1 Introduction
In [2] (see also [3]) we considered some aspects of harmonic analysis associated to the following
family of one-dimensional (A,ε)-operators:
(1.1)ΛA,εf(x)=f′(x)+A′(x)A(x)(f(x)-f(-x)2)-εϱf(-x),
where A is a so-called Chébli function on ℝ (i.e. A is a continuous
ℝ+-valued function on ℝ satisfying certain regularity and convexity hypotheses),
ϱ is the index of A and -1≤ε≤1. We note that ϱ is non-negative. The function A and the real number ε are the deformation parameters
giving three well-known cases (as special examples) when one of the following conditions holds:
A(x)=Aα(x)=|x|2α+1 and ε arbitrary (Dunkl’s operators [15]),
A(x)=Aα,β(x)=|sinhx|2α+1(coshx)2β+1 and ε=0 (Heckman’s operators [19]),
A(x)=Aα,β(x)=|sinhx|2α+1(coshx)2β+1 and ε=1 (Cherednik’s operators [12]).
We mention that a differential-reflection operator built in terms of a Chébli function was built for the first time in [21].
In [2] we proved that for λ∈ℂ the equation
ΛA,εf(x)=iλf(x),
where f:ℝ→ℂ, admits a unique solution ΨA,ε(λ,⋅)
satisfying ΨA,ε(λ,0)=1.
Moreover, we established in [2, Theorems 3.4 and 3.5] suitable estimates for the growth of the eigenfunction
ΨA,ε(λ,x) and of its partial derivatives.
In this paper we are concerned with a development of an Lp-harmonic analysis for a generalized Fourier transform ℱA,ε
when 0<p≤21+1-ε2. Here
ℱA,εf(λ)=∫ℝf(x)ΨA,ε(λ,-x)A(x)𝑑x
for f∈L1(ℝ,A(x)dx).
Using the estimates for the growth of ΨA,ε(λ,x), we get holomorphic properties of ℱA,ε on the space Lp(ℝ,A(x)dx) with
(1.2){p=1for ϱ=0 or [ϱ>0 and ε=0],1≤p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0}.
A Riemann–Lebesgue lemma is also obtained for p is as in (1.2).
We then turn our attention to an Lp-Schwartz space isomorphism theorem for ℱA,ε. In [18] Harish-Chandra proved an L2-Schwartz space isomorphism for the spherical Fourier transform on non-compact Riemannian symmetric spaces. This result was extended to Lp-Schwartz spaces with 0<p<2 by
Trombi and Varadarajan [26] (see also [16, 17, 13]). In the early nineties, Anker gave a new and simple proof of their result, based on the Paley–Wiener theorem for the spherical Fourier transform on Riemannian symmetric spaces [1]. Recently, Anker’s method was used in [22] to prove an Lp-Schwartz space isomorphism theorem for the Heckman–Opdam hypergeometric functions. Our approach is inspired by Anker’s paper [1]. More precisely, for -1≤ε≤1 and 0<p≤21+1-ε2, we define the tube domain
ℂp,ε:={λ∈ℂ∣|Imλ|≤ϱ((2/p)-1-1-ε2)}.
Denote by 𝒮p(ℝ) the Lp-Schwartz space on ℝ, and by 𝒮(ℂp,ε) the Schwartz space on the tube domain ℂp,ε.
We prove that ℱA,ε is a topological isomorphism between 𝒮p(ℝ) and
𝒮(ℂp,ε) with
{0<p≤1for ϱ=0 or [ϱ>0 and ε=0],0<p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0}
(see Theorem 3.12).
We close this paper by establishing a result in connection with pointwise multipliers of 𝒮(ℂp,ε). More precisely, for arbitrary γ≥0, a function ψ defined on the tube domain ℂγ:={λ∈ℂ∣|Imλ|≤γ}
is called a pointwise multiplier of 𝒮(ℂγ) if the mapping ϕ↦ψϕ is continuous from 𝒮(ℂγ) into itself. In [4] Betancor et al. characterized the set of pointwise multipliers of the Schwartz spaces 𝒮(ℂγ) for arbitrary γ≥0. Under the assumptions
{0<p≤1for ϱ=0 or [ϱ>0 and ε=0],22+1-ε2≤p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0},
we proved that if T is in the dual space 𝒮p′(ℝ) of 𝒮p(ℝ) such that ψ:=ℱA,ε(T) is
a pointwise multiplier of 𝒮(ℂp,ε), then for any s∈ℕ
there exist ℓ∈ℕ and continuous functions fm defined on ℝ, m=0,1,…,ℓ, such that
T=∑m=0ℓΛA,εmfm
and, for every such m,
supx∈ℝ(|x|+1)se(2p-1-ε2)ϱ|x||fm(x)|<∞.
The organization of this paper is as follows: In Section 2 we recapitulate some definitions and basic notations, as well as some results from literature. Further, we recall from [2] some results on ΨA,ε(λ,x). In Section 3 we develop an Lp-harmonic analysis for the Fourier transform ℱA,ε, where we mainly prove an Lp-Schwartz space isomorphism theorem for ℱA,ε. Finally, in Section 4 we characterize the distributions T∈𝒮p′(ℝ) so that ℱA,ε(T) are pointwise multipliers of the Schwartz space 𝒮(ℂp,ε).
3 Fourier transform of Lp-Schwartz spaces
Assume that -1≤ε≤1. For f∈L1(ℝ,A(x)dx) put
ℱA,εf(λ)=∫ℝf(x)ΨA,ε(λ,-x)A(x)𝑑x.
To state its alleged inverse transform, let us introduce the following Plancherel measure:
(3.1)πε(dλ)=|λ|λ2-(1-ε2)ϱ2|c(λ2-(1-ε2)ϱ2)|2𝟙ℝ∖]-1-ε2ϱ,1-ε2ϱ[(λ)dλ,
where c is Harish-Chandra’s function associated to the second order differential operator Δ (see Section 2 for more details on the c-function). In the sequel we set fˇ(x):=f(-x).
Theorem 3.1Let f be a smooth function with compact support on R. Then we have the following formulas:
(Inversion formula)
(3.2)f(x)=14∫ℝℱA,ε(f)(λ)ΨA,ε(λ,x)(1-εϱiλ)πε(dλ) almost everywhere.
(Plancherel formula)
(3.3)∫ℝ|f(x)|2A(x)𝑑x=14∫ℝℱA,ε(f)(λ)ℱA,ε(fˇ)(-λ)¯(1-εϱiλ)πε(dλ).
We may rewrite (3.3) for two smooth and compactly supported functions f and g as
∫ℝf(x)g(-x)A(x)𝑑x=14∫ℝℱA,ε(f)(λ)ℱA,ε(g)(λ)(1-εϱiλ)πε(dλ).
Proof.
For the sake of completeness, we provide a detailed proof.
(i) We set Jh(x):=∫-∞xh(t)𝑑t.
Using the superposition (2.8) of the eigenfunction Ψε(λ,x), we obtain
ℱA,εf(λ)=2ℱΔ(fe)(με)+2(iλ+εϱ)ℱΔ(Jfo)(με),
where FΔ is the Chébli transform (2.3) and με is so that με2=λ2+(ε2-1)ϱ2.
By the inversion formula (2.4) for ℱΔ we deduce that
(3.4)f(x)=∫ℝ+{ℱΔ(fe)(με)φμε(x)+ℱΔ(Jfo)(με)φμε′(x)}π(dμε).
Let us write φμε and φμε′ in terms of ΨA,ε as follows:
φμε(x)=12(ΨA,ε(-λ,-x)+ΨA,ε(-λ,x)),φμε′(x)=iλ+εϱ2(ΨA,ε(-λ,-x)-ΨA,ε(-λ,x)).
Consequently, formula (3.4) becomes
f(x)=12∫ℝ+ΨA,ε(-λ,-x){ℱΔ(fe)(με)+(iλ+εϱ)ℱΔ(Jfo)(με)}π(dμε)
+12∫ℝ+ΨA,ε(-λ,x){ℱΔ(fe)(με)-(iλ+εϱ)ℱΔ(Jfo)(με)}π(dμε)
(3.5)=14∫ℝ+{ΨA,ε(-λ,-x)ℱA,ε(f)(λ)+ΨA,ε(-λ,x)ℱA,ε(fˇ)(λ)}πε(dλ).
Further, it is easy to check that
(3.6)ΨA,ε(λ,x)=(1+εϱiλ)ΨA,ε(-λ,-x)-εϱiλΨA,ε(λ,-x),
and therefore
(3.7)ℱA,ε(fˇ)(λ)=(1+εϱiλ)ℱA,ε(f)(-λ)-εϱiλℱA,ε(f)(λ).
In view of (3.6) and (3.7) we obtain
∫ℝ+ℱA,ε(f)(λ)ΨA,ε(-λ,-x)πε(dλ)=∫ℝ+ΨA,ε(λ,x)ℱA,ε(f)(λ)(1-εϱiλ)πε(dλ)
(3.8)+∫ℝ+ΨA,ε(-λ,x)ℱA,ε(f)(λ)(εϱiλ)πε(dλ),
and
∫ℝ+ℱA,ε(fˇ)(λ)ΨA,ε(-λ,x)πε(dλ)=∫ℝ+ΨA,ε(-λ,x)ℱA,ε(f)(-λ)(1+εϱiλ)πε(dλ)
(3.9)+∫ℝ+ΨA,ε(-λ,x)ℱA,ε(f)(λ)(-εϱiλ)πε(dλ).
Substituting (3.8) and (3.9) into (3.5), we get the inversion formula (3.2).
(ii) Using the fact that
ΨA,ε(λ,x)¯=ΨA,ε(-λ,x) for λ∈ℝ,
we have
ℱA,ε(gˇ)(λ)¯=∫ℝg(x)¯Ψε(-λ,x)A(x)𝑑x.
Application of the identity (3.5) for f gives
(3.10)∫ℝf(x)g(x)¯A(x)𝑑x=14∫ℝ+{ℱA,ε(f)(λ)ℱA,ε(g)(λ)¯+ℱA,ε(fˇ)(λ)ℱA,ε(gˇ)(λ)¯}πε(dλ).
Further, from (3.6) it follows that
(3.11)ℱA,ε(fˇ)(-λ)¯=(1+εϱiλ)ℱA,ε(f)(λ)¯-εϱiλℱA,ε(f)(-λ)¯.
Therefore,
(3.12)(1-εϱiλ)ℱA,ε(f)(λ)ℱA,ε(fˇ)(-λ)¯=(1+ε2ϱ2λ2)|ℱA,ε(f)(λ)|2-εϱiλ(1-εϱiλ)ℱA,ε(f)(λ)ℱA,ε(f)(-λ)¯.
Now let us rewrite (3.11) as
ℱA,ε(f)(-λ)¯=iλiλ-εϱℱA,ε(fˇ)(λ)¯+εϱ-iλ+εϱℱA,ε(f)(λ)¯.
Hence
ℱA,ε(f)(λ)ℱA,ε(f)(-λ)¯=iλiλ-εϱℱA,ε(f)(λ)ℱA,ε(fˇ)(λ)¯+εϱ-iλ+εϱ|ℱA,ε(f)(λ)|2,
which implies
(-εϱiλ)(iλ-εϱiλ)ℱA,ε(f)(λ)ℱA,ε(f)(-λ)¯=-(εϱiλ)ℱA,ε(f)(λ)ℱA,ε(fˇ)(λ)¯-(ε2ϱ2λ2)|ℱA,ε(f)(λ)|2.
Thus (3.12) becomes
(3.13)(1-εϱiλ)ℱA,ε(f)(λ)ℱA,ε(fˇ)(-λ)¯=|ℱA,ε(f)(λ)|2-εϱiλℱA,ε(f)(λ)ℱA,ε(fˇ)(λ)¯,
which is the key identity towards the Plancherel formula (3.3). Moreover, from (3.13) we also have
(3.14)(1-εϱiλ)ℱA,ε(fˇ)(-λ)¯ℱA,ε(f)(λ)=|ℱA,ε(fˇ)(-λ)|2-εϱiλℱA,ε(fˇ)(-λ)¯ℱA,ε(f)(-λ).
Indeed, we obtain (3.14) in three steps:
Replace f by fˇ in (3.13).
Substitute λ by -λ in the resulting identity from step 1.
Take the complex conjugates in the resulting identity from step 2.
Putting the pieces together, we arrive at
∫ℝℱA,ε(f)(λ)ℱA,ε(fˇ)(-λ)¯(1-εϱiλ)πε(dλ)
=∫ℝ+|ℱA,ε(f)(λ)|2πε(dλ)-∫ℝ+ℱA,ε(f)(λ)ℱA,ε(fˇ)(λ)¯(εϱiλ)πε(dλ)
+∫ℝ-|ℱA,ε(fˇ)(-λ)|2πε(dλ)-∫ℝ-ℱA,ε(f)(-λ)ℱA,ε(fˇ)(-λ)¯(εϱiλ)πε(dλ)
=∫ℝ+{|ℱA,ε(f)(λ)|2+|ℱA,ε(fˇ)(λ)|2}πε(dλ),
which compares very well with 4∥f∥L2(ℝ,A(x)dx)2 (see (3.10)).
∎
Remarks 3.2Notice that:
For ε=1, the Plancherel formula (3.3) corrects [9, Theorem 5.13]
(stated without a proof).
For ε=0 we can prove the following (stronger) versions of the inversion and the Plancherel formulas:
If f∈L1(ℝ,A(x)dx) and ℱA,0(f)∈L1(ℝ,π0(dλ)), then
f(x)=14∫ℝℱA,0(f)(λ)ΨA,0(λ,x)π0(dλ) almost everywhere.
If f∈L1∩L2(ℝ,A(x)dx), then
ℱA,0f∈L2(ℝ,π0(dλ)) and ∥ℱA,0f∥L2(ℝ,π0(dλ))=2∥f∥L2(ℝ,A(x)dx).
There exists a unique isometry on L2(ℝ,A(x)dx) that coincides with (1/2)ℱA,0 on L1∩L2(ℝ,A(x)dx).
The following lemma will be needed in the proof of a Paley–Wiener theorem for ℱA,ε.
Lemma 3.3For R>0 denote by DR(R) the space of smooth functions with support inside [-R,R].
Then f∈DR(R) if and
only if VA,εtf∈DR(R).
Proof.
The direct statement follows from (2.10). The converse direction is more involved. On the one hand, one can prove that
VA,ε-1tg(y)=𝒜-1∘ℰεt(ge)(y)+(εϱ+ddy)𝒜-1∘ℰεt(Jgo)(y),
where g=ge+go and Jh(x):=∫-∞xh(t)𝑑t. On the other hand, from (2.11) and [6, Lemma 4.10] it follows that if ge∈𝒟R(ℝ), then
𝒜-1∘ℰεt(ge)∈𝒟R(ℝ). Further, one may check that go∈𝒟R(ℝ) if and only if Jgo∈𝒟R(ℝ). As Jgo is an even function, it follows from above that
𝒜-1∘ℰεt(Jgo)∈𝒟R(ℝ).∎
For R>0, let PWR(ℂ) be the space of entire functions h on ℂ which are of exponential type and rapidly decreasing, i.e. for all t∈ℕ we have
supλ∈ℂ(|λ|+1)te-R|Imλ||h(λ)|<∞.
Denote by PW(ℂ) the union of the spaces PWR(ℂ), R>0.
Theorem 3.4Theorem 3.4 (Paley–Wiener Theorem)
Assume that -1≤ε≤1. The Fourier transform FA,ε is a topological linear isomorphism
between D(R) and PW(C). More precisely, it is a topological linear isomorphism between DR(R) and
PWR(C) for all R>0.
Proof.
The proof is standard. We shall only indicate how to proceed towards the statement. On the one hand, the Fourier transform
ℱA,ε can be written as ℱA,ε(f)=ℱeuc∘VA,εt(fˇ), where ℱeuc
is the Euclidean Fourier transform and VA,εt is the intertwining operator (2.10). This is a direct consequence of (2.9). Now, in the light of Lemma 3.3, appealing to the Paley–Wiener theorem for the Euclidean
Fourier transform ℱeuc we get the desired statement.
∎
For -1≤ε≤1 and 0<p≤21+1-ε2 set
ϑp,ε:=2p-1-1-ε2. Observe that 1≤21+1-ε2≤2.
We introduce the tube domain
ℂp,ε:={λ∈ℂ∣|Imλ|≤ϱϑp,ε}.
Proposition 3.5For all λ∈C1,ε and for all x∈R we have |ΨA,ε(λ,x)|≤2.
Proof.
Let R>0 be arbitrary but fixed and let
ℝ1,ε:={ν∈ℝ∣|ν|≤ϱ(1-1-ε2)}.
Application of the maximum modulus principle together with the fact that |ΨA,ε(λ,x)|≤ΨA,ε(iImλ,x) in the domain
[-R,R]+iℝ1,ε implies that the maximum of |ΨA,ε(λ,x)| is obtained when λ belongs to the boundary of iℝ1,ε, that is
λ=iη with |η|=ϱ(1-1-ε2).
Now recall that ΨA,ε(iη,x)+ΨA,ε(iη,-x)=2φμε(x) when ε≠0,±1, and ΨA,ε(iη,x)+ΨA,ε(iη,-x)=2 when ε=0,±1. Further, the parameter με satisfies
με2=λ2-(1-ε2)ϱ2=-ϱ2(1-21-ε2{1-1-ε2})≤0,
and therefore με∈iℝ with |με|≤ϱ.
Using the fact that ΨA,ε(iη,x)>0 for all x∈ℝ, together with the fact that
φμε(x)≤1 for με as above (see Lemma 2.2), we obtain that ΨA,ε(iη,x)≤2 for all x∈ℝ and -1≤ε≤1.
∎
Corollary 3.6Assume that ϱ≥0 and ε∈[-1,1]. Let f∈L1(R,A(x)dx). Then the following properties hold:
The Fourier transform ℱA,ε(f)(λ) is well defined for all λ∈ℂ1,ε. Moreover,
|ℱA,ε(f)(λ)|≤2∥f∥L1(ℝ,A(x)dx) for all λ∈ℂ1,ε.
The function ℱA,ε(f) is holomorphic on ℂ̊1,ε, the interior of ℂ1,ε.
(Riemann–Lebesgue lemma.) We have
(3.15)limλ∈ℂ1,ε,|λ|→∞|ℱA,ε(f)(λ)|=0.
Proof.
The first two statements are direct consequences of Proposition 3.5, the fact that
ΨA,ε(λ,⋅) is holomorphic in λ, and Morera’s theorem. For the Riemann–Lebesgue lemma, a classical proof for the Euclidean Fourier transform carries over. More precisely, assume that f∈𝒟(ℝ) (the space of smooth functions with compact support on ℝ). Now use the Paley–Wiener Theorem 3.4 to conclude that the limit (3.15) holds for test functions; the general case then follows from the fact that 𝒟(ℝ) is dense in L1(ℝ,A(x)dx).
∎
Next we discuss some properties of the Fourier transform ℱA,ε on Lp(ℝ,A(x)dx) with p>1.
Observe that the set ℂ̊p,ε, the interior of ℂp,ε in ℂ, is empty whenever ϱ=0 or p=21+1-ε2. Observe also that when ε=0, we have 21+1-ε2=1.
Lemma 3.7Assume that ϱ>0, and f∈Lp(R,A(x)dx) with 1<p<21+1-ε2 and
ε∈[-1,1]∖{0}. Then the following properties hold:
The Fourier transform ℱA,ε(f)(λ) is well defined for all
λ
in
ℂ̊p,ε. Moreover,
|ℱA,ε(f)(λ)|≤c∥f∥Lp(ℝ,A(x)dx) for all λ∈ℂ̊p,ε.
The function ℱA,ε(f) is holomorphic on ℂ̊p,ε.
(Riemann–Lebesgue lemma.) We have
limλ∈ℂ̊p,ε,|λ|→∞|ℱA,ε(f)(λ)|=0.
Proof.
The first two statements follow easily from the estimate
|ΨA,ε(λ,x)|≤c(|x|+1)e|Imλ||x|e-ϱ|x|(1-1-ε2) for ϱ>0,
the fact that A(x)≤c|x|βe2ϱ|x| for some β>0 (a consequence of hypothesis (H4) on the function A), the fact that
ΨA,ε(λ,⋅) is holomorphic in λ, and Morera’s theorem. The Riemann–Lebesgue lemma is established exactly as for (3.15) by approximating any function in the space Lp(ℝ,A(x)dx) by compactly supported smooth functions.
∎
Henceforth, when ϱ=0, we will assume that the parameter α appearing in the hypothesis (H1) is strictly positive. This is due to the fact that when ϱ=0, estimates for the c-function |c(μ)| for |μ|≪1
are evaluable only when α>0 (see Section 2).
Theorem 3.8Under the same assumptions as in Corollary 3.6 and Lemma 3.7, the Fourier transform FA,ε is injective on
Lp(R,A(x)dx).
Proof.
Take q such that p+q=pq. For f∈Lp(ℝ,A(x)dx) and g∈𝒟(ℝ) we have
|(f,g)A|:=|∫ℝf(x)g(-x)A(x)𝑑x|≤∥f∥Lp(ℝ,A(x)dx)∥g∥Lq(ℝ,A(x)dx)
and
|(ℱA,ε(f),ℱA,ε(g))πε|:=|∫ℝℱA,ε(f)(λ)ℱA,ε(g)(λ)(1-εϱiλ)πε(dλ)|
≤supλ|ℱA,ε(f)(λ)|∥ℱA,ε(g)∥L1(ℝ,π~ε(dλ))
(3.16)≤c∥f∥Lp(ℝ,A(x)dx)∥ℱA,ε(g)∥L1(ℝ,π~ε(dλ)),
where π~ε(dλ):=|1-εϱiλ|πε(dλ).
Above we have used Corollary 3.6 and Lemma 3.7 to get (3.16).
Therefore, the mappings f↦(f,g)A and f↦(ℱA,ε(f),ℱA,ε(g))πε are continuous functionals on Lp(ℝ,A(x)dx). Now,
(f,g)A=(1/4)(ℱA,ε(f),ℱA,ε(g))πε
for all f∈𝒟(ℝ) (see Theorem 3.1) and by continuity for all f∈Lp(ℝ,A(x)dx). Assume that f∈Lp(ℝ,A(x)dx) and that ℱA,ε(f)=0. Then for all g∈𝒟(ℝ) we have (f,g)A=(1/4)(ℱA,ε(f),ℱA,ε(g))πε=0 and therefore f=0.
∎
For -1≤ε≤1 and 0<p≤21+1-ε2 let 𝒮p(ℝ) be the space consisting of all functions
f∈C∞(ℝ) such that
supx∈ℝ(|x|+1)sφ0(x)-2/p|f(k)(x)|<∞
for any s∈ℕ and any k∈ℕ. The topology of 𝒮p(ℝ) is defined by
the seminorms
σs,k(p)(f)=supx∈ℝ(|x|+1)sφ0(x)-2/p|f(k)(x)|.
We pin down that 𝒮p(ℝ) is a dense subspace of Lq(ℝ,A(x)dx)
for all q≥p.
The following facts are standard; see for instance [14, Appendix A].
Lemma 3.9𝒮p(ℝ) is a Fréchet space with respect to the seminorms σs,k(p).
𝒟(ℝ) is a dense subspace of 𝒮p(ℝ).
Recall from above the tube domain
ℂp,ε:={λ∈ℂ∣|Imλ|≤ϱϑp,ε},
where
ϑp,ε=2p-1-1-ε2.
The Schwartz space 𝒮(ℂp,ε) consists of all complex valued
functions h that are analytic in the interior
of ℂp,ε and such that h and all its derivatives extend
continuously to ℂp,ε and satisfy
supλ∈ℂp,ε(|λ|+1)t|h(ℓ)(λ)|<∞
for any t∈ℕ and any ℓ∈ℕ.
The topology of 𝒮(ℂp,ε) is defined by the seminorms
(3.17)τt,ℓ(ϑp,ε)(h):=supλ∈ℂp,ε(|λ|+1)t|h(ℓ)(λ)|.
For ϑp,ε=0 or ϱ=0, the space 𝒮(ℂp,ε) is the classical Schwartz space on ℝ.
By [6, Lemma 4.17] the Paley–Wiener space PW(ℂ) is dense in the Schwartz space
𝒮(ℂp,ε).
Lemma 3.10The Fourier transform FA,ε maps Sp(R) continuously into
S(Cp,ε) and is injective with
{0<p≤1for ϱ=0 or [ϱ>0 and ε=0],0<p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0}.
Proof.
Let f∈𝒮p(ℝ). For λ∈ℂp,ε we have
|ℱA,ε(f)(λ)|≤∫ℝ|f(x)||ΨA,ε(λ,-x)|A(x)𝑑x
≤∫ℝ|f(x)|φ0(x)-2/pφ0(x)2/pΨA,ε(0,-x)e|Imλ||x|A(x)𝑑x
≤c1∫ℝ|f(x)|φ0(x)-2/p(|x|+1)2/p+1e-2ϱ|x|A(x)𝑑x.
Under hypothesis (H4) on Chébli’s function A, there exists a β>0 such that A(x)≤c|x|βe2ϱ|x|.
Hence,
|ℱA,ε(f)(λ)|≤c2∫ℝ|f(x)|φ0(x)-2/p(|x|+1)2/p+1|x|β𝑑x<∞.
This proves that ℱA,ε(f) is well defined for all f∈𝒮p(ℝ) when -1≤ε≤1 and 0<p≤21+1-ε2. Moreover, since the map λ↦ΨA,ε(λ,x) is holomorphic on ℂ, it follows that for all f∈𝒮p(ℝ), the function ℱA,ε(f) is analytic in the interior of ℂp,ε and continuous on ℂp,ε. Furthermore, by Theorem 2.7 (ii), we have
|ℱA,ε(f)(λ)(k)|≤c3∫ℝ|f(x)|φ0(x)-2/p(|x|+1)2/p+k+1|x|β𝑑x<∞.
Thus, all derivatives of ℱA,ε(f) extend continuously to ℂp,ε. Next, we will prove that given a continuous
seminorm τ on 𝒮(ℂp,ε), there exists a continuous seminorm σ on 𝒮p(ℝ) such that
τ(ℱA,ε(f))≤c4σ(f) for all f∈𝒮p(ℝ).
Note that the space 𝒮(ℂp,ε) and its topology are also determined by the seminorms
(3.18)h↦τ~t,ℓ(ϑp,ε)(h):=supλ∈ℂp,ε|{(λ+1)th(λ)}(ℓ)|,
where t and ℓ are two arbitrary positive integers. Invoking the identity (2.6), we have
(iλ)rℱA,ε(f)(λ)=(iλ)r∫ℝfˇ(x)ΨA,ε(λ,x)A(x)𝑑x=∫ℝfˇ(x)ΛA,εrΨA,ε(λ,x)A(x)𝑑x
=(-1)r∫ℝ(ΛA,ε+2εϱS)rfˇ(x)ΨA,ε(λ,x)A(x)𝑑x=∫ℝΛA,εrf(-x)ΨA,ε(λ,x)A(x)𝑑x
=ℱA,ε(ΛA,εrf)(λ)
for r∈ℕ, where S denotes the symmetry Sf(x)=f(-x). Above we have used (ΛA,ε+2εϱS)r∘S=(-1)rS∘ΛA,εr.
Thus,
{(iλ)rℱA,ε(f)(λ)}(ℓ)=∫ℝΛA,εrf(x)∂λℓΨA,ε(λ,-x)A(x)𝑑x.
On the one hand, using Theorem 2.7 (ii), we obtain
|{(iλ)rℱA,ε(f)(λ)}(ℓ)|≤c5∫ℝ|ΛA,εrf(x)|(|x|+1)ℓφi1-ε2ϱ(x)e|Imλ||x|A(x)𝑑x
=c5∫|x|≤a|ΛA,εrf(x)|(|x|+1)ℓφi1-ε2ϱ(x)e|Imλ||x|A(x)𝑑x
+c5∫|x|>a|ΛA,εrf(x)|(|x|+1)ℓφi1-ε2ϱ(x)e|Imλ||x|A(x)𝑑x
≤c6∫|x|≤a|ΛA,εrf(x)|φ0(x)-2/p(|x|+1)2/p+ℓ+1e-2ϱ|x|A(x)𝑑x
+c6∫|x|>a|ΛA,εrf(x)|φ0(x)-2/p(|x|+1)2/p+ℓ+1e-2ϱ|x|A(x)𝑑x.
On the other hand, by mimicking the proof of [6, Lemma 4.18], the following holds:
For |x|≤a we have
|ΛA,εrf(x)|≤c7(∑i=0r|f(i)(x)|+∑i=0r-1|f(i)(-x)|+∑i=0r∑m=0nr|f(i)(ξm)|),
where ξm=ξm(x,r)∈]-|x|,|x|[.
For |x|>a we have
|ΛA,εrf(x)|≤c7′(∑i=0r|f(i)(x)|+∑i=0r-1|f(i)(-x)|).
The estimate
τ(ℱA,ε(f))≤c8∑finiteσ(f) for all f∈𝒮p(ℝ)
is now a matter of putting the pieces together.
We obtain the injectivity of the transform ℱA,ε on 𝒮p(ℝ) from the facts that ℱA,ε is injective on Lq(ℝ,A(x)dx) for a certain range of q (see Theorem 3.8) and that 𝒮p(ℝ) is a dense subspace of Lq(ℝ,A(x)dx) for q≥p.
This concludes the proof of Lemma 3.10.
∎
Lemma 3.11Let -1≤ε≤1 and 0<p≤21+1-ε2. The inverse Fourier transform
FA,ε-1:PW(C)→D(R) given by
ℱA,ε-1h(x)=14∫ℝh(λ)ΨA,ε(λ,x)(1-εϱiλ)πε(dλ)
is continuous for the topologies induced by S(Cp,ε) and Sp(R).
Proof.
Let f∈𝒟(ℝ) and h∈PW(ℂ) so that f=ℱA,ε-1(h). Given a seminorm
σ on 𝒮p(ℝ), we should find a continuous seminorm τ on 𝒮(ℂp,ε)
such that σ(f)≤cτ(h).
Denote by g the image of h by the inverse Euclidean Fourier
transform ℱeuc-1. Making use of the Paley–Wiener Theorem 3.4 for ℱA,ε and
the classical Paley–Wiener theorem for ℱeuc, we have the following support
conservation property:
supp(f)⊂IR:=[-R,R]⇔supp(g)⊂IR.
For j∈ℕ≥1 let ωj∈C∞(ℝ) with ωj=0 on Ij-1 and ωj=1
outside of Ij. Assume that ωj and all its derivatives are bounded uniformly in j.
We will write gj=ωjg, and define hj:=ℱeuc(gj) and fj:=ℱA,ε-1(hj).
Note that gj=g outside Ij. Hence, by the above support property,
fj=f outside Ij. We shall estimate the function
x↦(|x|+1)sφ0(x)-2/p|fj(k)(x)|
on Ij+1∖Ij with j∈ℕ≥1.
Recall that fj=f on Ij+1∖Ij. In view of Theorem 2.7 (i) we have
|fj(k)(x)|≤14∫ℝ|hj(λ)||∂xkΨA,ε(λ,x)||1-εϱiλ|πε(dλ)
≤c1φi1-ε2ϱ(x)∫ℝ|hj(λ)|(|λ|+1)k|1-εϱiλ|πε(dλ),
where
|1-εϱiλ|πε(dλ)=λ2+ε2ϱ2λ2-(1-ε2)ϱ21|c(λ2-(1-ε2)ϱ2)|2𝟙ℝ∖]-1-ε2ϱ,1-ε2ϱ[(λ)dλ.
By knowing about the asymptotic behavior of the c-function (see Section 2), one comes to
|fj(k)(x)|≤c2φi1-ε2ϱ(x)τt1,0(0)(hj)
for some integer t1>0. It follows that
supx∈Ij+1∖Ij(|x|+1)sφ0(x)-2/p|fj(k)(x)|≤c3js+1eϱj(2p-1+1-ε2)τt1,0(0)(hj).
Recall that the two seminorms τt,ℓ(ϑp,ε) (see (3.17)) and τ~t,ℓ(ϑp,ε) (see (3.18))
are equivalent on 𝒮(ℂp,ε). Since hj=ℱeuc(gj), it follows that
(1+λ)t1hj(λ)=∑ℓ=0t1(t1ℓ)λℓℱeuc(gj)(λ).
Denote by C and by Cj the compact supports of g and gj, respectively.
Thus,
τ~t1,0(0)(hj)≤∑ℓ=0t1(t1ℓ)∫Cj|gj(ℓ)(y)|𝑑y
≤c4∑ℓ=0t1supy∈Cj(|y|+1)2|gj(ℓ)(y)|
=c4∑ℓ=0t1supw∈{±1}supy∈Cj,+(y+1)2|gj(ℓ)(wy)|,
where Cj,+:=Cj∩ℝ+. Now one uses the Leibniz rule to compute the derivatives of gj=ωjg.
Since
ωj=0 on Ij-1 and since ωj and all its derivatives are bounded uniformly in j,
we then have
τ~t1,0(0)(hj)≤c5∑ℓ=0t1supw∈{±1}supy∈C+∖Ij-1(y+1)2|g(ℓ)(wy)|,
where C+:=C∩ℝ+. Hence,
js+1eϱj(2p-1+1-ε2)τ~t1,0(0)(hj)≤c6∑ℓ=0t1supw∈{±1}supy∈C+∖Ij-1(y+1)s+3eϱy(2p-1+1-ε2)|g(ℓ)(wy)|
≤c7∑ℓ=0t1supw∈{±1}supy∈C+∖Ij-1(y+1)s+3eϱy(2p-1-1-ε2)|g(ℓ)(wy)|.
Recall that g(x)=ℱeuc-1(h)(x), where ℱeuc is the Euclidean Fourier
transform and h∈PW(ℂ). By Cauchy’s integral theorem, it is known that
p(u)eαug(ℓ)(u)=c∫ℝp(i∂λ){(iλ-α)ℓh(λ+iα)}eiλu𝑑λ
for any polynomial p∈ℝ[u]. Hence,
∑ℓ=0t1supw∈{±1}supy∈C+∖Ij-1(y+1)s+3eϱy(2p-1-1-ε2)|g(ℓ)(wy)|≤c8∑r=0s+3sup|Imλ|≤ϱϑp,ε(|λ|+1)t2|h(r)(λ)|
=c8∑r=0s+3τt2,r(ϑp,ε)(h)
for some integer t2>0.
It remains for us to estimate the function
x↦(|x|+1)sφ0(x)-2/p|f(k)(x)|
on I1=[-1,1]. First, it is not hard to prove that for |x|≤1 there is a positive constant c and an integer mk≥1 such that
|∂k∂xkΨA,ε(λ,x)|≤c(|λ|+1)mk|iλ-εϱ|φ0(x)
for λ∈ℝ such that |λ|≥1-ε2ϱ. Now, arguing as above, we have
|f(k)(x)|≤c1φ0(x)∫ℝ|h(λ)|(|λ|+1)mk|iλ-εϱ||1-εϱiλ|πε(dλ).
Since I1 is compact, it follows that
supx∈I1(|x|+1)sφ0(x)-2/p|f(k)(x)|≤c2∫ℝ|h(λ)|(|λ|+1)mk|iλ-εϱ||1-εϱiλ|πε(dλ)≤c3τt,0(0)(h)
for some integer t>0.
This finishes the proof of Lemma 3.11.
∎
In summary, we have proved the following theorem.
Theorem 3.12The Fourier transform FA,ε is a topological isomorphism between Sp(R) and S(Cp,ε) with
(3.19){0<p≤1for ϱ=0 or [ϱ>0 and ε=0],0<p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0}.
4 Pointwise multipliers
For -1≤ε≤1 and p as in (3.19) denote by 𝒮p′(ℝ) and by 𝒮′(ℂp,ε) the topological dual spaces of
𝒮p(ℝ) and 𝒮(ℂp,ε), respectively.
Let f be a Lebesgue measurable function on ℝ such that
∫ℝ|f(x)|φ0(x)2/p(|x|+1)-ℓA(x)𝑑x<∞
for some ℓ∈ℕ. Then the functional
Tf defined on 𝒮p(ℝ) by
〈Tf,ϕ〉=4∫ℝf(x)ϕ(-x)A(x)𝑑x,ϕ∈𝒮p(ℝ),
is in 𝒮p′(ℝ). Indeed,
|〈Tf,ϕ〉|≤cσℓ,0(p)(ϕ)∫ℝ|f(x)|φ0(x)2/p(|x|+1)-ℓA(x)𝑑x<∞.
Further, since p≤21+1-ε2≤2, the Schwartz space 𝒮p(ℝ) can be seen as a subspace of
𝒮p′(ℝ) by
identifying f∈𝒮p(ℝ) with Tf∈𝒮p′(ℝ).
Now let h be a measurable function on ℝ such that
∫ℝ|h(λ)|(|λ|+1)-ℓ|1-εϱiλ|πε(dλ)<∞
for some ℓ∈ℕ. Here πε(dλ) denotes the Plancherel measure (3.1),
πε(dλ)=|λ|λ2-(1-ε2)ϱ2|c(λ2-(1-ε2)ϱ2)|2𝟙ℝ∖]-1-ε2ϱ,1-ε2ϱ[(λ)dλ,
where c is Harish-Chandra’s function associated to the operator Δ (see Section 2). Then the functional 𝒯h defined on 𝒮(ℂp,ε) by
〈𝒯h,ψ〉=∫ℝh(λ)ψ(λ)(1-εϱiλ)πε(dλ),ψ∈𝒮(ℂp,ε),
is in the dual space 𝒮′(ℂp,ε).
In fact,
|〈𝒯h,ψ〉|≤cτt,0(0)(ψ)∫ℝ|h(λ)|(|λ|+1)-t|1-εϱiλ|πε(dλ)<∞.
Moreover, since |c(μ)|-2∼|μ|2α+1 for |μ| large (with α>-1/2) and
|c(μ)|-2∼{|μ|2for |μ|≪1 and ϱ>0,|μ|2α+1for |μ|≪1 and ϱ=0 (with α>0),
it follows that the Schwartz space 𝒮(ℂp,ε) can be identified with a subspace of 𝒮′(ℂp,ε).
For T in 𝒮p′(ℝ) we define the distributional Fourier
transform ℱA,ε(T) of T on 𝒮(ℂp,ε)=ℱA,ε(𝒮p(ℝ)) by
〈ℱA,ε(T),ℱA,ε(ϕ)〉=〈T,ϕ〉,ϕ∈𝒮p(ℝ).
This can be written as
〈ℱA,ε(T),ψ〉=〈T,ℱA,ε-1(ψ)〉,ψ∈𝒮(ℂp,ε).
This definition is an extension of the Fourier transform on 𝒮p(ℝ). Indeed,
let f∈𝒮p(ℝ). Applying Fubini’s theorem, for every ϕ∈𝒮p(ℝ), we have
〈𝒯ℱA,ε(f),ℱA,ε(ϕ)〉=∫ℝℱA,ε(f)(λ)ℱA,ε(ϕ)(λ)(1-εϱiλ)πε(dλ)
=∫ℝf(x){∫ℝℱA,ε(ϕ)(λ)ΨA,ε(λ,-x)(1-εϱiλ)πε(dλ)}A(x)𝑑x
=4∫ℝf(x)ϕ(-x)A(x)𝑑x
=〈Tf,ϕ〉.
Hence, ℱA,ε(Tf)=𝒯ℱA,ε(f).
A function ψ defined on the tube domain ℂp,ε is called a pointwise multiplier of 𝒮(ℂp,ε)
if the mapping ϕ↦ψϕ is continuous from 𝒮(ℂp,ε) into itself.
The following statement comes from [4, Proposition 3.2], with changes appropriate to our setting.
Lemma 4.1Let ψ be a function defined on Cp,ε. Then ψ
is a pointwise multiplier of S(Cp,ε) if and only if ψ satisfies the following three conditions:
ψ
is holomorphic in the interior of ℂp,ε.
For every t∈ℕ the derivatives ψ(t) extend continuously to ℂp,ε.
For every t∈ℕ there exists nt∈ℕ such that
supλ∈ℂp,ε(|λ|+1)-nt|ψ(t)(λ)|<∞.
Theorem 4.2Suppose that
{0<p≤1for ϱ=0 or [ϱ>0 and ε=0],22+1-ε2≤p<21+1-ε2for ϱ>0 and ε∈[-1,1]∖{0}.
If T∈Sp′(R) such that ψ:=FA,ε(T) is
a pointwise multiplier of S(Cp,ε), then for any s∈N
there exist ℓ∈N and continuous functions fm defined on R, m=0,1,…,ℓ, such that
T=∑m=0ℓΛA,εmfm
and, for every such m, we have
(4.1)supx∈ℝ(|x|+1)sφ0(x)-2p+1-ε2|fm(x)|<∞.
Here
ΛA,ε is the differential-reflection operator (1.1).
Proof.
It is assumed that ψ=ℱA,ε(T) is a pointwise multiplier of 𝒮(ℂp,ε). Then by Lemma 4.1, for all t∈ℕ there is an integer nt∈ℕ such that
(4.2)supλ∈ℂp,ε(|λ|+1)-nt|ψ(t)(λ)|<∞.
Fix s∈ℕ and consider an integer ℓ that will be specified later. Define the function
κ on ℂp,ε by
κ(λ)=(iλ+ϱ+1)-ℓψ(λ).
In view of our assumption on p, the function κ satisfies the first and the second conditions in the definition of the space 𝒮(ℂp,ε).
Further, since |ΨA,ε(λ,x)|≤2 for all λ∈ℝ, we have
|ℱA,ε-1(κ)(x)|:=|c∫ℝκ(λ)ΨA,ε(λ,x)(1-εϱiλ)πε(dλ)|≤c1∫ℝ|κ(λ)||1-εϱiλ|πε(dλ),
where
|1-εϱiλ|πε(dλ)=λ2+ε2ϱ2λ2-(1-ε2)ϱ21|c(λ2-(1-ε2)ϱ2)|2𝟙ℝ∖]-1-ε2ϱ,1-ε2ϱ[(λ)dλ.
Thus, in view of the estimate (4.2) and the behavior of |c(μ)|-2 for small and large |μ|, it follows that ℱA,ε-1(κ)(x) exists for all x∈ℝ provided that ℓ>n0+2α+2.
Here the parameter n0 comes from (4.2).
Moreover, for all ϕ∈𝒮p(ℝ), Fubini’s theorem leads to
∫ℝϕ(-x)ℱA,ε-1(κ)(x)A(x)𝑑x=c1∫ℝϕ(-x)(∫ℝκ(λ)ΨA,ε(λ,x)(1-εϱiλ)πε(dλ))A(x)𝑑x
=c1∫ℝκ(λ)(∫ℝϕ(-x)ΨA,ε(λ,x)A(x)𝑑x)(1-εϱiλ)πε(dλ)
=c1∫ℝκ(λ)ℱA,ε(ϕ)(λ)(1-εϱiλ)πε(dλ).
It follows that the inverse Fourier transform ℱA,ε-1(κ) of κ as an element of
𝒮′(ℂp,ε)
concurs with the classical Fourier transform of κ.
Further,
T=ℱA,ε-1((iλ+ϱ+1)ℓκ)=∑m=0ℓ(ℓm)(ϱ+1)ℓ-mΛA,εmℱA,ε-1(κ):=∑m=0ℓΛA,εmfm.
It remains for us to show that, given s∈ℕ, the functions fm satisfy (4.1), provided that ℓ is large enough. To do so, we will use a similar approach to that in the proof of Lemma 3.11.
Set ξ:=ℱA,ε-1(κ) and g:=ℱeuc-1(κ),
where ℱeuc denotes the
Euclidean Fourier transform. Observe that if ℓ is large enough, then g is well defined.
For j∈ℕ≥1 let ωj∈C∞(ℝ) such that ωj=0
on Ij-1:=[-(j-1),j-1] and ωj=1 outside of Ij. We shall assume that ωj
and all its derivatives are bounded uniformly in j.
We set gj:=ωjg, and define
κj:=ℱeuc(gj) and ξj=ℱA,ε-1(κj). Since ωj=1 outside of Ij,
it follows that gj-g=0 outside of Ij, that is supp(gj-g)⊂Ij.
Using the support conservation property from the proof of Lemma 3.11, we deduce that
ξ may differ from ξj only inside Ij. Now we will estimate the function
(4.3)x↦(|x|+1)sφ0(x)-2p+1-ε2ξ(x),
first on I1
and next on Ij+1∖Ij for j∈ℕ≥1.
We claim that
|ΨA,ε(λ,x)|≤c2(|λ|+1)φ0(x)
for λ∈ℝ such that |λ|≥1-ε2ϱ. Indeed, as λ∈ℝ is such that |λ|≥1-ε2ϱ, it follows that με∈ℝ. Thus, the claim follows from the superposition (2.8) of ΨA,ε(λ,x) and the facts that |φμε(x)|≤φ0(x) and |φμε′(x)|≤c(με2+ϱ2)φ0(x) (see Lemma 2.2).
From the claim above we have
|ξ(x)|≤c3∫ℝ|κ(λ)||ΨA,ε(λ,x)||1-εϱiλ|πε(dλ)
≤c4φ0(x)∫ℝ|κ(λ)|(|λ|+1)|1-εϱiλ|πε(dλ).
Since I1 is compact, we deduce that for every s∈ℕ we have
supx∈I1(|x|+1)sφ0(x)-2p+1-ε2|ξ(x)|<∞
whenever ℓ>n0+2α+3. Here the parameter n0 comes from (4.2).
Now we consider the estimate of function (4.3) on Ij+1∖Ij for j∈ℕ≥1.
Recall that ξ=ξj outside of Ij.
Arguing as above, we obtain
|ξj(x)|≤c5φ0(x)supλ∈ℝ∖]-1-ε2ϱ,1-ε2ϱ[|(λ+1)t1κj(λ)|
for some integer t1>2α+3. It follows that
supx∈Ij+1∖Ij(|x|+1)sφ0(x)-2p+1-ε2|ξj(x)|≤c6jse(2p-1-1-ε2)ϱjsupλ∈ℝ∖]-1-ε2ϱ,1-ε2ϱ[|(λ+1)t1κj(λ)|.
Since κj=ℱeuc(gj)
with gj=ωjg, we claim that
(4.4)|(λ+1)t1κj(λ)|≤c7∑q=0t1supw∈{±1}supx∈ℝ+∖Ij-1(x+1)2|g(q)(wx)|.
Indeed, on the one hand, we have
(4.5)(λ+1)t1κj(λ)=∑r=0t1cr∫ℝgj(x)∂xreiλxdx=∑r=0t1cr∫ℝωj(x)g(x)∂xreiλxdx.
On the other hand, we have
(4.6)(ωjg)(r)(x)=∑q=0rcqg(q)(x)ωj(r-q)(x)→0 as |x|→+∞.
In fact, starting from g=ℱeuc-1(κ), we obtain
(4.7)g(q)(x)=c∫ℝκ(λ)(iλ)qeiλx𝑑λ.
Thus, if ℓ>n0+t1+1 then by the Riemann–Lebesgue lemma for the Euclidean Fourier transform, we obtain
g(q)(x)→0
as |x|→∞. Thus (4.6) holds true.
Now in view of (4.6) we may rewrite (4.5) as
(λ+1)t1κj(λ)=∑r=0t1∑q=0rcq,r∫ℝg(q)(x)ωj(r-q)(x)eiλx𝑑x.
Recall that the function
ωj vanishes on Ij-1 and is bounded, together with all its derivatives, uniformly in j.
Therefore,
|(λ+1)t1κj(λ)|≤c∑q=0t1∫ℝ∖Ij-1|g(q)(x)|𝑑x
≤c∑q=0t1supx∈ℝ∖Ij-1(|x|+1)2|g(q)(x)|.
This finishes the proof of claim (4.4).
It follows that
jse(2p-1-1-ε2)ϱjsupλ∈ℝ∖]-1-ε2ϱ,1-ε2ϱ[|(λ+1)t1κj(λ)|
≤c7∑q=0t1supw∈{±1}supx∈ℝ+∖Ij-1(x+1)s+2e(2p-1-1-ε2)ϱx|g(q)(wx)|.
Next we shall prove that the right-hand side is finite. Assume first that ϱ=0. By (4.7) we have
(4.8)(x+1)s+2g(q)(wx)=∑r=0s+2cq,r∫ℝκ(λ)λq∂λreiλwxdλ.
We claim that
(4.9)(κ(λ)λq)(r)→0 as |λ|→+∞,
provided that ℓ is large enough. Indeed, this claim follows immediately from the fact that
(κ(λ)λq)(r)=∑a=0rcaλq-r+aκ(a)(λ) (with r-a≤q)
(4.10)=∑a=0r∑b=0aca,bλq-r+a(iλ+1)-ℓ-a+bψ(b)(λ),
together with the fact that ψ satisfies (4.2). Thus, by (4.9) we may rewrite (4.8) as
(4.11)(x+1)s+2g(q)(wx)=∑r=0s+2cq,r′∫ℝ(κ(λ)λq)(r)eiλwx𝑑λ.
Using again the fact that ψ satisfies (4.2) together with the double sum (4.10), we obtain from (4.11) that
supw∈{±1}supx∈ℝ+∖Ij-1(x+1)s+2|g(q)(wx)|<∞
for ϱ=0, provided that ℓ is large enough.
Now assume that ϱ>0.
Since g=ℱeuc-1(κ) and κ is holomorphic in the interior of ℂp,ε, Cauchy’s integral theorem gives
p(u)eαug(q)(u)=cst∫ℝp(i∂λ){(iλ-α)qκ(λ+iα)}eiλu𝑑λ,
with p(x)=(x+1)s+2 and
α=(2p-1-1-ε2)ϱ. The same argument as above implies that
supw∈{±1}supx∈ℝ+∖Ij-1(x+1)s+2e(2p-1-1-ε2)ϱx|g(q)(wx)|<∞,
provided that ℓ is large enough.
Putting the pieces together, we conclude that
supx∈Ij+1∖Ij(|x|+1)sφ0(x)-2p+1-ε2|ξj(x)|<∞
for ℓ large enough.
∎
,
Asma Boussen
