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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.

35R11Fractional partial differential equations
35A23Inequalities involving derivatives etc. (PDE)
35B53Liouville theorems, Phragmén-Lindelöf theorems (PDE)
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
Full Text: DOI arXiv
[1] Andrews, G. E.; Askey, R.; Roy, R.: Special functions, Encyclopedia math. Appl. 71 (1999) · Zbl 0920.33001
[2] Bliedtner, J.; Hansen, W.: Potential theory -- an analytic and probabilistic approach to balayage, Universitext (1986) · Zbl 0706.31001
[3] Caffarelli, L.; Silvestre, L.: An extension problem related to the fractional Laplacian, Commun. partial differ. Equ. 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[4] Ferrari, F.: Ground state solutions for k-th Hessian operators, Boll. unione mat. Ital. B (7) 9, 553-586 (1995) · Zbl 0854.35034
[5] F. Ferrari, B. Franchi, I. Verbitsky, Hessian inequalities and the fractional Laplacian, J. Reine Angew. Math. (Crelle’s Journal), http://dx.doi.org/10.1515/crelle.2011.116, in press. · Zbl 1273.49047
[6] Hardy, G. H.; Littlewood, J. E.; Polya, G.: Inequalities, (1934) · Zbl 0010.10703
[7] Hartman, P.: Ordinary differential equations, (2002) · Zbl 1009.34001
[8] Karp, D.; Sitnik, S. M.: Inequalities and monotonicity of ratios of hypergeometric like functions, J. approx. Theory 161, 337-352 (2009) · Zbl 1185.33008 · doi:10.1016/j.jat.2008.10.002
[9] Karp, D.; Sitnik, S. M.: Log-convexity and log-concavity for generalized hypergeometric functions, J. math. Anal. appl. 364, 384-394 (2010) · Zbl 1226.33003 · doi:10.1016/j.jmaa.2009.10.057
[10] Landkof, N. S.: Foundations of modern potential theory, Grundlehren math. Wiss. 180 (1972) · Zbl 0253.31001
[11] Magnus, W.; Oberhettinger, F.; Soni, R. P.: Formulas and theorems for the special functions of mathematical physics, Grundlehren math. Wiss. 52 (1966) · Zbl 0143.08502
[12] Trudinger, N. S.; Wang, X. -J.: Hessian measures I, Topol. methods nonlinear anal. 10, 225-239 (1997) · Zbl 0915.35039
[13] Trudinger, N. S.; Wang, X. -J.: Hessian measures II, Ann. math. 150, 579-604 (1999) · Zbl 0947.35055 · doi:10.2307/121089 · http://www.math.princeton.edu/~annals/issues/1999/150_2.html
[14] Verbitsky, I. E.: Hessian Sobolev and Poincaré inequalities, Math. forschungsinst. Oberwolfach 36, 33-35 (2011)
[15] Wang, X. -J.: A class of fully nonlinear elliptic equations and related functionals, Indiana univ. Math. J. 43, 25-54 (1994) · Zbl 0805.35036 · doi:10.1512/iumj.1994.43.43002
[16] Wang, X. -J.: The k-Hessian equation, Lecture notes in math. 1977 (2009)