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Dirichlet Spaces In Balls And Half-spaces of $\R^n$

Yan Yang, Tao Qian

TL;DR

This work extends the classical Douglas–Ahlfors characterizations of the harmonic Dirichlet norm from the 2D unit disk to higher-dimensional balls and upper-half spaces. By combining real harmonic analysis with Clifford analysis, it proves that the gradient form, holomorphic/monogenic derivative form, Fourier-integral form, and double-singular integral (Ahlfors) form are identical in these settings, identifying precise equivalences and the role of fractional Sobolev spaces on boundaries. The results cover the unit ball $B_n$, the upper-half spaces ${b R}_+^{n+1}$, and the quaternionic unit ball, with explicit formulas linking interior energy to boundary data via Fourier transforms and boundary double-integrals; an exceptional behavior is noted for unit balls $B_{n+1}$ with $n>1$, where the Ahlfors quantity is equivalent to the Dirichlet norm but not to the semi-norm. Overall, the paper unifies several representations of Dirichlet norms across real, complex, and hypercomplex frameworks, enriching the Besov-space viewpoint $oldsymbol{ abla u}$ versus $g$-functions and their boundary traces.

Abstract

The present paper studies the Dirichlet spaces in balls and upper-half Euclidean spaces. As main results, we give identical characterizations of the Dirichlet norms in the respective contexts as for the classical 2-D disc case proved by Douglas and Ahlfors.

Dirichlet Spaces In Balls And Half-spaces of $\R^n$

TL;DR

This work extends the classical Douglas–Ahlfors characterizations of the harmonic Dirichlet norm from the 2D unit disk to higher-dimensional balls and upper-half spaces. By combining real harmonic analysis with Clifford analysis, it proves that the gradient form, holomorphic/monogenic derivative form, Fourier-integral form, and double-singular integral (Ahlfors) form are identical in these settings, identifying precise equivalences and the role of fractional Sobolev spaces on boundaries. The results cover the unit ball , the upper-half spaces , and the quaternionic unit ball, with explicit formulas linking interior energy to boundary data via Fourier transforms and boundary double-integrals; an exceptional behavior is noted for unit balls with , where the Ahlfors quantity is equivalent to the Dirichlet norm but not to the semi-norm. Overall, the paper unifies several representations of Dirichlet norms across real, complex, and hypercomplex frameworks, enriching the Besov-space viewpoint versus -functions and their boundary traces.

Abstract

The present paper studies the Dirichlet spaces in balls and upper-half Euclidean spaces. As main results, we give identical characterizations of the Dirichlet norms in the respective contexts as for the classical 2-D disc case proved by Douglas and Ahlfors.

Paper Structure

This paper contains 8 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The Dirichlet space in the unit ball $B_n$
  3. Dirichlet spaces in upper-half spaces: The real harmonic analysis methods
  4. The complex and hyper-complex settings for the Ahlfors form quantity
  5. The Dirichlet space in the upper-half complex plane
  6. The Dirichlet space in ${\mathbb{R}}^{n+1}_+$
  7. The quaternionic upper-space case
  8. Balls cases revised

Key Result

Theorem 1.1

$^{\hbox{\scriptsize }WZAhlfors}$ If $u\in h^2({\mathbb D})$, $f$ is its boundary function and $F$ is the canonical holomorphic function with the real part $u$, then where $a_k=\frac{1}{\pi}\int_0^{2\pi}f(e^{{\bf i}\theta})\cos(k\theta)d\theta$ and $b_k=\frac{1}{\pi}\int_0^{2\pi}f(e^{{\bf i}\theta})\sin(k\theta)d\theta$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 16 more