Integral operators defined “up to a polynomial”

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