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

Sławomir Dinew

doi:10.5802/afst.1711

m

-subharmonic and

m

-plurisubharmonic functions: on two problems of Sadullaev

Sławomir Dinew ¹

¹ Department of Mathematics and Computer Science, Jagiellonian University, 30-409 Kraków, ul. Lojasiewicza 6, Poland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 995-1009.

Abstract
Abstract

We show that the spaces of $A$ - $m$ -subharmonic and $B$ - $m$ -subharmonic functions differ in sufficiently high dimensions. We also prove that the Monge–Ampère type operator $ℳ_{m}$ 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.

Nous montrons que les fonctions $A$ - $m$ -sousharmoniques et $B$ - $m$ -sousharmoniques diffèrent en dimension suffisamment grande. Nous prouvons que l’opérateur de type Monge–Ampère $ℳ_{m}$ associé à l’espace des fonctions $m$ -plurisousharmoniques ne permet pas un principe de comparaison intégral sauf dans les cas classiques $m = 1$ et $m = n$ . Cela répond par la négative à deux problèmes posés par A. Sadullaev.

Published online: 2022-07-01

DOI: 10.5802/afst.1711

Classification: 32W20, 32U15, 32Q15

Author's affiliations:

Sławomir Dinew ¹

¹ Department of Mathematics and Computer Science, Jagiellonian University, 30-409 Kraków, ul. Lojasiewicza 6, Poland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {S{\l}awomir Dinew},
     title = {$m$-subharmonic and $m$-plurisubharmonic functions: on two problems of {Sadullaev}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {995--1009},
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Sławomir Dinew. $m$-subharmonic and $m$-plurisubharmonic functions: on two problems of Sadullaev. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 995-1009. doi : 10.5802/afst.1711. https://afst.centre-mersenne.org/articles/10.5802/afst.1711/

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