The kernel theorem for Laplace ultradistributions (Q2773301)

I write a word of appreciation for a brief, yet a well organized paper, in which the author stated and proved a kernel theorem for spaces of Laplace ultradistributions, supported by an \(n\)-dimensional cone of product type. Relevant definitions and notations have been given in Section 1 (if it could be called so), while in Section 2 the main result, the kernel theorem is proved. Sufficient informations can be obtained from the ad hoc references. I presume that this work can be extended to very many distribution theories and even to Boehmian spaces (the generalization of the distribution theory).NEWLINENEWLINENEWLINEQuoting the reference of L. Schwartz (1950) where he proved the existence of a unique distribution \(K\in D'(\Omega\times \Omega)\), called the distributional kernel, the author has given the kernel theorem (said above) for the space \(L^{\prime (M_p)}_{(\omega)}(\Gamma)\) of Laplace ultradistributions supported by an \(n\)-dimensional cone \(\Gamma\) of product type, i.e. \(\Gamma= \upsilon+(\overline R_+)^n\). Vector notations are used in the entire paper. A linear continuous extension mapping is constructed to analyze the problem, and then a lemma is stated which generalizes a theorem on the change of order of integration, required in the proof of the kernel theorem. Construction of a continuous inverse transformation, use of Hahn-Banach theorem and the Hadamard factorization theorem are employed in the proof.