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Description

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 present a sharp version of the Schauder estimates in this framework and a Liouville Theorem. The results presented have been recently obtained in collaboration with Serena Dipierro and Ovidiu Savin.