Structure of singular sets of some classes of subharmonic functions

Autor: B.I. Abdullaev, S.A. Imomkulov, R.A. Sharipov
Rok vydání: 2021
Předmět:
Zdroj: Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki. 31:519-535
ISSN: 2076-5959
1994-9197
DOI: 10.35634/vm210401
Popis: In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.
Databáze: OpenAIRE