Double ball property: an overview and the case of step two Carnot groups

Giulio Tralli


We investigate the notion of the so-called Double Ball Property, which concerns the nonnegative sub-solutions of some differential operators. Thanks to the axiomatic approach developed in [6], this is an important tool in order to solve the Krylov-Safonov's Harnack inequality problem for this kind of operators. In particular, we are interested in linear second order horizontally-elliptic operators in non-divergence formand with measurable coefficients. In the setting of homogeneous Carnot groups, we would like to stress the relation between the Double Ball Property and a kind of solvability of the Dirichlet problem for the operator in the exterior of some homogeneous balls. We present a recent result obtained in [15], where the double ball property has been proved in a generic Carnot group of step two.


Degenerate-elliptic equations; invariant Harnack inequality; homogeneous Carnot groups.

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A. Bonglioli, E. Lanconelli, F. Uguzzoni. Stratied Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin (2007).

L.A. Caffarelli. Interior a priori estimates for solutions of fully non-linear equations. Ann. of Math. (2), 130 (1989) 189-213.

L.A. Caffarelli, C.E. Gutierrez. Properties of the solutions of the linearized Monge-Ampere equation. Amer. J. Math., 119 (1997) 423-465.

G. Citti, E. Lanconelli, A. Montanari. Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature. Acta Math., 188 (2002) 87-128.

F. Da Lio, A. Montanari. Existence and Uniqueness of Lipschitz Continuous Graphs with Prescribed Levi Curvature. Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006) 1-28.

G. Di Fazio, C.E. Gutierrez, E. Lanconelli. Covering theorems, inequalities on metric spaces and applications to pde's. Math. Ann., 341 (2008) 255-291.

C.E. Gutierrez. The Monge-Ampere equation, Progress in Nonlinear Dierential Equations and their Applications, Birkhauser, Boston (2001).

C.E. Gutierrez, F. Tournier. Harnack inequality for a degenerate elliptic equation. Comm. Partial Dierential Equations, 36 (2011) 2103-2116.

N.V. Krylov, M.V. Safonov. A property of the solutions of parabolic equations with measurable coffiecients. Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980) 161-175.

V. Martino, A. Montanari. Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems. Forum Math., 22 (2010) 255-267.

A. Montanari, E. Lanconelli. Pseudoconvex fully nonlinear partial dierential operators. Strong comparison Theorems. J. Differential Equations, 202 (2004) 306-331.

Z. Slodkowski, G. Tomassini. Weak solutions for the Levi equation and envelope of holomorphy. J. Funct. Anal., 101 (1991) 392-407.

Z. Slodkowski, G. Tomassini. The Levi equation in higher dimension and relationship to the envelope of holomorphy. Amer. J. Math., 116 (1994) 479-499.

G. Tomassini. Geometric properties of solutions of the Levi-equation. Ann. Mat. Pura Appl. (4), 152 (1988) 331-344.

G. Tralli. Double Ball Property for non-divergence horizontally elliptic operators on step two Carnot groups. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., to appear.

DOI: 10.6092/issn.2240-2829/3417


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