Open AccessStructure of singular sets of some classes of subharmonic functions
Author(s) -
B. I. Abdullaev,
S. A. Imomkulov,
Руслан Шарипов
Publication year - 2021
Publication title -
vestnik udmurtskogo universiteta. matematika, mehanika, kompʹûternye nauki
Language(s) - English
Resource type - Journals
eISSN - 2076-5959
pISSN - 1994-9197
DOI - 10.35634/vm210401
Subject(s) - subharmonic , mathematics , subharmonic function , harmonic function , norm (philosophy) , mathematical analysis , space (punctuation) , backslash , class (philosophy) , pure mathematics , combinatorics , physics , nonlinear system , computer science , quantum mechanics , artificial intelligence , political science , law , operating system
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.