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

Abstract It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution distributions a well-defined bilinear mapping. The are systematically endowed with topologies bornologies hypocontinuous whenever defined. Largest modules smallest algebras for semigroups constructed along same lines. fact exist within this fundamentally related quantale theory. theoretic residual formed from two characterized as largest subspace corresponding space convolutors. theory applied obtain maximal distributional fractional integrals derivatives, Laplacians, Riesz potentials Hilbert transform. Further, joint families these operators obtained such their composition laws preserved.