Radial fractional Laplace operators and Hessian inequalities (Q423602)

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 NEWLINE\[NEWLINE\int_0^{+\infty} \frac{|u(r)|}{(1+r)^{n+2s}}\,r^{n-1}\,dr<+\infty\,,NEWLINE\]NEWLINE the following formula holds NEWLINE\[NEWLINE\begin{multlined} (-\Delta)^su(r)=\\ c_{s,n}r^{-2s}\int_1^{+\infty}\left(u(r)-u(r\tau)+\left(u(r)-u(r/ \tau)\right)\tau^{-n+2s}\right)\tau (\tau^2-1)^{-1-2s}H(\tau)\,d\tau,\end{multlined}NEWLINE\]NEWLINE where \(r=|x|>0\), \(x\in \mathbb R^n\), \(c_{s,n}\) is a positive normalization constant and NEWLINE\[NEWLINEH(\tau)=2\pi \alpha_n\int_0^\pi\sin^{n-2}\theta\,\frac{(\sqrt{\tau^2-\sin^2\theta}+\cos \theta)^{1+2s}}{\sqrt{\tau^2-\sin^2\theta}}\,d\theta,\, \tau\geq 1,\, \alpha_n=\frac{\pi^{\frac{n-3}{2}}}{\Gamma(\frac{n-1}{2})}\,.NEWLINE\]NEWLINE 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.