Local compactness for families of \(\mathcal A\)-harmonic functions (Q1425769)

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