Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-improving estimates of growth of subharmonic and analytic functions

Glenier Bello, Dmitry Yakubovich

Abstract

Given a bounded open subset $Ω$ and closed subsets $A,B$ of $\mathbb{R}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\inΩ\setminus(A\cup B)$, for a function $u$ subharmonic on $Ω\setminus B$, implies that $u(x)\le h(dist(x,B))$, $x\inΩ\setminus B$, 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.

Self-improving estimates of growth of subharmonic and analytic functions

Abstract

Given a bounded open subset and closed subsets of , we discuss when an estimate , , for a function subharmonic on , implies that , , where are decreasing functions and . We seek for explicit expressions of in terms of . 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.

Paper Structure

This paper contains 8 sections, 18 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Our main results
  3. Proof of Theorem \ref{['thm:mainP1']}
  4. Quantitative versions of Domar's theorems
  5. Our version of Domar's Theorem A
  6. Our version of Domar's Theorem B
  7. Application of quantitative Domar's results to our problem. Proof of Theorem \ref{['thm-Domar-s^-b']}
  8. Appendix

Key Result

Theorem 2.3

Let $\Omega\subset\mathbb{R}^k$ be a bounded open set. Let $A\subset \Omega$ be a closed subset of a Lipschitz curve (that is, a 1-dimensional Lipschitz surface) and $B$ be a closed subset of $\mathbb{R}^k$ such that $B\cap\Omega$ is non-empty. Let $\psi\colon(\beta,\infty)\to(0,\infty)$ be a concav then there exist positive constants $c_1,c_2$ and $\tau$, which do not depend on the function $u$,

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Remark 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 28 more