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Extension Problem and Harnack's Inequality for Some Fractional Operators

Stinga, Pablo Raúl ; Torrea, José Luis

Communications in partial differential equations, 2010-10, Vol.35 (11), p.2092-2122 [Periódico revisado por pares]

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Título:
Extension Problem and Harnack's Inequality for Some Fractional Operators
Autor: Stinga, Pablo Raúl ; Torrea, José Luis
Assuntos: Degenerate Schrödinger equation ; Fractional Laplacian ; Harmonic oscillator ; Harnack's inequality ; Heat semigroup ; Inequality ; Partial differential equations
É parte de: Communications in partial differential equations, 2010-10, Vol.35 (11), p.2092-2122
Descrição: 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.
Editor: Philadelphia: Taylor & Francis Group
Idioma: Inglês

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