Quasiharmonic removability of the unbounded sets in Example 2.16(a) and (b)

Determine whether the unbounded closed sets \(F_1\) and \(F_2\) constructed in Example 2.16(a) and (b) are removable for bounded \(Q\)-quasiharmonic functions in the corresponding domains \(G_1\) and \(G_2\), respectively.

Background

Example 2.16 constructs a weighted real-line space that is pp-hyperbolic and exhibits removability phenomena for bounded pp-harmonic functions when the removed set is unbounded and closed. In parts (a) and (b), the authors prove removability for bounded pp-harmonic functions, but the corresponding assertion for the broader class of bounded QQ-quasiharmonic functions is left unresolved.

The question is significant because the paper’s main compact-set characterization does not extend directly to unbounded closed sets, and the examples demonstrate that several necessary conditions for compact-set removability can fail in the unbounded setting.

References

Since $X$ is $p$-hyperbolic, neither in (a) nor in (b) of Example~\ref{ex-hyp-R} do we know whether $F_j$ is removable for bounded $Q$-quasiharmonic functions in $G_j$, $j=1,2$.

— Liouville theorems and removable sets for bounded $p$-harmonic and quasiharmonic functions on metric spaces under local assumptions  (2608.26878 - Björn et al., 27 Aug 2026) in Example 2.16, immediately after part (b)