Abstract
In a recent paper by M. Măntoiu and M. Ruzhansky, a global pseudo-differential calculus has been developed for unimodular groups of type I. In the present article we generalize the main results to arbitrary locally compact groups of type I. Our methods involve the use of Plancherel’s theorem for non-unimodular groups. We also make connections with a \(C^*\)-algebraic formalism, involving dynamical systems, and give explicit constructions for the group of affine transformations of the real line.
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References
Bahouri, H., Fermanian-Kammerer, C., Gallagher, I.: Phase space analysis and pseudo-differential calculus on the Heisenberg group. Astérisque 342 (2012)
Bruhat, F.: Distributions sur un groupe localement compact et applications a l’étude des représentations des groupes \(p\)-adiques. Bull. Soc. Math. France 89, 43–75 (1961)
Bustos, H., Măntoiu, M.: Twisted pseudo-differential operators on type I locally compact groups. Illinois J. Math. 60(2), 365–390 (2016)
Christ, M., Geller, D., Glowacki, P., Polin, L.: Pseudo-differential operators on groups with dilations. Duke Math. J. 68(1), 31–65 (1992)
Delgado, J., Ruzhansky, M.: \(L^p\)-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. 102(1), 153–172 (2014)
Derighetti, A.: Convolution Operators on Groups. Lecture Notes of the Unione Matematica Italiana, vol. 11, Springer, Heidelberg; UMI, Bologna (2011)
Dixmier, J.: Les \(C^*\)-algébres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Cie, Paris (1964)
Duflo, M., Moore, C.C.: On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21(2), 209–243 (1976)
Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups. Progress in Mathematics. Birkhäuser, Basel (2016)
Folland, G.B.: A Course in Abstract Harmonic Analysis, 2nd edn. Textbooks in Mathematics. CRC Press, Boca Raton, FL (2016)
Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863. Springer-Verlag, Berlin (2005)
Glowacki, P.: Invertibility of convolution operators on homogeneous groups. Rev. Mat. Iberoam. 28(1), 141–156 (2012)
Glowacki, P.: The Melin calculus for general homogeneous groups. Ark. Mat. 45(1), 31–48 (2007)
Iftimie, V., Măntoiu, M., Purice, R.: Magnetic pseudo-differential operators. Publ. RIMS. 43, 585–623 (2007)
Mackey, G.W.: The Theory of Unitary Group Representations. University of Chicago Press, Chicago, London (1976)
Măntoiu, M.: Essential spectrum and Fredholm properties for operators on locally compact groups. J. Oper. Theory 77(2), 481–501 (2017)
Melin, A.: Parametrix constructions for right invariant differential operators on nilpotent groups. Ann. Global Anal. Geom. 1(1), 79–130 (1983)
Măntoiu, M., Purice, R.: The magnetic Weyl calculus. J. Math. Phys. 45(4), 1394–1417 (2004)
Măntoiu, M., Purice, R., Richard, S.: Spectral and propagation results for magnetic Schrödinger operators; A \(C^*\)-algebraic framework. J. Funct. Anal. 250, 42–67 (2007)
Măntoiu, M., Ruzhansky, M.: Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups. Doc, Math (2017)
Măntoiu, M., Ruzhansky, M.: Quantizations on nilpotent Lie groups and algebras having flat coadjoint orbits, Preprint ArXiV and submitted
Ruzhansky, M. and Turunen, V.: Pseudo-differential Operators and Symmetries Pseudo-differential Operators. Theory and Applications, vol. 2. Birkhäuser, Basel (2010)
Ruzhansky, M., Turunen, V.: Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl. 16, 943–982 (2010)
Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces. Int. Math. Res. Not. IMRN 11, 2439–2496 (2013)
Ruzhansky, M., Turunen, V., Wirth, J.: Hörmander-class of pseudo-differential operators on compact Lie groups and global hypoellipticity. J. Fourier Anal. Appl. 20, 476–499 (2014)
Ruzhansky, M., Wirth, J.: Global functional calculus for operators on compact Lie groups. J. Funct. Anal. 267, 144–172 (2014)
Tatsuuma, N.: Plancherel formula for non-unimodular locally compact groups. J. Math. Kyoto Univ. 12, 179–261 (1972)
Taylor, M.E.: Pseudo-differential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton, NJ (1981)
Taylor, M.E.: Noncommutative Microlocal Analysis I. American Mathematical Soc., vol. 52, no. 313 (1984)
Williams, D.P.: Crossed Products of \(C^*\)-Algebras, Mathematical Surveys and Monographs, vol. 34. American Mathematical Society, Providence, RI (2007)
Acknowledgement
M. Sandoval has been supported by Beca de Magister Nacional 2016 Conicyt and partially supported by Núcleo Milenio de Física Matemática RC120002. M. Măntoiu is supported by the Fondecyt Project 1160359.
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Măntoiu, M., Sandoval, M. (2019). Pseudo-differential Operators Associated to General Type I Locally Compact Groups. In: Delgado, J., Ruzhansky, M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-030-05657-5_11
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