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2020•arXiv (Cornell University)Open access

m-subharmonic and m-plurisubharmonic functions -- on two problems of Sadullaev

Sławomir Dinew

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Abstract

We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.

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What this paper is about

We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.

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Available abstract

We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.

Key concepts: Subharmonic, Mathematics, Pure mathematics, Space (punctuation), Subharmonic function, Operator (biology), Mathematical analysis, Type (biology)

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