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New distribution spaces associated to translation-invariant Banach spaces

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Abstract

We introduce and study new distribution spaces, the test function space \({\mathcal {D}}_E\) and its strong dual \({\mathcal {D}}'_{E'_{*}}\). These spaces generalize the Schwartz spaces \({\mathcal {D}}_{L^{q}}\), \({\mathcal {D}}'_{L^{p}}\), \({\mathcal {B}}'\) and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions \(E\), which turns out to be a convolution module over a Beurling algebra \(L^{1}_{\omega }\). The Banach space \(E'_{*}\) stands for \(L_{\check{\omega }}^1*E'\). We also study convolution and multiplicative products on \({\mathcal {D}}'_{E'_{*}}\).

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Authors and Affiliations

  1. Faculty of Technology and Metallurgy, Ss. Cyril and Methodius University, Ruger Boskovic 16, 1000, Skopje, Republic of Macedonia

    Pavel Dimovski

  2. Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000, Novi Sad, Serbia

    Stevan Pilipović

  3. Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000, Ghent, Belgium

    Jasson Vindas

Authors
  1. Pavel Dimovski
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  2. Stevan Pilipović
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  3. Jasson Vindas
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Corresponding author

Correspondence to Stevan Pilipović.

Additional information

Communicated by K. Gröchenig.

The research of S. Pilipović is supported by the Serbian Ministry of Education, Science and Technological Development, through grant 174024.

J. Vindas gratefully acknowledges support by Ghent University, through the BOF-grant number 01N010114.

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Dimovski, P., Pilipović, S. & Vindas, J. New distribution spaces associated to translation-invariant Banach spaces. Monatsh Math 177, 495–515 (2015). https://doi.org/10.1007/s00605-014-0706-3

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