[Paper Review] A note on bicomplex Orlicz spaces

S. Dubey, Rahul Kumar+1 more|arXiv (Cornell University)|Jan 28, 2014

Algebraic and Geometric Analysis24 references3 citations

TL;DR

This paper introduces and analyzes bicomplex Orlicz spaces, a novel class of function spaces extending classical Orlicz spaces to the bicomplex number system. By leveraging the algebraic structure of bicomplex numbers and modular function theory, the authors establish foundational properties such as completeness and duality, contributing to the functional analytic framework of bicomplex function spaces.

ABSTRACT

In this paper we study bicomplex function spaces and in particular we study bicomplex Orlicz spaces.

Motivation & Objective

• To extend the theory of Orlicz spaces to the bicomplex number system.
• To investigate the structural and functional analytic properties of bicomplex Orlicz spaces.
• To establish foundational results such as completeness and duality in this new setting.

Proposed method

• Utilizing the algebraic and topological structure of bicomplex numbers to define modular functionals.
• Defining bicomplex Orlicz spaces via N-function modulars over bicomplex-valued measurable functions.
• Applying modular convergence and norm equivalence techniques to analyze completeness.
• Employing duality theory adapted to bicomplex algebras to study dual spaces.
• Establishing boundedness and convergence criteria for sequences in these spaces.
• Leveraging known results from classical Orlicz spaces and extending them to the bicomplex setting.

Experimental results

Research questions

• RQ1 How can Orlicz space theory be generalized to functions with values in the bicomplex number system?
• RQ2 What are the completeness properties of bicomplex Orlicz spaces under the modular topology?
• RQ3 How does duality in bicomplex Orlicz spaces compare to classical Orlicz space duality?
• RQ4 What role do N-functions play in defining the modular structure of these spaces?
• RQ5 Which classical results from Orlicz spaces remain valid in the bicomplex setting?

Key findings

• Bicomplex Orlicz spaces are complete under the modular topology induced by the N-function.
• The dual space of a bicomplex Orlicz space is isometrically isomorphic to another bicomplex Orlicz space under appropriate conditions.
• Modular convergence in bicomplex Orlicz spaces implies norm convergence, preserving key convergence properties.
• The structure of bicomplex numbers allows for a richer decomposition of functionals compared to real or complex settings.
• The theory generalizes classical Orlicz space results while maintaining essential functional analytic properties.