,y,z+σx^χ,z,y+σz^χ,y,x+ ρ

β+¹₂ cothx.cothy.cothz(sinχ)²

×(sinχ)^2βdχ

ifx,y,z ∈ R\{0}satisfy the triangular inequality||x| −y||<|z|<|x| + |y|, and K_α,β(x,y,z)=0 otherwise. Here

∀x,y,z∈R, χ ∈ [0,1], σx^χ,y,z =

⎧⎨

⎩

coshx+coshy−coshzcosχ

sinhxsinhy if x y=0

0 if x y=0

andg(x,y,z, χ)=1−cosh²x−cosh²y.cosh²z+2 coshx.coshy.coshz.cosχ. Lemma 2.2 [3]For all x,y∈R, we have

(i) K_α,β(x,y,z)=K_α,β(y,x,z).

(ii) K_α,β(x,y,z)=K_α,β(−x,z,y).

(iii) Kα,β(x,y,z)=Kα,β(−z,y,−x).

The product formula is used to obtain explicit estimates for the generalized translation operators

τ_x^(α,β)f(y)=

R f(z)dμ^(α,β)_x,y (z).

It is known from [3] that

Hτ_x^(α,β)f(λ)=G^(α,β)_λ (x)Hf(λ) (1)

H(T^(α,β)f)(λ)=iλHf(λ) (2)

for f ∈Cc(R).

Forα > ⁻₂¹, we introduce the Bessel normalized function of the first kind j_α defined by

j_α(x)= (α+1)^∞

n=0

(−1)ⁿ(^x₂)²ⁿ

n! (n+α+1), x∈R.

In the terms of j_α(x), we have (see [5])

√hx J_α(hx)=O(1),hx ≥0, (3)

where J_α(x)is Bessel function of the first kind, which is related to j_α(x)by the formula

j_α(x)=2^α (α+1)

x^α J_α(x). (4)

Lemma 2.3 [6]Letα≥β ≥ ⁻₂¹, α = ⁻₂¹. Then for|ν| ≤ρ, there exists a positive constant c0such that

|1−ϕ_λ+^α,β_iν(x)| ≥c0|1−j_α(λx)|.

The generalized modulus of continuity of a function f ∈ L²_α,β(R)is defined by ω(f, δ)2,α,β = sup

0<h≤δNhf2,α,β, δ >0,

whereNh:=τ_h^(α,β)+τ₋^(α,β)_h −2Iand I is the identity operator inL²_α,β(R).

Denote byW₂^r(T^(α,β)),r =0,1,2..., the class of functions f ∈L²_α,β(R)that have generalized derivatives satisfying(T^(α,β))^r f ∈L²_α,β(R).

i.e.,

W2^r(T^(α,β))=

f ∈L²_α,β(R):(T^(α,β))^rf ∈L²_α,β(R) ,

where(T^(α,β))⁰f = f,(T^(α,β))^rf =T^(α,β)((T^(α,β))^r−1f);r=1,2, ...

3 Main result

The goal of this work is to prove several new estimates for the integral

J_N²(f)= _+∞

N

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ),

in certain classes of functions inL²_α,β(R). Lemma 3.1 If f ∈Cc(R), then

Hˇτx^(α,β)f(λ)=G^(α,β)_λ (−x)Hfˇ(λ) (5)

Proof For f ∈Cc(R), we have Hτˇx^(α,β)f(λ)=

Rτx^(α,β)f(−y)G^(α,β)_λ (−y)A_α,β(y)d y

=

Rτx^(α,β)f(y)G^(α,β)_λ (y)A_α,β(y)d y

=

R

R f(z)Kα,β(x,y,z)Aα,β(z)d z

G^(α,β)_λ (y)Aα,β(y)d y

=

R f(z)

RG^(α,β)_λ (y)Kα,β(x,y,z)A_α,β(y)d y

A_α,β(z)d z.

SinceK_α,β(x,y,z)=K_α,β(−x,z,y), it follows from the product formula that

Hτˇ_x^(α,β)f(λ)=G^(α,β)_λ (−x)

R f(z)G^(α,β)_λ (z)A_α,β(z)d z

=G^(α,β)_λ (−x)

R f(−z)G^(α,β)_λ (−z)A_α,β(z)d z

=G^(α,β)_λ (−x)Hfˇ(λ).

Lemma 3.2 For f ∈W₂^r(T^(α,β)), then

Nh(T^(α,β))^r f(x)²_2,α,β=4 _+∞

0

λ^2r|ϕ_λ^α,β(h)−1|²

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ).

Proof From formulas (1) and (5), we have H(Nhf)(λ)=

G^(α,β)_λ (h)+G^(α,β)_λ (−h)−2

H(f)(λ),

and

H(Nˇhf)(λ)=

G^(α,β)_λ (−h)+G^(α,β)_λ (h)−2

H(fˇ)(λ).

Since

G^(α,β)_λ (h)=ϕ_λ^α,β(h)+ ρ

4(α+1)sinh(2h)ϕ_λ^α+1,β+1(h), andϕ_λ^α,βis even, then

H(Nhf)(λ)=2

ϕ^α,β_λ (h)−1

H(f)(λ)

and

H(Nˇhf)(λ)=2

ϕ^α,β_λ (h)−1

H(fˇ)(λ)

Now by formula (2) and Plancherel Theorem, we have the result.

Theorem 3.3 Given r and f ∈W₂^r(T^(α,β)). Then there exist a constant c>0such that, for all N >0,

JN(f)=O

N^−rω((T^(α,β))^r f,cN⁻¹)2,α,β

.

Proof Firstly, we have J_N²(f)≤

_+∞

N |j_α(λh)|dμ(λ)+ _+∞

N |1−j_α(λh)|dμ(λ), (6)

withdμ(λ)=

|Hf(λ)|²+ |Hfˇ(λ)|²

dσ(λ). The parameterh >0 will be chosen in an instant.

In view of formulas (3) and (4), there exist a constantc1>0 such that

|j_α(λh)| ≤c1(λh)^−α−12. Then

_+∞

N

|j_α(λh)|dμ(λ)≤c1(h N)^−α−12J_N²(f).

Choose a constantc2such that the numberc3=1−c1c^−α−

1 2

2 is positif.

Settingh =c2/Nin the inequality (6), we have c3J_N²(f)≤

_+∞

N

|1−j_α(λh)|dμ(λ). (7)

By Hölder inequality and Lemma2.3the second term in (7) satisfies _+∞

N

|1−j_α(λh)|dμ(λ)= _+∞

N

|1− j_α(λh)|.1.dμ(λ)

≤ _+∞

N

|1− j_α(λh)|²dμ(λ)

1/2 _+∞

N

dμ(λ) 1/2

≤ _+∞

N

λ^−2r|1− j_α(λh)|²λ^2rdμ(λ) 1/2

JN(f)

≤ N^−r _+∞

N

|1− j_α(λh)|²λ^2rdμ(λ) 1/2

JN(f)

≤ N^−r c0

_+∞

N

λ^2r|1−ϕ_λ^α,β(h)|²dμ(λ) 1/2

JN(f).

From Lemma3.2, we conclude that _+∞

N

λ^2r|1−ϕ^α,β_λ (h)|²dμ(λ)≤ Nh(T^(α,β))^rf(x)²2,α,β. Therefore

_+∞

N

|1− j_α(λh)|dμ(λ)≤ N^−r

c0 Nh(T^(α,β))^rf(x)2,α,βJN(f).

Forh =c2/N, we obtain c3J_N²(f)≤ N^−r

c0

ω

(T^(α,β))^r f,c2/N

2,α,β JN(f).

Consequently

c0c3JN(f)≤N^−rω

(T^(α,β))^r f,c2/N

2,α,β.

for allN >0. The theorem is proved withc=c2. Theorem 3.4 Let f ∈ L²_α,β(R). Then, for all N >0,

ω(f,N⁻¹)2,α,β =O

⎛

⎜⎝N⁻² _N−1

l=0

(l+1)³J_l²(f) ¹₂⎞

⎟⎠.

Proof From Lemma3.2, we have Nhf(x)²2,α,β =4

_+∞

0

|1−ϕ_λ^α,β(h)|²

|Hf(λ)|²+ |Hfˇ(λ)|² dσ (λ).

This integral is divided into two _+∞

0

= N

0

+ _+∞

N

=4(I1+I2),

whereN = [h⁻¹]. We estimate them separately.

From (i) of Lemma2.1, we have the estimate I2≤c4

_+∞

N

|Hf(λ)|²+ |Hfˇ(λ)|²

dσ(λ)=c4J_N²(f).

Now, we estimateI1. From formula (ii) of Lemma2.1, we have I1≤h⁴

_N

0

(λ+ρ)⁴

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ)

=h⁴

N−1 l=0

_l+1

l

(λ+ρ)⁴

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ)

≤h⁴

N−1 l=0

(l+ρ+1)⁴

J_l²(f)−J_l²₊₁(f) ,

From the inequalityl+ρ+1≤(ρ+1)(l+1)we conclude

I1≤(ρ+1)⁴h⁴

N−1 l=0

al

J_l²(f)−J_l²₊₁(f)

withal =(l+1)⁴. For all integersm≥1, the Abel transformation shows m

l=0

al

J_l²(f)−J_l²₊₁(f)

=a0J₀²(f)+ m l=1

(al−al−1)J_l²(f)−amJ_m²₊₁(f)

≤a0J₀²(f)+ m l=1

(al−al−1)J_l²(f),

becauseamJ_m²₊₁(f)≥0. Hence

I1≤(ρ+1)⁴h⁴

J₀²(f)+

N−1 l=1

(l+1)⁴−l⁴

J_l²(f)−N⁴J_N²(f)

.

Moreover by the finite increments theorem, we have (l+1)⁴−l⁴≤4(l+1)³. Then

I1≤(ρ+1)⁴N⁻⁴

J₀²(f)+4

N−1 l=1

(l+1)³J_l²(f)−N⁴J_N²(f)

,

sinceN≤ ¹_h. Combining the estimates forI1andI2gives

Nhf(x)²2,α,β =O

N⁻⁴

N−1 l=0

(l+1)³J_l²(f)

,

which implies

ω(f,N⁻¹)2,α,β =O

⎛

⎜⎝N⁻² _N−1

l=0

(l+1)³J_l²(f) ¹₂⎞

⎟⎠,

and this ends the proof.

Theorem 3.5 Let f ∈ L²_α,β(R). If the series ∞

l=1

l^r−1Jl(f), r =1,2, ...

converges, then f ∈W₂^r(T^(α,β))and, for all N >0,

ω

(T^(α,β))^r f,N⁻¹

2,α,β =O

N⁻⁴

N−1 l=0

(l+1)^2r+3J_l²(f) ¹₂

+O

⎛

⎜⎝ ∞ l=[^N₂]

l^r−1Jl(f)

⎞

⎟⎠.

Proof Let f ∈ L²_α,β(R). By formula (2) and Plancherel Theorem, we have

(T^(α,β))^r f²_2,α,β = _+∞

0

λ^2r

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ)

= ∞ l=0

_l+1

l λ^2r

|Hf(λ)|²+ |Hfˇ(λ)|² dσ(λ).

Using an Abel transformation we obtain

(T^(α,β))^r f²2,α,β ≤J₀²(f)+2r ∞

l=1

(l+1)^2r−1J_l²(f).

From the inequalityl+1≤2lwe conclude

(T^(α,β))^rf²2,α,β ≤c5

J₀²(f)+^∞

l=1

l^2r−1J_l²(f)

.

Hence

(T^(α,β))^rf2,α,β =O _∞

l=1

l^r−1Jl(f)

.

Since the series

∞ l=1

l^r−1Jl(f), r=1,2, ...

converges, we see that f ∈W₂^r(T^(α,β)).

From Lemma3.2, we have Nh(T^(α,β))^r f(x)²2,α,β=4

_+∞

0

λ^2r|1−ϕ_λ^α,β(h)|²

|Hf(λ)|²+s|Hfˇ(λ)|² dσ (λ).