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Distributions in superspace constitute a very useful tool for establishing an integration theory. In particular, distributions have been used to obtain a suitable extension of the Cauchy formula to superspace and to define integration over the superball and the supersphere through the Heaviside and Dirac distributions, respectively. In this paper, we extend the distributional approach to integration over more general domains and surfaces in superspace. The notions of domain and surface in superspace are defined by smooth bosonic phase functions g. This allows to define domain integrals and oriented (as well as non-oriented) surface integrals in terms of the Heaviside and Dirac distributions of the superfunction g. It will be shown that the presented definition for the integrals does not depend on the choice of the phase function g defining the corresponding domain or surface. In addition, some examples of integration over a super-paraboloid and a super-hyperboloid will be presented. Finally, a new distributional Cauchy-Pompeiu formula will be obtained, which generalizes and unifies the previously known approaches.

In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace.

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO 0 leaving invariant such an inner product turns out to be an extension of SO(m) × Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO 0 by means of the representation h(s)[x] = sxs. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m) × U(n) and is isomorphic to the subset SO J 0 of SO 0 of elements that commute with the complex structure J. The realization of SO J 0 within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

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