### Abstract

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 (in addition to the various results that we prove here) can also 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 Hölder norm of the solution is controlled in terms of the seminorm of the forcing term. 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.

Original language | English |
---|---|

Pages (from-to) | 1079-1122 |

Number of pages | 44 |

Journal | Revista Matematica Iberoamericana |

Volume | 35 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Cite this

*Revista Matematica Iberoamericana*,

*35*(4), 1079-1122. https://doi.org/10.4171/rmi/1079

}

*Revista Matematica Iberoamericana*, vol. 35, no. 4, pp. 1079-1122. https://doi.org/10.4171/rmi/1079

**Definition of fractional Laplacian for functions with polynomial growth.** / Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Definition of fractional Laplacian for functions with polynomial growth

AU - Dipierro, Serena

AU - Savin, Ovidiu

AU - Valdinoci, Enrico

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 (in addition to the various results that we prove here) can also 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 Hölder norm of the solution is controlled in terms of the seminorm of the forcing term. 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.

AB - 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 (in addition to the various results that we prove here) can also 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 Hölder norm of the solution is controlled in terms of the seminorm of the forcing term. 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.

KW - Fractional Laplacian

KW - Liouville theorems

KW - Polynomial growth

KW - Regularity results

KW - Schauder estimates

UR - http://www.scopus.com/inward/record.url?scp=85074585767&partnerID=8YFLogxK

U2 - 10.4171/rmi/1079

DO - 10.4171/rmi/1079

M3 - Article

VL - 35

SP - 1079

EP - 1122

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 4

ER -