Bochner‐Martinelli formula in superspace

Abstract

In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner-Martinelli formula to superspace by means of the corresponding projections on the space of spinor-valued superfunctions. KEYWORDS Bochner-Martinelli integral formula, Hermitian Clifford analysis, superanalysis, several complex variables mand · c denotes the complex conjugation. The form (Z, U) is the so-called Bochner-Martinelli kernel. When m = 1, this kernel reduces to the Cauchy kernel (2 i) −1 (z − u) −1 dz, whence formula (1) reduces to the traditional Cauchy integral formula in one complex variable. For m > 1, (Z, U) fails to be holomorphic but it still remains harmonic, see, eg, Kytmanov. 1 Formula (1) was obtained independently and through different methods by Martinelli and Bochner, see, eg, Krantz 2 for a detailed description. The interest for proving different generalizations of the classical Bochner-Martinelli formula has emerged as a successful research topic. Math Meth Appl Sci. 2018;41:9449-9476. wileyonlinelibrary.com/journal/mma|x− | 2m is the so-called Cauchy kernel, |S 2m−1 | is the area of the unit sphere S 2m−1 in R 2m , n(x) denotes the exterior normal vector to Ω at the point x ∈ Ω, and dS x is the Lebesgue surface measure in Ω. This formula has been a cornerstone in the development of the monogenic function theory.Both integral representations above were proven to be related when one considers so-called Hermitian Clifford analysis, which constitutes yet a refinement of the Euclidean case. This refinement focusses on the simultaneous null solutions of the complex Hermitian Dirac operators Z and Z † , which decomposes the Laplace operator in the sense that 4( Z Z † + Z † Z ) = Δ 2m . We refer the reader to literature 6-9 for a general overview. Indeed, in Brackx et al, 10 a Cauchy integral formula for Hermitian monogenic functions was obtained by passing to the framework of circulant (2 × 2) matrix functions. This Hermitian Cauchy integral representation was proven to reduce to the traditional Bochner-Martinelli formula (1) when considering the special case of functions taking values in the zero homogeneous part of complex spinor space. This means that the theory of Hermitian monogenic functions not only refines Euclidean Clifford analysis (and thus harmonic analysis as well) but also has strong connections with the theory of functions of several complex variables, even encompassing some of its results.Our main goal in the current paper is to extend the Bochner-Martinelli formula (1) to superspace by exploiting the above-described relation with Clifford analysis. Superspaces play an important role in contemporary theoretical physics, eg, in the particle theory of supersymmetry, supergravity, or superstring theories. Traditionally, they have been studi...