Another way to say harmonic

Abstract

It is known that solutions of − Δ ∞ u = − ∑ i , j = 1 n u x i u x j u x i x j = 0 -\Delta _\infty u=-\sum _{i,j=1}^nu_{x_i} u_{x_j}u_{x_ix_j}=0 , that is, the ∞ \infty -harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions G ( x ) = a | x | G(x)=a|x| . We establish a more difficult linear result: a function in R n {\mathbb R^n} is harmonic if it has the comparison property with respect to sums of n n translates of the radial harmonic functions G ( x ) = a | x | 2 − n G(x)=a|x|^{2-n} for n ≠ 2 not =2 and G ( x ) = b ln ⁡ ( | x | ) G(x)=b\ln (|x|) for n = 2 n=2 . An attempt to generalize these results for − Δ ∞ u = 0 -\Delta _\infty u=0 ( p = ∞ p=\infty ) and − Δ u = 0 -\Delta u=0 ( p = 2 p=2 ) to the general p p -Laplacian leads to the fascinating discovery that certain sums of translates of radial p p -superharmonic functions are again p p -superharmonic. Mystery remains: the class of p p -superharmonic functions so constructed for p ∉ { 2 , ∞ } pot \in \{2,\infty \} does not suffice to characterize p p -subharmonic functions.