Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure

Abstract

We study the mean-value harmonic functions on open subsets of $\mathbb{R}^n$ equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming Sobolev regularity of weight $w \in W^{l,\infty}$ we show that strongly harmonic functions are as well in $W^{l,\infty}$ and that they are analytic, whenever the weight is analytic. The analysis is illustrated by finding all mean-value harmonic functions in $\mathbb{R}^2$ for the $l^p$-distance $1 \leq p \leq \infty$. The essential outcome is a certain discontinuity with respect to $p$, i.e. that for all $p \ne 2$ there are only finitely many linearly independent mean-value harmonic functions, while for $p=2$ there are infinitely many of them. We conclude with a remarkable observation that strongly harmonic functions in $\mathbb{R}^n$ possess the mean value property with respect to infinitely many weight functions obtained from a given weight.