Linear Functional Equations and their Solutions in Lorentz Spaces

TL;DR

This paper investigates the existence and uniqueness of solutions to linear functional equations within Lorentz spaces, extending classical analysis to more general function spaces with potential applications in various mathematical fields.

Contribution

It introduces conditions for solutions of linear functional equations in Lorentz spaces, a generalization beyond traditional Lebesgue spaces, with a focus on existence and uniqueness.

Findings

Established criteria for solution existence

Proved uniqueness under certain conditions

Extended analysis to Lorentz spaces

Abstract

Assume that $\Omega\subset \mathbb{R}^k$ is an open set, $V$ is a separable Banach space over a field $\mathbb K\in\{\mathbb R,\mathbb C\}$ and $f_1,\ldots,f_N \colon\Omega\to \Omega$, $g_1,\ldots, g_N\colon\Omega\to \mathbb{K}$, $h_0\colon \Omega\to V$ are given functions. We are interested in the existence and uniqueness of solutions $\varphi\colon \Omega\to V$ of the linear functional equation $\varphi=\sum_{k=1}^{N}g_k\cdot(\varphi\circ f_k)+h_0$ in Lorentz spaces.