Extension problem and Harnack's inequality for some fractional operators

Details Extension problem and Harnack's inequality for some fractional operators Extension problem and Harnack's inequality for some fractional operators

2009-10-14

arxiv.org/abs/0910.2569v2

Mathematics

Analysis of PDEs

24 pages. Main results improved to full generality thanks to referee comments. To appear in Communications in Partial Differen

Scientific paper

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy-Riemann equations for the extension. The method is applied to the fractional harmonic oscillator $H^\sigma=(-\Delta+|x|^2)^\sigma$ to deduce a Harnack's inequality. A pointwise formula for $H^\sigma f(x)$ and some maximum and comparison principles are derived.

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