Geodesic
An explicit Carleman formula for the Dolbeault cohomology
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 3 (2010) no. 4, pp. 450-460

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We study formulas which recover a Dolbeault cohomology class in a domain of $\mathbb C^n$ through its values on an open part of the boundary. These are called Carleman formulas after the mathematician who first used such a formula for a simple problem of analytic continuation. For functions of several complex variables our approach gives the simplest formula of analytic continuation from a part of the boundary. The extension problem for the Dolbeault cohomology proves surprisingly to be stable at positive steps if the data are given on a concave piece of the boundary. In this case we construct an explicit extension formula.
Keywords: $\bar\partial$-operator, cohomology, integral formulas. 
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     title = {An explicit {Carleman} formula for the {Dolbeault} cohomology},
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     url = {http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a2/}
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Nikolai Tarkhanov. An explicit Carleman formula for the Dolbeault cohomology. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 3 (2010) no. 4, pp. 450-460. http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a2/