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Surjectivity of Hadamard type operators on spaces of smooth functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We obtain a complete characterization of surjective Hadamard type operators \(H_T,T\in C^\infty (\mathbb {R}^d)'\) (i.e. of multiplicative convolution operators) on \(C^\infty (\mathbb {R}^d)\) using a restrictive slowly decreasing condition and a division property both new and valid for the Mellin transform \(\mathscr {M}(T)\). We also characterize bijectivity and calculate the spectrum of Hadamard type operators on \(C^\infty (\mathbb {R}^d)\). We prove a Theorem of Supports for the multiplicative convolution. The Mellin transform \(\mathscr {M}\) is defined on the space of all distributions with compact support, providing a topological isomorphism onto a certain weighted space of holomorphic germs.

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Acknowledgements

The research was supported by the National Center of Science (Poland), Grant no. UMO-2013/10/A/ST1/00091.

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Authors and Affiliations

  1. Faculty of Mathematics and Comp. Sci., A. Mickiewicz University Poznań, Umultowska 87, 61-614, Poznań, Poland

    Paweł Domański

  2. Department of Mathematics, University of Oldenburg, 26111, Oldenburg, Germany

    Michael Langenbruch

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  1. Paweł Domański
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  2. Michael Langenbruch
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Correspondence to Michael Langenbruch.

Additional information

The original version of this article was revised: The presentation of Equation (\(\mathscr {E}^{2(\mathbf{k}+\ell )+\sigma }\)) was incorrect in the proof section of the Theorem 2.18.

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Domański, P., Langenbruch, M. Surjectivity of Hadamard type operators on spaces of smooth functions. RACSAM 113, 1625–1676 (2019). https://doi.org/10.1007/s13398-018-0560-6

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