Extension Problem and Harnack's Inequality for Some Fractional Operators 论文

详细信息

发表期刊/会议

Communications in Partial Differential Equations

发表日期

2010-10-08

发表年份

2010

摘要

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 σ = (− Δ + |x|2)σ to deduce a Harnack's inequality. A pointwise formula for H σ f(x) and some maximum and comparison principles are derived.