Perturbations of surjective convolution operators
HTML articles powered by AMS MathViewer
- by C. Fernández, A. Galbis and D. Jornet PDF
- Proc. Amer. Math. Soc. 130 (2002), 2377-2381 Request permission
Abstract:
Let $\mu _1$ and $\mu _2$ be (ultra)distributions with compact support which have disjoint singular supports. We assume that the convolution operator $f \rightarrow \mu _1 * f$ is surjective when it acts on a space of functions or (ultra)distribu- tions, and we investigate whether the perturbed convolution operator $f\rightarrow$ $(\mu _1 + \mu _2)* f$ is surjective. In particular we solve in the negative a question asked by Abramczuk in 1984.References
- Wojciech Abramczuk, A class of surjective convolution operators, Pacific J. Math. 110 (1984), no. 1, 1–7. MR 722732
- J. Bonet, A. Galbis, and R. Meise, On the range of convolution operators on non-quasianalytic ultradifferentiable functions, Studia Math. 126 (1997), no. 2, 171–198. MR 1472697
- R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3-4, 206–237. MR 1052587, DOI 10.1007/BF03322459
- R. W. Braun, R. Meise, and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultra-differentiable functions, Proc. London Math. Soc. (3) 61 (1990), no. 2, 344–370. MR 1063049, DOI 10.1112/plms/s3-61.2.344
- L. Ehrenpreis, Solution of some problems of division. IV. Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522–588. MR 119082, DOI 10.2307/2372971
- Léon Ehrenpreis and Paul Malliavin, Invertible operators and interpolation in $AU$ spaces, J. Math. Pures Appl. (9) 53 (1974), 165–182 (English, with French summary). MR 402497
- Lars Hörmander, On the range of convolution operators, Ann. of Math. (2) 76 (1962), 148–170. MR 141984, DOI 10.2307/1970269
- Lars Hörmander, Supports and singular supports of convolutions, Acta Math. 110 (1963), 279–302. MR 154112, DOI 10.1007/BF02391861
Additional Information
- C. Fernández
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, E-46100 Burjassot (Valencia), Spain
- Email: Carmen.Fdez-Rosell@uv.es
- A. Galbis
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, E-46100 Burjassot (Valencia), Spain
- Email: Antonio.Galbis@uv.es
- D. Jornet
- Affiliation: Departamento de Matemática Aplicada, E.T.S. Arquitectura, Universidad Politéc- nica de Valencia, Camino de Vera, E-46071 Valencia, Spain
- Email: dajorca@mat.upv.es
- Received by editor(s): July 24, 2000
- Received by editor(s) in revised form: March 22, 2001
- Published electronically: February 12, 2002
- Additional Notes: This work was completed with the support of DGESIC under Proyecto PB97-0333.
The third author was also supported by Ministerio de Educación y Cultura, grant FP98 48285420.
The authors want to express their gratitude to the referee for helpful suggestions. - Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2377-2381
- MSC (2000): Primary 46F05; Secondary 46F10
- DOI: https://doi.org/10.1090/S0002-9939-02-06359-1
- MathSciNet review: 1897463