Spectral theorems associated with the \((k,a)\)-generalized wavelet multipliers. (English) Zbl 06979693
Summary: We introduce the notion of the \((k,a)\)-generalized wavelet multipliers. Particular cases of such generalized wavelet multipliers are the classical and Dunkl wavelet multipliers. The restriction of the \((k,a)\)-generalized wavelet multipliers to radial functions is given by the generalized Hankel wavelet multiplier. We study the boundedness, Schatten class properties of the \((k,a)\)-generalized wavelet multipliers and we give them trace formula. We prove that the generalized Landau-Pollak-Slepian operator is a \((k,a)\)-generalized wavelet multiplier. Next, we give results on the boundedness and compactness of \((k,a)\)-generalized wavelet multipliers on \(L^{p}_{k,a}(\mathbb {R}^{d})\), \(1 \leq p \leq \infty\).
MSC:
| 47G10 | Integral operators |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 47G30 | Pseudodifferential operators |
Keywords:
\((k, a)\)-Laguerre semigroup; \((k, a)\)-generalized Fourier transform; \((k, a)\)-generalized multipliers; \((k, a)\)-generalized two-wavelet multipliers; Schatten-von Neumann classReferences:
| [1] | Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Cambridge (1988) · Zbl 0647.46057 |
| [2] | Ben Said, S.; Kobayashi, T.; Ørsted, B., Laguerre semigroup and Dunkl operators, Compos. Math., 148, 1265-1336, (2012) · Zbl 1255.43004 · doi:10.1112/S0010437X11007445 |
| [3] | Ben Said, S., Strichartz estimates for Schrödinger-Laguerre operators, Semigroup Forum, 90, 251-269, (2015) · Zbl 1316.35056 · doi:10.1007/s00233-014-9617-9 |
| [4] | Calderon, JP, Intermediate spaces and interpolation, the complex method, Studia Mathematica, 24, 113-190, (1964) · Zbl 0204.13703 · doi:10.4064/sm-24-2-113-190 |
| [5] | Constales, D.; Bie, H.; Lian, P., Explicit formulas for the Dunkl dihedral kernel and the \((\kappa , a)\)-generalized Fourier kernel, J. Math. Anal. Appl., 460, 900-926, (2018) · Zbl 1382.65474 · doi:10.1016/j.jmaa.2017.12.018 |
| [6] | Bie, H., The kernel of the radially deformed Fourier transform, Integr. Transform. Spec. Funct., 24, 1000-1008, (2013) · Zbl 1285.42011 · doi:10.1080/10652469.2013.799467 |
| [7] | Bie, H., Clifford algebras, Fourier transforms, and quantum mechanics, Math. Methods Appl. Sci., 35, 2198-2228, (2012) · Zbl 1259.42001 · doi:10.1002/mma.2679 |
| [8] | Bie, H.; Ørsted, B.; Somberg, P.; Souček, V., Dunkl operators and a family of realizations of \(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)\), Trans. Am. Math. Soc., 364, 3875-3902, (2012) · Zbl 1276.33021 · doi:10.1090/S0002-9947-2012-05608-X |
| [9] | Du, J.; Wong, MW, Traces of wavelet multipliers, C. R. Math. Rep. Acad. Sci. Can., 23, 148-152, (2001) · Zbl 0996.42022 |
| [10] | Dunkl, CF, Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc., 311, 167-183, (1989) · Zbl 0652.33004 · doi:10.1090/S0002-9947-1989-0951883-8 |
| [11] | Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of the Special Session on Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, FL, 1991), Contemp. Math. 138, 123-138 (1992) · Zbl 0789.33008 |
| [12] | Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995) · Zbl 0841.35001 |
| [13] | Gorbachev, D.; Ivanov, V.; Tikhonov, S., Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in \(L^{2}\), Int. Math. Res. Not., 2016, 7179-7200, (2016) · Zbl 1334.42020 · doi:10.1093/imrn/rnv398 |
| [14] | He, Z.; Wong, MW, Wavelet multipliers and signals, J. Austral. Math. Soc. Ser. B, 40, 437-446, (1999) · Zbl 0937.35196 · doi:10.1017/S0334270000010523 |
| [15] | Howe, R.: The oscillator semigroup. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 61-132, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, RI (1988) · Zbl 0687.47034 |
| [16] | Johansen, T-R, Weighted inequalities and uncertainty principles for the \((k, a)\)-generalized Fourier transform, Int. J. Math., 27, 1650019, (2016) · Zbl 1355.43005 · doi:10.1142/S0129167X16500191 |
| [17] | Kobayashi, T.; Mano, G., The inversion formula an holomorphic extension of the minimal representation of the conformal group, harmonic analysis, group representations, automorphic forms and invariant theory: In honor of Roger Howe, Word Sci., 2007, 159-223, (2007) |
| [18] | Kobayashi, T., Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group \(O(p,q)\). vi + pp. 132, Mem. Amer. Math. Soc. vol. 212(1000), (2011) |
| [19] | Liu, L., A trace class operator inequality, J. Math. Anal. Appl., 328, 1484-1486, (2007) · Zbl 1114.47020 · doi:10.1016/j.jmaa.2006.04.092 |
| [20] | Rösler, M., Positivity of Dunkl’s intertwining operator, Duke Math. J., 98, 445-463, (1999) · Zbl 0947.33013 · doi:10.1215/S0012-7094-99-09813-7 |
| [21] | Stein, EM, Interpolation of linear operators, Trans. Am. Math. Soc., 83, 482-492, (1956) · Zbl 0072.32402 · doi:10.1090/S0002-9947-1956-0082586-0 |
| [22] | Wong, M.W.: Wavelet Transforms and Localization Operators, vol. 136. Springer, Berlin (2002) · Zbl 1016.42017 · doi:10.1007/978-3-0348-8217-0 |
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.
