Boundedness and dimension for weighted average functions

Abstract:

The paper considers a weighted average property of the type $u({x_o}) = ({\smallint _B}uwdx)/({\smallint _B}wdx)$, $B$ a ball in ${E^n}$ with center ${x_o}$. A lemma constructing such functions is presented from which it follows that if $n = 1$ and the weight function $w$ is continuously differentiable but is not an eigenfunction of the $1$-dimensional Laplace operator, then $u$ is constant. It is also shown that if $w$ is integrable on ${E^n}$ and $u$ is bounded above or below, $u$ is constant.