The aim of this paper is to generalize the q-Heisenberg uncertainty principles
studied by Bettaibi et al. (2007), to state local uncertainty principles for the q-Fourier-cosine, the q-Fourier-sine,
and the q-Bessel-Fourier transforms, then to provide an inequality of Heisenberg-Weyl-type for the q-Bessel-Fourier transform.
1. Introduction
The uncertainty principle is a metatheorem in harmonic
analysis that asserts, with the use of some inequalities, that a function and
its Fourier transform cannot be sharply localized. We refer to the survey
article by Folland and Sitaram [1] and the book of Havin and Jöricke [2] for various classical
uncertainty principles of different nature which may be found in the literature.
In [3], the authors gave q-analogues of the Heisenberg uncertainty
principle for the q-Fourier-cosine and the q-Fourier-sine transforms. One of the aims of
this paper is to provide a generalization of their work next to state local
uncertainty principles for various q-Fourier transforms.
This paper is organized as follows. In Section 2, we
present some preliminaries results and notations that will be useful in the
sequel. In Section 3, we prove a density theorem and a q-analogue of the Hausdorff-Young inequality.
Then, we state a generalization of the q-Heisenberg uncertainty principle for the q-Fourier-cosine and the q-Fourier-sine transforms. In Section 4, we
state local uncertainty principles for the q-Fourier-cosine, q-Fourier-sine, and q-Bessel-Fourier transforms. Then, we give a
Heisenberg-Weyl-type inequality for some q-Bessel-Fourier transform.
2. Notations and Preliminaries
Throughout
this paper, we assume q∈]0,1[.
We recall some usual notions and notations used in the q-theory (see [4, 5]). We refer to the book by
Gasper and Rahman [4] for the definitions, notations, and properties of the q-shifted factorials and the q-hypergeometric functions.
We write ℝq={±qn:n∈ℤ}, ℝq,+={qn:n∈ℤ},
and[x]q=1−qx1−q,x∈ℂ,[n]q!=(q;q)n(1−q)n,n∈ℕ.The q-derivative of a function f is given by(Dqf)(x)=f(x)−f(qx)(1−q)xifx≠0,(Dqf)(0)=limk→+∞(Dqf)(qk),
provided that the limit exists.
The q-Jackson integrals from 0 to a and from 0 to ∞,
of a function f,
are (see [6])∫0af(x)dqx=(1−q)a∑n=0∞f(aqn)qn,∫0∞f(x)dqx=(1−q)∑n=−∞∞f(qn)qn,provided that the sums converge
absolutely.
The q-Jackson integral in a generic interval [a,b] is given by (see [6])∫abf(x)dqx=∫0bf(x)dqx−∫0af(x)dqx.The q -integration by parts rule is given, for
suitable functions f and g, by∫abg(x)Dqf(x)dqx=f(b)g(b)−f(a)g(a)−∫abf(qx)Dqg(x)dqx.
Jackson (see
[6]) defined a q-analogue of the Gamma function
byΓq(x)=(q;q)∞(qx;q)∞(1−q)1−x,x≠0,−1,−2,….
The third Jackson (2.7)
q-Bessel function (see [,]) isJν(z;q2)=zν(1−q2)νΓq2(ν+1)1φ1(0;q2ν+2;q2,q2z2), and the q-trigonometric functions (q-cosine and q-sine) are defined by (see [9])cos(x;q2)=Γq2(1/2)q(1+q−1)1/2x1/2J−1/2(1−qqx;q2)=∑n=0∞(−1)nqn(n−1)x2n[2n]q!,sin(x;q2)=Γq2(1/2)(1+q−1)1/2x1/2J1/2(1−qqx;q2)=∑n=0∞(−1)nqn(n−1)x2n+1[2n+1]q!.They verify Dqcos(x;q2)=−1qsin(qx;q2),Dqsin(x;q2)=cos(x;q2).We need the following spaces and
norms.
𝒮*q(ℝq) is the space of even functions f on ℝq satisfying ∀n,m∈N,Pn,m,q(f)=supx∈Rq;0≤k≤n|(1+x2)mDqkf(x)|<+∞.
Lqn(ℝq,+,x2ν+1dqx), n≥1, ν≥−1/2,
is the set of all functions defined on ℝq,+ such that∥f∥n,ν,q={∫0∞|f(x)|nx2ν+1dqx}1/n<∞.
Lqn(ℝq,+)=Lqn(ℝq,+,dqx), n≥1,
and ∥⋅∥n,q=∥⋅∥n,−1/2,q.
Lq∞(ℝq,+) is the set of all bounded functions on ℝq,+. We write ∥f∥∞,q=supx∈ℝq,+|f(x)|.
3. Generalization of the Heisenberg Uncertainty Principle
The q-Fourier-cosine and the q-Fourier-sine transforms are defined as (see
[8, 9])ℱq(f)(x)=cq∫0∞f(t)cos(xt;q2)dqt,ℱq(f)(x)=cq∫0∞f(t)sin(xt;q2)dqt,wherecq=(1+q−1)1/2Γq2(1/2).Letting q↑1 subject to the condition (Log(1−q)/Log(q))∈ℤ gives, at least formally, the classical
Fourier transforms (see [3, 10]). In the remainder of the present paper, we assume
that this condition holds.
It was shown in [8, 9] that we have
the following result.
Proposition 3.1.
(1) For f∈Lq1(ℝq,+),
one has ℱq(f)∈Lq∞(ℝq,+) and ∥ℱq(f)∥∞,q≤(1+q−1)1/2Γq2(1/2)(q;q)∞2∥f∥1,q.
(2) ℱq is an isomorphism of Lq2(ℝq,+) (resp., S*,q(ℝq)) onto itself. Moreover, one
has ℱq−1=ℱq and the following Plancherel formula:∥ℱq(f)∥2,q=∥f∥2,q,f∈Lq2(Rq,+).
Similarly, it was shown in [3, 8] that the q-Fourier-sine transform verifies the following
properties.
Proposition 3.2.
(1) For f∈Lq1(ℝq,+),
one has ℱq(f)∈Lq∞(ℝq,+) and ∥ℱq(f)∥∞,q≤(1+q−1)1/2Γq2(1/2)(q;q)∞2∥f∥1,q.
(2)ℱq is an isomorphism of Lq2(ℝq,+) onto itself; its inverse is given by ℱq−1=(1/q2)qℱ. One has the following Plancherel formula:∥qℱ(f)∥2,q=q∥f∥2,q,f∈Lq2(Rq,+).
Let us now
state the following useful density result.
Proposition 3.3.
For all n≥1, S*,q(ℝq) is dense in Lqn(ℝq,+).
Proof.
Let n≥1 and f∈Lqn(ℝq,+).
For p∈ℕ,
put fp=f⋅χ[qp,q−p],
where χ[qp,q−p] is the characteristic function of [qp,q−p].
It is clear that for all p∈ℕ, fp∈S*,q(ℝq) and |f−fp|n≤|f|n.
So, the Lebesgue theorem implies that (fp)p converges to f in Lqn(ℝq,+).
Remark 3.4.
Using the density of S*,q(ℝq) in Lqn(ℝq,+) (n≥1), one can see that the q-Fourier-cosine (resp., q-Fourier-sine) transform has a unique
continuous extension on Lqn(ℝq,+),
that will also be denoted as ℱq (resp., ℱq). We have the following q-analogue of the Hausdorff-Young inequality.
Theorem 3.5.
Let n∈]1,2] (resp., n=1) and m=n/(n−1) (resp., m=∞) be the dual exponent of n.
For all f in Lqn(ℝq,+),
the functions ℱq(f) and ℱq(f) belong to Lqm(ℝq,+),
and one has∥ℱq(f)∥m,q≤C1∥f∥n,q,∥ℱq(f)∥m,q≤C2∥f∥n,q,whereC1=((1+q−1)1/2Γq2(1/2)(q;q)∞2)1−2((n−1)/n),C2=((1+q−1)1/2Γq2(1/2)(q;q)∞2)1−2((n−1)/n)q2((n−1)/n).
Proof.
The result is a
direct consequence of [11, Theorem 1.3.4, page 35], and Propositions 3.1 and 3.2, by
taking S*,q(ℝq) as a set of simple functions.
The following lemma gives relations between the two
Fourier q-trigonometric transforms.
Lemma 3.6.
(1) For f∈Lq2(ℝq,+) such that Dqf∈Lq2(ℝq,+),
one hasℱq(Dqf)(λ)=−λqℱq(f)(λq),λ∈Rq,+.
(2) Additionally, if limn→+∞f(qn)=0,
thenℱq(Dqf)(λ)=λq2ℱq(f)(λ),λ∈Rq,+.
Proof.
The same steps as in the
proof of [3, Lemma 2]; the q-integration by parts rule and the fact that∫0∞f(t)dqt=limn→+∞∫qnq−nf(t)dqtgive the result.
In [3], the authors proved the following q-analogues of the Heisenberg uncertainty
principle.
Theorem 3.7.
Let f be in Lq2(ℝq,+) such that Dqf is in Lq2(ℝq,+).
Then,∥tf∥2,q∥λℱq(f)∥2,q≥qq3/2+1∥f∥2,q2.In addition, if limn→+∞f(qn)=0,
one has∥tf∥2,q∥λℱq(f)∥2,q≥qq−3/2+1∥f∥2,q2.
Now, we are in a position to generalize Theorem 3.7. One obvious way to
generalize it is to replace the Lq2 norms by Lqn norms. This is the purpose of the following
result.
Theorem 3.8.
For 1≤n≤2 and f∈Lq2(ℝq,+),
one has∥f∥2,q2≤C1′∥xf∥n,q∥λℱq(f)∥n,q,∥f∥2,q2≤C2′∥xf∥n,q∥λℱq(f)∥n,q,whereC1′=q−1+1/n(1+q−(n+1)/n)C2,C2′=q−1(1+q−(n+1)/n)C1,with C1 and C2 being given by (3.8).
Proof.
The case n=2 has been dealt with in Theorem 3.7. Now, assume 1≤n<2 and let m be the dual exponent of n.
Let f∈S*,q(ℝq) such that limt→0f(t)=0.
From the relationDq(ff¯)(t)=Dqf(t)f¯(t)+f(qt)Dqf¯(t),the q-integration by parts rule, and the Hölder
inequality, we have, since t|f(t)|2 tends to 0 as t tends to ∞ in ℝq,+,1q∫0∞|f(t)|2dqt=|∫0∞tDq(ff¯)(t)dqt|≤∫0∞|tDqf(t)f¯(t)|dqt+∫0∞|tf(qt)Dqf¯(t)|dqt≤(∫0∞|tf¯(t)|ndqt)1/n(∫0∞|Dqf(t)|mdqt)1/m+(∫0∞|tf(qt)|ndqt)1/n(∫0∞|Dqf¯(t)|mdqt)1/m.However, the change of variable u=qt gives(∫0∞|tf(qt)|ndqt)1/n=q−(n+1)/n(∫0∞|tf(t)|ndqt)1/n.So,1q∫0∞|f(t)|2dqt≤(1+q−(n+1)/n)∥tf∥n,q∥Dq(f)∥m,q.On the other hand, we have Dq(f)=ℱq[ℱq(Dq(f))]=q−2ℱq[ℱq(Dq(f))] since Dq(f) is in Lq2(ℝq,+).
Then, by using Lemma 3.6 and the q-analogue of the Hausdorff-Young inequality,
we obtain∥Dq(f)∥m,q≤C1∥ℱq(Dq(f))∥n,q=C1q2∥λℱq(f)∥n,q,∥Dq(f)∥m,q≤q−2C2∥qℱ(Dq(f))∥n,q=q−2C2∥λqℱq(f)(λq)∥n,q=q−2+1/nC2∥λℱq(f)∥n,q. Thus,∥f∥2,q2≤q−1(1+q−(n+1)/n)C1∥tf∥n,q∥λℱq(f)∥n,q,∥f∥2,q2≤q−1+1/n(1+q−(n+1)/n)C2∥tf∥n,q∥λℱq(f)∥n,q.Now, let f∈Lq2(ℝq,+);
it is easy to see that for all p∈ℕ, fp=fχ[qp,q−p]∈S*,q(ℝq), limt→0fp(t)=0,
and (fp)p converges to f in Lq2(ℝq,+).
Moreover, if the right-hand side of (3.14) (resp., (3.15)) is finite, then the
functions tf and λℱq(f) (resp., λℱq(f)) are in Lqn(ℝq,+), and they are limits in Lqn(ℝq,+) (as p tends to ∞) of tfp and λℱq(fp) (resp., λℱq(fp)), respectively. Finally, the substitution of fp in (3.22) and a passage to the limit
when p tends to ∞ complete the proof.
4. Local Uncertainty Principles
In the literature, the first classical local
inequalities were obtained by Faris (see [12]) in 1978, and they were
generalized by Price (see [13, 14]) in 1983 and 1987. In this section, we will
generalize Price's results by giving their q-analogues.
4.1. Local Uncertainty Principles for the q-Fourier Trigonometric Transforms
Theorem 4.1.
If 0<a<1/2,
there is a constant K=K(a,q) such that for all bounded subset E of ℝq,+ and all f∈Lq2(ℝq,+),
one has∫E|ℱq(f)(λ)|2dqλ≤K|E|2a∥xaf∥2,q2.Here, |E|=∫0∞χE(x)dqx and K=((c˜q/[1−2a]q)((1−2a)/2a))4a(1/(1−2a)2),
where c˜q=(1+q−1)1/2/Γq2(1/2)(q;q)∞2.
Proof.
For r>0,
let χr=χ[0,r] be the characteristic function of [0,r] and χ˜r=1−χr.
Then, for r>0,
we have, since f⋅χr∈Lq1(ℝq,+),(∫E|ℱq(f)(λ)|2dqλ)1/2=∥ℱq(f)χE∥2,q≤∥ℱq(f⋅χr)χE∥2,q+∥ℱq(f⋅χ˜r)χE∥2,q≤|E|1/2∥ℱq(f⋅χr)∥∞,q+∥ℱq(f⋅χ˜r)∥2,q,and by the use of the Hölder
inequality, we obtain∥ℱq(f⋅χr)∥∞,q≤c˜q∥f⋅χr∥1,q=c˜q∥x−aχr⋅xaf∥1,q≤c˜q∥x−aχr∥2,q∥xaf∥2,q≤c˜q[1−2a]qr1/2−a∥xaf∥2,q.On the other hand, since f∈Lq2(ℝq,+), we have f⋅χ˜r∈Lq2(ℝq,+),
and by the Plancherel formula, we get∥ℱq(f⋅χ˜r)∥2,q=∥f⋅χ˜r∥2,q=∥x−aχ˜r.xaf∥2,q≤∥x−aχ˜r∥∞,q∥xaf∥2,q≤r−a∥xaf∥2,q.So,(∫E|ℱq(f)(λ)|2dqλ)1/2≤(c˜q[1−2a]q|E|1/2r1/2−a+r−a)∥xaf∥2,q.The desired result is obtained
by minimizing the right-hand side of the previous inequality over r>0.
Corollary 4.2.
For 0<a<1/2 and b>0,
there is a constant Ka,b such that for all f∈Lq2(ℝq,+),
one has ∥f∥2,q(a+b)≤Ka,b∥xaf∥2,qb∥λbℱq(f)∥2,qa.
Proof.
For r>0,
put Er=[0,r[∩ℝq,+ and E˜r=[r,+∞[∩ℝq,+.
It is easy to see that Er is a bounded subset of ℝq,+ and |Er|≤r.
Then, from the Plancherel formula and Theorem 4.1, we
have∥f∥2,q2=∥ℱq(f)∥2,q2=∫Er|ℱq(f)|2(λ)dqλ+∫E˜r|ℱq(f)|2(λ)dqλ≤Kr2a∥xaf∥2,q2+r−2b∥λbℱq(f)∥2,q2.Choosing r>0 so as to minimize the right-hand side of the
inequality, we obtain ∥f∥2,q2≤(Ka,b∥xaf∥2,qb∥λbℱq(f)∥2,qa)2/(a+b),
with Ka,b=((a/b)b/(a+b)+(b/a)a/(a+b))(a+b)/2Kb/2,
and K is the constant given in Theorem 4.1.
In the same way, one can prove the following local
uncertainty principle for the q-Fourier-sine transform.
Theorem 4.3.
If 0<a<1/2,
there is a constant K′=K′(a,q) such that for all bounded subset E of ℝq,+ and all f∈Lq2(ℝq,+),
one has∫E|qℱ(f)(λ)|2dqλ≤K′|E|2a∥xaf∥2,q2,where K′=((c˜q/[1−2a]q)((1−2a)/2qa))4a[1+2qa/(1−2a)]2.
Corollary 4.4.
For 0<a<1/2 and b>0,
there is a constant Ka,b′ such that for all f∈Lq2(ℝq,+),
one has∥f∥2,q(a+b)≤Ka,b′∥xaf∥2,qb∥λbFq(f)∥2,qa,with Ka,b′=((a/b)b/(a+b)+(b/a)a/(a+b))(a+b)/2(K′)b/2q−(a+b).
Proof.
The same steps of Corollary 4.2 give the
result.
Theorem 4.5.
If a>1/2,
there is a constant K1=K1(a,q) such that for all bounded subset E of ℝq,+ and f∈Lq2(ℝq,+),
one has
The proof of this result needs the following lemmas.
Lemma 4.6.
Suppose a>1/2,
then for all f∈Lq2(ℝq,+), such that xaf∈Lq2(ℝq,+),∥f∥1,q2≤K2[∥f∥2,q2+∥xaf∥2,q2],where K2=K2(a,q)=(1−q)((q2a,q2a,−q,−q2a−1;q2a)∞/(q,q2a−1,−q2a,−1;q2a)∞).
Proof.
From
[15, Example 1], and the Hölder inequality, we have∥f∥1,q2=[∫0+∞(1+x2a)1/2|f(x)|(1+x2a)−1/2dqx]2≤K2[∥f∥2,q2+∥xaf∥2,q2],where K2=∫0+∞(1+x2a)−1dqx=(1−q)((q2a,q2a,−q,−q2a−1;q2a)∞/(q,q2a−1,−q2a,−1;q2a)∞).
Lemma 4.7.
Suppose a>1/2,
then for all f∈Lq2(ℝq,+),
such that xaf∈Lq2(ℝq,+),
one has∥f∥1,q≤K3∥f∥2,q(1−1/2a)∥xaf∥2,q1/2a,where K3=K3(a,q)=[2aK2(2aq−q)1/2a−1]1/2.
Proof.
For s∈ℝq,+, define the function fs by fs(x)=f(sx),x∈ℝq,+.
We have ∥fs∥1,q=s−1∥f∥1,q, ∥xafs∥2,q2=s−2a−1∥xaf∥2,q2.
Replacement of f by fs in Lemma 4.6 gives∥f∥1,q2≤K2[s∥f∥2,q2+s−2a+1∥xaf∥2,q2].Now, for all r>0, put α(r)=Log(r)/Log(q)−E(Log(r)/Log(q)). We have s=(r/qα(r))∈ℝq,+ and r≤s<r/q. Then, for all r>0,∥f∥1,q2≤K2[rq∥f∥2,q2+r−2a+1∥xaf∥2,q2].The right-hand side of this
inequality is minimized by choosingr=(2a−1)1/2aq1/2a∥f∥2,q−1/a∥xaf∥2,q1/a.When this is done, we obtain the
result.
Proof of Theorem 4.5.
Since the proofs of the two statements are similar, it is sufficient to
prove (4.11).
Let E be a bounded subset of ℝq,+.
When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that f∈Lq1(ℝq,+);
so ℱq(f) is defined and bounded on ℝq,+.
Using Proposition 3.1, Lemma 4.7, and the fact that∫E|ℱq(f)(λ)|2dqλ≤|E|∥ℱq(f)∥∞,q2,we obtain the result with K1=((1+q−1)/Γq22(1/2)(q;q)∞4)K32.
Remark 4.8.
By the same technique as in the proof of Corollary 4.2,
we can show that Theorem 4.5 leads to inequalities (4.6) and (4.9) with some
different constants.
4.2. Local Uncertainty Principles for the q-Bessel-Fourier Transform
The q-Bessel-Fourier transform is defined (see
[16]) for f∈Lq1(ℝq,+,x2ν+1dqx) byℱν,q(f)(λ)=cν,q∫0∞f(x)jν(λx;q2)x2ν+1dqx,wherejν(z;q2)=(1−q2)νΓq2(ν+1)((1−q)q−1z)−νJν((1−q)q−1z;q2)is the normalized third Jackson q-Bessel function, andcν,q=(1+q−1)−νΓq2(ν+1).It was shown in [10] that for ν≥−1/2,
we have the following result.
Theorem 4.9.
(1) For f∈Lq1(ℝq,+,x2ν+1dqx),
one has ℱν,q(f)∈Lq∞(ℝq,+) and∥ℱν,q(f)∥∞,q≤cν,q(q;q2)∞2∥f∥1,ν,q.
(2) ℱν,q is an isomorphism of Lq2(ℝq,+,x2ν+1dqx) onto itself, ℱν,q−1=q4ν+2ℱν,q,
and one has the following Plancherel formula:∀f∈Lq2(Rq,+,x2ν+1dqx),∥ℱν,q∥2,ν,q=q2ν+1∥f∥2,ν,q.
The following
result states a local uncertainty principle for the q-Bessel-Fourier transform.
Theorem 4.10.
For ν≥−1/2 and 0<a<ν+1,
there is a constant Ka,ν=K(a,ν,q) such that for all f∈Lq2(ℝq,+,x2ν+1dqx) and all bounded subset E of ℝq,+,
one has∫E|ℱν,q(f)(λ)|2λ2ν+1dqλ≤Ka,ν|E|νa/(ν+1)∥xaf∥2,ν,q2.Here, |E|ν=∫0∞χE(x)x2ν+1dqx, c˜ν,q=cν,q/(q;q2)∞2,
and
Ka,ν=(c˜ν,q[2ν+2−2a]q)2a/(ν+1)[(aq2ν+1ν+1−a)1−a/(ν+1)+q2ν+1(aq2ν+1ν+1−a)−a/(ν+1)]2.
Proof
Let (4.25)
ν≥−1/2, (4.25)
0<a<ν+1, (4.25)
f∈Lq2(Rq,+,x2ν+1dqx),
and let (4.25)
E be a bounded subset of (4.25)
Rq,+.
For (4.25)
r>0,
we have, since (4.25)
f.χr∈Lq1(Rq,+,x2ν+1dqx),(∫E|ℱν,q(f)(λ)|2λ2ν+1dqλ)1/2=∥ℱν,q(f)χE∥2,ν,q≤∥ℱν,q(f⋅χr)χE∥2,ν,q+∥ℱν,q(f⋅χ˜r)χE∥2,ν,q≤|E|ν1/2∥ℱν,q(f⋅χr)∥∞,q+∥ℱν,q(f⋅χ˜r)∥2,ν,q.
Proof.
Let ν≥−1/2, 0<a<ν+1, f∈Lq2(ℝq,+,x2ν+1dqx),
and let E be a bounded subset of ℝq,+.
For r>0,
we have, since f.χr∈Lq1(ℝq,+,x2ν+1dqx),∥ℱν,q(f⋅χr)∥∞,q≤c˜ν,q∥f⋅χr∥1,q=c˜q∥x−aχr.xaf∥1,ν,q≤c˜ν,q∥x−aχr∥2,ν,q∥xaf∥2,ν,q.However, by the use of the
Hölder inequality, we obtain∥x−aχr∥2,ν,q2=∫0∞x−2aχr(x)x2ν+1dqx=∫0qkx2ν+1−2adqx=q2k(ν+1−a)[2ν+2−2a]q≤r2(ν+1−a)[2ν+2−2a]q.Now, if k is the integer such that qk≤r<qk−1,
we get, since a<ν+1,∥ℱν,q(f⋅χr)∥∞,q≤c˜ν,q[2ν+2−2a]qr(ν+1−a)∥xaf∥2,ν,q.Then,∥ℱν,q(f⋅χ˜r)∥2,ν,q=q2ν+1∥f⋅χ˜r∥2,ν,q=q2ν+1∥x−aχ˜r⋅xaf∥2,ν,q≤q2ν+1∥x−aχ˜r∥∞,q∥xaf∥2,q≤q2ν+1r−a∥xaf∥2,ν,q.On the other hand, since f∈Lq2(ℝq,+,x2ν+1dqx),
we have f.χ˜r∈Lq2(ℝq,+,x2ν+1dqx), and by the Plancherel formula (4.23), we
obtain(∫E|ℱν,q(f)(λ)|2λ2ν+1dqλ)1/2≤(c˜ν,q[2ν+2−2a]q|E|ν1/2r(ν+1−a)+q2ν+1r−a)∥xaf∥2,ν,q.So,∫E|ℱν,q(f)(λ)|2λ2ν+1dqλ≤Ka,ν′|E|∥f∥2,ν,q2(1−(ν+1)/a)∥xaf∥2,ν,q2((ν+1)/a).By minimization of the
right-hand side of the previous inequality over r>0 and by easy computation, we obtain the desired
result.
Theorem 4.11.
For ν≥−1/2 and a>ν+1,
there exists a constant Ka,ν′ such that for all bounded subset E of ℝq,+ and all f in Lq2(ℝq,+,x2ν+1dqx),
one hasKa,ν′=(q2a,q2a,−q2ν+2,−q2(a−ν−1);q2a)∞(q2ν+2,q2(a−ν−1),−q2a,−1;q2a)∞cν,q′,cν,q′=(1−q)(cν,q(q;q2)∞2)2(aν+1−1)(ν+1)/a(aa−ν−1)q−2(ν+1)((a−ν−1)/a).
Proof.
Since a>ν+1,
the same steps as in the proof of Theorem 4.5 and the relation (4.22) give the
result with∥f∥2,ν,q(a+b)≤Ka,b,ν∥xaf∥2,ν,qb∥λbℱν,q(f)∥2,ν,qa,
Corollary 4.12.
For ν≥−1/2 and a,b>0,
there is a constant Ka,b,ν=K(a,b,ν,q) such that for all f∈Lq2(ℝq,+,x2ν+1dqx),
one hasKa,b,ν={[(ba)a/(a+b)+(ab)b/(a+b)](a+b)/2(Ka,ν)b/2q−(2ν+1)(a+b)([2ν+2]q)ab/2(ν+1)ifa<ν+1,(Ka,ν′[2ν+2]q)ab/(2ν+2)(q−(4ν+2)[(bν+1)(ν+1)/(ν+b+1)+(bν+1)−b/(ν+b+1)])a(ν+b+1)/2(ν+1)ifa>ν+1,with q4ν+2∥f∥2,ν,q2=∥ℱν,q(f)∥2,ν,q2=∫Er|ℱν,q(f)|2(λ)λ2ν+1dqλ+∫E˜r|ℱν,q(f)|2(λ)λ2ν+1dqλ≤{Ka,ν|Er|νa/(ν+1)∥xaf∥2,ν,q2+r−2b∥λbℱν,q(f)∥2,ν,q2ifa<ν+1,Ka,ν′|Er|∥f∥2,ν,q2(a−ν−1)/a∥xaf∥2,ν,q2(ν+1)/a+r−2b∥λbℱν,q(f)∥2,ν,q2ifa>ν+1,≤{Ka,ν[2ν+2]qa/(ν+1)r2a∥xaf∥2,ν,q2+r−2b∥λbℱν,q(f)∥2,ν,q2ifa<ν+1,Ka,ν′r2ν+2[2ν+2]q∥f∥2,ν,q2(a−ν−1)/a∥xaf∥2,ν,q2(ν+1)/a+r−2b∥λbℱν,q(f)∥2,ν,q2ifa>ν+1. where Ka,ν (resp., Ka,ν′) is the constant given in Theorem 4.10 (resp.,
Theorem 4.11).
Proof.
For r>0,
we put Er=[0,r[∩ℝq,+ and E˜r=[r,+∞[∩ℝq,+.
We have Er is a bounded subset of ℝq,+ and |Er|ν≤r2ν+2/[2ν+2]q. Then, the Plancherel formula (4.23) and Theorems
4.10 and 4.11 lead to ∥f∥2,ν,q2≤K1,1,ν∥xf∥2,ν,q∥λℱν,q(f)∥2,ν,q.The desired result follows by
minimizing the right expressions over r>0.
Remark that when a=b=1,
we obtain a Heisenberg-Weyl-type inequality for the q-Bessel-Fourier transform.
Corollary 4.13.
For ν≥−1/2,ν≠0,
one has for all f∈Lq2(ℝq,+,x2ν+1dqx),∥f∥2,ν,q2≤K1,1,ν∥xf∥2,ν,q∥λℱν,q(f)∥2,ν,q.
Acknowledgment
The authors would like to
thank the reviewers for their helpful remarks and constructive criticism.
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