Abstract
In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.
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References
ALBRECHT, E. and M. NEUMANN: Automatic continuity of generalized local linear operators. Preprint 1980.
BJÖRCK, G.: Linear partial differential operators and generalized distributions. Ark. Mat.6, 351–406 (1966).
CIORĂNESCU, J. and L. ZSIDÓ: ω-ultradistributions and their applications in operator theory. INCREST Preprint series in Math.17 (1978).
DROSTE, B.: Beitrag zum Divisionsproblem für Ultra-distributionen und ein Fortsetzungssatz. Manuscripta math.27, 259–278 (1979).
FLORET, K. and J. WLOKA: Einführung in die Theorie der lokalkonvexen Räume. Springer Lecture Notes Math.56 (1968).
KOMATSU, H.: Ultradistributions I, Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo20, 25–105 (1973).
—: Ultradistributions II, The kernel theorem and ultradistributions with support in a submanifold. J. Fac. Sci. Univ. Tokyo24, 607–628 (1977).
KÖTHE, G.: Topological vector spaces I. New York - Heidelberg - Berlin: Springer 1969.
—: Topological vector spaces II. New York - Heidelberg - Berlin: Springer 1979.
MEISE, R.: Räume holomorpher Vektorfunktionen mit Wachstumsbedingungen und topologische Tensorprodukte. Math. Ann.119, 293–312 (1972).
PEETRE, J.: Rectification á l'article “Une caractérisation abstraite des opérateurs différentiels”. Math. Scand.8, 116–120 (1960).
PVIETSCH, A.: Nuclear locally convex spaces. Berlin - Heidelberg - New York: Springer 1972.
RAIKOV, D.A.: Double closed-graph theorem for topological linear spaces. Siberian Math. J.7, 287–300 (1966).
SCHAEFER, H.H.: Topological vector spaces. New York - London: Macmillan 1966.
SCHWARTZ, L.: Théorie des distributions á valeurs vectorielles I. Ann. Inst. Fourier7, 1–142 (1957).
TREVES, F.: Topological vector spaces, distributions and kernels. New York - London: Academic Press 1967.
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Albrecht, E., Neumann, M. Local operators between spaces of ultradifferentiable functions and ultradistributions. Manuscripta Math 38, 131–161 (1982). https://doi.org/10.1007/BF01168588
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DOI: https://doi.org/10.1007/BF01168588