Local compactness for families of {$\scr A$}-harmonic functions

Abstract

We show that if a family of $\mathcal{A}$-harmonic functions that admits a common growth condition is closed in $L^p_{\operatorname{loc}}$, then this family is locally compact on a dense open set under a family of topologies, all generated by norms. This implies that when this family of functions is a vector space, then such a vector space of $\mathcal{A}$-harmonic functions is finite dimensional if and only if it is closed in $L^p_{\operatorname{loc}}$. We then apply our theorem to the family of all $p$-harmonic functions on the plane with polynomial growth at most $d$ to show that this family is essentially small.