Liouville theorems and removable sets for bounded $p$-harmonic and quasiharmonic functions on metric spaces under local assumptions

Abstract

For connected proper metric spaces $X$, equipped with a locally doubling measure supporting a local $p$-Poincar\'e inequality, we completely characterize which compact sets $K$ with positive capacity are removable for bounded $p$-harmonic functions, $p>1$. Similar results are proved also for bounded quasiharmonic functions. The characterization is both in geometric and analytic terms. In particular, removability is shown to be equivalent to the validity of a Liouville type theorem in $X\setminus K$. Properties such as local connectedness, sequential annular quasiconvexity, concentration of capacity and $p$-parabolicity are identified as crucial for removability. Along the way, we give a rather elementary proof of the Liouville theorem for quasisuperharmonic functions in $p$-parabolic spaces. Our results apply in particular to manifolds and $\mathbf{R}^n$ equipped with (locally) $p$-admissible weights.