## Illinois Journal of Mathematics

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

K. Rogovin

#### 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.

#### Article information

Source
Illinois J. Math., Volume 48, Number 1 (2004), 71-87.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136174

Digital Object Identifier
doi:10.1215/ijm/1258136174

Mathematical Reviews number (MathSciNet)
MR2048215

Zentralblatt MATH identifier
1042.30008

#### Citation

Rogovin, K. Local compactness for families of {$\scr A$}-harmonic functions. Illinois J. Math. 48 (2004), no. 1, 71--87. doi:10.1215/ijm/1258136174. https://projecteuclid.org/euclid.ijm/1258136174