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Kolodziej's subsolution theorem for unbounded pseudoconvex domains

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Åhag P., Czyż R.

Universitatis Iagellonicae Acta Mathematica

2012

vol. 50

7-23

In this paper we generalize Kolodziej's subsolution theorem to bounded and unbounded pseudoconvex domains, and in that way we are able to solve complex Monge-Ampère equations on general pseudoconvex domains. We then give a negative answer to a question of Cegrell and Kolodziej by constructing a compactly supported Radon measure $\mu$ that vanishes on all pluripolar sets in $C^n$ such that $\lambda(C^n)=(2\pi)^n$, and for which there is no function $u$ in $\mathcal L_+$ such that $(dd^cu)^n=\mu$. We end this paper by solving a Monge_Amp±re type equation. Furthermore, we prove uniqueness and stability of the solution.

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1 Universitatis Iagellonicae Acta Mathematica

1 2012

1 Czyż R.

1 Åhag P.

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Kolodziej's subsolution theorem for unbounded pseudoconvex domains