Atlas: A perturbation result for surjective convolution operators on the spaces of all non-quasianalytic functions by David Jornet

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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A perturbation result for surjective convolution operators on the spaces of all non-quasianalytic functions
by
David Jornet
Universidad Politécnica de Valencia

Let E_(\omega)(R^N) be the space of all non-quasianalytic functions of Beurling type on R^N. An ultradistribution \mu in E'_(\omega)(R^N) with compact support satisfies that the convolution operator f --> f*\mu is surjective in E_(\omega)(R^N) if and only if \mu is (\omega)-slowly decreasing.

We prove that if \mu is (\omega)-slowly decreasing, then \mu+\nu is also (\omega)-slowly decreasing for each \nu in E'_(\omega)(R^N) whose (\omega)-singular support does not intersect the one of \mu. The same result holds for ultradistributions of Roumieu type. This extends a result of Hörmander [2] (see also [1] for a different proof).

References

[1] W. Abramczuk, A class of surjective convolution operators, Pacific J. Math., 110 No. 1 (1984), 1-7. [2] L. Hörmander, Supports and singular supports of convolutions, Acta Math., 110 (1963), 279-302.

We report on research done under the advice of C. Fernández and A. Galbis.

(P)

Date received: April 13, 2000