Hardy-Littlewood Type Theorems and a Hopf Type Lemma

The main aim of this paper is to investigate Hardy-Littlewood type Theorems and a Hopf type lemma on functions induced by a differential operator. We first prove more general Hardy-Littlewood type theorems for the Dirichlet solution of a differential operator which depends on \(\alpha \in (-1,\infty )\) $\alpha \in (-1,\infty )$ over the unit ball \(\mathbb {B}^n\) ${\mathbb{B}}^n$ of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ with \(n\ge 2\) $n\ge 2$ , related to the Lipschitz type space defined by a majorant which satisfies some assumption. We find that the case \(\alpha \in (0,\infty )\) $\alpha \in (0,\infty )$ is completely different from the case \(\alpha =0\) $\alpha =0$ due to Dyakonov (Adv. Math. 187 (2004), 146–172). Then a more general Hopf type lemma for the Dirichlet solution of a differential operator will also be established in the case \(\alpha >n-2\) $\alpha >n-2$ .