Distributions and Integration in superspace

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.Comment: 25 page

Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the bi-axial Dirac operator. In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.Comment: 27 page

Pizzetti and Cauchy formulae for higher dimensional surfaces : a distributional approach

In this paper, we study Pizzetti-type formulas for Stiefel manifolds and Cauchy-type formulas for the tangential Dirac operator from a distributional perspective. First we illustrate a general distributional method for integration over manifolds in R^m defined by means of k equations phi(1)(x) = ... = phi(k)(x) = 0. Next, we apply this method to derive an alternative proof of the Pizzetti formulae for the real Stiefel manifolds SO(m)/SO(m - k). Besides, a distributional interpretation of invariant oriented integration is provided. In particular, we obtain a distributional Cauchy theorem for the tangential Dirac operator on an embedded (m - k)-dimensional smooth surface

Higher order Borel-Pompeiu representations

We establish a higher order Borel-Pompeiu formula for monogenic functions associated to an arbitrary orthogonal basis (called structural set) of Euclidean space, or combinations of such structural sets

Higher order Borel–Pompeiu representations in Clifford analysis

In this paper, we show that a higher order Borel–Pompeiu (Cauchy–Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of a Euclidean space, can be extended to the framework of generalized Clifford analysis. Furthermore, in lower dimensional cases, as well as for combinations of standard structural sets, explicit expressions of the kernel functions are derived

Hypermonogenic plane wave solutions of the dirac equation in superspace

In this paper, we obtain Cauchy-Kovalevskaya theorems for hypermonogenic superfunctions depending only on purely bosonic and fermionic vector variables. In addition, we use these results to construct plane wave examples of such functions

Spin actions in Euclidean and Hermitian Clifford analysis in 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) x 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] = sx\bar{s}. 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) X U(n) and is isomorphic to the subset SO_0^J of SO_0 of elements that commute with the complex structure J. The realization of SO_0^J 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