Extension Problem and Harnack's Inequality for Some Fractional Operators
2010; Taylor & Francis; Volume: 35; Issue: 11 Linguagem: Inglês
10.1080/03605301003735680
ISSN1532-4133
AutoresPablo Raúl Stinga, José L. Torrea,
Tópico(s)Fractional Differential Equations Solutions
ResumoThe 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.
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