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International Journal of Mathematics and Mathematical Sciences
Volumeย 2008ย (2008), Article IDย 465909, 13 pages
http://dx.doi.org/10.1155/2008/465909
Research Article

On Some Inequalities of Uncertainty Principles Type in Quantum Calculus

1Facultรฉ des Sciences de Tunis, Universitรฉ de Tunis El Manar, 1060 Tunis, Tunisia
2Institut Prรฉparatoire aux ร‰tudes d'Ingรฉnieur de Monastir, Universitรฉ de Monastir, 5000 Monastir, Tunisia
3Institut de Biotechnologie, Universitรฉ de Jendouba, 9000 Bรฉja, Tunisia

Received 19 July 2007; Revised 31 January 2008; Accepted 14 April 2008

Academic Editor: Wolfgangย Castell

Copyright ยฉ 2008 Ahmed Fitouhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to generalize the ๐‘ž-Heisenberg uncertainty principles studied by Bettaibi et al. (2007), to state local uncertainty principles for the ๐‘ž-Fourier-cosine, the ๐‘ž-Fourier-sine, and the ๐‘ž-Bessel-Fourier transforms, then to provide an inequality of Heisenberg-Weyl-type for the ๐‘ž-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 Jรถricke [2] for various classical uncertainty principles of different nature which may be found in the literature.

In [3], the authors gave ๐‘ž-analogues of the Heisenberg uncertainty principle for the ๐‘ž-Fourier-cosine and the ๐‘ž-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 ๐‘ž-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 ๐‘ž-analogue of the Hausdorff-Young inequality. Then, we state a generalization of the ๐‘ž-Heisenberg uncertainty principle for the ๐‘ž-Fourier-cosine and the ๐‘ž-Fourier-sine transforms. In Section 4, we state local uncertainty principles for the ๐‘ž-Fourier-cosine, ๐‘ž-Fourier-sine, and ๐‘ž-Bessel-Fourier transforms. Then, we give a Heisenberg-Weyl-type inequality for some ๐‘ž-Bessel-Fourier transform.

2. Notations and Preliminaries

Throughout this paper, we assume ๐‘žโˆˆ]0,1[. We recall some usual notions and notations used in the ๐‘ž-theory (see [4, 5]). We refer to the book by Gasper and Rahman [4] for the definitions, notations, and properties of the ๐‘ž-shifted factorials and the ๐‘ž-hypergeometric functions.

We write โ„๐‘ž={ยฑ๐‘ž๐‘›โˆถ๐‘›โˆˆโ„ค}, โ„๐‘ž,+={๐‘ž๐‘›โˆถ๐‘›โˆˆโ„ค}, and[๐‘ฅ]๐‘ž=1โˆ’๐‘ž๐‘ฅ1โˆ’๐‘ž,๐‘ฅโˆˆโ„‚,[๐‘›]๐‘ž!=(๐‘ž;๐‘ž)๐‘›(1โˆ’๐‘ž)๐‘›,๐‘›โˆˆโ„•.(2.1)The ๐‘ž-derivative of a function ๐‘“ is given by๎€ท๐ท๐‘ž๐‘“๎€ธ(๐‘ฅ)=๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ž๐‘ฅ)(1โˆ’๐‘ž)๐‘ฅif๐‘ฅโ‰ 0,(2.2)(๐ท๐‘ž๐‘“)(0)=lim๐‘˜โ†’+โˆž(๐ท๐‘ž๐‘“)(๐‘ž๐‘˜), provided that the limit exists.

The ๐‘ž-Jackson integrals from 0 to ๐‘Ž and from 0 to โˆž, of a function ๐‘“, are (see [6])๎€œ๐‘Ž0๐‘“(๐‘ฅ)๐‘‘๐‘ž๐‘ฅ=(1โˆ’๐‘ž)๐‘Žโˆž๎“๐‘›=0๐‘“๎€ท๐‘Ž๐‘ž๐‘›๎€ธ๐‘ž๐‘›,๎€œโˆž0๐‘“(๐‘ฅ)๐‘‘๐‘ž๐‘ฅ=(1โˆ’๐‘ž)โˆž๎“๐‘›=โˆ’โˆž๐‘“๎€ท๐‘ž๐‘›๎€ธ๐‘ž๐‘›,(2.3)provided that the sums converge absolutely.

The ๐‘ž-Jackson integral in a generic interval [๐‘Ž,๐‘] is given by (see [6])๎€œ๐‘๐‘Ž๐‘“(๐‘ฅ)๐‘‘๐‘ž๎€œ๐‘ฅ=๐‘0๐‘“(๐‘ฅ)๐‘‘๐‘ž๎€œ๐‘ฅโˆ’๐‘Ž0๐‘“(๐‘ฅ)๐‘‘๐‘ž๐‘ฅ.(2.4)The ๐‘ž -integration by parts rule is given, for suitable functions ๐‘“ and ๐‘”, by๎€œ๐‘๐‘Ž๐‘”(๐‘ฅ)๐ท๐‘ž๐‘“(๐‘ฅ)๐‘‘๐‘ž๎€œ๐‘ฅ=๐‘“(๐‘)๐‘”(๐‘)โˆ’๐‘“(๐‘Ž)๐‘”(๐‘Ž)โˆ’๐‘๐‘Ž๐‘“(๐‘ž๐‘ฅ)๐ท๐‘ž๐‘”(๐‘ฅ)๐‘‘๐‘ž๐‘ฅ.(2.5)

Jackson (see [6]) defined a ๐‘ž-analogue of the Gamma function byฮ“๐‘ž(๐‘ฅ)=(๐‘ž;๐‘ž)โˆž๎€ท๐‘ž๐‘ฅ๎€ธ;๐‘žโˆž(1โˆ’๐‘ž)1โˆ’๐‘ฅ,๐‘ฅโ‰ 0,โˆ’1,โˆ’2,โ€ฆ.(2.6)

The third Jackson (2.6)๐‘ž-Bessel function (see [,]) is๐ฝ๐œˆ๎€ท๐‘ง;๐‘ž2๎€ธ=๐‘ง๐œˆ(1โˆ’๐‘ž2)๐œˆฮ“๐‘ž2(๐œˆ+1)1๐œ‘1๎€ท0;๐‘ž2๐œˆ+2;๐‘ž2,๐‘ž2๐‘ง2๎€ธ,(2.6) and the ๐‘ž-trigonometric functions (๐‘ž-cosine and ๐‘ž-sine) are defined by (see [9])๎€ทcos๐‘ฅ;๐‘ž2๎€ธ=ฮ“๐‘ž2(1/2)๐‘ž(1+๐‘žโˆ’1)1/2๐‘ฅ1/2๐ฝโˆ’1/2๎‚ต1โˆ’๐‘ž๐‘ž๐‘ฅ;๐‘ž2๎‚ถ=โˆž๎“๐‘›=0(โˆ’1)๐‘›๐‘ž๐‘›(๐‘›โˆ’1)๐‘ฅ2๐‘›[2๐‘›]๐‘ž!,๎€ทsin๐‘ฅ;๐‘ž2๎€ธ=ฮ“๐‘ž2(1/2)(1+๐‘žโˆ’1)1/2๐‘ฅ1/2๐ฝ1/2๎‚ต1โˆ’๐‘ž๐‘ž๐‘ฅ;๐‘ž2๎‚ถ=โˆž๎“๐‘›=0(โˆ’1)๐‘›๐‘ž๐‘›(๐‘›โˆ’1)๐‘ฅ2๐‘›+1[2๐‘›+1]๐‘ž!.(2.8)They verify ๐ท๐‘ž๎€ทcos๐‘ฅ;๐‘ž2๎€ธ1=โˆ’๐‘ž๎€ทsin๐‘ž๐‘ฅ;๐‘ž2๎€ธ,๐ท๐‘ž๎€ทsin๐‘ฅ;๐‘ž2๎€ธ๎€ท=cos๐‘ฅ;๐‘ž2๎€ธ.(2.9)We need the following spaces and norms.

(i)๐’ฎโˆ—๐‘ž(โ„๐‘ž) is the space of even functions ๐‘“ on โ„๐‘ž satisfying โˆ€๐‘›,๐‘šโˆˆโ„•,๐‘ƒ๐‘›,๐‘š,๐‘ž(๐‘“)=sup๐‘ฅโˆˆโ„๐‘ž;0โ‰ค๐‘˜โ‰ค๐‘›||๎€ท1+๐‘ฅ2๎€ธ๐‘š๐ท๐‘˜๐‘ž||๐‘“(๐‘ฅ)<+โˆž.(2.10)(ii)๐ฟ๐‘›๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), ๐‘›โ‰ฅ1, ๐œˆโ‰ฅโˆ’1/2, is the set of all functions defined on โ„๐‘ž,+ such thatโ€–๐‘“โ€–๐‘›,๐œˆ,๐‘ž=๎‚†๎€œโˆž0||||๐‘“(๐‘ฅ)๐‘›๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ๎‚‡1/๐‘›<โˆž.(2.11)(iii)๐ฟ๐‘›๐‘ž(โ„๐‘ž,+)=๐ฟ๐‘›๐‘ž(โ„๐‘ž,+,๐‘‘๐‘ž๐‘ฅ), ๐‘›โ‰ฅ1, and โ€–โ‹…โ€–๐‘›,๐‘ž=โ€–โ‹…โ€–๐‘›,โˆ’1/2,๐‘ž.(iv)๐ฟโˆž๐‘ž(โ„๐‘ž,+) is the set of all bounded functions on โ„๐‘ž,+. We write โ€–๐‘“โ€–โˆž,๐‘ž=sup๐‘ฅโˆˆโ„๐‘ž,+|๐‘“(๐‘ฅ)|.

3. Generalization of the Heisenberg Uncertainty Principle

The ๐‘ž-Fourier-cosine and the ๐‘ž-Fourier-sine transforms are defined as (see [8, 9])โ„ฑ๐‘ž(๐‘“)(๐‘ฅ)=๐‘๐‘ž๎€œโˆž0๎€ท๐‘“(๐‘ก)cos๐‘ฅ๐‘ก;๐‘ž2๎€ธ๐‘‘๐‘ž๐‘ก,๐‘žโ„ฑ(๐‘“)(๐‘ฅ)=๐‘๐‘ž๎€œโˆž0๎€ท๐‘“(๐‘ก)sin๐‘ฅ๐‘ก;๐‘ž2๎€ธ๐‘‘๐‘ž๐‘ก,(3.1)where๐‘๐‘ž=(1+๐‘žโˆ’1)1/2ฮ“๐‘ž2(1/2).(3.2)Letting ๐‘žโ†‘1 subject to the condition (Log(1โˆ’๐‘ž)/Log(๐‘ž))โˆˆโ„ค 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 ๐‘“โˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+), one has โ„ฑ๐‘ž(๐‘“)โˆˆ๐ฟโˆž๐‘ž(โ„๐‘ž,+) and โ€–โ€–โ„ฑ๐‘žโ€–โ€–(๐‘“)โˆž,๐‘žโ‰ค(1+๐‘žโˆ’1)1/2ฮ“๐‘ž2(1/2)(๐‘ž;๐‘ž)2โˆžโ€–๐‘“โ€–1,๐‘ž.(3.3)
(2) โ„ฑ๐‘ž is an isomorphism of ๐ฟ2๐‘ž(โ„๐‘ž,+) (resp., ๐‘†โˆ—,๐‘ž(โ„๐‘ž)) onto itself. Moreover, one has โ„ฑ๐‘žโˆ’1=โ„ฑ๐‘ž and the following Plancherel formula: โ€–โ„ฑ๐‘ž(๐‘“)โ€–2,๐‘ž=โ€–๐‘“โ€–2,๐‘ž,๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+).(3.4)

Similarly, it was shown in [3, 8] that the ๐‘ž-Fourier-sine transform verifies the following properties.

Proposition 3.2. (1) For ๐‘“โˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+), one has ๐‘žโ„ฑ(๐‘“)โˆˆ๐ฟโˆž๐‘ž(โ„๐‘ž,+) and โ€–๐‘žโ„ฑ(๐‘“)โ€–โˆž,๐‘žโ‰ค(1+๐‘žโˆ’1)1/2ฮ“๐‘ž2(1/2)(๐‘ž;๐‘ž)2โˆžโ€–๐‘“โ€–1,๐‘ž.(3.5)
(2)๐‘žโ„ฑ is an isomorphism of ๐ฟ2๐‘ž(โ„๐‘ž,+) onto itself; its inverse is given by ๐‘žโ„ฑโˆ’1=(1/๐‘ž2)๐‘žโ„ฑ. One has the following Plancherel formula: โ€–๐‘žโ„ฑ(๐‘“)โ€–2,๐‘ž=๐‘žโ€–๐‘“โ€–2,๐‘ž,๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+).(3.6)

Let us now state the following useful density result.

Proposition 3.3. For all ๐‘›โ‰ฅ1, ๐‘†โˆ—,๐‘ž(โ„๐‘ž) is dense in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+).

Proof. Let ๐‘›โ‰ฅ1 and ๐‘“โˆˆ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+). For ๐‘โˆˆโ„•, put ๐‘“๐‘=๐‘“โ‹…๐œ’[๐‘ž๐‘,๐‘žโˆ’๐‘], where ๐œ’[๐‘ž๐‘,๐‘žโˆ’๐‘] is the characteristic function of [๐‘ž๐‘,๐‘žโˆ’๐‘].
It is clear that for all ๐‘โˆˆโ„•, ๐‘“๐‘โˆˆ๐‘†โˆ—,๐‘ž(โ„๐‘ž) and |๐‘“โˆ’๐‘“๐‘|๐‘›โ‰ค|๐‘“|๐‘›. So, the Lebesgue theorem implies that (๐‘“๐‘)๐‘ converges to ๐‘“ in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+).

Remark 3.4. Using the density of ๐‘†โˆ—,๐‘ž(โ„๐‘ž) in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+) (๐‘›โ‰ฅ1), one can see that the ๐‘ž-Fourier-cosine (resp., ๐‘ž-Fourier-sine) transform has a unique continuous extension on ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+), that will also be denoted as โ„ฑ๐‘ž (resp., ๐‘žโ„ฑ). We have the following ๐‘ž-analogue of the Hausdorff-Young inequality.

Theorem 3.5. Let ๐‘›โˆˆ]1,2] (resp., ๐‘›=1) and ๐‘š=๐‘›/(๐‘›โˆ’1) (resp., ๐‘š=โˆž) be the dual exponent of ๐‘›. For all ๐‘“ in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+), the functions โ„ฑ๐‘ž(๐‘“) and ๐‘žโ„ฑ(๐‘“) belong to ๐ฟ๐‘š๐‘ž(โ„๐‘ž,+), and one has โ€–โ„ฑ๐‘ž(๐‘“)โ€–๐‘š,๐‘žโ‰ค๐ถ1โ€–๐‘“โ€–๐‘›,๐‘ž,โ€–๐‘žโ„ฑ(๐‘“)โ€–๐‘š,๐‘žโ‰ค๐ถ2โ€–๐‘“โ€–๐‘›,๐‘ž,(3.7)where ๐ถ1=๎‚ต(1+๐‘žโˆ’1)1/2ฮ“๐‘ž2(1/2)(๐‘ž;๐‘ž)2โˆž๎‚ถ1โˆ’2((๐‘›โˆ’1)/๐‘›),๐ถ2=๎‚ต(1+๐‘žโˆ’1)1/2ฮ“๐‘ž2(1/2)(๐‘ž;๐‘ž)2โˆž๎‚ถ1โˆ’2((๐‘›โˆ’1)/๐‘›)๐‘ž2((๐‘›โˆ’1)/๐‘›).(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 ๐‘†โˆ—,๐‘ž(โ„๐‘ž) as a set of simple functions.

The following lemma gives relations between the two Fourier ๐‘ž-trigonometric transforms.

Lemma 3.6. (1) For ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+) such that ๐ท๐‘ž๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has ๐‘žโ„ฑ๎€ท๐ท๐‘ž๐‘“๎€ธ๐œ†(๐œ†)=โˆ’๐‘žโ„ฑ๐‘ž๎‚ต๐œ†(๐‘“)๐‘ž๎‚ถ,๐œ†โˆˆโ„๐‘ž,+.(3.9)
(2) Additionally, if lim๐‘›โ†’+โˆž๐‘“(๐‘ž๐‘›)=0, then โ„ฑ๐‘ž๎€ท๐ท๐‘ž๐‘“๎€ธ๐œ†(๐œ†)=๐‘ž2๐‘žโ„ฑ(๐‘“)(๐œ†),๐œ†โˆˆโ„๐‘ž,+.(3.10)

Proof. The same steps as in the proof of [3, Lemma 2]; the ๐‘ž-integration by parts rule and the fact that๎€œโˆž0๐‘“(๐‘ก)๐‘‘๐‘ž๐‘ก=lim๐‘›โ†’+โˆž๎€œ๐‘žโˆ’๐‘›๐‘ž๐‘›๐‘“(๐‘ก)๐‘‘๐‘ž๐‘ก(3.11)give the result.

In [3], the authors proved the following ๐‘ž-analogues of the Heisenberg uncertainty principle.

Theorem 3.7. Let ๐‘“ be in ๐ฟ2๐‘ž(โ„๐‘ž,+) such that ๐ท๐‘ž๐‘“ is in ๐ฟ2๐‘ž(โ„๐‘ž,+). Then, โ€–๐‘ก๐‘“โ€–2,๐‘žโ€–๐œ†โ„ฑ๐‘ž(๐‘“)โ€–2,๐‘žโ‰ฅ๐‘ž๐‘ž3/2+1โ€–๐‘“โ€–22,๐‘ž.(3.12)In addition, if lim๐‘›โ†’+โˆž๐‘“(๐‘ž๐‘›)=0, one has โ€–๐‘ก๐‘“โ€–2,๐‘žโ€–๐œ†๐‘žโ„ฑ(๐‘“)โ€–2,๐‘žโ‰ฅ๐‘ž๐‘žโˆ’3/2+1โ€–๐‘“โ€–22,๐‘ž.(3.13)

Now, we are in a position to generalize Theorem 3.7. One obvious way to generalize it is to replace the ๐ฟ2๐‘ž norms by ๐ฟ๐‘›๐‘ž norms. This is the purpose of the following result.

Theorem 3.8. For 1โ‰ค๐‘›โ‰ค2 and ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has โ€–๐‘“โ€–22,๐‘žโ‰ค๐ถ๎…ž1โ€–๐‘ฅ๐‘“โ€–๐‘›,๐‘žโ€–๐œ†โ„ฑ๐‘ž(๐‘“)โ€–๐‘›,๐‘ž,(3.14)โ€–๐‘“โ€–22,๐‘žโ‰ค๐ถ๎…ž2โ€–๐‘ฅ๐‘“โ€–๐‘›,๐‘žโ€–๐œ†๐‘žโ„ฑ(๐‘“)โ€–๐‘›,๐‘ž,(3.15)where ๐ถ๎…ž1=๐‘žโˆ’1+1/๐‘›๎€ท1+๐‘žโˆ’(๐‘›+1)/๐‘›๎€ธ๐ถ2,๐ถ๎…ž2=๐‘žโˆ’1๎€ท1+๐‘žโˆ’(๐‘›+1)/๐‘›๎€ธ๐ถ1,(3.16)with ๐ถ1 and ๐ถ2 being given by (3.8).

Proof. The case ๐‘›=2 has been dealt with in Theorem 3.7. Now, assume 1โ‰ค๐‘›<2 and let ๐‘š be the dual exponent of ๐‘›. Let ๐‘“โˆˆ๐‘†โˆ—,๐‘ž(โ„๐‘ž) such that lim๐‘กโ†’0๐‘“(๐‘ก)=0. From the relation๐ท๐‘ž๎€ท๐‘“๐‘“๎€ธ(๐‘ก)=๐ท๐‘ž๐‘“(๐‘ก)๐‘“(๐‘ก)+๐‘“(๐‘ž๐‘ก)๐ท๐‘ž๐‘“(๐‘ก),(3.17)the ๐‘ž-integration by parts rule, and the Hรถlder inequality, we have, since ๐‘ก|๐‘“(๐‘ก)|2 tends to 0 as ๐‘ก tends to โˆž in โ„๐‘ž,+,1๐‘ž๎€œโˆž0||||๐‘“(๐‘ก)2๐‘‘๐‘ž|||๎€œ๐‘ก=โˆž0๐‘ก๐ท๐‘ž(๐‘“๐‘“)(๐‘ก)๐‘‘๐‘ž๐‘ก|||โ‰ค๎€œโˆž0||๐‘ก๐ท๐‘ž๐‘“(๐‘ก)||๐‘‘๐‘“(๐‘ก)๐‘ž๎€œ๐‘ก+โˆž0||๐‘ก๐‘“(๐‘ž๐‘ก)๐ท๐‘ž||๐‘‘๐‘“(๐‘ก)๐‘ž๐‘กโ‰ค๎‚€๎€œโˆž0||๐‘ก||๐‘“(๐‘ก)๐‘›๐‘‘๐‘ž๐‘ก๎‚1/๐‘›๎‚€๎€œโˆž0||๐ท๐‘ž||๐‘“(๐‘ก)๐‘š๐‘‘๐‘ž๐‘ก๎‚1/๐‘š+๎‚€๎€œโˆž0||||๐‘ก๐‘“(๐‘ž๐‘ก)๐‘›๐‘‘๐‘ž๐‘ก๎‚1/๐‘›๎‚€๎€œโˆž0||๐ท๐‘ž||๐‘“(๐‘ก)๐‘š๐‘‘๐‘ž๐‘ก๎‚1/๐‘š.(3.18)However, the change of variable ๐‘ข=๐‘ž๐‘ก gives๎‚€๎€œโˆž0||||๐‘ก๐‘“(๐‘ž๐‘ก)๐‘›๐‘‘๐‘ž๐‘ก๎‚1/๐‘›=๐‘žโˆ’(๐‘›+1)/๐‘›๎‚€๎€œโˆž0||||๐‘ก๐‘“(๐‘ก)๐‘›๐‘‘๐‘ž๐‘ก๎‚1/๐‘›.(3.19)So,1๐‘ž๎€œโˆž0||||๐‘“(๐‘ก)2๐‘‘๐‘ž๎€ท๐‘กโ‰ค1+๐‘žโˆ’(๐‘›+1)/๐‘›๎€ธโ€–๐‘ก๐‘“โ€–๐‘›,๐‘žโ€–๐ท๐‘ž(๐‘“)โ€–๐‘š,๐‘ž.(3.20)On the other hand, we have ๐ท๐‘ž(๐‘“)=โ„ฑ๐‘ž[โ„ฑ๐‘ž(๐ท๐‘ž(๐‘“))]=๐‘ž๐‘žโˆ’2โ„ฑ[๐‘žโ„ฑ(๐ท๐‘ž(๐‘“))] since ๐ท๐‘ž(๐‘“) is in ๐ฟ2๐‘ž(โ„๐‘ž,+). Then, by using Lemma 3.6 and the ๐‘ž-analogue of the Hausdorff-Young inequality, we obtainโ€–โ€–๐ท๐‘žโ€–โ€–(๐‘“)๐‘š,๐‘žโ‰ค๐ถ1โ€–โ€–โ„ฑ๐‘ž๎€ท๐ท๐‘ž๎€ธโ€–โ€–(๐‘“)๐‘›,๐‘ž=๐ถ1๐‘ž2โ€–โ€–๐œ†๐‘žโ€–โ€–โ„ฑ(๐‘“)๐‘›,๐‘ž,โ€–โ€–๐ท๐‘žโ€–โ€–(๐‘“)๐‘š,๐‘žโ‰ค๐‘žโˆ’2๐ถ2โ€–โ€–๐‘žโ„ฑ๎€ท๐ท๐‘ž๎€ธโ€–โ€–(๐‘“)๐‘›,๐‘ž=๐‘žโˆ’2๐ถ2โ€–โ€–โ€–๐œ†๐‘žโ„ฑ๐‘ž๎‚€๐œ†(๐‘“)๐‘ž๎‚โ€–โ€–โ€–๐‘›,๐‘ž=๐‘žโˆ’2+1/๐‘›๐ถ2โ€–โ€–๐œ†โ„ฑ๐‘žโ€–โ€–(๐‘“)๐‘›,๐‘ž.(3.21) Thus,โ€–๐‘“โ€–22,๐‘žโ‰ค๐‘žโˆ’1๎€ท1+๐‘žโˆ’(๐‘›+1)/๐‘›๎€ธ๐ถ1โ€–๐‘ก๐‘“โ€–๐‘›,๐‘žโ€–โ€–๐œ†๐‘žโ€–โ€–โ„ฑ(๐‘“)๐‘›,๐‘ž,(3.22)โ€–๐‘“โ€–22,๐‘žโ‰ค๐‘žโˆ’1+1/๐‘›๎€ท1+๐‘žโˆ’(๐‘›+1)/๐‘›๎€ธ๐ถ2โ€–๐‘ก๐‘“โ€–๐‘›,๐‘žโ€–โ€–๐œ†โ„ฑ๐‘žโ€–โ€–(๐‘“)๐‘›,๐‘ž.(1)Now, let ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+); it is easy to see that for all ๐‘โˆˆโ„•, ๐‘“๐‘=๐‘“๐œ’[๐‘ž๐‘,๐‘žโˆ’๐‘]โˆˆ๐‘†โˆ—,๐‘ž(โ„๐‘ž), lim๐‘กโ†’0๐‘“๐‘(๐‘ก)=0, and (๐‘“๐‘)๐‘ converges to ๐‘“ in ๐ฟ2๐‘ž(โ„๐‘ž,+). Moreover, if the right-hand side of (3.14) (resp., (3.15)) is finite, then the functions ๐‘ก๐‘“ and ๐œ†โ„ฑ๐‘ž(๐‘“) (resp., ๐œ†๐‘žโ„ฑ(๐‘“)) are in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+), and they are limits in ๐ฟ๐‘›๐‘ž(โ„๐‘ž,+) (as ๐‘ tends to โˆž) of ๐‘ก๐‘“๐‘ and ๐œ†โ„ฑ๐‘ž(๐‘“๐‘) (resp., ๐œ†๐‘žโ„ฑ(๐‘“๐‘)), respectively. Finally, the substitution of ๐‘“๐‘ in (3.22) and a passage to the limit when ๐‘ tends to โˆž 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 ๐‘ž-analogues.

4.1. Local Uncertainty Principles for the ๐‘ž-Fourier Trigonometric Transforms

Theorem 4.1. If 0<๐‘Ž<1/2, there is a constant ๐พ=๐พ(๐‘Ž,๐‘ž) such that for all bounded subset ๐ธ of โ„๐‘ž,+ and all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has ๎€œ๐ธ||โ„ฑ๐‘ž||(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†โ‰ค๐พ|๐ธ|2๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐‘ž.(4.1)Here, โˆซ|๐ธ|=โˆž0๐œ’๐ธ(๐‘ฅ)๐‘‘๐‘ž๐‘ฅ and ๐พ=((๎‚๐‘๐‘ž/โˆš[1โˆ’2๐‘Ž]๐‘ž)((1โˆ’2๐‘Ž)/2๐‘Ž))4๐‘Ž(1/(1โˆ’2๐‘Ž)2), where ๎‚๐‘๐‘ž=(1+๐‘žโˆ’1)1/2/ฮ“๐‘ž2(1/2)(๐‘ž;๐‘ž)2โˆž.Proof. For ๐‘Ÿ>0, let ๐œ’๐‘Ÿ=๐œ’[0,๐‘Ÿ] be the characteristic function of [0,๐‘Ÿ] and ๎‚๐œ’๐‘Ÿ=1โˆ’๐œ’๐‘Ÿ.
Then, for ๐‘Ÿ>0, we have, since ๐‘“โ‹…๐œ’๐‘Ÿโˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+),๎‚€๎€œ๐ธ||โ„ฑ๐‘ž||(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†๎‚1/2=โ€–โ€–โ„ฑ๐‘ž(๐‘“)๐œ’๐ธโ€–โ€–2,๐‘žโ‰คโ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)๐œ’๐ธโ€–โ€–2,๐‘ž+โ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๎‚๐œ’๐‘Ÿ)๐œ’๐ธโ€–โ€–2,๐‘žโ‰ค|๐ธ|1/2โ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)โ€–โ€–โˆž,๐‘ž+โ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๎‚๐œ’๐‘Ÿ)โ€–โ€–2,๐‘ž,(4.2)and by the use of the Hรถlder inequality, we obtainโ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)โ€–โ€–โˆž,๐‘žโ‰ค๎‚๐‘๐‘žโ€–โ€–๐‘“โ‹…๐œ’๐‘Ÿโ€–โ€–1,๐‘ž=๎‚๐‘๐‘žโ€–โ€–๐‘ฅโˆ’๐‘Ž๐œ’๐‘Ÿโ‹…๐‘ฅ๐‘Ž๐‘“โ€–โ€–1,๐‘žโ‰ค๎‚๐‘๐‘žโ€–โ€–๐‘ฅโˆ’๐‘Ž๐œ’๐‘Ÿโ€–โ€–2,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘žโ‰ค๎‚๐‘๐‘žโˆš[1โˆ’2๐‘Ž]๐‘ž๐‘Ÿ1/2โˆ’๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘ž.(4.3)On the other hand, since ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), we have ๐‘“โ‹…๎‚๐œ’๐‘Ÿโˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), and by the Plancherel formula, we getโ€–โ€–โ„ฑ๐‘ž(๐‘“โ‹…๎‚๐œ’๐‘Ÿ)โ€–โ€–2,๐‘ž=โ€–โ€–๐‘“โ‹…๎‚๐œ’๐‘Ÿโ€–โ€–2,๐‘ž=โ€–โ€–๐‘ฅโˆ’๐‘Ž๎‚๐œ’๐‘Ÿ.๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘žโ‰คโ€–โ€–๐‘ฅโˆ’๐‘Ž๎‚๐œ’๐‘Ÿโ€–โ€–โˆž,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘žโ‰ค๐‘Ÿโˆ’๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘ž.(4.4)So,๎‚€๎€œ๐ธ||โ„ฑ๐‘ž||(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†๎‚1/2โ‰ค๎ƒฉ๎‚๐‘๐‘žโˆš[1โˆ’2๐‘Ž]๐‘ž|๐ธ|1/2๐‘Ÿ1/2โˆ’๐‘Ž+๐‘Ÿโˆ’๐‘Ž๎ƒชโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘ž.(4.5)The desired result is obtained by minimizing the right-hand side of the previous inequality over ๐‘Ÿ>0.

Corollary 4.2. For 0<๐‘Ž<1/2 and ๐‘>0, there is a constant ๐พ๐‘Ž,๐‘ such that for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has โ€–๐‘“โ€–(๐‘Ž+๐‘)2,๐‘žโ‰ค๐พ๐‘Ž,๐‘โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–๐‘2,๐‘žโ€–โ€–๐œ†๐‘โ„ฑ๐‘žโ€–โ€–(๐‘“)๐‘Ž2,๐‘ž.(4.6)

Proof. For ๐‘Ÿ>0, put ๐ธ๐‘Ÿ=[0,๐‘Ÿ[โˆฉโ„๐‘ž,+ and ๎‚‹๐ธ๐‘Ÿ=[๐‘Ÿ,+โˆž[โˆฉโ„๐‘ž,+. It is easy to see that ๐ธ๐‘Ÿ is a bounded subset of โ„๐‘ž,+ and |๐ธ๐‘Ÿ|โ‰ค๐‘Ÿ.
Then, from the Plancherel formula and Theorem 4.1, we haveโ€–๐‘“โ€–22,๐‘ž=โ€–โ€–โ„ฑ๐‘žโ€–โ€–(๐‘“)22,๐‘ž=๎€œ๐ธ๐‘Ÿ||โ„ฑ๐‘ž||(๐‘“)2(๐œ†)๐‘‘๐‘ž๎€œ๐œ†+๎‚‹๐ธ๐‘Ÿ||โ„ฑ๐‘ž||(๐‘“)2(๐œ†)๐‘‘๐‘ž๐œ†โ‰ค๐พ๐‘Ÿ2๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐‘ž+๐‘Ÿโˆ’2๐‘โ€–โ€–๐œ†๐‘โ„ฑ๐‘žโ€–โ€–(๐‘“)22,๐‘ž.(4.7)Choosing ๐‘Ÿ>0 so as to minimize the right-hand side of the inequality, we obtain โ€–๐‘“โ€–22,๐‘žโ‰ค(๐พ๐‘Ž,๐‘โ€–๐‘ฅ๐‘Ž๐‘“โ€–๐‘2,๐‘žโ€–๐œ†๐‘โ„ฑ๐‘ž(๐‘“)โ€–๐‘Ž2,๐‘ž)2/(๐‘Ž+๐‘), with ๐พ๐‘Ž,๐‘=((๐‘Ž/๐‘)๐‘/(๐‘Ž+๐‘)+(๐‘/๐‘Ž)๐‘Ž/(๐‘Ž+๐‘))(๐‘Ž+๐‘)/2๐พ๐‘/2, and ๐พ is the constant given in Theorem 4.1.

In the same way, one can prove the following local uncertainty principle for the ๐‘ž-Fourier-sine transform.

Theorem 4.3. If 0<๐‘Ž<1/2, there is a constant ๐พ๎…ž=๐พ๎…ž(๐‘Ž,๐‘ž) such that for all bounded subset ๐ธ of โ„๐‘ž,+ and all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has ๎€œ๐ธ||๐‘ž||โ„ฑ(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†โ‰ค๐พ๎…ž|๐ธ|2๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐‘ž,(4.8)where ๐พ๎…ž=((๎‚๐‘๐‘ž/โˆš[1โˆ’2๐‘Ž]๐‘ž)((1โˆ’2๐‘Ž)/2๐‘ž๐‘Ž))4๐‘Ž[1+2๐‘ž๐‘Ž/(1โˆ’2๐‘Ž)]2.

Corollary 4.4. For 0<๐‘Ž<1/2 and ๐‘>0, there is a constant ๐พ๎…ž๐‘Ž,๐‘ such that for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has โ€–๐‘“โ€–(๐‘Ž+๐‘)2,๐‘žโ‰ค๐พ๎…ž๐‘Ž,๐‘โ€–๐‘ฅ๐‘Ž๐‘“โ€–๐‘2,๐‘žโ€–๐œ†๐‘๐‘žโ„ฑ(๐‘“)โ€–๐‘Ž2,๐‘ž,(4.9)with ๐พ๎…ž๐‘Ž,๐‘=((๐‘Ž/๐‘)๐‘/(๐‘Ž+๐‘)+(๐‘/๐‘Ž)๐‘Ž/(๐‘Ž+๐‘))(๐‘Ž+๐‘)/2(๐พ๎…ž)๐‘/2๐‘žโˆ’(๐‘Ž+๐‘).

Proof. The same steps of Corollary 4.2 give the result.

Theorem 4.5. If ๐‘Ž>1/2, there is a constant ๐พ1=๐พ1(๐‘Ž,๐‘ž) such that for all bounded subset ๐ธ of โ„๐‘ž,+ and ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has
๎€œ๐ธ||โ„ฑ๐‘ž||(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†โ‰ค๐พ1|๐ธ|โ€–๐‘“โ€–(2โˆ’1/๐‘Ž)2,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–1/๐‘Ž2,๐‘ž,(4.10)๎€œ๐ธ||โ„ฑ๐‘ž||(๐‘“)(๐œ†)2๐‘‘๐‘ž๐œ†โ‰ค๐พ1|๐ธ|โ€–๐‘“โ€–(2โˆ’1/๐‘Ž)2,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–1/๐‘Ž2,๐‘ž.(4.11)

The proof of this result needs the following lemmas.

Lemma 4.6. Suppose ๐‘Ž>1/2, then for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), such that ๐‘ฅ๐‘Ž๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), โ€–๐‘“โ€–21,๐‘žโ‰ค๐พ2๎€บโ€–๐‘“โ€–22,๐‘ž+โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐‘ž๎€ป,(4.12)where ๐พ2=๐พ2(๐‘Ž,๐‘ž)=(1โˆ’๐‘ž)((๐‘ž2๐‘Ž,๐‘ž2๐‘Ž,โˆ’๐‘ž,โˆ’๐‘ž2๐‘Žโˆ’1;๐‘ž2๐‘Ž)โˆž/(๐‘ž,๐‘ž2๐‘Žโˆ’1,โˆ’๐‘ž2๐‘Ž,โˆ’1;๐‘ž2๐‘Ž)โˆž).

Proof. From [15, Example 1], and the Hรถlder inequality, we haveโ€–๐‘“โ€–21,๐‘ž=๎‚ƒ๎€œ0+โˆž๎€ท1+๐‘ฅ2๐‘Ž๎€ธ1/2๎€ท|๐‘“(๐‘ฅ)|1+๐‘ฅ2๐‘Ž๎€ธโˆ’1/2๐‘‘๐‘ž๐‘ฅ๎‚„2โ‰ค๐พ2[โ€–๐‘“โ€–22,๐‘ž+โ€–๐‘ฅ๐‘Ž๐‘“โ€–22,๐‘ž],(4.13)where ๐พ2=โˆซ0+โˆž(1+๐‘ฅ2๐‘Ž)โˆ’1๐‘‘๐‘ž๐‘ฅ=(1โˆ’๐‘ž)((๐‘ž2๐‘Ž,๐‘ž2๐‘Ž,โˆ’๐‘ž,โˆ’๐‘ž2๐‘Žโˆ’1;๐‘ž2๐‘Ž)โˆž/(๐‘ž,๐‘ž2๐‘Žโˆ’1,โˆ’๐‘ž2๐‘Ž,โˆ’1;๐‘ž2๐‘Ž)โˆž).

Lemma 4.7. Suppose ๐‘Ž>1/2, then for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), such that ๐‘ฅ๐‘Ž๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+), one has โ€–๐‘“โ€–1,๐‘žโ‰ค๐พ3โ€–๐‘“โ€–(1โˆ’1/2๐‘Ž)2,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–1/2๐‘Ž2,๐‘ž,(4.14)where ๐พ3=๐พ3(๐‘Ž,๐‘ž)=[2๐‘Ž๐พ2(2๐‘Ž๐‘žโˆ’๐‘ž)1/2๐‘Žโˆ’1]1/2.

Proof. For ๐‘ โˆˆโ„๐‘ž,+, define the function ๐‘“๐‘  by ๐‘“๐‘ (๐‘ฅ)=๐‘“(๐‘ ๐‘ฅ),๐‘ฅโˆˆโ„๐‘ž,+.
We have โ€–๐‘“๐‘ โ€–1,๐‘ž=๐‘ โˆ’1โ€–๐‘“โ€–1,๐‘ž, โ€–๐‘ฅ๐‘Ž๐‘“๐‘ โ€–22,๐‘ž=๐‘ โˆ’2๐‘Žโˆ’1โ€–๐‘ฅ๐‘Ž๐‘“โ€–22,๐‘ž.
Replacement of ๐‘“ by ๐‘“๐‘  in Lemma 4.6 givesโ€–๐‘“โ€–21,๐‘žโ‰ค๐พ2[๐‘ โ€–๐‘“โ€–22,๐‘ž+๐‘ โˆ’2๐‘Ž+1โ€–๐‘ฅ๐‘Ž๐‘“โ€–22,๐‘ž].(4.15)Now, for all ๐‘Ÿ>0, put ๐›ผ(๐‘Ÿ)=Log(๐‘Ÿ)/Log(๐‘ž)โˆ’๐ธ(Log(๐‘Ÿ)/Log(๐‘ž)). We have ๐‘ =(๐‘Ÿ/๐‘ž๐›ผ(๐‘Ÿ))โˆˆโ„๐‘ž,+ and ๐‘Ÿโ‰ค๐‘ <๐‘Ÿ/๐‘ž. Then, for all ๐‘Ÿ>0,โ€–๐‘“โ€–21,๐‘žโ‰ค๐พ2๎‚ƒ๐‘Ÿ๐‘žโ€–๐‘“โ€–22,๐‘ž+๐‘Ÿโˆ’2๐‘Ž+1โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐‘ž๎‚„.(4.16)The right-hand side of this inequality is minimized by choosing๐‘Ÿ=(2๐‘Žโˆ’1)1/2๐‘Ž๐‘ž1/2๐‘Žโ€–๐‘“โ€–โˆ’1/๐‘Ž2,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–1/๐‘Ž2,๐‘ž.(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 ๐ธ be a bounded subset of โ„๐‘ž,+. When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that ๐‘“โˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+); so โ„ฑ๐‘ž(๐‘“) is defined and bounded on โ„๐‘ž,+. Using Proposition 3.1, Lemma 4.7, and the fact that๎€œ๐ธ|โ„ฑ๐‘ž(๐‘“)(๐œ†)|2๐‘‘๐‘ž๐œ†โ‰ค|๐ธ|โ€–โ„ฑ๐‘ž(๐‘“)โ€–2โˆž,๐‘ž,(4.18)we obtain the result with ๐พ1=((1+๐‘žโˆ’1)/ฮ“2๐‘ž2(1/2)(๐‘ž;๐‘ž)4โˆž)๐พ23.

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 ๐‘ž-Bessel-Fourier Transform

The ๐‘ž-Bessel-Fourier transform is defined (see [16]) for ๐‘“โˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ) byโ„ฑ๐œˆ,๐‘ž(๐‘“)(๐œ†)=๐‘๐œˆ,๐‘ž๎€œโˆž0๐‘“(๐‘ฅ)๐‘—๐œˆ๎€ท๐œ†๐‘ฅ;๐‘ž2๎€ธ๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ,(4.19)where๐‘—๐œˆ๎€ท๐‘ง;๐‘ž2๎€ธ=๎€ท1โˆ’๐‘ž2๎€ธ๐œˆฮ“๐‘ž2๎€ท(๐œˆ+1)(1โˆ’๐‘ž)๐‘žโˆ’1๐‘ง๎€ธโˆ’๐œˆ๐ฝ๐œˆ๎€ท(1โˆ’๐‘ž)๐‘žโˆ’1๐‘ง;๐‘ž2๎€ธ(4.20)is the normalized third Jackson ๐‘ž-Bessel function, and๐‘๐œˆ,๐‘ž=๎€ท1+๐‘žโˆ’1๎€ธโˆ’๐œˆฮ“๐‘ž2(๐œˆ+1).(4.21)It was shown in [10] that for ๐œˆโ‰ฅโˆ’1/2, we have the following result.

Theorem 4.9. (1) For ๐‘“โˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), one has โ„ฑ๐œˆ,๐‘ž(๐‘“)โˆˆ๐ฟโˆž๐‘ž(โ„๐‘ž,+) and โ€–โ€–โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)โˆž,๐‘žโ‰ค๐‘๐œˆ,๐‘ž(๐‘ž;๐‘ž2)2โˆžโ€–๐‘“โ€–1,๐œˆ,๐‘ž.(4.22)
(2) โ„ฑ๐œˆ,๐‘ž is an isomorphism of ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ) onto itself, โ„ฑโˆ’1๐œˆ,๐‘ž=๐‘ž4๐œˆ+2โ„ฑ๐œˆ,๐‘ž, and one has the following Plancherel formula: โˆ€๐‘“โˆˆ๐ฟ2๐‘ž๎€ทโ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ๎€ธ,โ€–โ€–โ„ฑ๐œˆ,๐‘žโ€–โ€–2,๐œˆ,๐‘ž=๐‘ž2๐œˆ+1โ€–๐‘“โ€–2,๐œˆ,๐‘ž.(4.23)
The following result states a local uncertainty principle for the ๐‘ž-Bessel-Fourier transform.

Theorem 4.10. For ๐œˆโ‰ฅโˆ’1/2 and 0<๐‘Ž<๐œˆ+1, there is a constant ๐พ๐‘Ž,๐œˆ=๐พ(๐‘Ž,๐œˆ,๐‘ž) such that for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ) and all bounded subset ๐ธ of โ„๐‘ž,+, one has ๎€œ๐ธ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)(๐œ†)2๐œ†2๐œˆ+1๐‘‘๐‘ž๐œ†โ‰ค๐พ๐‘Ž,๐œˆ|๐ธ|๐œˆ๐‘Ž/(๐œˆ+1)โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐œˆ,๐‘ž.(4.24)Here, |๐ธ|๐œˆ=โˆซโˆž0๐œ’๐ธ(๐‘ฅ)๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ, ๎‚‹๐‘๐œˆ,๐‘ž=๐‘๐œˆ,๐‘ž/(๐‘ž;๐‘ž2)2โˆž, and ๐พ๐‘Ž,๐œˆ=๎ƒฉ๎‚‹๐‘๐œˆ,๐‘žโˆš[2๐œˆ+2โˆ’2๐‘Ž]๐‘ž๎ƒช2๐‘Ž/(๐œˆ+1)๎‚ธ๎‚ต๐‘Ž๐‘ž2๐œˆ+1๎‚ถ๐œˆ+1โˆ’๐‘Ž1โˆ’๐‘Ž/(๐œˆ+1)+๐‘ž2๐œˆ+1๎‚ต๐‘Ž๐‘ž2๐œˆ+1๎‚ถ๐œˆ+1โˆ’๐‘Žโˆ’๐‘Ž/(๐œˆ+1)๎‚น2.(4.25)

ProofLet (4.25)๐œˆโ‰ฅโˆ’1/2, (4.25)0<๐‘Ž<๐œˆ+1, (4.25)๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), and let (4.25)๐ธ be a bounded subset of (4.25)โ„๐‘ž,+.
For (4.25)๐‘Ÿ>0, we have, since (4.25)๐‘“.๐œ’๐‘Ÿโˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ),๎‚€๎€œ๐ธ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)(๐œ†)2๐œ†2๐œˆ+1๐‘‘๐‘ž๐œ†๎‚1/2=โ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“)๐œ’๐ธโ€–โ€–2,๐œˆ,๐‘žโ‰คโ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)๐œ’๐ธโ€–โ€–2,๐œˆ,๐‘ž+โ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“โ‹…๎‚๐œ’๐‘Ÿ)๐œ’๐ธโ€–โ€–2,๐œˆ,๐‘žโ‰ค|๐ธ|๐œˆ1/2โ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)โ€–โ€–โˆž,๐‘ž+โ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“โ‹…๎‚๐œ’๐‘Ÿ)โ€–โ€–2,๐œˆ,๐‘ž.(4.25)

Proof. Let ๐œˆโ‰ฅโˆ’1/2, 0<๐‘Ž<๐œˆ+1, ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), and let ๐ธ be a bounded subset of โ„๐‘ž,+.
For ๐‘Ÿ>0, we have, since ๐‘“.๐œ’๐‘Ÿโˆˆ๐ฟ1๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ),โ€–โ€–โ„ฑ๐œˆ,๐‘ž๎€ท๐‘“โ‹…๐œ’๐‘Ÿ๎€ธโ€–โ€–โˆž,๐‘žโ‰ค๎‚‹๐‘๐œˆ,๐‘žโ€–โ€–๐‘“โ‹…๐œ’๐‘Ÿโ€–โ€–1,๐‘ž=๎‚๐‘๐‘žโ€–โ€–๐‘ฅโˆ’๐‘Ž๐œ’๐‘Ÿ.๐‘ฅ๐‘Ž๐‘“โ€–โ€–1,๐œˆ,๐‘žโ‰ค๎‚‹๐‘๐œˆ,๐‘žโ€–โ€–๐‘ฅโˆ’๐‘Ž๐œ’๐‘Ÿโ€–โ€–2,๐œˆ,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐œˆ,๐‘ž.(4.26)However, by the use of the Hรถlder inequality, we obtainโ€–โ€–๐‘ฅโˆ’๐‘Ž๐œ’๐‘Ÿโ€–โ€–22,๐œˆ,๐‘ž=๎€œโˆž0๐‘ฅโˆ’2๐‘Ž๐œ’๐‘Ÿ(๐‘ฅ)๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๎€œ๐‘ฅ=๐‘ž๐‘˜0๐‘ฅ2๐œˆ+1โˆ’2๐‘Ž๐‘‘๐‘ž๐‘ž๐‘ฅ=2๐‘˜(๐œˆ+1โˆ’๐‘Ž)[2๐œˆ+2โˆ’2๐‘Ž]๐‘žโ‰ค๐‘Ÿ2(๐œˆ+1โˆ’๐‘Ž)[2๐œˆ+2โˆ’2๐‘Ž]๐‘ž.(4.27)Now, if ๐‘˜ is the integer such that ๐‘ž๐‘˜โ‰ค๐‘Ÿ<๐‘ž๐‘˜โˆ’1, we get, since ๐‘Ž<๐œˆ+1,โ€–โ€–โ„ฑ๐œˆ,๐‘ž(๐‘“โ‹…๐œ’๐‘Ÿ)โ€–โ€–โˆž,๐‘žโ‰ค๎‚‹๐‘๐œˆ,๐‘žโˆš[2๐œˆ+2โˆ’2๐‘Ž]๐‘ž๐‘Ÿ(๐œˆ+1โˆ’๐‘Ž)โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐œˆ,๐‘ž.(4.28)Then,โ€–โ€–โ„ฑ๐œˆ,๐‘ž๎€ท๐‘“โ‹…๎‚๐œ’๐‘Ÿ๎€ธโ€–โ€–2,๐œˆ,๐‘ž=๐‘ž2๐œˆ+1โ€–โ€–๐‘“โ‹…๎‚๐œ’๐‘Ÿโ€–โ€–2,๐œˆ,๐‘ž=๐‘ž2๐œˆ+1โ€–โ€–๐‘ฅโˆ’๐‘Ž๎‚๐œ’๐‘Ÿโ‹…๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐œˆ,๐‘žโ‰ค๐‘ž2๐œˆ+1โ€–โ€–๐‘ฅโˆ’๐‘Ž๎‚๐œ’๐‘Ÿโ€–โ€–โˆž,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐‘žโ‰ค๐‘ž2๐œˆ+1๐‘Ÿโˆ’๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐œˆ,๐‘ž.(4.29)On the other hand, since ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), we have ๐‘“.๎‚๐œ’๐‘Ÿโˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), and by the Plancherel formula (4.23), we obtain๎‚€๎€œ๐ธ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)(๐œ†)2๐œ†2๐œˆ+1๐‘‘๐‘ž๐œ†๎‚1/2โ‰ค๎ƒฉ๎‚‹๐‘๐œˆ,๐‘žโˆš[2๐œˆ+2โˆ’2๐‘Ž]๐‘ž|๐ธ|๐œˆ1/2๐‘Ÿ(๐œˆ+1โˆ’๐‘Ž)+๐‘ž2๐œˆ+1๐‘Ÿโˆ’๐‘Ž๎ƒชโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2,๐œˆ,๐‘ž.(4.30)So,๎€œ๐ธ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)(๐œ†)2๐œ†2๐œˆ+1๐‘‘๐‘ž๐œ†โ‰ค๐พ๎…ž๐‘Ž,๐œˆ|๐ธ|โ€–๐‘“โ€–2(1โˆ’(๐œˆ+1)/๐‘Ž)2,๐œˆ,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2((๐œˆ+1)/๐‘Ž)2,๐œˆ,๐‘ž.(4.31)By minimization of the right-hand side of the previous inequality over ๐‘Ÿ>0 and by easy computation, we obtain the desired result.

Theorem 4.11. For ๐œˆโ‰ฅโˆ’1/2 and ๐‘Ž>๐œˆ+1, there exists a constant ๐พ๎…ž๐‘Ž,๐œˆ such that for all bounded subset ๐ธ of โ„๐‘ž,+ and all ๐‘“ in ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), one has ๐พ๎…ž๐‘Ž,๐œˆ=(๐‘ž2๐‘Ž,๐‘ž2๐‘Ž,โˆ’๐‘ž2๐œˆ+2,โˆ’๐‘ž2(๐‘Žโˆ’๐œˆโˆ’1);๐‘ž2๐‘Ž)โˆž(๐‘ž2๐œˆ+2,๐‘ž2(๐‘Žโˆ’๐œˆโˆ’1),โˆ’๐‘ž2๐‘Ž,โˆ’1;๐‘ž2๐‘Ž)โˆž๐‘๎…ž๐œˆ,๐‘ž,๐‘๎…ž๐œˆ,๐‘ž๎‚ต๐‘=(1โˆ’๐‘ž)๐œˆ,๐‘ž(๐‘ž;๐‘ž2)2โˆž๎‚ถ2๎‚ต๐‘Ž๎‚ถ๐œˆ+1โˆ’1(๐œˆ+1)/๐‘Ž๎‚ต๐‘Ž๎‚ถ๐‘ž๐‘Žโˆ’๐œˆโˆ’1โˆ’2(๐œˆ+1)((๐‘Žโˆ’๐œˆโˆ’1)/๐‘Ž).(4.32)

Proof. Since ๐‘Ž>๐œˆ+1, the same steps as in the proof of Theorem 4.5 and the relation (4.22) give the result withโ€–๐‘“โ€–(๐‘Ž+๐‘)2,๐œˆ,๐‘žโ‰ค๐พ๐‘Ž,๐‘,๐œˆโ€–๐‘ฅ๐‘Ž๐‘“โ€–๐‘2,๐œˆ,๐‘žโ€–๐œ†๐‘โ„ฑ๐œˆ,๐‘ž(๐‘“)โ€–๐‘Ž2,๐œˆ,๐‘ž,(4.33)

Corollary 4.12. For ๐œˆโ‰ฅโˆ’1/2 and ๐‘Ž,๐‘>0, there is a constant ๐พ๐‘Ž,๐‘,๐œˆ=๐พ(๐‘Ž,๐‘,๐œˆ,๐‘ž) such that for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), one has ๐พ๐‘Ž,๐‘,๐œˆ=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๎‚ธ๎‚€๐‘๐‘Ž๎‚๐‘Ž/(๐‘Ž+๐‘)+๎‚€๐‘Ž๐‘๎‚๐‘/(๐‘Ž+๐‘)๎‚น(๐‘Ž+๐‘)/2(๐พ๐‘Ž,๐œˆ)๐‘/2๐‘žโˆ’(2๐œˆ+1)(๐‘Ž+๐‘)([2๐œˆ+2]๐‘ž)๐‘Ž๐‘/2(๐œˆ+1)๎‚ต๐พif๐‘Ž<๐œˆ+1,๎…ž๐‘Ž,๐œˆ[2๐œˆ+2]๐‘ž๎‚ถ๐‘Ž๐‘/(2๐œˆ+2)๎‚ต๐‘žโˆ’(4๐œˆ+2)๎‚ธ๎‚€๐‘๎‚๐œˆ+1(๐œˆ+1)/(๐œˆ+๐‘+1)+๎‚€๐‘๎‚๐œˆ+1โˆ’๐‘/(๐œˆ+๐‘+1)๎‚น๎‚ถ๐‘Ž(๐œˆ+๐‘+1)/2(๐œˆ+1)if๐‘Ž>๐œˆ+1,(4.34)with ๐‘ž4๐œˆ+2โ€–๐‘“โ€–22,๐œˆ,๐‘ž=โ€–โ€–โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)22,๐œˆ,๐‘ž=๎€œ๐ธ๐‘Ÿ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)2(๐œ†)๐œ†2๐œˆ+1๐‘‘๐‘ž๎€œ๐œ†+๎‚‹๐ธ๐‘Ÿ||โ„ฑ๐œˆ,๐‘ž||(๐‘“)2(๐œ†)๐œ†2๐œˆ+1๐‘‘๐‘ž๐œ†โ‰ค๎ƒฏ๐พ๐‘Ž,๐œˆ||๐ธ๐‘Ÿ||๐œˆ๐‘Ž/(๐œˆ+1)โ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐œˆ,๐‘ž+๐‘Ÿโˆ’2๐‘โ€–โ€–๐œ†๐‘โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)22,๐œˆ,๐‘ž๐พif๐‘Ž<๐œˆ+1,๎…ž๐‘Ž,๐œˆ||๐ธ๐‘Ÿ||โ€–๐‘“โ€–2(๐‘Žโˆ’๐œˆโˆ’1)/๐‘Ž2,๐œˆ,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2(๐œˆ+1)/๐‘Ž2,๐œˆ,๐‘ž+๐‘Ÿโˆ’2๐‘โ€–โ€–๐œ†๐‘โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)22,๐œˆ,๐‘žโ‰คโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐พif๐‘Ž>๐œˆ+1,๐‘Ž,๐œˆ[2๐œˆ+2]๐‘ž๐‘Ž/(๐œˆ+1)๐‘Ÿ2๐‘Žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–22,๐œˆ,๐‘ž+๐‘Ÿโˆ’2๐‘โ€–โ€–๐œ†๐‘โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)22,๐œˆ,๐‘ž๐พif๐‘Ž<๐œˆ+1,๎…ž๐‘Ž,๐œˆ๐‘Ÿ2๐œˆ+2[2๐œˆ+2]๐‘žโ€–๐‘“โ€–2(๐‘Žโˆ’๐œˆโˆ’1)/๐‘Ž2,๐œˆ,๐‘žโ€–โ€–๐‘ฅ๐‘Ž๐‘“โ€–โ€–2(๐œˆ+1)/๐‘Ž2,๐œˆ,๐‘ž+๐‘Ÿโˆ’2๐‘โ€–โ€–๐œ†๐‘โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)22,๐œˆ,๐‘žif๐‘Ž>๐œˆ+1.(4.35) where ๐พ๐‘Ž,๐œˆ (resp., ๐พ๎…ž๐‘Ž,๐œˆ) is the constant given in Theorem 4.10 (resp., Theorem 4.11).

Proof. For ๐‘Ÿ>0, we put ๐ธ๐‘Ÿ=[0,๐‘Ÿ[โˆฉโ„๐‘ž,+ and ๎‚‹๐ธ๐‘Ÿ=[๐‘Ÿ,+โˆž[โˆฉโ„๐‘ž,+.
We have ๐ธ๐‘Ÿ is a bounded subset of โ„๐‘ž,+ and |๐ธ๐‘Ÿ|๐œˆโ‰ค๐‘Ÿ2๐œˆ+2/[2๐œˆ+2]๐‘ž. Then, the Plancherel formula (4.23) and Theorems 4.10 and 4.11 lead to โ€–๐‘“โ€–22,๐œˆ,๐‘žโ‰ค๐พ1,1,๐œˆโ€–๐‘ฅ๐‘“โ€–2,๐œˆ,๐‘žโ€–โ€–๐œ†โ„ฑ๐œˆ,๐‘žโ€–โ€–(๐‘“)2,๐œˆ,๐‘ž.(4.36)The desired result follows by minimizing the right expressions over ๐‘Ÿ>0.

Remark that when ๐‘Ž=๐‘=1, we obtain a Heisenberg-Weyl-type inequality for the ๐‘ž-Bessel-Fourier transform.

Corollary 4.13. For ๐œˆโ‰ฅโˆ’1/2,๐œˆโ‰ 0, one has for all ๐‘“โˆˆ๐ฟ2๐‘ž(โ„๐‘ž,+,๐‘ฅ2๐œˆ+1๐‘‘๐‘ž๐‘ฅ), โ€–๐‘“โ€–22,๐œˆ,๐‘žโ‰ค๐พ1,1,๐œˆโ€–๐‘ฅ๐‘“โ€–2,๐œˆ,๐‘žโ€–๐œ†โ„ฑ๐œˆ,๐‘ž(๐‘“)โ€–2,๐œˆ,๐‘ž.(4.37)

Acknowledgment

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

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