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arXiv · 2509.23944

The Range of the Monge-Amp\`ere operator $(\omega + dd^c .)^n$ in bounded domains

Omar Alehyane·Fatima Zahra Assila·Mohammed Salouf
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Abstract

Let $\Omega$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n = \mu$ in the generalized Cegrell classes $\mathcal{K}(\Omega,\omega,H)$, where $H \in \mathcal{E}(\Omega)$ is maximal, $\omega$ is a smooth real $(1,1)$-form defined in a neighborhood of $\bar\Omega$ and $\mu$ is a positive Radon measure. This generalizes the previous work of the last author \cite{Sal25} to the case of non-continuous functions $H$ and also to the case of measures $\mu$ which do not vanish on pluripolar sets.

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Omar Alehyane, Fatima Zahra Assila, Mohammed Salouf. 2025-09-28. The Range of the Monge-Amp\`ere operator $(\omega + dd^c .)^n$ in bounded domains. https://arxiv.org/abs/2509.23944

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