Fundamental theorems for Clifford algebra-valued distributions in elliptic Clifford analysis
By a fundamental theorem in Clifford analysis (CA) one usually means a relation between a pair of Clifford algebra-valued functions, defined over an open subset \(U\) of a base manifold \(X^n\), and expressing a type of integrated reciprocity relation holding over a compact and oriented subset \(\overline{c}\) of \(U\). Such a theorem is constructed as a generalization to Stokes' theorem and can thus be regarded as a far generalization of the concept of partial integration, to be used in CA. In this paper, a fundamental theorem in elliptic Clifford analysis (ECA), with the standard vector Dirac operator, is presented that is valid for Clifford algebra-valued distributions based on \(X^n\). A Clifford algebra-valued distribution based on \(X^n\) is a continuous linear functional, defined on the space of real-valued test functions with compact support in \(X^n\) and taking values in a Clifford algebra. This theorem holds under fairly general conditions on the allowed singularities of the right-hand side distributions and on the region of integration \(\overline{c}\). Next, a specialization of this fundamental theorem is proved that forms the starting point for solving boundary value problems with distributional sources in ECA. Finally, distributional equivalents of the residue theorem, Cauchy's theorem and Cauchy's integral theorem are stated.
scientific article; zbMATH DE number 6021827