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Radial fractional Laplace operators and Hessian inequalities. (English) Zbl 1306.35139
In this paper the authors study the fractional Laplace operator $(-\Delta)^s$ on radially symmetric functions and in the main theorem they prove the following result for $s\in (0,1)$: for every radial $C^2$ function such that $$\int_0^{+\infty} \frac{|u(r)|}{(1+r)^{n+2s}}\,r^{n-1}\,dr<+\infty\,,$$ the following formula holds $$\multline (-\Delta)^su(r)=\\ c_{s,n}r^{-2s}\int_1^{+\infty}\left(u(r)-u(r\tau)+\left(u(r)-u(r/ \tau)\right)\tau^{-n+2s}\right)\tau (\tau^2-1)^{-1-2s}H(\tau)\,d\tau,\endmultline$$ where $r=|x|>0$, $x\in \mathbb R^n$, $c_{s,n}$ is a positive normalization constant and $$H(\tau)=2\pi \alpha_n\int_0^\pi\sin^{n-2}\theta\,\frac{(\sqrt{\tau^2-\sin^2\theta}+\cos \theta)^{1+2s}}{\sqrt{\tau^2-\sin^2\theta}}\,d\theta,\, \tau\geq 1,\, \alpha_n=\frac{\pi^{\frac{n-3}{2}}}{\Gamma(\frac{n-1}{2})}\,.$$ Using this result, they give a sufficient condition for radially symmetric functions to be $s$-subharmonic when $s\in (0,1)$. Moreover, as a consequence of the main result of the paper, they prove a Liouville theorem, the maximum principle for radial $s$-subharmonic functions and a derivative formula involving the fractional Laplacian.

MSC:
35R11Fractional partial differential equations
35A23Inequalities involving derivatives etc. (PDE)
35B53Liouville theorems, Phragmén-Lindelöf theorems (PDE)
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
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References:
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