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Boundary behavior of alppha-harmonic functions and their Riesz-Fejer inequalities

Bo-Yong Long

Abstract

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. In this paper, the boundary correspondence and boundary behavior of alpha-harmonic functions are studied, and the corresponding Dirichlet problem is solved. As one of its applications, an asymptotic optimal Riesz-Fejer inequality for alpha-harmonic functions is obtained. In addition, the subharmonic properties of alpha-harmonic functions is explored and an optimal radius is obtained.

Boundary behavior of alppha-harmonic functions and their Riesz-Fejer inequalities

Abstract

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. In this paper, the boundary correspondence and boundary behavior of alpha-harmonic functions are studied, and the corresponding Dirichlet problem is solved. As one of its applications, an asymptotic optimal Riesz-Fejer inequality for alpha-harmonic functions is obtained. In addition, the subharmonic properties of alpha-harmonic functions is explored and an optimal radius is obtained.

Paper Structure

This paper contains 3 sections, 9 theorems, 107 equations.

Table of Contents

  1. Introduction and main results
  2. Prliminaries
  3. Proofs

Key Result

Theorem 1.1

Let function $f$ be piecewise continuous on the unit circumference $\mathbb{T}$. Then for any $\alpha>-1$, $u(z)=P_{\alpha}[f](z)$ is a bounded $\alpha$-harmonic function on the unit disk $\mathbb{D}$, and at the continuous point $e^{it_{0}}$ of $f$, it holds that

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Definition 1
  • ...and 7 more