Cauchy integral decomposition of multi-vector valued functions on hypersurfaces
Let \(\Gamma\) be a Jordan curve with certain smoothness and let \(f\) be a Hölder function on \(\Gamma\), then one of the Sokhotski--Plemelj formulas can be interpreted as a decomposition of \(f\) into the sum of two functions which are holomorphically extendable inside and outside \(\Gamma\). Trying to generalize such a decomposition to higher dimensions is closely related with the problem of an adequate multidimensional analog of holomorphic function theory in the plane. By obvious reason, complex analysis in \(\mathbb C^m\), \(m>1\), does not serve for that purpose. \textit{E. Dyn'kin} in [Complex Variables, Theory Appl. 31, No.2, 165--176 (1996; Zbl 0865.30056)], [J. Anal. Math. 73, 165--186 (1997; Zbl 0899.58001)] worked out the decomposition of a continuous \(k\)--form \(\eta_k\) on a \(C_1\)--smooth hypersurface \(\sum\) in \(\mathbb R^m\) as a sum of continuous \(k\)-- forms \(\eta_k^{\pm}\) on \(\sum\): \[ \eta_k = \eta_k^++\eta_k^- \tag{Juan} \] where (i) \(\sum\) is the boundary of an open domain \(G\) in \(\mathbb R^m\); (ii) \(\eta_k^{\pm}\) are extendable to harmonic \(k\)--forms of class \(H^s\), \(s>0\), in respectively \(G_+:=G\) and \(G_-:=\mathbb R^m\setminus\overline G\) with \(\eta_k^-(\infty)=0\). Dyn'kin gave necessary and sufficient conditions under which \(\eta_k\) can be represented by (ref {Juan}), his so--called condition (A) and condition (B). As was pointed out in [\textit{E. Dyn'kin}, J. Anal. Math. 73, 165--186 (1997; Zbl 0899.58001)], for an open domain in \(\mathbb R^m\) and \(0<s<1\), the class \(H^s\) of harmonic \(k\)-forms appearing in (B) consists of all harmonic \(k\)-forms in that domain satisfying a Hölder condition of order \(s\) in the closure of the domain. One of the main aims of the paper under review is to show that Condition (A) for harmonic \(k\)-forms is equivalent to a so-called conservation law (CL) for harmonic \(k\)-multi-vector fields. The latter is being established in Section 4, after Section 2 where some preliminaries concerning Clifford algebras and Clifford analysis are given and Section 3 where some results are listed about the Cauchy transform \(C_{\sum}\) and the Hilbert transform \(H_{\sum}\) on \(C_{\infty(\sum)}\). The term ``Conservation Law'' is borrowed from electrodynamics; the conservation laws in the paper correspond to the conservation of electromagnetic charge \(J\), i.e., div\(J=0.\) It's appeared that for a function \(f\) to be two-sided monogenic in a domain \(\Omega\) is equivalent to have the boundary value satisfying the (CL). This allows the authors to prove the main result, Theorem 4.3, giving a list of equivalent assertions about the possibility of decomposing a given \(k\)-- multivector--valued function \(F_k\in C_\infty(\sum)\) into a sum of type \[ F_k=F_k^++F_k^- \] where (i) \(\sum\) is as above; \noindent (ii) \(F_k^\pm \) are extendable to harmonic \(k\)-multivector fields \(F_k^\pm\) in, respectively, \(\Omega_+:=\Omega\) and \(\Omega_-:=\mathbb R^m\setminus\overline\Omega\) with \(F_k^-(\infty)=0\). Here a series of interesting remarks and comments explains many subtle points concerning the relations between the harmonic differential forms and Clifford analysis serving as a crucial tool in treating the former. In Section 5, it is shown that Dyn'kin's Condition (A) on the boundary value of a harmonic \(k\)-form is equivalent to the conservation law for the boundary value of the corresponding \(k\)-multi-vector field.
scientific article; zbMATH DE number 5011339