A variation of the \(\mathcal{L}_{q, p}\)-uncertainty inequalities of Heisenberg-type for symmetric \(q\)-integral transforms. (English) Zbl 1537.44006
Summary: The aim of this paper is to prove new uncertainty inequalities of Heisenberg-type for a \(q\)-integral operator \(\mathcal{T}_q\) with a bounded kernel. To do so, we prove a Nash and Carlson’s inequalities for this transformation on \(\mathcal{L}_{q, 1} \cap \mathcal{L}_{q, p}(\Omega, | \omega(x) | d_q x)\) for \(1 < p \leq 2\), on \(\mathcal{L}_{q, 2} \cap \mathcal{L}_{q, p}(\Omega, | \omega(x) | d_q x)\) for \(1 < p< 2\), and on \(\mathcal{L}_{q, p_1} \cap \mathcal{L}_{q, p_2}(\Omega, | \omega(x) | d_q x)\) for \(1 < p_1 < p_2 \leq 2\). Our results can be applied to the the \(q\)-Fourier-cosine transform, the \(q\)-Dunkl transform, and the \(q\)-Bessel-Fourier transform.
{© 2022 John Wiley & Sons, Ltd.}
{© 2022 John Wiley & Sons, Ltd.}
MSC:
| 44A20 | Integral transforms of special functions |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 33B15 | Gamma, beta and polygamma functions |
| 33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |
Keywords:
Carlson’s inequality; Heisenberg uncertainty inequality; Nash’s inequality; symmetric \(q\)-integral transformsReferences:
| [1] | GasperG, RahmanM. Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications: Cambridge University Press; 1990;35. · Zbl 0695.33001 |
| [2] | KacVG, CheungP. Quantum Calculus. Universitext. New York: Springer‐Verlag; 2002. · Zbl 0986.05001 |
| [3] | AnnabyMH, MansourZS, CalculusF. Equations Lecture Notes in Mathematics: Springer; 2056;2012. |
| [4] | JacksonFH. On a q‐definite integrals. Quart J Pure Appl Math. 1910;41:193‐203. · JFM 41.0317.04 |
| [5] | SteinEM. Interpolation of linear operators. Trans Amer Math Soc. 1956;83:482‐492. · Zbl 0072.32402 |
| [6] | SteinEM, WeissG. Introduction to Fourier Analysis on Euclidean Spaces. Princeton NJ: Princeton University Press; 1971. · Zbl 0232.42007 |
| [7] | DhaouadiL. On the q‐Bessel‐Fourier transform. Bull Math Anal Appl. 2013;5(2):42‐60. · Zbl 1314.33016 |
| [8] | DhaouadiL. Heisenberg uncertainty principle for the q‐Bessel‐Fourier transform preprint; 2007. |
| [9] | DhaouadiL, FitouhiA, El KamelJ. Inequalities in q‐Fourier analysis. J Inequalities Pure Appl Math. 2006;7(5):171. · Zbl 1232.26025 |
| [10] | BettaibiNN, BettaiebRH. q‐Aanalogue of the Dunkl transform on the real line, Tamsui Oxford. J Mathematical Sciences Arxiv08010069v1 [mathQA], 29; 2007. |
| [11] | BouzeffourF. q‐Analyse harmonique. Ph.D. thesis. Tunis, Tunisia: Faculty of Sciences of Tunis; 2002. |
| [12] | KoornwinderTH, SwarttouwRF. On q‐analogues of the Fourier and Hankel transforms. Trans Amer Math Soc. 1992;333(1):445‐461. · Zbl 0759.33007 |
| [13] | FitouhiA, BouzeffourF. The q‐cosine Fourier transform and the q‐heat equation. The Ramanujan J. 2012;28(3):443‐461. · Zbl 1255.33009 |
| [14] | FollandGB, SitaramA. The uncertainty principle: a mathematical survey. J Fourier Anal Appl. 1997;3(3):207‐238. · Zbl 0885.42006 |
| [15] | BettaibiN. Uncertainty principles in q^2‐analogue Fourier analysis. Math Sci Res J. 2007;11:590‐602. · Zbl 1155.42001 |
| [16] | BrahimK, NefziB, TefjeniE. Uncertainty principles for the continuous quaternion shearlet transform. Adv Appl Clifford Algebras. 2019:29‐43. · Zbl 1431.42017 |
| [17] | FitouhiA, NouriF, GuesmiS. On Heisenberg and uncertainty principles for the q‐Dunkl transform. JIPAM J Inequal Pure Appl Math. 2009;10(2):42‐10. · Zbl 1167.33312 |
| [18] | GhobberS. Uncertainty principles involving L^1‐norms for the Dunkl transform. Int Trans Spec Funct. 2013;24(6):491‐501. · Zbl 1275.42011 |
| [19] | GhobberS. Variations on uncertainty principles for integral operators. Appl Anal. 2014;93(5):1057‐1072. · Zbl 1290.42026 |
| [20] | HleiliM, NefziB, BsaissaA. A variation on uncertainty principles for the generalized q‐Bessel‐Fourier transform. J Math Anal Appl. 2016;440:823‐832. · Zbl 1342.42004 |
| [21] | MejjaoliH. Uncertainty principles for the generalized Fourier transform associated to a Dunkl‐type operator. Appl Anal. 2016;95(9):1930‐1956. · Zbl 1348.42011 |
| [22] | MejjaoliH, OmriS. Quantitative uncertainty principles associated with the directional short‐time Fourier transform. Integral Transforms and Special Functions. doi:10.1080/10652469.2020.1843166 · Zbl 1489.43005 |
| [23] | MejjaoliH, OthmaniY. L^p Quantitative uncertainty principles for the generalized Fourier transform associated with the spherical mean operator. Commun Math Anal. 2015;18(1):83‐99. · Zbl 1327.43005 |
| [24] | NefziB. Uncertainty principles for the q‐Bessel‐Fourier transform. Integr Trans Special Funct. 2019;30(11):920‐939. · Zbl 1454.33015 |
| [25] | NefziB, BrahimK. Calderón’s reproducing formula uncertainty principle for the continuous wavelet transform associated with the q‐Bessel operator. J Pseudo‐Differ Oper Appl. 2018;9(3):495‐522. · Zbl 1402.42007 |
| [26] | NefziB, BrahimK, FitouhiA, ForUP. The multivariate continuous shearlet transform. J Pseudo‐Differ Oper Appl. 2020;11(2):517‐542. · Zbl 1440.42037 |
| [27] | RöslerM, VoitM. Uncertainty principle for Hankel transforms. Proc Amer Math Soc. 1999;127:183‐194. · Zbl 0910.44003 |
| [28] | SoltaniF, GhazwaniJ. A variation of the L^p uncertainty principles for the Fourier transform. Proceedings of the Institute of Mathematics and Mechanics. Natl Acad Sci Azerbaijan. 2016;42(1):10‐24. · Zbl 1358.42007 |
| [29] | HleiliM, BrahimK. On Nash and Carlson’s inequalities for symmetric q‐integral transforms. J Math Anal Appl. 2016;440:823‐832. |
| [30] | NashJ. Continuity of solutions of parabolic and elliptic equations. Amer J Math. 1958;80:931‐954. · Zbl 0096.06902 |
| [31] | CarlenEA, LossM. Sharp constant in Nash’s inequality. Amer J Math. 1993;7:213‐215. · Zbl 0822.35018 |
| [32] | CarlsonF. Une inégalité. Ark Mat Astron Fys. 1934;25(B):1‐5. · Zbl 0009.34202 |
| [33] | LarssonL, MaligrandaL, Pec̆aric̆J, PerssonLE. Multiplicative Inequalities of Carlson Type and Interpolation. Hackensack, NJ: World Scientific Publishing Co. Pty. Ltd.; 2006. · Zbl 1102.41001 |
| [34] | FitouhiA, BettaibiN, BettaiebRH. On Hardy’s inequality for symmetric integral transforms and analogous. Appl Math Comput. 2008;198:346‐354. · Zbl 1132.26356 |
| [35] | FitouhiA, DhaouadiL. Positivity of the generalized translation associated with the q‐Hankel transform. Constr Approx. 2011;34:435‐472. · Zbl 1247.33030 |
| [36] | SwarttouwRF. The Hahn‐Exton q‐Bessel functions. Ph. D Thesis: Delft Technical University; 1992. |
| [37] | KoelinkHT, Van AsscheW. Orthogonal Polynomials and Laurent Polynomials related to the Hahn‐Exton q‐Bessel function. Constr Approx. 1995;11:477‐512. · Zbl 0855.33011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
