Windowed-Kontorovich-Lebedev transforms

Abstract

The aim of this paper is to study the boundedness of the windowed-Kontorovich-Lebedev transforms. For this purpose, we first define the translation associated to the Kontorovich-Lebedev transform and a generalized convolution product, then obtain some harmonic analysis results. We present a sufficient and necessary condition for the boundedness of the windowed-Kontorovich-Lebedev transform. Finally, we define the corresponding Weyl operator, and study the boundedness and compactedness of the Weyl operator with symbols in L^q (q ∈ [1, 2]) acting on L^p.

Keywords

Kontorovich-Lebedev transform / translation / Weyl operator / Windowed-Kontorovich-Lebedev transform

References

[1] Bennett C, Sharpley R. Interpolation of Operators. New York: Academic Press, 1988

[2] Bloom W R, Heyer H. Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics, Vol 20. Berlin: Walter de Gruyter Co, 1995

[3] Boggiatto P, Buzano E, Rodino L. Global Hypoellipticity and Spectral Theory. Berlin: Akademie-Verlag, 1996

[4] Boggiatto P, De Donno G, Oliaro A. Weyl quantization of Lebesgue spaces. Math Nachr, 2009, 282(12): 1656-1663

[5] Boggiatto P, Rodino L. Quantization and pseudo-differential operators. Cubo Mat Educ, 2003, 5(1): 237-272

[6] Bouattour L, Trimeche K. Beurling-Hörmander’s theorem for the Chébli-Trimèche transform. Glob J Pure Appl Math, 2005, 1(3): 342-357

[7] Dachraoui A. Weyl-Bessel transforms. J Comput Appl Math, 2001, 133(1-2): 263-276

[8] Gröchenig K H. Foundations of Time-Frequency Analysis. Boston: Birkhäuser, 2001

[9] Lebedev N N. Special Functions and Their Applications. Moscow: Gosudarstv Izdat Tehn-Teor Lit, 1953

[10] Oberhettinger F. Tables of Bessel Transforms. New York: Springer, 1972

[11] Peng L Z, Zhao J M. Weyl transforms on the upper half plane. Rev Mat Complut, 2010, 23: 77-95

[12] Rachdi L T, Trimèche K. Weyl transforms associated with the spherical mean operator. Anal Appl (Singap), 2003, 1(2): 141-164

[13] Stein E M. Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

[14] Terras A. Harmonic Analysis on Symmetric Space and Application, I. New York: Springer-Verlag, 1985

[15] Toft J. Continuity properties for modulation spaces with applications to pseudodifferential calculus, I. J Funct Anal, 2004, 207(2): 399-429

[16] Toft J. Continuity properties for modulation spaces with applications to pseudodifferential calculus, II. Ann Global Anal Geom, 2004, 26(1): 73-106

[17] Watson G N. A Treatise on the Theory of Bessel Functions. 2nd ed. London: Cambridge Univ Press, 1966

[18] Wong M W. Weyl Transform. New York: Springer-Verlag, 1998

[19] Zhao J M, Peng L Z. Wavelet and Weyl transform associated with the spherical mean operator. Integral Equation and Operator Theory, 2004, 50: 279-290

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg