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 stating that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming the Sobolev regularity of the weight $w \in W^{l,\infty}$ we show that strongly harmonic functions are also in $W^{l,\infty}$ and that they are analytic, whenever the weight is analytic.

Keywords

• harmonic function
• mean value property
• metric measure space
• minkowski functional
• norm induced metric
• pizzetti formula
• weighted lebesgue measure