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

Definition of fractional Laplacian for functions with polynomial growth\tha