Extension Problem and Harnack's Inequality for Some Fractional Operators |
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Authors: | Pablo Raúl Stinga José Luis Torrea |
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Institution: | 1. Departamento de Matemáticas , Universidad Autónoma de Madrid , Madrid, Spain pablo.stinga@uam.es;3. Departamento de Matemáticas , Universidad Autónoma de Madrid , Madrid, Spain |
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Abstract: | 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. |
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Keywords: | Degenerate Schrödinger equation Fractional Laplacian Harmonic oscillator Harnack's inequality Heat semigroup |
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