CUBO A Mathematical Journal Vol.12, N°03, (187–202). October 2010

 

Modulation Spaces with A ^loc_∞ -Weights

 

YOSHIHIRO SAWANO

Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan, email: yosihiro@math.kyoto-u.ac.jp

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ABSTRACT

In this paper we describe the function space M ^s,w_p,q with w ∈ A ^loc_∞ together with some related results of weighted modulation spaces.

Key words and phrases: Modulation spaces, Exponential weights.

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RESUMEN

En este artículo describimos el espacio de la funciones M ^s,w_p,q con w ∈ A ^loc_∞ junto con algunos resultados relacionados a espacios de modulación con peso.

Math. Subj. Class.: 41A17,42B35.

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References

[1] BALAN, R., CASAZZA, P.G., HEIL, C. AND LANDAU, Z., Density, overcompleteness, and localization of frames, II. Gabor systems, J. Fourier Anal. Appl., 12 (3), 309–344, 2006.

[2] BÉNYI, A., GRÖCHENIG, K., OKOUDJOU, K.A. AND ROGERS, L.G., Unimodular Fourier multipliers for modulation spaces (English summary), J. Funct. Anal., 246, no. 2, 366– 384, 2007.

[3] BAOXIANG, W. AND CHUNYAN, H., Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239(2007), no 1, 213–250.

[4] BAOXIANG, W., LIFENG, Z. AND BOLING, G., Isometric decomposition operators, function spaces E^λ _p,q and applications to nonlinear evolution equations, J. Funct. Anal., 233(1), 1–39, 2006.

[5] FEICHTINGER, H., Modulation spaces on locally compact abelian groups, Technical report, University of Vienna.        [ Links ]

[6] FEICHTINGER, H., Atomic characterization of modulation spaces through Gabor-type representation, In Proc. Conf. Constructive Function Theory, Edmonton, July (1989), 113–126.

[7] FEICHTINGER, H., Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, II, 188 (8-810), 451–471, 1979.

[8] FEICHTINGER, H. AND GRÖCHENIG, K., GABOR WAVELETS AND THE HEISENBERG GROUP: GABOR EXPANSIONS AND SHORT TIME FOURIER TRANSFORM FROM THE GROUP THEORETICAL POINT OF VIEW, In Charles K. Chui, editor, Wavelets :A tutorial in theory and applications, 359–398, Academic Press, Boston, MA, 1992.

[9] FEICHTINGER, H. AND GRÖCHENIG, K., Gabor frames and time-frequency analysis of distributions, J. Functional. Anal., 146(2) (1997), 464–495.

[10] GALPERIN, Y.V. AND SAMARAH, S., Time-frequency analysis on modulation spaces M^p,q _m , 0 < p, q ≤∞, Appl. Comput. Harmon. Anal., 16 (2004), 1–18.

[11] GRÖCHENIG, K., Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2001.        [ Links ]

[12] GRÖCHENIG, K., Time-frequency analysis of Sjöstrands class, RevistaMat. Iberoam., 22 (2), 703–724, (2006), arXiv:math.FA/0409280v1.

[13] GRÖCHENIG, K., Weight functions in time-frequency analysis. (English summary) Pseudo-differential operators: partial differential equations and time-frequency analysis, Fields Inst. Commun., bf52, Amer. Math. Soc., Providence, RI, (2007), 343–366.

[14] GRÖCHENIG, K. AND HEIL, C., Modulation spaces and pseudo-differential operators, Integral Equations Operator Theory, 34, 439–457, 1999.

[15] GRÖCHENIG, K. AND RZESZOTNIK, Z., Almost diagonalization of pseudodifferential operators, Ann. Inst. Fourier, (2008), to appear        [ Links ]

[16] HASUMI, M., Note on the n-dimension Tempered Ultradistributions, Tohoku Math. J., 13, 94–104, 1961.

[17] IZUKI, M. AND SAWANO, Y., Greedy bases in weighted modulation spaces, to appear in J. Nonlinear Analysis Series A: Theory, Methods and Applications.        [ Links ]

[18] KOBAYASHI, M., Modulation spaces M^p,q for 0 < p, q ≤∞, J. Function Spaces Appl, 4(3) (2006), 329–341.

[19] KOBAYASHI, M. AND SAWANO, Y., Molecular decomposition of the modulation spaces M^p,q and its application to the pseudo-differential operators, to appear in Osaka Mathematical Journal.        [ Links ]

[20] SAWANO, Y., Atomic decomposition for the modulation space M^s_p,qq with 0 < p, q ≤∞, s∈ R , Proceedings of A. Razmadze Mathematical Institute, 145, 63–68, 2007.

[21] SAWANO, Y., Weighted modulation space M^s_p,q (w) with w ∈ A ^loc_p , J. Math. Anal. Appl, 345, 615–627, 2008.

[22] SJÖSTRAND, J., An algebra of pseudodifferential operators, Math. Res. Lett., 1, no.2, 185–192, 1994.

[23] SJÖSTRAND, J., Wiener type algebras of pseudodifferential operators, In Séminaire sur les équations aux dérivées partielles, 1994–1995, pages Exp. No. IV, 21. École Polytech., Palaiseau, 1995.

[24] SJÖSTRAND, J., Pseudodifferential operators and weighted normed symbol spaces, Preprint, 2007. arXiv:0704.1230v1.        [ Links ]

[25] TACHIZAWA, K., The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr., 168, 263–277, 1994.