Another way to say harmonic

Abstract:

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