Abstract. We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the L ∞ −norm of its intrinsic gradient.

We study the symmetry properties for solutions of elliptic systems of the typewhere F ∈ C 1,1 loc R 2 , s 1 , s 2 ∈ (0, 1) and the operator (−∆) s is the socalled fractional Laplacian. We obtain some Poincaré-type formulas for the αharmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.

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Abstract. We prove a general magnetic Bourgain-Brezis-Mironescu formula which extends the one obtained in [37] in the Hilbert case setting. In particular, after developing a rather complete theory of magnetic bounded variation functions, we prove the validity of the formula in this class.

We characterize those mappings from a compact subset of R into the Heisenberg group H n which can be extended to a C m horizontal curve in H n . The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.