Abstract
We consider distributions on a closed compact manifold \(M\) as maps on smoothing operators. Thus spaces of maps between \({{\Psi }^{\!-\!\infty }}(M)\) and \(\mathcal{C ^{\infty }}(M)\) are considered as generalized functions. For any collection of regularizing processes we produce various algebras of generalized functions and equivariant embeddings of distributions into such algebras. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This also recovers the singularity behaviour of distributions (singular support/wavefront sets) in terms of certain subalgebras of the algebra of generalized functions. This notion of regularity is compared with the regularity in Colombeau algebras in the \(\mathcal{G }^{\infty }\) sense.
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Notes
A map \(f:A\rightarrow B\) that respects equivalence relations, that is \(x\sim y\in A\Rightarrow f(x)\sim f(y)\in B\) induces a map on the quotients which we refer to by the same letter by abuse of notation and say \(f\) descends to \(A/\!\sim \rightarrow B/\!\sim \).
As already noted in the Sect. 2, we assume a smooth dependence on \(\varepsilon \) throughout this paper.
By action of a group \(G\) on a set \(S\) is always meant a group homomorphism \(\rho :G\rightarrow Auto(S)\) and as usual convention we often write \(g\cdot s:=\rho (g)(s)\). All actions considered in this paper are left actions, that is for \(g,h\in G\) we have \((g\cdot h)\cdot s=g\cdot (h\cdot s)\).
Throughout this paper the sign of \(\Delta \) is chosen so as to make it a positive operator.
References
Aragona, J., Biagioni, H.A.: Intrinsic definition of the Colombeau algebra of generalized functions. Anal. Math. 17(2), 75–132 (1991)
Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. North Holland, Amsterdam (1984)
de Roever, J.W., Damsma, M.: Colombeau algebras on a \(C^\infty \)-manifold. Indag. Math. (N.S.) 2(3), 341–358 (1991)
Dave, S.: Geometrical embeddings of distributions into algebras of generalized functions. Mathematische Nachrichten 283, 1575–1588 (2010)
Garetto, C.: Topological structures in Colombeau algebras: topological \(\tilde{\mathbb{C}}\)-modules and duality theory. Acta Appl. Math. 88, 81–123 (2005)
Garetto, C., Hörmann, G.: Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc (2) 48(3), 603–629 (2005)
Gröchenig, K.: Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis Birkhäuser Boston Inc, Boston (2001)
Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions, vol. 537 of Mathematics and its Applications 537. Kluwer Academic Publishers, Dordrecht (2001)
Grosser, M., Kunzinger, M., Vickers, J.: A global theory of algebras of generalized functions. Adv. Math. 166(1), 50–72 (2002)
Guillemin, V.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55, 131–160 (1985)
Hamilton, R.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7(1), 65–222 (1982)
Hörmann, G., Oberguggenberger, M.: Elliptic regularity and solvability for partial differential equations with Colombeau coefficients. Electron. J. Differ. Equ. 2004, 1–30 (2004)
Hörmann, G., Oberguggenberger, M., Pilipović, S.: Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. Trans. Amer. Math. Soc. 358(8), 3363–3383 (2006)
Kriegl, A., Michor, P.: The convenient setting of global analysis, Mathematical Surveys and Monographs 53. American Mathematical Society, Providence (1997)
Kunzinger, M., Steinbauer, R., Vickers, J.: Sheaves of nonlinear generalized functions and manifold-valued distributions. Trans. Amer. Math. Soc. 361, 5177–5192 (2009)
Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics vol. 259. Longman, Harlow (1992)
Nedeljkov, M., Pilipović, S., Scarpalezos, D.: The linear theory of Colombeau generalized functions, Pitman Research Notes in Mathematics Series 385. Longman, Harlow (1998)
Scarpalézos, D.: Some remarks on functoriality of Colombeau’s construction; topological and microlocal aspects and applications, Integral Transform. Spec. Funct. 6, 295–307 (1998). Generalized functions–linear and nonlinear problems (Novi Sad, 1996)
Schwartz, L.: Sur l’impossibilit de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)
Wong, M.: Weyl transforms, heat kernels, Green functions and Riemann zeta functions on compact Lie groups, Modern trends in pseudo-differential operators. Oper. Theory Adv. Appl. 172, 67–85 (2007)
Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)
Acknowledgments
I would like to thank Michael Kunzinger for his encouragement and support with this work. I am very grateful to the referee for many valuable comments and corrections.
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Communicated by A. Constantin.
Supported by FWF grants Y237-N13 and P20525 of the Austrian Science Fund.
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Dave, S. Rapidly converging approximations and regularity theory. Monatsh Math 170, 121–145 (2013). https://doi.org/10.1007/s00605-013-0480-7
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DOI: https://doi.org/10.1007/s00605-013-0480-7