Maximal functions of plurisubharmonic functions (Q1202786)

Let \(B\) denote the unit ball in \(\mathbb{C}^ n\) \((n\geq 1)\) with boundary \(S\). For a function \(u:B\to\mathbb{C}\), the radial maximal function \({\mathcal M}u\) on \(S\) is defined by \[ {\mathcal M}u(\eta)=\sup\{| u(r\eta)|:0\leq r<1\}. \] For \(\alpha>1\), \(\eta\in S\), let \(D_ \alpha(\eta)=\{z:| 1-\langle z,\eta\rangle|<{\alpha\over 2}(1- | z|^ 2)\}\). The admissible maximal function \({\mathcal M}_ \alpha u\) is defined by \[ {\mathcal M}_ \alpha u(\eta)=\sup\{| u(z)|:z\in D_ \alpha(\eta)\}. \] In the paper the authors prove the following: Theorem I. For \(0<p<\infty\), there is a positive constant \(C\) depending only on \(n,p,\alpha\) such that if \(u\geq 0\) is plurisubharmonic in \(B\), then \[ \int_ S{\mathcal M}_ \alpha u(\eta)^ pd\sigma(\eta)\leq C\int_ S{\mathcal M}u(\eta)^ pd\sigma(\eta), \] where \(\sigma\) is a normalized Lebesgue measure on \(S\).