Abstract
We first give a short survey on the methods of Microlocal Analysis. In particular we recall some basic facts concerning the theory of pseudodifferential operators. We then present two applications. We first discuss lower bounds for operators with multiple characteristics. Then we give a new formula for the composition of Wick operators.
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References
H. Ando and Y. Morimoto, Wick calculus and the Cauchy problem for some dispersive equationsOsaka J. Math. 39(2002), 123–147.
R. Beals, A general calculus of pseudo-differential operatorsDuke Math. J. 42(1975), 1–42.
R. Beals and C. Fefferman, Spatially inhomogeneous pseudo-differential operatorsI Comm. Pure Appl. Math. 27(1974), 1–24.
F.A. Berezin, Wick and anti-Wick symbols of operatorsMath. Sb. (N.S.)86(128) (1971), 578–610.
P. Boggiatto, E. Buzano and L. RodinoGlobal Hypoellipticity and Spectral TheoryAkademie Verlag, Berlin, 1996.
P. Boggiatto, E. Cordero and K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spacesIntegral Equations Operator Theoryto appear.
P. Boggiatto and L. Rodino, Quantization and pseudo-differential operatorsCubo Mat Ed. 5(2003), 237–272.
J.-M. Bony, Calcul symbolique et propagations des singularités pour les équations aux dérivées partielles non linéairesAnn. Sci. École Norm. Sup. 14(1981), 209–246.
J.-M. Bony, Sur l’inégalité de Fefferman-Phong, inSéminaire sur les Équations aux Dérivé Partielles1998–1999, Exp. No. III, École Polytech., Palaiseau, 1998.
L. Boutet de Monvel, A. Grigis and B. Helfer, Paramétrixes d’opérateurs pseudo-différentiels à caractéristiques multiplesAstérique 34–35(1976), 93–121.
A.P. Calderón and R. Vaillancourt, A class of bounded pseudo-differential operatorsProc. Nat. Acad. Sci. USA 69(1972), 1185–1187.
L. CohenTime-Frequency AnalysisPrentice Hall, Englewood Cliffs, NJ, 1995.
E. Cordero and K. Gröchenig, Time-frequency analysis of localization operatorsJ. Funct. Anal. 205(2003), 107–131
E. Cordero and L. Rodino, Wick calculus: a time-frequency approachOsaka J. Math.to appear.
A. Córdoba and C. Fefferman, Wave packets and Fourier integral operatorsComm. Partial Differential Equations 3(1978), 979–1005, 1978.
I.Daubechies, Time-frequency localization operators: a geometric phase space approachIEEE Trans. Inform. Theory 34(1988), 605–612.
J. Du and M.W. Wong, A product formula for localization operatorsBull. Korean Math. Soc. 37(2000), 77–84.
C. Fefferman and D.H. Phong, On positivity of pseudo-differential operatorsProc. Natl. Acad. Sci. USA 75(1978), 4673–4674.
H.G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, inAdvances in Gabor AnalysisEditors: H. G. Feichtinger and T. Strohmer, Birkhäuser, Boston, 2002, 99–128.
G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989.
L. Gárding, Dirichlet’s problem for linear elliptic partial differential equationsMath. Scand. 1(1953), 55–72.
K. GröchenigFoundations of Time-Frequency AnalysisBirkhäuser, Boston, 2001.
F. HérauOpérateurs Pseudo-Differentieles Semi-BornésPh.D. Dissertation, University of Rennes, 1999.
F. Hérau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebraArk. Mat. 39(2001), 311–338.
F. Hérau, Melin inequality for paradifferential operators and applicationsComm. Partial Differential Equations 27(2002), 1659–1680.
L. Hörmander, Pseudo-differential operatorsComm. Pure Appl. Math. 18(1965), 501–517.
L. Hörmander, Pseudo-differential operators and hypoelliptic equations, in Singular IntegralsProc. Sympos. Pure Math.Amer. Math. Soc.10(1966), 138–183.
L. Hörmander, Fourier integral operators IActa Math. 127(1971), 79–183.
L. Hörmander, The Cauchy problem for differential equations with double characteristicsJ. Analyse Math. 32(1977), 118–196.
L. Hörmander, The Weyl calculus of pseudo-differential operatorsComm. Pure Appl. Math. 32(1979), 359–443.
L. HörmanderThe Analysis of Linear Partial Differential Operators I—IVSpringer-Verlag, Berlin, 1983–1985.
Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, New York, 1968.
J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operatorsComm. Pure Appl. Math. 18(1965), 269–305.
N. Lerner, The Wick calculus of pseudo-differential operators and energy estimates, inNew Trends in Microlocal AnalysisSpringer, 1997, 23–37.
O. Liess and L. Rodino, Linear partial differential equations with multiple involutive characteristics, inMicrolocal Analysis and Spectral TheoryEditor: L. Rodino, Kluwer Academic Publishers, 1997, 227–250.
M. Mascarello and L. RodinoLinear Partial Differential Operators with Multiple CharacteristicsWiley-Akademie Verlag, Berlin, 1997.
A. Melin, Lower bounds for pseudo-differential operatorsArk. Mat.9 (1971), 117–140.
] M. Mughetti and F. Nicola, A counterexample to a lower bound for a class of pseudodifferential operatorsProc. Amer. Math. Soc.to appear.
M. Mughetti and F. Nicola, Ageneralization of Hörmander’s inequality IIin preparation.
F. Nicola and L. RodinoRemarks on lower bounds for pseudo-differential operatorspreprint.
C. Parenti and A. Parmeggiani, Lower bounds for pseudo-differential operators, inMicrolocal Analysis and Spectral TheoryEditor: L. Rodino, Kluwer Academic Publishers, 1997, 227–250.
C. Parenti and A. Parmeggiani, Some remarks on almost-positivity of pseudodifferential operatorsBoll. Un. Mat./t.1-B 8 (1998), 187–215.
C. Parenti and A. Parmeggiani, A generalization of Hörmander’s inequality IComm. Partial Differential Equations25 (2000), 457–506.
C. Parenti and A. Parmeggiani, Lower bounds for systems with double characteristicsJ. Analyse. Math.86 (2002), 49–91.
J. Ramanathan and P. Topiwala, Time-frequency localization via the Weyl correspondenceSIAM J. Math. Anal.24 (1993), 1378–1393.
L. Schwartz, Théorie des Distributions, Hermann, Paris, 1966.
M.A. ShubinPseudo-Differential Operators and Spectral TheorySecond Edition, Springer-Verlag, Berlin, 2001.
J. Sjöstrand, Parametrices for pseudo-differential operators with multiple characteristicsArk. Mat.12 (1974), 85–130.
D. Tataru, On the Fefferman-Phong inequality and related problemsComm. Partial Differential Equations27 (2002), 2101–2138.
J. ToftModulation spaces and pseudo-differential operatorspreprint.
M.W. WongLocalization OperatorsSeoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1999.
M.W. Wong, Wavelet Transforms and Localization Operators, Birkhäuser, Basel, 2002.
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Cordero, E., Nicola, F., Rodino, L. (2004). Microlocal Analysis and Applications. In: Ashino, R., Boggiatto, P., Wong, M.W. (eds) Advances in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7840-1_1
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DOI: https://doi.org/10.1007/978-3-0348-7840-1_1
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