Distribution Function and Nonincreasing Rearrangement of \({\mathbb {B}}{\mathbb {C}}\)-Valued Functions with \({\mathbb {B}} {\mathbb {C}}\)-Measure

This paper investigates the distribution function and nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) $\mathbb{B}\mathbb{C}$ -valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for \( \mathbb{B}\mathbb{C}\) $\mathbb{B}\mathbb{C}$ -valued functions, which characterizes valuable insights into the behavior and structure of \(\mathbb{B}\mathbb{C}\) $\mathbb{B}\mathbb{C}$ -valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) $\mathbb{B}\mathbb{C}$ -valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) $\mathbb{B}\mathbb{C}$ -valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties.