Some relations between fractional Laplace operators and Hessian operators
DOI:
https://doi.org/10.6092/issn.2240-2829/2668Abstract
After recalling the many representations of the fractional Laplace operator and some of its important properties, some recent results (proved in a joint work with Bruno Franchi and Igor Verbitsky) about the relations between the k-Hessian energy of the k-Hessian operator of a k convex function vanishing at infinity and the fractional energy of a particular fractional operator will be introduced.
Moreover we shall recall an integration by parts formula for the fractional Laplace operator giving a new simpler proof.
References
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260.
P. Constantin, Statistics in turbolence, SIAM Review, 36, 73-98 (1994).
P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2 − D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994), 1495-1533.
D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math. 148 (1998), 1135-1152.
S. Eilertsen, On weighted positivity and the Wiener regularity of aboundary point for the fractional Laplacian, Ark. Mat., 38 (2000), 53-75.
F. Ferrari, B. Franchi, Harnack inequality for fractional operators in Carnot groups, in preparation.
F. Ferrari, B. Franchi, I. Verbitsky, Hessian inequalities and the fractional Laplacian, accepted paper: to appear in J. Reine Angew. Math.
F. Ferrari, I. Verbitsky, in preparation.
Q.-Y. Guan, Z.-M. Ma, Boundary problems for fractional Laplacian in Stoch. Dyn. 5 (2005), 385-424.
Q.-Y. Guan, Z.-M. Ma, Reflected symmetric −stable processes and regional fractional Laplacian, Probab. Theory Relat. Fields 134, 649-694 (2006)
N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren der mathematischen Wissenschaften 180, Springer–Verlag, New York–Heidelberg, 1972.
J. Pedlosky, Geophysical fluid dynamics, New York: Springer 1987
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. PhD thesis, University of Texas at Austin, 2005.
E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, New Jersey 1970
N. S. Trudinger and X.-J. Wang, Hessian measures II, Ann. Math. 150 (1999), 579–604.
E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA No. 49 (2009), 3344.
I. Verbitsky, The Hessian Sobolev inequality and its extensions, preprint (2008).
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