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 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 ∈ W l,∞ we show that strongly harmonic functions are as well in W l,∞ and that they are analytic, whenever the weight is analytic.The analysis is illustrated by finding all mean-value harmonic functions in R 2 for the l p -distance 1 ≤ p ≤ ∞. The essential outcome is a certain discontinuity with respect to p, i.e. that for all p = 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 R n possess the mean value property with respect to infinitely many weight functions obtained from a given weight.