For the hypoelliptic differential operators $P={\partial }_x^2+{\left(x^k{\partial }_y-x^l{\partial }_t\right)}^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

It is shown that the $(1,p)$-fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the $p$-Laplace equation$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=0$$continuous. Fine limits of quasiregular and BLD mappings are also studied.