Distributions and Integration in superspace

Abstract

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.

arXiv comment25 pages

References in corpus (7)

• Spherical harmonics and integration in superspace
• A smooth introduction to the wavefront set
• Integration in superspace using distribution theory
• Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
• A Clifford analysis approach to superspace
• Orthosymplectically invariant functions in superspace
• A Cauchy integral formula in superspace

Cited by in corpus (3)

• Generalized Grassmann algebras and applications to stochastic processes
• Pizzetti and Cauchy formulae for higher dimensional surfaces: a distributional approach
• Bochner-Martinelli formula in superspace