Boundary limits and non-integrability of $\mathcal M$-subharmonic functions in the unit ball of $\mathbb C^n (n\ge 1)$

Abstract:

In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball $B$ in ${{\mathbb {C}}^{\vphantom {P}}}^{n}$ that are subharmonic with respect to the Laplace-Beltrami operator $\widetilde {\varDelta }$ on $B$. Since the operator $\widetilde {\varDelta }$ is invariant under the group $\mathcal {M}$ of holomorphic automorphisms of $B$, functions that are subharmonic with respect to $\widetilde {\varDelta }$ are usually referred to as $\mathcal {M}$-subharmonic functions. Our main result is as follows: Let $f$ be a non-negative $\mathcal {M}$-subharmonic function on $B$ satisfying \begin{equation*}\int _{B} (1-|z|^{2})^{\gamma }f^{p}(z) d\lambda (z)< \infty \end{equation*} for some $p> 0$ and some $\gamma >\min \{n,pn\}$, where $\lambda$ is the $\mathcal {M}$-invariant measure on $B$. Suppose $\tau \ge 1$. Then for a.e. $\zeta \in S$, \begin{equation*}f^{p}(z)= o\left ((1-|z|^{2})^{n/\tau -\gamma }\right ) \end{equation*} uniformly as $z\to \zeta$ in each $\mathcal {T}_{\tau ,\alpha }(\zeta )$, where for $\alpha >0$ ($\alpha >\frac {1}{2}$ when $\tau =1$) \begin{equation*}\mathcal {T}_{\tau ,\alpha }(\zeta ) = \{z\in B: |1-\langle z,\zeta \rangle |^{\tau } <\alpha (1-|z|^{2}) \}. \end{equation*} We also prove that for $\gamma \le \min \{n,pn\}$ the only non-negative $\mathcal {M}$-subharmonic function satisfying the above integrability criteria is the zero function.