On extremal domains and codomains for convolution of distributions and fractional calculus

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever d...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2022, Vol.198 (1), p.121-152
Main Authors: Kleiner, T., Hilfer, R.
Format: Article
Language:English
Subjects:
Convolution
Domains
Fractional calculus
Hilbert transformation
Mathematical analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
Topology
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Summary:It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz potentials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-021-01646-1