Harnack inequalities for non-local operators of variable order

Bass, Richard; Kassmann, Moritz

doi:10.1090/S0002-9947-04-03549-4

Harnack inequalities for non-local operators of variable order
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by Richard F. Bass and Moritz Kassmann PDF

Trans. Amer. Math. Soc. 357 (2005), 837-850 Request permission

Abstract:

We consider harmonic functions with respect to the operator \[ \mathcal {L} u(x)=\int [u(x+h)-u(x)-1_{(|h|\leq 1)} h\cdot \nabla u(x)] n(x,h) dh. \] Under suitable conditions on $n(x,h)$ we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator $\mathcal {L}$ is allowed to be anisotropic and of variable order.

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