Novel Results on Homogenous Bicomplex Banach Modules with Hyperbolic-Valued Norm

Abstract

The initial phase of this research involves the examination of the properties of space \(L_{\mathbb {B}\mathbb {C}}^{p}\left( G,\lambda \right) \), for \(1\le p<\infty \), using the \(\mathbb {D}\)-Haar measure \(\lambda \) on the locally compact abelian group G. It is demonstrated that \(L_{\mathbb {B}\mathbb {C}}^{1}\left( G,\lambda \right) \) is a \(\mathbb {D}\)-normed bicomplex Banach algebra with bounded approximate identity. Then, as an extension of the well-known Lipschitz spaces, bicomplex Lipschitz spaces are presented. The bicomplex versions of semi homogeneous and homogeneous Banach spaces are defined as semi-homogeneous bicomplex Banach-modules and homogeneous bicomplex Banach-modules, respectively. Numerous properties of these modules are investigated.