Classe $L$ Log $L$ et densité le l’intégrale d’aire dans $\mathbf{R}_{+}^{n+1}$

Abstract

Let $u$ be a harmonic function defined on the half-space $\mathbf{R}_{+}^{n+1}$. We denote by $D^0$ the value at $0$ of the density of the area integral associated with $u$. We obtain a characterization of the class $L$ Log $L$ as a subspace of $H^1$, using the condition
\[
\int_{\mathbf{R}^n} D^0(x)\mathrm{Log}^+ D^0(x) dx < + \infty. \]