  We introduce a suitable notion of integral operators (comprising the
fractional Laplacian as a particular case) acting on functions with minimal
requirements at infinity. For these functions, the classical definition would
lead to divergent expressions, thus we replace it with an appropriate framework
obtained by a cut-off procedure. The notion obtained in this way quotients out
the polynomials which produce the divergent pattern once the cut-off is
removed.
  We also present results of stability under the appropriate notion of
convergence and compatibility results between polynomials of different orders.
Additionally, we address the solvability of the Dirichlet problem.
  The theory is developed in general in the pointwise sense. A viscosity
counterpart is also presented under the additional assumption that the
interaction kernel has a sign, in conformity with the maximum principle
structure.
