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Peak functions on convex domains

Kolář, Martin (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

Let Ω ⊂ ℂ n be a domain with smooth boundary and p ∈ ∂ Ω . A holomorphic function f on Ω is called a C k ( k = 0 , 1 , 2 , ⋯ ) peak function at p if f ∈ C k ( Ω ¯ ) , f ( p ) = 1 , and | f ( q ) | < 1 for all q ∈ Ω ¯ ∖ { p } . If Ω is strongly pseudoconvex, then C ∞ peak functions exist. On the other hand, J. E. Fornaess constructed an example in ℂ 2 to show that this result fails, even for C 1 functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function on a...

Penrose transform and monogenic sections

Tomáš Salač (2012)

Archivum Mathematicum

The Penrose transform gives an isomorphism between the kernel of the 2 -Dirac operator over an affine subset and the third sheaf cohomology group on the twistor space. In the paper we give an integral formula which realizes the isomorphism and decompose the kernel as a module of the Levi factor of the parabolic subgroup. This gives a new insight into the structure of the kernel of the operator.

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