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Boundary limits and non-integrability of )
Authors:Manfred Stoll
Affiliation:Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
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),dlambda (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)= oleft ((1-|z|^{2})^{n/tau -gamma }right ) end{equation*}

uniformly as $zto 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 ) = {zin 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.

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