Research and Innovation Articles
Published 2019-07-29
Keywords
- Fourier integral operator,
- nuclear operator,
- nuclear trace,
- spectral trace,
- compact
- homogeneous manifold ...More
How to Cite
Cardona, D. (2019). On the nuclear trace of Fourier integral operators. Revista Integración, Temas De matemáticas, 37(2), 219–249. https://doi.org/10.18273/revint.v37n2-2019002
Abstract
In this paper we characterise the r-nuclearity of Fourier integral
operators on Lebesgue spaces. Fourier integral operators will be considered in Rn, the discrete group Zn, the n-dimensional torus and symmetric spaces (compact homogeneous manifolds). We also give formulae for the nuclear trace of these operators. Explicit examples will be given on Zn, the torus Tn, the special unitary group SU(2), and the projective complex plane CP2. Our main theorems will be applied to the characterization of r-nuclear pseudodifferential operators defined by the Weyl quantization procedure.
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References
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[16] Delgado J. and Ruzhansky M., “Lp-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups” , J. Math. Pures Appl. (9), 102 (2014), No. 1, 153–172.
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[18] Delgado J. and Ruzhansky M., “Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds”, C. R. Acad. Sci. Paris. Ser. I. 352 (2014), No. 10, 779–784.
[19] Delgado J. and Ruzhansky M., “Fourier multipliers, symbols and nuclearity on compact manifolds”, J. Anal. Math. 135 (2018), No. 2, 757–800.
[20] Delgado J. and Ruzhansky M., “The bounded approximation property of variable Lebesgue spaces and nuclearity”, Math. Scand. 122 (2018), No.2, 299–319.
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[24] Delgado J., Ruzhansky M. and Tokmagambetov N., “Schatten classes, nuclearity and non-harmonic analysis on compact manifolds with boundary”, J. Math. Pures Appl. (9) 107 (2017), No. 6, 758–783.
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[26] Eskin G. I., “Degenerate elliptic pseudodifferential equations of principal type”, Mat. Sb.(N.S.), 82(124) (1970), 585–628.
[27] Fegan H., Introduction to compact Lie groups. Series in Pure Mathematics, 13. World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
[28] Fujiwara A., “construction of the fundamental solution for the Schrödinger equations”, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), No. 1, 10–14.
[29] Ghaemi M.B., Jamalpour Birgani M. and Wong M.W., “Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products”, J. Pseudo-Differ. Oper. Appl. 8 (2017), No. 2, 191–201.
[30] Ghaemi M.B., Jamalpour Birgani M. and Wong M.W., “Characterization, adjoints and products of nuclear pseudo-differential operators on compact and Hausdorff groups”, U.P.B. Sci. Bull., Series A 79 (2017), No. 4, 207–220.
[31] Grothendieck A., Produits tensoriels topologiques et espaces nucléaires, Memoirs Amer. Math. Soc. 16, Providence, 1955.
[32] Hörmander L., “Fourier integral operators I”, Acta Math. 127 (1971), No. 1–2, 79–183.
[33] Hörmander L., “Pseudo-differential Operators and Hypo-elliptic equations”, Proc. Symposium on Singular Integrals, Amer. Math. Soc. 10 (1967), 138–183.
[34] Hörmander L., The Analysis of the linear partial differential operators, III, Springer-Verlag, Berlin, 1985.
[35] Kumano-go H., “A calculus of Fourier integral operators on Rn and the fundamental solution for an operator of hyperbolic type”, Comm. Partial Differential Equations 1 (1976), No. 1, 1–44.
[36] Jamalpour Birgani M., “Characterizations of Nuclear Pseudo-differential Operators on Z with some Applications”, Math. Model. of Nat. Phenom. 13 (2018), No. 4, 13–30.
[37] Miyachi A., “On some estimates for the wave equation in Lp and Hp”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), No. 2, 331–354.
[38] Molahajloo S., “Pseudo-differential Operators on Z”, in textitPseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory:Advances and Applications vol. 205 (eds. Schulze B.W. and Wong M.W.), Birkhäuser Basel (2009),213–221.
[39] Peral J.C., “Lp-estimates for the wave equation”, J. Funct. Anal. 36 (1980), No. 1, 114–145.
[40] Pietsch A., Operator ideals, Mathematische Monographien, 16. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
[41] Pietsch A., History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007.
[42] Reinov O.I. and Latif Q., “Grothendieck-Lidskii theorem for subspaces of Lp-spaces”, Math. Nachr. 286 (2013), No. 2–3, 279–282.
[43] Nicola F. and Rodino L., “Dixmier traceability for general pseudo-differential operators”, in C∗-algebras and elliptic theory II, Trends Math. (eds. Burghelea D., Melrose R., Mishchenko A. and Troitsky E.), Birkhäuser Basel (2008), 227–237.
[44] Ruzhansky M. and Sugimoto M., “Global L2-boundedness theorems for a class of Fourier integral operators”, Comm. Partial Differential Equations 31 (2006), No. 4, 547–569.
[45] Ruzhansky M. and Sugimoto M., “A smoothing property of Schrödinger equations in the critical case”, Math. Ann. 335 (2006), No. 3, 645–673.
[46] Ruzhansky M. and Sugimoto M., “Weighted Sobolev L2 estimates for a class of Fourier integral operators”, Math. Nachr. 284 (2011), No. 13, 1715–1738.
[47] Ruzhansky M. and Sugimoto M., “Global regularity properties for a class of Fourier integral operators”, arxiv: 1510.03807v1.
[48] Ruzhansky M. and Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhaüser-Verlag, Basel, 2010.
[49] Ruzhansky M. and Wirth J., Dispersive type estimates for Fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, (Special), 1263–1270. doi:10.3934/proc.2011.2011.1263
[50] Ruzhansky M., Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, volume 131 of CWI Tract. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2001.
[51] Seeger A., Sogge C.D. and Stein E.M., “Regularity properties of Fourier integral operators”, Ann. of Math. 134 (1991), No. 2, 231–251.
[52] Tao T., “The weak-type (1, 1) of Fourier integral operators of order −(n − 1)/2”, J. Aust. Math. Soc. 76 (2004), No. 1, 1–21.
[2] Asada K. and Fujiwara D., “On some oscillatory integral transformations in L2(Rn)”, Japan J. Math. (N.S.) 4 (1978), No. 2, 299–361.
[3] Barraza E.S. and Cardona D., “On nuclear Lp-multipliers associated to the Harmonic oscillator”, in Analysis in Developing Countries, Springer Proceedings in Mathematics & Statistics (eds. Ruzhansky M. and Delgado J.), Springer (2018), 31–41.
[4] Bronzan J. B., “Parametrization of SU(3)”, Phys. Rev. D. 38 (1988), No. 6, 1994–1999.
[5] Botchway L., Kibiti G. and Ruzhansky M., “Difference equations and pseudo-differential operators on Zn”, arXiv:1705.07564.
[6] Buzano E. and Toft J., “Schatten-von Neumann properties in the Weyl calculus”, J. Funct. Anal. 259 (2010), No. 12, 3080–3114.
[7] Cardona D. and Barraza E.S., “Characterization of nuclear pseudo-multipliers associated to the harmonic oscillator”, UPB, Scientific Bulletin, Series A: Applied Mathematics and Physics 80 (2018), No. 4, 163–172.
[8] Cardona D., “A brief description of operators associated to the quantum harmonic oscillator on Schatten-von Neumann classes”, Rev. Integr. temas mat. 36 (2018), No. 1, 49–57.
[9] Cardona D., “Nuclear pseudo-differential operators in Besov spaces on compact Lie groups”, J. Fourier Anal. Appl. 23 (2017), No. 5, 1238–1262.
[10] Coriasco S. and Ruzhansky M., “On the boundedness of Fourier integral operators on Lp(Rn)”, C. R. Math. Acad. Sci. Paris 348 (2010), No. 15–16, 847–851.
[11] Coriasco S. and Ruzhansky M., “Global Lp continuity of Fourier integral operators”, Trans. Amer. Math. Soc. 366 (2014), No. 5, 2575–2596.
[12] Delgado J., “Estimations Lp pour une classe d’opŕateurs pseudo-differentiels dans le cadre du calcul de Weyl-Hörmander”, J. Anal. Math. 100 (2006), 337–374.
[13] Delgado J., “A trace formula for nuclear operators on Lp”, in textitPseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory:Advances and Applications vol. 205 (eds. Schulze B.W. and Wong M.W.), Birkhäuser Basel (2009), 181–193.
[14] Delgado J. and Wong M.W.,“Lp-nuclear pseudo-differential operators on Z and S1”, Proc. Amer. Math. Soc. 141 (2013), No. 11, 3935–394.
[15] Delgado J.,“The trace of nuclear operators on Lp(μ) for -finite Borel measures on second countable spaces”, Integral Equations Operator Theory 68 (2010), No. 1, 61–74.
[16] Delgado J. and Ruzhansky M., “Lp-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups” , J. Math. Pures Appl. (9), 102 (2014), No. 1, 153–172.
[17] Delgado J. and RuzhanskyM., “Schatten classes on compact manifolds: Kernel conditions”, J. Funct. Anal. 267 (2014), No. 3, 772–798.
[18] Delgado J. and Ruzhansky M., “Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds”, C. R. Acad. Sci. Paris. Ser. I. 352 (2014), No. 10, 779–784.
[19] Delgado J. and Ruzhansky M., “Fourier multipliers, symbols and nuclearity on compact manifolds”, J. Anal. Math. 135 (2018), No. 2, 757–800.
[20] Delgado J. and Ruzhansky M., “The bounded approximation property of variable Lebesgue spaces and nuclearity”, Math. Scand. 122 (2018), No.2, 299–319.
[21] Delgado J. and Ruzhansky M., “Schatten classes and traces on compact groups”, Math. Res. Lett. 24 (2017), No. 4, 979–1003.
[22] Delgado J., RuzhanskyM. andWang B., “Approximation property and nuclearity on mixed-norm Lp, modulation and Wiener amalgam spaces”, J. Lond. Math. Soc. 94 (2016), No. 2, 391–408.
[23] Delgado J., Ruzhansky M. and Wang B., “Grothendieck-Lidskii trace formula for mixed- norm and variable Lebesgue spaces”, J. Spectr. Theory 6 (2016), No. 4, 781–791.
[24] Delgado J., Ruzhansky M. and Tokmagambetov N., “Schatten classes, nuclearity and non-harmonic analysis on compact manifolds with boundary”, J. Math. Pures Appl. (9) 107 (2017), No. 6, 758–783.
[25] Duistermaat J.J. and Hörmander L., “Fourier integral operators. II”, Acta Math. 128 (1972), No. 3–4, 183–269.
[26] Eskin G. I., “Degenerate elliptic pseudodifferential equations of principal type”, Mat. Sb.(N.S.), 82(124) (1970), 585–628.
[27] Fegan H., Introduction to compact Lie groups. Series in Pure Mathematics, 13. World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
[28] Fujiwara A., “construction of the fundamental solution for the Schrödinger equations”, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), No. 1, 10–14.
[29] Ghaemi M.B., Jamalpour Birgani M. and Wong M.W., “Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products”, J. Pseudo-Differ. Oper. Appl. 8 (2017), No. 2, 191–201.
[30] Ghaemi M.B., Jamalpour Birgani M. and Wong M.W., “Characterization, adjoints and products of nuclear pseudo-differential operators on compact and Hausdorff groups”, U.P.B. Sci. Bull., Series A 79 (2017), No. 4, 207–220.
[31] Grothendieck A., Produits tensoriels topologiques et espaces nucléaires, Memoirs Amer. Math. Soc. 16, Providence, 1955.
[32] Hörmander L., “Fourier integral operators I”, Acta Math. 127 (1971), No. 1–2, 79–183.
[33] Hörmander L., “Pseudo-differential Operators and Hypo-elliptic equations”, Proc. Symposium on Singular Integrals, Amer. Math. Soc. 10 (1967), 138–183.
[34] Hörmander L., The Analysis of the linear partial differential operators, III, Springer-Verlag, Berlin, 1985.
[35] Kumano-go H., “A calculus of Fourier integral operators on Rn and the fundamental solution for an operator of hyperbolic type”, Comm. Partial Differential Equations 1 (1976), No. 1, 1–44.
[36] Jamalpour Birgani M., “Characterizations of Nuclear Pseudo-differential Operators on Z with some Applications”, Math. Model. of Nat. Phenom. 13 (2018), No. 4, 13–30.
[37] Miyachi A., “On some estimates for the wave equation in Lp and Hp”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), No. 2, 331–354.
[38] Molahajloo S., “Pseudo-differential Operators on Z”, in textitPseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory:Advances and Applications vol. 205 (eds. Schulze B.W. and Wong M.W.), Birkhäuser Basel (2009),213–221.
[39] Peral J.C., “Lp-estimates for the wave equation”, J. Funct. Anal. 36 (1980), No. 1, 114–145.
[40] Pietsch A., Operator ideals, Mathematische Monographien, 16. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
[41] Pietsch A., History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007.
[42] Reinov O.I. and Latif Q., “Grothendieck-Lidskii theorem for subspaces of Lp-spaces”, Math. Nachr. 286 (2013), No. 2–3, 279–282.
[43] Nicola F. and Rodino L., “Dixmier traceability for general pseudo-differential operators”, in C∗-algebras and elliptic theory II, Trends Math. (eds. Burghelea D., Melrose R., Mishchenko A. and Troitsky E.), Birkhäuser Basel (2008), 227–237.
[44] Ruzhansky M. and Sugimoto M., “Global L2-boundedness theorems for a class of Fourier integral operators”, Comm. Partial Differential Equations 31 (2006), No. 4, 547–569.
[45] Ruzhansky M. and Sugimoto M., “A smoothing property of Schrödinger equations in the critical case”, Math. Ann. 335 (2006), No. 3, 645–673.
[46] Ruzhansky M. and Sugimoto M., “Weighted Sobolev L2 estimates for a class of Fourier integral operators”, Math. Nachr. 284 (2011), No. 13, 1715–1738.
[47] Ruzhansky M. and Sugimoto M., “Global regularity properties for a class of Fourier integral operators”, arxiv: 1510.03807v1.
[48] Ruzhansky M. and Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhaüser-Verlag, Basel, 2010.
[49] Ruzhansky M. and Wirth J., Dispersive type estimates for Fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, (Special), 1263–1270. doi:10.3934/proc.2011.2011.1263
[50] Ruzhansky M., Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, volume 131 of CWI Tract. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2001.
[51] Seeger A., Sogge C.D. and Stein E.M., “Regularity properties of Fourier integral operators”, Ann. of Math. 134 (1991), No. 2, 231–251.
[52] Tao T., “The weak-type (1, 1) of Fourier integral operators of order −(n − 1)/2”, J. Aust. Math. Soc. 76 (2004), No. 1, 1–21.