Radial growth, Lipschitz and Dirichlet spaces on solutions to the Yukawa equation

TL;DR

This paper explores properties of solutions to the Yukawa PDE in complex unit balls, establishing positive answers to open problems, and analyzing their relationships with Lipschitz, BMO, and Dirichlet spaces, with applications to energy integrals.

Contribution

It proves a positive solution to an open problem regarding Yukawa PDE solutions and investigates their connections with Lipschitz and BMO spaces, along with Dirichlet energy analysis.

Findings

Confirmed positive answer to Girela and Peláez's open problem.

Established relationships between Yukawa solutions and Lipschitz/BMO spaces.

Analyzed Dirichlet-type energy integrals for these solutions.

Abstract

In this paper, we investigate some properties to solutions $f$ to the Yukawa PDE: $\Delta f=\lambda f$ in the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$, where $\lambda$ is a nonnegative constant. First, we prove that the answer to an open problem of Girela and Pel\'{a}ez, concerning such solutions, is positive. Then we study relationships on such solutions between the bounded mean oscillation and Lipschitz-type spaces. At last, we discuss Dirichlet-type energy integrals on such solutions in the unit ball of $\mathbb{C}^{n}$ and give an application.