On Some Inequalities of Uncertainty Principles Type in Quantum Calculus

Abstract

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 ℝ_𝑞={±𝑞^𝑛∶𝑛∈ℤ}, ℝ_𝑞,+={𝑞^𝑛∶𝑛∈ℤ}, and[𝑥]_𝑞=1−𝑞^𝑥1−𝑞,𝑥∈ℂ,[𝑛]^𝑞!=(𝑞;𝑞)^𝑛(1−𝑞)^𝑛,𝑛∈ℕ.(2.1)The 𝑞-derivative of a function 𝑓 is given by𝐷_𝑞𝑓(𝑥)=𝑓(𝑥(1)−−𝑞𝑓)(𝑥𝑞𝑥)if𝑥≠0,(2.2)(𝐷_𝑞𝑓)(0)=lim_𝑘→+∞(𝐷_𝑞𝑓)(𝑞^𝑘), provided that the limit exists.

The 𝑞-Jackson integrals from 0 to 𝑎 and from 0 to ∞, of a function 𝑓, are (see [6])^𝑎0𝑓(𝑥)𝑑_𝑞𝑥=(1−𝑞)𝑎^∞_𝑛=0𝑓𝑎𝑞^𝑛𝑞^𝑛,^∞0𝑓(𝑥)𝑑_𝑞𝑥=(1−𝑞)^∞_𝑛_=−∞𝑓𝑞^𝑛𝑞^𝑛,(2.3)provided that the sums converge absolutely.

The 𝑞-Jackson integral in a generic interval [𝑎,𝑏] is given by (see [6])^𝑏𝑎𝑓(𝑥)𝑑_𝑞𝑥=^𝑏0𝑓(𝑥)𝑑_𝑞𝑥−^𝑎0𝑓(𝑥)𝑑_𝑞𝑥.(2.4)The 𝑞 -integration by parts rule is given, for suitable functions 𝑓 and 𝑔, by^𝑏𝑎𝑔(𝑥)𝐷_𝑞𝑓(𝑥)𝑑_𝑞𝑥=𝑓(𝑏)𝑔(𝑏)−𝑓(𝑎)𝑔(𝑎)−^𝑏𝑎𝑓(𝑞𝑥)𝐷_𝑞𝑔(𝑥)𝑑_𝑞𝑥.(2.5)

Jackson (see [6]) defined a 𝑞-analogue of the Gamma function byΓ_𝑞(𝑥)=(𝑞;𝑞)^∞𝑞^𝑥;𝑞_∞(1−𝑞)^1−𝑥,𝑥≠0,−1,−2,….(2.6)

The third Jackson (2.6)𝑞-Bessel function (see [,]) is𝐽_𝜈𝑧;𝑞²=𝑧^𝜈(1−𝑞²)^𝜈Γ_𝑞2(𝜈+1)¹𝜑₁0;𝑞^2𝜈+2;𝑞²,𝑞²𝑧²,(2.6) and the 𝑞-trigonometric functions (𝑞-cosine and 𝑞-sine) are defined by (see [9])cos𝑥;𝑞²=Γ^𝑞2(1/2)𝑞(1+𝑞⁻¹)^1/2𝑥^1/2𝐽_−1/21−𝑞𝑞𝑥;𝑞²=^∞_𝑛=0(−1)^𝑛𝑞^{𝑛(𝑛−1)}𝑥^2𝑛sin𝑥;𝑞₂[2𝑛]_𝑞!,=Γ^𝑞2(1/2)(1+𝑞⁻¹)^1/2𝑥^1/2𝐽_1/21−𝑞𝑞𝑥;𝑞²=^∞_𝑛=0(−1)^𝑛𝑞^{𝑛(𝑛−1)}𝑥^2𝑛+1[2𝑛+1]_𝑞!.(2.8)They verify 𝐷_𝑞cos𝑥;𝑞²=−1𝑞sin𝑞𝑥;𝑞²,𝐷_𝑞sin𝑥;𝑞²=cos𝑥;𝑞².(2.9)We need the following spaces and norms.

(i)𝒮_∗𝑞(ℝ_𝑞) is the space of even functions 𝑓 on ℝ_𝑞 satisfying ∀𝑛,𝑚∈ℕ,𝑃_{𝑛,𝑚,𝑞}(𝑓)=sup_{𝑥∈ℝ𝑞;0≤𝑘≤𝑛}||1+𝑥²^𝑚𝐷^𝑘𝑞𝑓(𝑥)||<+∞.(2.10)(ii)𝐿^𝑛𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), 𝑛≥1, 𝜈≥−1/2, is the set of all functions defined on ℝ_𝑞,+ such that‖𝑓‖_{𝑛,𝜈,𝑞}=^∞0||𝑓(𝑥)||^𝑛𝑥^2𝜈+1𝑑_𝑞𝑥^1/𝑛<∞.(2.11)(iii)𝐿^𝑛𝑞(ℝ_𝑞,+)=𝐿^𝑛𝑞(ℝ_𝑞,+,𝑑_𝑞𝑥), 𝑛≥1, and ‖⋅‖_𝑛,𝑞=‖⋅‖_{𝑛,−1/2,𝑞}.(iv)𝐿^∞𝑞(ℝ_𝑞,+) is the set of all bounded functions on ℝ_𝑞,+. We write ‖𝑓‖_∞,𝑞=sup_{𝑥∈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])ℱ_𝑞(𝑓)(𝑥)=𝑐_𝑞^∞0𝑓(𝑡)cos𝑥𝑡;𝑞²𝑑_𝑞𝑡,_𝑞ℱ(𝑓)(𝑥)=𝑐_𝑞^∞0𝑓(𝑡)sin𝑥𝑡;𝑞²𝑑_𝑞𝑡,(3.1)where𝑐_𝑞=(1+𝑞⁻¹)^1/2Γ_𝑞2(1/2).(3.2)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∈𝐿^1𝑞(ℝ_𝑞,+), one has ℱ_𝑞(𝑓)∈𝐿^∞𝑞(ℝ_𝑞,+) and ‖‖ℱ_𝑞(𝑓)‖‖_∞,𝑞≤(1+𝑞⁻¹)^1/2Γ_𝑞2(1/2)(𝑞;𝑞)^2∞‖𝑓‖_1,𝑞.(3.3)
(2) ℱ_𝑞 is an isomorphism of 𝐿^2𝑞(ℝ_𝑞,+) (resp., 𝑆_∗,𝑞(ℝ_𝑞)) onto itself. Moreover, one has ℱ^𝑞−1=ℱ_𝑞 and the following Plancherel formula: ‖ℱ_𝑞(𝑓)‖_2,𝑞=‖𝑓‖_2,𝑞,𝑓∈𝐿^2𝑞(ℝ_𝑞,+).(3.4)

Similarly, it was shown in [3, 8] that the 𝑞-Fourier-sine transform verifies the following properties.

Proposition 3.2. (1) For 𝑓∈𝐿^1𝑞(ℝ_𝑞,+), one has _𝑞ℱ(𝑓)∈𝐿^∞𝑞(ℝ_𝑞,+) and ‖_𝑞ℱ(𝑓)‖_∞,𝑞≤(1+𝑞⁻¹)^1/2Γ_𝑞2(1/2)(𝑞;𝑞)^2∞‖𝑓‖_1,𝑞.(3.5)
(2)_𝑞ℱ is an isomorphism of 𝐿^2𝑞(ℝ_𝑞,+) onto itself; its inverse is given by _𝑞ℱ⁻¹=(1/𝑞²)_𝑞ℱ. One has the following Plancherel formula: ‖_𝑞ℱ(𝑓)‖_2,𝑞=𝑞‖𝑓‖_2,𝑞,𝑓∈𝐿^2𝑞(ℝ_𝑞,+).(3.6)

Let us now state the following useful density result.

Proposition 3.3. For all 𝑛≥1, 𝑆_∗,𝑞(ℝ_𝑞) is dense in 𝐿^𝑛𝑞(ℝ_𝑞,+).

Proof. Let 𝑛≥1 and 𝑓∈𝐿^𝑛𝑞(ℝ_𝑞,+). For 𝑝∈ℕ, put 𝑓_𝑝=𝑓⋅𝜒_{[𝑞𝑝,𝑞−𝑝]}, where 𝜒_{[𝑞𝑝,𝑞−𝑝]} is the characteristic function of [𝑞^𝑝,𝑞^−𝑝].
It is clear that for all 𝑝∈ℕ, 𝑓_𝑝∈𝑆_∗,𝑞(ℝ_𝑞) and |𝑓−𝑓_𝑝|^𝑛≤|𝑓|^𝑛. So, the Lebesgue theorem implies that (𝑓_𝑝)_𝑝 converges to 𝑓 in 𝐿^𝑛𝑞(ℝ_𝑞,+).

Remark 3.4. Using the density of 𝑆_∗,𝑞(ℝ_𝑞) in 𝐿^𝑛𝑞(ℝ_𝑞,+) (𝑛≥1), one can see that the 𝑞-Fourier-cosine (resp., 𝑞-Fourier-sine) transform has a unique continuous extension on 𝐿^𝑛𝑞(ℝ_𝑞,+), that will also be denoted as ℱ_𝑞 (resp., _𝑞ℱ). 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 𝐿^𝑛𝑞(ℝ_𝑞,+), the functions ℱ_𝑞(𝑓) and _𝑞ℱ(𝑓) belong to 𝐿^𝑚𝑞(ℝ_𝑞,+), and one has ‖ℱ_𝑞(𝑓)‖_𝑚,𝑞≤𝐶₁‖𝑓‖_𝑛,𝑞,‖_𝑞ℱ(𝑓)‖_𝑚,𝑞≤𝐶₂‖𝑓‖_𝑛,𝑞,(3.7)where 𝐶₁=(1+𝑞⁻¹)^1/2Γ_𝑞2(1/2)(𝑞;𝑞)^2∞^{1−2((𝑛−1)/𝑛)},𝐶₂=(1+𝑞⁻¹)^1/2Γ_𝑞2(1/2)(𝑞;𝑞)^2∞^{1−2((𝑛−1)/𝑛)}𝑞^{2((𝑛−1)/𝑛)}.(3.8)

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(ℝ_𝑞) as a set of simple functions.

The following lemma gives relations between the two Fourier 𝑞-trigonometric transforms.

Lemma 3.6. (1) For 𝑓∈𝐿^2𝑞(ℝ_𝑞,+) such that 𝐷_𝑞𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has _𝑞ℱ𝐷_𝑞𝑓(𝜆)=−𝜆𝑞ℱ^𝑞(𝑓)𝜆𝑞,𝜆∈ℝ^𝑞,+.(3.9)
(2) Additionally, if lim_𝑛→+∞𝑓(𝑞^𝑛)=0, then ℱ_𝑞𝐷_𝑞𝑓(𝜆)=𝜆𝑞^2𝑞ℱ(𝑓)(𝜆),𝜆∈ℝ_𝑞,+.(3.10)

Proof. The same steps as in the proof of [3, Lemma 2]; the 𝑞-integration by parts rule and the fact that^∞0𝑓(𝑡)𝑑_𝑞𝑡=lim_𝑛→+∞^𝑞−𝑛_𝑞𝑛𝑓(𝑡)𝑑_𝑞𝑡(3.11)give the result.

In [3], the authors proved the following 𝑞-analogues of the Heisenberg uncertainty principle.

Theorem 3.7. Let 𝑓 be in 𝐿^2𝑞(ℝ_𝑞,+) such that 𝐷_𝑞𝑓 is in 𝐿^2𝑞(ℝ_𝑞,+). Then, ‖𝑡𝑓‖_2,𝑞‖𝜆ℱ_𝑞(𝑓)‖_2,𝑞≥𝑞𝑞^3/2+1‖𝑓‖^22,𝑞.(3.12)In addition, if lim_𝑛→+∞𝑓(𝑞^𝑛)=0, one has ‖𝑡𝑓‖_2,𝑞‖𝜆_𝑞ℱ(𝑓)‖_2,𝑞≥𝑞𝑞^−3/2+1‖𝑓‖^22,𝑞.(3.13)

Now, we are in a position to generalize Theorem 3.7. One obvious way to generalize it is to replace the 𝐿^2𝑞 norms by 𝐿^𝑛𝑞 norms. This is the purpose of the following result.

Theorem 3.8. For 1≤𝑛≤2 and 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has ‖‖𝑓𝑓‖‖^22,𝑞_22,𝑞≤𝐶^1‖𝑥𝑓‖_𝑛,𝑞‖𝜆ℱ_𝑞(𝑓)‖_𝑛,𝑞,(3.14)≤𝐶^2‖𝑥𝑓‖_𝑛,𝑞‖𝜆_𝑞ℱ(𝑓)‖_𝑛,𝑞,(3.15)where 𝐶^1=𝑞^−1+1/𝑛1+𝑞^{−(𝑛+1)/𝑛}𝐶₂,𝐶^2=𝑞⁻¹1+𝑞^{−(𝑛+1)/𝑛}𝐶₁,(3.16)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(ℝ_𝑞) such that lim_𝑡→0𝑓(𝑡)=0. From the relation𝐷_𝑞𝑓𝑓(𝑡)=𝐷_𝑞𝑓(𝑡)𝑓(𝑡)+𝑓(𝑞𝑡)𝐷_𝑞𝑓(𝑡),(3.17)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 ℝ_𝑞,+,1𝑞^∞0||𝑓(𝑡)||²𝑑_𝑞𝑡=≤|||^∞0_∞0𝑡𝐷_𝑞(𝑓𝑓)(𝑡)𝑑_𝑞𝑡|||≤_∞0||𝑡𝐷_𝑞𝑓(𝑡)𝑓(𝑡)||𝑑_𝑞𝑡+^∞0||𝑡𝑓(𝑞𝑡)𝐷_𝑞𝑓(𝑡)||𝑑_𝑞𝑡||𝑡𝑓(𝑡)||^𝑛𝑑_𝑞𝑡^1/𝑛^∞0||𝐷_𝑞𝑓(𝑡)||^𝑚𝑑_𝑞𝑡^1/𝑚+^∞0||𝑡𝑓(𝑞𝑡)||^𝑛𝑑_𝑞𝑡^1/𝑛^∞0||𝐷_𝑞𝑓(𝑡)||^𝑚𝑑_𝑞𝑡^1/𝑚.(3.18)However, the change of variable 𝑢=𝑞𝑡 gives^∞0||𝑡𝑓(𝑞𝑡)||^𝑛𝑑_𝑞𝑡^1/𝑛=𝑞^{−(𝑛+1)/𝑛}^∞0||𝑡𝑓(𝑡)||^𝑛𝑑_𝑞𝑡^1/𝑛.(3.19)So,1𝑞^∞0||𝑓(𝑡)||²𝑑_𝑞𝑡≤1+𝑞^{−(𝑛+1)/𝑛}‖𝑡𝑓‖_𝑛,𝑞‖𝐷_𝑞(𝑓)‖_𝑚,𝑞.(3.20)On the other hand, we have 𝐷_𝑞(𝑓)=ℱ_𝑞[ℱ_𝑞(𝐷_𝑞(𝑓))]=𝑞^𝑞−2ℱ[_𝑞ℱ(𝐷_𝑞(𝑓))] since 𝐷_𝑞(𝑓) is in 𝐿^2𝑞(ℝ_𝑞,+). Then, by using Lemma 3.6 and the 𝑞-analogue of the Hausdorff-Young inequality, we obtain‖‖𝐷_𝑞(𝑓)‖‖_𝑚,𝑞≤𝐶₁‖‖ℱ_𝑞𝐷_𝑞(𝑓)‖‖_𝑛,𝑞=𝐶¹𝑞²‖‖𝐷_𝑞‖‖𝜆_𝑞ℱ(𝑓)‖‖_𝑛,𝑞,(𝑓)‖‖_𝑚,𝑞≤𝑞⁻²𝐶₂‖‖_𝑞ℱ𝐷_𝑞(𝑓)‖‖_𝑛,𝑞=𝑞⁻²𝐶₂‖‖‖𝜆𝑞ℱ^𝑞(𝑓)𝜆𝑞‖‖‖_𝑛,𝑞=𝑞^−2+1/𝑛𝐶₂‖‖𝜆ℱ_𝑞(𝑓)‖‖_𝑛,𝑞.(3.21) Thus,‖‖𝑓𝑓‖‖^22,𝑞_22,𝑞≤𝑞⁻¹1+𝑞^{−(𝑛+1)/𝑛}𝐶₁‖𝑡𝑓‖_𝑛,𝑞‖‖𝜆_𝑞ℱ(𝑓)‖‖_𝑛,𝑞,(3.22)≤𝑞^−1+1/𝑛1+𝑞^{−(𝑛+1)/𝑛}𝐶₂‖𝑡𝑓‖_𝑛,𝑞‖‖𝜆ℱ_𝑞(𝑓)‖‖_𝑛,𝑞.(1)Now, let 𝑓∈𝐿^2𝑞(ℝ_𝑞,+); it is easy to see that for all 𝑝∈ℕ, 𝑓_𝑝=𝑓𝜒_{[𝑞𝑝,𝑞−𝑝]}∈𝑆_∗,𝑞(ℝ_𝑞), lim_𝑡→0𝑓_𝑝(𝑡)=0, and (𝑓_𝑝)_𝑝 converges to 𝑓 in 𝐿^2𝑞(ℝ_𝑞,+). Moreover, if the right-hand side of (3.14) (resp., (3.15)) is finite, then the functions 𝑡𝑓 and 𝜆ℱ_𝑞(𝑓) (resp., 𝜆_𝑞ℱ(𝑓)) are in 𝐿^𝑛𝑞(ℝ_𝑞,+), and they are limits in 𝐿^𝑛𝑞(ℝ_𝑞,+) (as 𝑝 tends to ∞) of 𝑡𝑓_𝑝 and 𝜆ℱ_𝑞(𝑓_𝑝) (resp., 𝜆_𝑞ℱ(𝑓_𝑝)), respectively. Finally, the substitution of 𝑓_𝑝 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 ℝ_𝑞,+ and all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has _𝐸||ℱ_𝑞(𝑓)(𝜆)||²𝑑_𝑞𝜆≤𝐾|𝐸|^2𝑎‖‖𝑥^𝑎𝑓‖‖^22,𝑞.(4.1)Here, |𝐸|=∫^∞0𝜒_𝐸(𝑥)𝑑_𝑞𝑥 and 𝐾=((𝑐_𝑞/√[1−2𝑎]_𝑞)((1−2𝑎)/2𝑎))^4𝑎(1/(1−2𝑎)²), where 𝑐_𝑞=(1+𝑞⁻¹)^1/2/Γ_𝑞2(1/2)(𝑞;𝑞)^2∞.Proof. For 𝑟>0, let 𝜒_𝑟=𝜒_[0,𝑟] be the characteristic function of [0,𝑟] and 𝜒_𝑟=1−𝜒_𝑟.
Then, for 𝑟>0, we have, since 𝑓⋅𝜒_𝑟∈𝐿^1𝑞(ℝ_𝑞,+),_𝐸||ℱ_𝑞(𝑓)(𝜆)||²𝑑_𝑞𝜆^1/2=‖‖ℱ_𝑞(𝑓)𝜒_𝐸‖‖_2,𝑞≤‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)𝜒_𝐸‖‖_2,𝑞+‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)𝜒_𝐸‖‖_2,𝑞≤|𝐸|^1/2‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)‖‖_∞,𝑞+‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)‖‖_2,𝑞,(4.2)and by the use of the Hölder inequality, we obtain‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)‖‖_∞,𝑞≤𝑐_𝑞‖‖𝑓⋅𝜒_𝑟‖‖_1,𝑞=𝑐_𝑞‖‖𝑥^−𝑎𝜒_𝑟⋅𝑥^𝑎𝑓‖‖_1,𝑞≤𝑐_𝑞‖‖𝑥^−𝑎𝜒_𝑟‖‖_2,𝑞‖‖𝑥^𝑎𝑓‖‖_2,𝑞≤𝑐^𝑞√[1−2𝑎]_𝑞𝑟^1/2−𝑎‖‖𝑥^𝑎𝑓‖‖_2,𝑞.(4.3)On the other hand, since 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), we have 𝑓⋅𝜒_𝑟∈𝐿^2𝑞(ℝ_𝑞,+), and by the Plancherel formula, we get‖‖ℱ_𝑞(𝑓⋅𝜒_𝑟)‖‖_2,𝑞=‖‖𝑓⋅𝜒_𝑟‖‖_2,𝑞=‖‖𝑥^−𝑎𝜒_𝑟.𝑥^𝑎𝑓‖‖_2,𝑞≤‖‖𝑥^−𝑎𝜒_𝑟‖‖_∞,𝑞‖‖𝑥^𝑎𝑓‖‖_2,𝑞≤𝑟^−𝑎‖‖𝑥^𝑎𝑓‖‖_2,𝑞.(4.4)So,_𝐸||ℱ_𝑞(𝑓)(𝜆)||²𝑑_𝑞𝜆^1/2≤𝑐^𝑞√[1−2𝑎]_𝑞|𝐸|^1/2𝑟^1/2−𝑎+𝑟^−𝑎‖‖𝑥^𝑎𝑓‖‖_2,𝑞.(4.5)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 𝐾_𝑎,𝑏 such that for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has ‖𝑓‖^{(𝑎+𝑏)2,𝑞}≤𝐾_𝑎,𝑏‖‖𝑥^𝑎𝑓‖‖^𝑏2,𝑞‖‖𝜆^𝑏ℱ_𝑞(𝑓)‖‖^𝑎2,𝑞.(4.6)

Proof. For 𝑟>0, put 𝐸_𝑟=[0,𝑟[∩ℝ_𝑞,+ and 𝐸_𝑟=[𝑟,+∞[∩ℝ_𝑞,+. It is easy to see that 𝐸_𝑟 is a bounded subset of ℝ_𝑞,+ and |𝐸_𝑟|≤𝑟.
Then, from the Plancherel formula and Theorem 4.1, we have‖𝑓‖^22,𝑞=‖‖ℱ_𝑞(𝑓)‖‖^22,𝑞≤=𝐾𝑟_𝐸𝑟||_2𝑎ℱ_𝑞(𝑓)||²(𝜆)𝑑_𝑞𝜆+_𝐸𝑟||ℱ_𝑞(𝑓)||²(𝜆)𝑑_𝑞𝜆‖‖𝑥^𝑎𝑓‖‖^22,𝑞+𝑟^−2𝑏‖‖𝜆^𝑏ℱ_𝑞(𝑓)‖‖^22,𝑞.(4.7)Choosing 𝑟>0 so as to minimize the right-hand side of the inequality, we obtain ‖𝑓‖^22,𝑞≤(𝐾_𝑎,𝑏‖𝑥^𝑎𝑓‖^𝑏2,𝑞‖𝜆^𝑏ℱ_𝑞(𝑓)‖^𝑎2,𝑞)^{2/(𝑎+𝑏)}, with 𝐾_𝑎,𝑏=((𝑎/𝑏)^{𝑏/(𝑎+𝑏)}+(𝑏/𝑎)^{𝑎/(𝑎+𝑏)})^{(𝑎+𝑏)/2}𝐾^𝑏/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 ℝ_𝑞,+ and all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has _𝐸||_𝑞ℱ(𝑓)(𝜆)||²𝑑_𝑞𝜆≤𝐾^|𝐸|^2𝑎‖‖𝑥^𝑎𝑓‖‖^22,𝑞,(4.8)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 𝐾^𝑎,𝑏 such that for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has ‖𝑓‖^{(𝑎+𝑏)2,𝑞}≤𝐾^𝑎,𝑏‖𝑥^𝑎𝑓‖^𝑏2,𝑞‖𝜆^𝑏𝑞ℱ(𝑓)‖^𝑎2,𝑞,(4.9)with 𝐾^𝑎,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 ℝ_𝑞,+ and 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has
_𝐸||ℱ_𝑞(𝑓)(𝜆)||²𝑑_𝑞𝜆≤𝐾₁|𝐸|‖𝑓‖^{(2−1/𝑎)2,𝑞}‖‖𝑥^𝑎𝑓‖‖^1/𝑎2,𝑞,(4.10)_𝐸||ℱ_𝑞(𝑓)(𝜆)||²𝑑_𝑞𝜆≤𝐾₁|𝐸|‖𝑓‖^{(2−1/𝑎)2,𝑞}‖‖𝑥^𝑎𝑓‖‖^1/𝑎2,𝑞.(4.11)

The proof of this result needs the following lemmas.

Lemma 4.6. Suppose 𝑎>1/2, then for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), such that 𝑥^𝑎𝑓∈𝐿^2𝑞(ℝ_𝑞,+), ‖𝑓‖^21,𝑞≤𝐾₂‖𝑓‖^22,𝑞+‖‖𝑥^𝑎𝑓‖‖^22,𝑞,(4.12)where 𝐾₂=𝐾₂(𝑎,𝑞)=(1−𝑞)((𝑞^2𝑎,𝑞^2𝑎,−𝑞,−𝑞^2𝑎−1;𝑞^2𝑎)_∞/(𝑞,𝑞^2𝑎−1,−𝑞^2𝑎,−1;𝑞^2𝑎)_∞).

Proof. From [15, Example 1], and the Hölder inequality, we have‖𝑓‖^21,𝑞=^0+∞1+𝑥^2𝑎^1/2|𝑓(𝑥)|1+𝑥^2𝑎^−1/2𝑑_𝑞𝑥²≤𝐾₂[‖𝑓‖^22,𝑞+‖𝑥^𝑎𝑓‖^22,𝑞],(4.13)where 𝐾₂=∫^0+∞(1+𝑥^2𝑎)⁻¹𝑑_𝑞𝑥=(1−𝑞)((𝑞^2𝑎,𝑞^2𝑎,−𝑞,−𝑞^2𝑎−1;𝑞^2𝑎)_∞/(𝑞,𝑞^2𝑎−1,−𝑞^2𝑎,−1;𝑞^2𝑎)_∞).

Lemma 4.7. Suppose 𝑎>1/2, then for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+), such that 𝑥^𝑎𝑓∈𝐿^2𝑞(ℝ_𝑞,+), one has ‖𝑓‖_1,𝑞≤𝐾₃‖𝑓‖^{(1−1/2𝑎)2,𝑞}‖‖𝑥^𝑎𝑓‖‖^{1/2𝑎2,𝑞},(4.14)where 𝐾₃=𝐾₃(𝑎,𝑞)=[2𝑎𝐾₂(2𝑎𝑞−𝑞)^1/2𝑎−1]^1/2.

Proof. For 𝑠∈ℝ_𝑞,+, define the function 𝑓_𝑠 by 𝑓_𝑠(𝑥)=𝑓(𝑠𝑥),𝑥∈ℝ_𝑞,+.
We have ‖𝑓_𝑠‖_1,𝑞=𝑠⁻¹‖𝑓‖_1,𝑞, ‖𝑥^𝑎𝑓_𝑠‖^22,𝑞=𝑠^−2𝑎−1‖𝑥^𝑎𝑓‖^22,𝑞.
Replacement of 𝑓 by 𝑓_𝑠 in Lemma 4.6 gives‖𝑓‖^21,𝑞≤𝐾₂[𝑠‖𝑓‖^22,𝑞+𝑠^−2𝑎+1‖𝑥^𝑎𝑓‖^22,𝑞].(4.15)Now, for all 𝑟>0, put 𝛼(𝑟)=Log(𝑟)/Log(𝑞)−𝐸(Log(𝑟)/Log(𝑞)). We have 𝑠=(𝑟/𝑞^𝛼(𝑟))∈ℝ_𝑞,+ and 𝑟≤𝑠<𝑟/𝑞. Then, for all 𝑟>0,‖𝑓‖^21,𝑞≤𝐾₂𝑟𝑞‖𝑓‖^22,𝑞+𝑟^−2𝑎+1‖‖𝑥^𝑎𝑓‖‖^22,𝑞.(4.16)The right-hand side of this inequality is minimized by choosing𝑟=(2𝑎−1)^1/2𝑎𝑞^1/2𝑎‖𝑓‖^{−1/𝑎2,𝑞}‖‖𝑥^𝑎𝑓‖‖^1/𝑎2,𝑞.(4.17)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 ℝ_𝑞,+. When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that 𝑓∈𝐿^1𝑞(ℝ_𝑞,+); so ℱ_𝑞(𝑓) is defined and bounded on ℝ_𝑞,+. Using Proposition 3.1, Lemma 4.7, and the fact that_𝐸|ℱ_𝑞(𝑓)(𝜆)|²𝑑_𝑞𝜆≤|𝐸|‖ℱ_𝑞(𝑓)‖^2∞,𝑞,(4.18)we obtain the result with 𝐾₁=((1+𝑞⁻¹)/Γ^2𝑞₂(1/2)(𝑞;𝑞)^4∞)𝐾²³.

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∈𝐿^1𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥) byℱ_𝜈,𝑞(𝑓)(𝜆)=𝑐_𝜈,𝑞^∞0𝑓(𝑥)𝑗_𝜈𝜆𝑥;𝑞²𝑥^2𝜈+1𝑑_𝑞𝑥,(4.19)where𝑗_𝜈𝑧;𝑞²=1−𝑞²^𝜈Γ_𝑞2(𝜈+1)(1−𝑞)𝑞⁻¹𝑧^−𝜈𝐽_𝜈(1−𝑞)𝑞⁻¹𝑧;𝑞²(4.20)is the normalized third Jackson 𝑞-Bessel function, and𝑐_𝜈,𝑞=1+𝑞⁻¹^−𝜈Γ_𝑞2(𝜈+1).(4.21)It was shown in [10] that for 𝜈≥−1/2, we have the following result.

Theorem 4.9. (1) For 𝑓∈𝐿^1𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), one has ℱ_𝜈,𝑞(𝑓)∈𝐿^∞𝑞(ℝ_𝑞,+) and ‖‖ℱ_𝜈,𝑞(𝑓)‖‖_∞,𝑞≤𝑐^𝜈,𝑞(𝑞;𝑞²)^2∞‖𝑓‖_1,𝜈,𝑞.(4.22)
(2) ℱ_𝜈,𝑞 is an isomorphism of 𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥) onto itself, ℱ^{−1𝜈,𝑞}=𝑞^4𝜈+2ℱ_𝜈,𝑞, and one has the following Plancherel formula: ∀𝑓∈𝐿^2𝑞ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥,‖‖ℱ_𝜈,𝑞‖‖_2,𝜈,𝑞=𝑞^2𝜈+1‖𝑓‖_2,𝜈,𝑞.(4.23)
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 𝐾_𝑎,𝜈=𝐾(𝑎,𝜈,𝑞) such that for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥) and all bounded subset 𝐸 of ℝ_𝑞,+, one has _𝐸||ℱ_𝜈,𝑞(𝑓)(𝜆)||²𝜆^2𝜈+1𝑑_𝑞𝜆≤𝐾_𝑎,𝜈|𝐸|^{𝜈𝑎/(𝜈+1)}‖‖𝑥^𝑎𝑓‖‖^22,𝜈,𝑞.(4.24)Here, |𝐸|_𝜈=∫^∞0𝜒_𝐸(𝑥)𝑥^2𝜈+1𝑑_𝑞𝑥, 𝑐_𝜈,𝑞=𝑐_𝜈,𝑞/(𝑞;𝑞²)^2∞, and 𝐾_𝑎,𝜈=𝑐^𝜈,𝑞√[2𝜈+2−2𝑎]_𝑞^{2𝑎/(𝜈+1)}𝜈𝑎+𝑞1−^2𝜈+1𝑎^{1−𝑎/(𝜈+1)}+𝑞^2𝜈+1𝜈𝑎+𝑞1−^2𝜈+1𝑎^{−𝑎/(𝜈+1)}².(4.25)

ProofLet (4.25)𝜈≥−1/2, (4.25)0<𝑎<𝜈+1, (4.25)𝑓∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), and let (4.25)𝐸 be a bounded subset of (4.25)ℝ_𝑞,+.
For (4.25)𝑟>0, we have, since (4.25)𝑓.𝜒_𝑟∈𝐿^1𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥),_𝐸||ℱ_𝜈,𝑞(𝑓)(𝜆)||²𝜆^2𝜈+1𝑑_𝑞𝜆^1/2=‖‖ℱ_𝜈,𝑞(𝑓)𝜒_𝐸‖‖_2,𝜈,𝑞≤‖‖ℱ_𝜈,𝑞(𝑓⋅𝜒_𝑟)𝜒_𝐸‖‖_2,𝜈,𝑞+‖‖ℱ_𝜈,𝑞(𝑓⋅𝜒_𝑟)𝜒_𝐸‖‖_2,𝜈,𝑞≤|𝐸|^𝜈1/2‖‖ℱ_𝜈,𝑞(𝑓⋅𝜒_𝑟)‖‖_∞,𝑞+‖‖ℱ_𝜈,𝑞(𝑓⋅𝜒_𝑟)‖‖_2,𝜈,𝑞.(4.25)

Proof. Let 𝜈≥−1/2, 0<𝑎<𝜈+1, 𝑓∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), and let 𝐸 be a bounded subset of ℝ_𝑞,+.
For 𝑟>0, we have, since 𝑓.𝜒_𝑟∈𝐿^1𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥),‖‖ℱ_𝜈,𝑞𝑓⋅𝜒_𝑟‖‖_∞,𝑞≤𝑐_𝜈,𝑞‖‖𝑓⋅𝜒_𝑟‖‖_1,𝑞=𝑐_𝑞‖‖𝑥^−𝑎𝜒_𝑟.𝑥^𝑎𝑓‖‖_1,𝜈,𝑞≤𝑐_𝜈,𝑞‖‖𝑥^−𝑎𝜒_𝑟‖‖_2,𝜈,𝑞‖‖𝑥^𝑎𝑓‖‖_2,𝜈,𝑞.(4.26)However, by the use of the Hölder inequality, we obtain‖‖𝑥^−𝑎𝜒_𝑟‖‖^22,𝜈,𝑞=^∞0𝑥^−2𝑎𝜒_𝑟(𝑥)𝑥^2𝜈+1𝑑_𝑞𝑥=^𝑞𝑘₀𝑥^{2𝜈+1−2𝑎}𝑑_𝑞𝑥=𝑞^{2𝑘(𝜈+1−𝑎)}[2𝜈+2−2𝑎]_𝑞≤𝑟^{2(𝜈+1−𝑎)}[2𝜈+2−2𝑎]_𝑞.(4.27)Now, if 𝑘 is the integer such that 𝑞^𝑘≤𝑟<𝑞^𝑘−1, we get, since 𝑎<𝜈+1,‖‖ℱ_𝜈,𝑞(𝑓⋅𝜒_𝑟)‖‖_∞,𝑞≤𝑐^𝜈,𝑞√[2𝜈+2−2𝑎]_𝑞𝑟^{(𝜈+1−𝑎)}‖‖𝑥^𝑎𝑓‖‖_2,𝜈,𝑞.(4.28)Then,‖‖ℱ_𝜈,𝑞𝑓⋅𝜒_𝑟‖‖_2,𝜈,𝑞=𝑞^2𝜈+1‖‖𝑓⋅𝜒_𝑟‖‖_2,𝜈,𝑞=𝑞^2𝜈+1‖‖𝑥^−𝑎𝜒_𝑟⋅𝑥^𝑎𝑓‖‖_2,𝜈,𝑞≤𝑞^2𝜈+1‖‖𝑥^−𝑎𝜒_𝑟‖‖_∞,𝑞‖‖𝑥^𝑎𝑓‖‖_2,𝑞≤𝑞^2𝜈+1𝑟^−𝑎‖‖𝑥^𝑎𝑓‖‖_2,𝜈,𝑞.(4.29)On the other hand, since 𝑓∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), we have 𝑓.𝜒_𝑟∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), and by the Plancherel formula (4.23), we obtain_𝐸||ℱ_𝜈,𝑞(𝑓)(𝜆)||²𝜆^2𝜈+1𝑑_𝑞𝜆^1/2≤𝑐^𝜈,𝑞√[2𝜈+2−2𝑎]_𝑞|𝐸|^𝜈1/2𝑟^{(𝜈+1−𝑎)}+𝑞^2𝜈+1𝑟^−𝑎‖‖𝑥^𝑎𝑓‖‖_2,𝜈,𝑞.(4.30)So,_𝐸||ℱ_𝜈,𝑞(𝑓)(𝜆)||²𝜆^2𝜈+1𝑑_𝑞𝜆≤𝐾^𝑎,𝜈|𝐸|‖𝑓‖^{2(1−(𝜈+1)/𝑎)2,𝜈,𝑞}‖‖𝑥^𝑎𝑓‖‖^{2((𝜈+1)/𝑎)2,𝜈,𝑞}.(4.31)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 𝐾^𝑎,𝜈 such that for all bounded subset 𝐸 of ℝ_𝑞,+ and all 𝑓 in 𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), one has 𝐾^𝑎,𝜈=(𝑞^2𝑎,𝑞^2𝑎,−𝑞^2𝜈+2,−𝑞^{2(𝑎−𝜈−1)};𝑞^2𝑎)_∞(𝑞^2𝜈+2,𝑞^{2(𝑎−𝜈−1)},−𝑞^2𝑎,−1;𝑞^2𝑎)_∞𝑐_𝜈,𝑞𝑐^𝜈,𝑞,=(1−𝑞)𝑐^𝜈,𝑞(𝑞;𝑞²)^2∞²𝜈+𝑎1−1^{(𝜈+1)/𝑎}𝑎−𝑎𝜈−1𝑞^{−2(𝜈+1)((𝑎−𝜈−1)/𝑎)}.(4.32)

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‖^{(𝑎+𝑏)2,𝜈,𝑞}≤𝐾_{𝑎,𝑏,𝜈}‖𝑥^𝑎𝑓‖^{𝑏2,𝜈,𝑞}‖𝜆^𝑏ℱ_𝜈,𝑞(𝑓)‖^{𝑎2,𝜈,𝑞},(4.33)

Corollary 4.12. For 𝜈≥−1/2 and 𝑎,𝑏>0, there is a constant 𝐾_{𝑎,𝑏,𝜈}=𝐾(𝑎,𝑏,𝜈,𝑞) such that for all 𝑓∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), one has 𝐾_{𝑎,𝑏,𝜈}=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑏𝑎^{𝑎/(𝑎+𝑏)}+𝑎𝑏^{𝑏/(𝑎+𝑏)}^{(𝑎+𝑏)/2}(𝐾_𝑎,𝜈)^𝑏/2𝑞^{−(2𝜈+1)(𝑎+𝑏)}([2𝜈+2]_𝑞)^{𝑎𝑏/2(𝜈+1)}𝐾if𝑎<𝜈+1,_𝑎,𝜈[2𝜈+2]_𝑞^{𝑎𝑏/(2𝜈+2)}𝑞^−(4𝜈+2)𝜈+𝑏1^{(𝜈+1)/(𝜈+𝑏+1)}+𝑏𝜈+1^{−𝑏/(𝜈+𝑏+1)}^{𝑎(𝜈+𝑏+1)/2(𝜈+1)}if𝑎>𝜈+1,(4.34)with 𝑞^4𝜈+2‖𝑓‖^22,𝜈,𝑞=≤‖‖ℱ𝐾_𝜈,𝑞(_𝑎,𝜈𝑓)‖‖^22,𝜈,𝑞=_𝐸𝑟||ℱ_𝜈,𝑞(𝑓)||²(𝜆)𝜆^2𝜈+1𝑑_𝑞𝜆+_𝐸𝑟||ℱ_𝜈,𝑞(𝑓)||²(𝜆)𝜆^2𝜈+1𝑑_𝑞𝜆𝐾||𝐸_𝑟||^{𝜈𝑎/(𝜈+1)}‖‖𝑥^𝑎𝑓‖‖^22,𝜈,𝑞+𝑟^−2𝑏‖‖𝜆^𝑏ℱ_𝜈,𝑞(𝑓)‖‖^22,𝜈,𝑞if𝑎<𝜈+1,_𝑎,𝜈≤⎧⎪⎪⎨⎪⎪⎩𝐾||𝐸_𝑟||‖𝑓‖^{2(𝑎−𝜈−1)/𝑎2,𝜈,𝑞}‖‖𝑥^𝑎𝑓‖‖^{2(𝜈+1)/𝑎2,𝜈,𝑞}+𝑟^−2𝑏‖‖𝜆^𝑏ℱ_𝜈,𝑞(𝑓)‖‖^22,𝜈,𝑞if𝑎>𝜈+1,_𝑎,𝜈[2𝜈+2]^{𝑞𝑎/(𝜈+1)}𝐾𝑟^2𝑎‖‖𝑥^𝑎𝑓‖‖^22,𝜈,𝑞+𝑟^−2𝑏‖‖𝜆^𝑏ℱ_𝜈,𝑞(𝑓)‖‖^22,𝜈,𝑞if𝑎<𝜈+1,_𝑎,𝜈𝑟^2𝜈+2[2𝜈+2]_𝑞‖𝑓‖^{2(𝑎−𝜈−1)/𝑎2,𝜈,𝑞}‖‖𝑥^𝑎𝑓‖‖^{2(𝜈+1)/𝑎2,𝜈,𝑞}+𝑟^−2𝑏‖‖𝜆^𝑏ℱ_𝜈,𝑞(𝑓)‖‖^22,𝜈,𝑞if𝑎>𝜈+1.(4.35) where 𝐾_𝑎,𝜈 (resp., 𝐾^𝑎,𝜈) is the constant given in Theorem 4.10 (resp., Theorem 4.11).

Proof. For 𝑟>0, we put 𝐸_𝑟=[0,𝑟[∩ℝ_𝑞,+ and 𝐸_𝑟=[𝑟,+∞[∩ℝ_𝑞,+.
We have 𝐸_𝑟 is a bounded subset of ℝ_𝑞,+ and |𝐸_𝑟|_𝜈≤𝑟^2𝜈+2/[2𝜈+2]_𝑞. Then, the Plancherel formula (4.23) and Theorems 4.10 and 4.11 lead to ‖𝑓‖^22,𝜈,𝑞≤𝐾_1,1,𝜈‖𝑥𝑓‖_2,𝜈,𝑞‖‖𝜆ℱ_𝜈,𝑞(𝑓)‖‖_2,𝜈,𝑞.(4.36)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∈𝐿^2𝑞(ℝ_𝑞,+,𝑥^2𝜈+1𝑑_𝑞𝑥), ‖𝑓‖^22,𝜈,𝑞≤𝐾_1,1,𝜈‖𝑥𝑓‖_2,𝜈,𝑞‖𝜆ℱ_𝜈,𝑞(𝑓)‖_2,𝜈,𝑞.(4.37)

Acknowledgment

The authors would like to thank the reviewers for their helpful remarks and constructive criticism.