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Singular integral operators in Clifford analysis. (English) Zbl 0772.47016

Clifford algebras and their applications in mathematical physics, Proc. 2nd Workshop, Montpellier/Fr. 1989, Fundam. Theor. Phys. 47, 271-277 (1992).
[For the entire collection see Zbl 0745.00055.]
If \(D\) is a bounded simply connected domain of the complex plane \(C\) with smooth boundary \(\Gamma=\partial D\), and if the operator \(S\) with Cauchy kernel is defined by \(S(\varphi)(t)={1\over {\pi i}} \int_ \Gamma {{\varphi(s)} \over {s-t}}ds\), then it is known that \(P_ +={1\over 2}(I+S)\) is an orthogonal projection of \(L_ 2(\Gamma)\) onto Hardy space \(H_ 2(D)\), consisting of \(L_ 2\)-traces on \(\Gamma\) of holomorphic functions on \(D\).
In this paper, the authors state a multi-dimensional version of this result involving the quaternionic multi-dimensional singular operator \(^ \psi S\) defined by \(^ \psi S=\sum_{k=0}^ j \psi^ k S^ k\), where each real component \(S^ k\) is a singular integral of Mihlin-Calderón-Zygmund type (particularly the Riesz operators), and \(\psi\) is a four-tuple of quaternions \((\psi^ 1,\psi^ 2,\psi^ 3,\psi^ 4)\). It is stated that the multi-dimensional operator \(^ \psi P_ \pm={1\over 2}(I_ \pm ^ \psi S)\) is an orthogonal projection of \(L_ 2(\Gamma,\mathbb{H})\) onto the Hardy space \(H_ 2(\mathbb{D},\mathbb{H})\), consisting of \(L_ 2\)-traces of \(\mathbb{H}\)-valued holomorphic functions on \(\mathbb{D}\), where \(\mathbb{D}\) is a simply connected domain of \(R^ 4\) with smooth boundary \(\Gamma=\partial\mathbb{D}\), and \(\mathbb{H}\) is an algebra of real quaternions.
Other results of the paper include expressions for symbols of operators of the form \(A=M^ a I+M^ b{}^ \psi S+K\), where \(M^ a(f)(x)=f(x)a(x)\).

MSC:

47B38 Linear operators on function spaces (general)
47G10 Integral operators
45G05 Singular nonlinear integral equations
45E05 Integral equations with kernels of Cauchy type
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory

Citations:

Zbl 0745.00055
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