Self-improving estimates of growth of subharmonic and analytic functions

Abstract

Given a bounded open subset $\Omega$ and closed subsets $A,B$ of ${\mathbb{R}}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\in \Omega \setminus (A\cup B)$, for a function $u$ subharmonic on $\Omega \setminus B$, implies that $u(x)\le h(dist(x,B))$, $x\in \Omega \setminus B$, where $g,h:(0,\infty )\to (0,\infty )$ are decreasing functions and $g(0^{+})=h(0^{+})=\infty$. We seek for explicit expressions of $h$ in terms of $g$. We give some results of this type and show that Domar's work (On the existence of a largest subharmonic minorant of a given function, Ark. Mat., 3 (1957), pp. 429-440) permits one to deduce other results in this direction. Then we compare these two approaches. Similar results are deduced for estimates of analytic functions.

arXiv commentPublished version. Link: https://link.springer.com/article/10.1007/s00025-026-02606-7

References in corpus (1)

• An elliptic adaptation of ideas of Carleman and Domar from complex analysis related to Levinson's log log theorem