Properties of solutions of the \(\alpha\)-harmonic equation in the unit disk

Abstract

In this paper, we study Riesz-Fejér inequality, comparative growth of integral means and boundary behavior for solutions of the \(\alpha\)-harmonic equation in the unit disk \(\mathbb{D}\). For \(\alpha>\max\{-1,-\frac{2}{p}\}\) \(\alpha \geq 0\) and \(1<p<\infty\), we obtain a Riesz-Fejér inequality for functions in the real kernel \(\alpha\)-harmonic Hardy space consisting of solutions \(u\) of the \(\alpha\)-harmonic equation in \(\mathbb{D}\) with uniformly bounded integral mean \(M_{p}(r, u)\) with respect to \(r\in(0,1)\). Furthermore, for \(1\leq p<q\leq\infty\), we estimate the growth of \(M_{q}(r,u)\) if the growth of \(M_{p}(r,u)\) is known. Moreover, we consider the boundary behavior of real kernel \(\alpha\)-Poisson integrals in \(\mathbb{D}\), where \(\alpha>-1\). Our results generalize the related previous results.