Bicomplex hyperfunctions

Abstract

In this paper, we consider bicomplex holomorphic functions of several variables in BC n . We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space R n within the bicomplex space BC n , and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.

The Cauchy‐Kowalewski product for bicomplex holomorphic functions

“…This section contains a short summary of the results from [7] and [8]. To begin with, we note that Remark 2.11 implies:…”

“…More recently, we have studied some of the algebraic properties of the system of differential equations satisfied by such functions and, in the spirit of the methodology introduced and developed in [6], we have been able to deduce some new duality theorems which parallel the well-known result of Koethe-Martineau-Grothendieck for one and several complex variables. We refer the readers to [7], [8], [22] as well as to the more recent [23] for the most up-to-date description of the theory.…”

“…Specifically, in the case of classical hyperfunctions (see for example [13]), hyperfunctions on R n are defined as (suitable sums of) boundary values of holomorphic functions on C n , in the case of quaternionic hyperfunctions (see [5], [6] and [12]), hyperfunctions on R 3 are defined as boundary values of Cauchy-Fueter regular functions on H, while quaternionic hyperfunctions on suitable fivedimensional varieties in H 2 are defined as cohomology classes of Cauchy-Fueter regular functions in H 2 . Finally, we showed in [8] that bicomplex hyperfunctions on R n are defined as cohomology classes of bicomplex holomorphic functions on BC n .…”

“…Following [8] we will now clarify these comments, in order to put our contribution in an appropriate context. The definition of Sato's hyperfunctions relies on the extension (due to Martineau among others [13]) of the classical Kothe's duality theorem, which states that if K is a compact subset of C n , and if O denotes the sheaf of holomorphic functions, then…”

Sato’s Hyperfunctions and Boundary Values of Monogenic Functions

“…There is another possibility to look at hyperfunctions supported in R n , in fact one may also construct hypefunctions with values in the bicomplex numbers, see [6], [7]. These are constructed as n-relative cohomology class with values in the sheaf of bicomplex holomorphic functions.…”