One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$

TL;DR

This paper proves a stronger one-circle version of a classical result, showing that certain lower semicontinuous functions on b^2 with specific mean value properties must be constant, highlighting differences between dimensions 1 and 2.

Contribution

It establishes a new one-circle mean value characterization for supermedian functions on b^2, extending classical harmonic function results and revealing dimension-dependent behaviors.

Findings

Functions satisfying the mean value inequality are constant under given conditions.

A dimensional difference exists in the volume mean property between b^1 and b^2.

The result generalizes classical superharmonic function properties to a one-radius setting.

Abstract

A classical result states that every lower bounded superharmonic function on $\Bbb R^2$ is constant. In this paper the following (stronger) one-circle version is proven. If $f\colon \Bbb R^2\to (-\infty,\infty]$ is lower semicontinuous, $\liminf_{|x|\to\infty} f(x)/\ln|x|\ge 0$, and, for every $x\in\Bbb R^2$, $1/(2\pi) \int_0^{2\pi} f(x+r(x)e^{it}) dt\le f(x)$, where $r\colon \Bbb R^2\to (0,\infty)$ is continuous, $\sup_{x\in\Bbb R^2} (r(x)-|x|)<\infty$, and $\inf_{x\in\Bbb R^2} (r(x)-|x|)=-\infty$, then $f$ is constant. Moreover, it is shown that, with respect to the assumption $r\le c|\cdot|+M$ on $\Bbb R^d$, there is a striking difference between the restricted volume mean property for the cases $d=1$ and $d=2$.