Sheaves of slice regular functions

Abstract

Slice regular functions have been introduced in [20] as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see [21], to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions resp. the space of paravectors in R n + 1 . However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.

Entire Slice Regular Functions

“…The theory of slice regular functions was delevoped in the papers [47,104,106,107,109,110,112,113], in particular, the zeros were treated in [103,105,111] while further properties can be found in [33,44,85,92,93,102,140,141,150,151,152,154]. Slice monogenic functions with values in a Clifford algebra and their main properties were studied in [58,72,73,74,75,78,81,122,155]. Approximation of slice hyperholomorphic functions are collected in the works [77,94,95,96,97,98,99,100,145].…”

The C-property for slice regular functions and applications to the Bergman space

“…This result appeared first in Remark 3 in [16], in a more general setting, but it has been called Refined Splitting Lemma in [10]. Another useful consequence is:…”

“…Part 1 is immediate if one writes f (x+iy) = α(x, y)+iβ(x, y) with α(x, y) = α 0 (x, y)+ α 1 (x, y)i + α 2 (x, y)j + α 3 (x, y)ij, β(x, y) = β 0 (x, y) + β 1 (x, y)i + β 2 (x, y)j + β 3 (x, y)ij. The Cauchy-Riemann equations implies that the functions h ℓ = α ℓ + iβ ℓ (x, y) are holomorphic (this fact has been discussed also [10]); the conditions (7) imply that h ℓ ∈ Hol c (Λ i ) for all ℓ = 0, . .…”

Ideals of regular functions of a quaternionic variable

“…The papers [1], [22], [24], contain significative descriptions of ideals of holomorphic functions, in connection with their maximality. Sheaves of slice regular functions are introduced in [7].…”