  Wiener's criterion for the regularity of a boundary point with respect to the
Dirichlet problem for the Laplace equation has been extended to various classes
of elliptic and parabolic partial differential equations. They include linear
divergence and nondivergence equations with discontinuous coefficients,
equations with degenerate quadratic form, quasilinear and fully nonlinear
equations, as well as equations on Riemannian manifolds, graphs, groups, and
metric spaces. A common feature of these equations is that all of them are of
second order, and Wiener type characterizations for higher order equations have
been unknown so far. Indeed, the increase of the order results in the loss of
the maximum principle, Harnack's inequality, barrier techniques, and level
truncation arguments, which are ingredients in different proofs related to the
Wiener test for the second order equations.
  In the present work we extend Wiener's result to elliptic differential
operators $L(\partial)$ of order $2m$ in the Euclidean space ${\bf R}^n$ with
constant real coefficients
  $$L(\partial)=(-1)^m\sum_{|\alpha|=|\beta|=m}a_{\alpha\beta}
\partial^{\alpha+ \beta}.$$
  The results can be extended to equations with variable (for example, H\"older
continuous) coefficients in divergence form but we leave aside this
generalization to make exposition more lucid.
