We solve the Dirichlet problem for the quaternionic Monge-Amp\`ere equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is two which turns out to be less than an integrability exponent of the fundamental solution.

中文翻译：

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四元数 Monge-Ampère 方程的弱解

我们使用连续边界数据和 $L^p$ 中的右侧为 $p>2$ 解决四元数 Monge-Amp\`ere 方程的狄利克雷问题。这是 $p$ 的最佳界限。我们还证明了四元多次谐波函数的局部可积指数是 2，结果证明它小于基本解的可积指数。