Abstract
We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR 2d, which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol σ lies in the modulation spaceM ∞,1 (R 2d), then the corresponding pseudodifferential operator is bounded onL 2(R d) and, more generally, on the modulation spacesM p,p (R d) for 1≤p≤∞. If σ lies in the modulation spaceM s2,2 (R 2d)=L /2 s (R 2d)∩H s(R 2d), i.e., the intersection of a weightedL 2-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.
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References
[BS77] M. Š. Birman and M. Z. Solomjak,Estimates for the singular numbers of integral operators (Russian), Uspehi Mat. Nauk32 (1977), 17–84.
[CV72] A.-P. Calderón and R. Vaillancourt,A class of bounded pseudo-differential operators., Proc. Nat. Acad. Sci. U.S.A.69 (1972), 1185–1187.
[CM78] R. R. Coifman and Y. Meyer,Au delà des opérateurs pseudo-différentiels, Astérisque57 (1978).
[Dau80] I. Daubechies,On the distributions corresponding to bounded operators in the Weyl quantization, Comm. Math. Phys.75 (1980), 229–238.
[Dau83] I. Daubechies,Continuity statements and counterintuitive examples in connection with Weyl quantization, J. Math. Phys.24 (1983), 1453–1461.
[Dau90] I. Daubechies,The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory39 (1990), 961–1005.
[DG80] I. Daubechies and A. GrossmannAn integral transform related to quantization, J. Math. Phys.21 (1980), 2080–2090.
[DS88] N. Dunford and J. T. Schwartz,Linear Operators, Part II, Wiley, New York, 1988.
[Fei81] H. G. Feichtinger,Banach spaces of distributions of Wiener's type and interpolation, Functional Analysis and Approximation (Oberwolfach, 1980), Internat. Ser. Numer. Math., vol. 60, Birkhäuser, Basel, 1981, pp. 153–165.
[FG89a] H. G. Feichtinger and K. Gröchenig,Banach spaces related to integrable group representations and their atomic decompositions, I J. Funct Anal.86 (1989), 307–340.
[FG89b] H. G. Feichtinger and K. Gröchenig,Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math.108 (1989). 129–148.
[FG92] H. G. Feichtinger and K. Gröchenig,Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view, Wavelets: A Tutorial in Theory and Applications (C. K. Chui, ed.), Academic Press, Boston, 1992, pp. 359–398.
[FG97] H. G. Feichtinger and K. Gröchenig,Gabor frames and time-frequency analysis of distributions, J. Funct. Anal.146 (1997), 464–495.
[Fol89] G. B. Folland,Harmonic Analysis on Phase Space, Ann. of Math. Studies, Princeton University Press, Princeton, NJ, 1989.
[Grö91] K. Gröchenig,Describing functions: atomic decompositions versus frames, Monatsh. Math.112 (1991), 1–42.
[Grö96] K. Gröchenig,An uncertainty principle related to the Poisson summation formula, Studia Math.121 (1996), 87–104.
[GLS68] A. Grossmann, G. Loupias, and E. M. Stein,An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble)18 (2) (1968), 343–368.
[HRT97] C. Heil, J. Ramanathan, and P. Topiwala,Singular values of compact pseudodifferential operators, J. Funct. Anal.150 (1997), 426–452.
[HW89] C. Heil and D. Walnut,Continuous and discrete wavelet transforms, SIAM Review31 (1989), 628–666.
[Jan95] A. J. E. M. Janssen,Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl.1 (1995), 403–437.
[How80] R. Howe,Quantum mechanics and partial differential equations, J. Funct. Anal.38 (1980), 188–254.
[Kön86] H. König,Eigenvalue Distribution of Compact Operators, Birkhäuser, Boston, 1986.
[Mey90] Y. Meyer,Ondelettes et Opérateurs II, Hermann, Paris, 1990.
[Poo66] J. C. T. Pool,Mathematical aspects of the Weyl correspondence, J. Math. Phys.7 (1966), 66–76.
[RT97] R. Rochberg and K. Tachizawa,Pseudodifferential operators, Gabor frames, and local trigonometric bases, in Gabor Analysis and Algorithms: Theory and Applications (H. G. Feichtinger and T. Strohmer, eds.), Birkhäuser, Boston, 1997, pp. 171–192.
[Sei92] K. Seip,Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math.429 (1992), 91–106.
[SW92] K. Seip and R. Wallstén,Sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math.429 (1992), 107–113.
[Ste93] E. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
[Tac94] K. Tachizawa,The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr.168 (1994), 263–277.
[Zyg68] A. Zygmund,Trigonometric Series, Second ed., Cambridge University Press, Cambridge, 1968.
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Gröchenig, K., Heil, C. Modulation spaces and pseudodifferential operators. Integr equ oper theory 34, 439–457 (1999). https://doi.org/10.1007/BF01272884
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DOI: https://doi.org/10.1007/BF01272884