  We introduce a notion of fractional Laplacian for functions which grow more
than linearly at infinity. In such case, the operator is not defined in the
classical sense: nevertheless, we can give an ad-hoc definition which can be
useful for applications in various fields, such as blowup and free boundary
problems.
  In this setting, when the solution has a polynomial growth at infinity, the
right hand side of the equation is not just a function, but an equivalence
class of functions modulo polynomials of a fixed order.
  We also give a sharp version of the Schauder estimates in this framework, in
which the full smooth H\"older norm of the solution is controlled in terms of
the seminorm of the nonlinearity.
  Though the method presented is very general and potentially works for general
nonlocal operators, for clarity and concreteness we focus here on the case of
the fractional Laplacian.
