Eigenfunction expansions of ultradifferentiable functions and ultradistributions

Dasgupta, Aparajita; Ruzhansky, Michael

doi:10.1090/tran/6765

Eigenfunction expansions of ultradifferentiable functions and ultradistributions
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by Aparajita Dasgupta and Michael Ruzhansky PDF

Trans. Amer. Math. Soc. 368 (2016), 8481-8498 Request permission

Abstract:

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold $X$. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on $X$. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014).

References

Richard D. Carmichael, Andrzej Kamiński, and Stevan Pilipović, Boundary values and convolution in ultradistribution spaces, Series on Analysis, Applications and Computation, vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. MR 2347838, DOI 10.1142/9789812708786
Antoine Delcroix, Maximilian F. Hasler, Stevan Pilipović, and Vincent Valmorin, Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 697–708. MR 2103924, DOI 10.1017/S0305004104007923
Aparajita Dasgupta and Michael Ruzhansky, Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bull. Sci. Math. 138 (2014), no. 6, 756–782. MR 3251455, DOI 10.1016/j.bulsci.2013.12.001
J. Delgado and M. Ruzhansky, Fourier multipliers, symbols and nuclearity on compact manifolds, J. Anal. Math., to appear (arXiv:1404.6479).
Julio Delgado and Michael Ruzhansky, Schatten classes on compact manifolds: kernel conditions, J. Funct. Anal. 267 (2014), no. 3, 772–798. MR 3212723, DOI 10.1016/j.jfa.2014.04.016
V. I. Gorbachuk, Fourier series of periodic ultradistributions, Ukrain. Mat. Zh. 34 (1982), no. 2, 144–150, 267 (Russian). MR 656670, DOI 10.1007/BF01091513
Claudia Garetto and Michael Ruzhansky, Wave equation for sums of squares on compact Lie groups, J. Differential Equations 258 (2015), no. 12, 4324–4347. MR 3327555, DOI 10.1016/j.jde.2015.01.034
Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI 10.1007/BF02391913
Hikosaburo Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105. MR 320743
Hikosaburo Komatsu, Ultradistributions. II. The kernel theorem and ultradistributions with support in a submanifold, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 607–628. MR 477770
Hikosaburo Komatsu, Ultradistributions. III. Vector-valued ultradistributions and the theory of kernels, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 653–717. MR 687595
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Mats Neymark, On the Laplace transform of functionals on classes of infinitely differentiable functions, Ark. Mat. 7 (1969), 577–594. MR 241974, DOI 10.1007/BF02590896
Hans-Joachim Petzsche, Die Nuklearität der Ultradistributionsräume und der Satz vom Kern. II, Manuscripta Math. 27 (1979), no. 3, 221–251 (German, with English summary). MR 531139, DOI 10.1007/BF01309009
Stevan Pilipović and Bojan Prangoski, On the convolution of Roumieu ultradistributions through the $\epsilon$ tensor product, Monatsh. Math. 173 (2014), no. 1, 83–105. MR 3148662, DOI 10.1007/s00605-013-0503-4
Hans-Joachim Petzsche and Dietmar Vogt, Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions, Math. Ann. 267 (1984), no. 1, 17–35. MR 737333, DOI 10.1007/BF01458468
S. Pilipović and D. Scarpalezos, Regularity properties of distributions and ultradistributions, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3531–3537. MR 1860484, DOI 10.1090/S0002-9939-01-06013-0
Luigi Rodino, Linear partial differential operators in Gevrey spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1249275, DOI 10.1142/9789814360036
Charles Roumieu, Ultra-distributions définies sur $R^{n}$ et sur certaines classes de variétés différentiables, J. Analyse Math. 10 (1962/63), 153–192 (French). MR 158261, DOI 10.1007/BF02790307
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
R. T. Seeley, Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204. MR 173174, DOI 10.1090/S0002-9947-1965-0173174-1
R. T. Seeley, Eigenfunction expansions of analytic functions, Proc. Amer. Math. Soc. 21 (1969), 734–738. MR 240835, DOI 10.1090/S0002-9939-1969-0240835-4
Yoshiko Taguchi, Fourier coefficients of periodic functions of Gevrey classes and ultradistributions, Yokohama Math. J. 35 (1987), no. 1-2, 51–60. MR 928372