### Abstract

We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

Language | English |
---|---|

Pages | 21-44 |

Number of pages | 24 |

Journal | Potential Analysis |

Volume | 31 |

Issue number | 1 |

Early online date | 6 Mar 2009 |

DOIs | |

Publication status | Published - 31 Jul 2009 |

Externally published | Yes |

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### Keywords

- harmonic functions
- Harnack inequality
- integro-differential operators
- jump processes

### Cite this

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*Potential Analysis*, vol. 31, no. 1, pp. 21-44. https://doi.org/10.1007/s11118-009-9121-0

**Harmonic functions for a class of integro-differential operators.** / Foondun, Mohammud.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Harmonic functions for a class of integro-differential operators

AU - Foondun, Mohammud

PY - 2009/7/31

Y1 - 2009/7/31

N2 - We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

AB - We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

KW - harmonic functions

KW - Harnack inequality

KW - integro-differential operators

KW - jump processes

UR - http://www.scopus.com/inward/record.url?scp=67349199139&partnerID=8YFLogxK

UR - http://link.springer.com/journal/11118

U2 - 10.1007/s11118-009-9121-0

DO - 10.1007/s11118-009-9121-0

M3 - Article

VL - 31

SP - 21

EP - 44

JO - Potential Analysis

T2 - Potential Analysis

JF - Potential Analysis

SN - 0926-2601

IS - 1

ER -