For a function that is defined and continuous on Rn except from a C1-hypersurface V ⊂ Rn and that extends as a holomorphic function separately in each complex direction zj = xj + iyj to yj > 0, jointly continuous up to Rn \ V , we prove simultaneous holomorphic extension to the domain {z = x + iy ∈ Cn : yj > 0 for any j} provided that the conormal v =vx to V at any x∈V satisfies vj > 0 (or vj < 0) for any j. This is a generalization of the Ajrapetyan– Henkin “edge of the wedge theorem” [2] where singularities are not allowed that is V = ∅. Our statement has also a local variant and, moreover, applies to functions that are defined, when yj = 0 for any j, only on one side of V . There is a great amount of work that has been done on the problem of joint analyticity of separately holomorphic functions based on the method of the “pluripotential theory” whose use was initiated by Siciak. In absence of singularities, that is for V = ∅, we quote among others [5,16–19]; in case of V = ∅ analytic, we refer to [10,14]. Also, the above extension principle, in its formulation with a set of singularities V , has interesting applications to the range characterization of the exponential Radon transform (cf. [1, 8, 12]).

The edge of the wedge theorem for separately holomorphic functions with singularities.

BARACCO, LUCA;ZAMPIERI, GIUSEPPE
2009

Abstract

For a function that is defined and continuous on Rn except from a C1-hypersurface V ⊂ Rn and that extends as a holomorphic function separately in each complex direction zj = xj + iyj to yj > 0, jointly continuous up to Rn \ V , we prove simultaneous holomorphic extension to the domain {z = x + iy ∈ Cn : yj > 0 for any j} provided that the conormal v =vx to V at any x∈V satisfies vj > 0 (or vj < 0) for any j. This is a generalization of the Ajrapetyan– Henkin “edge of the wedge theorem” [2] where singularities are not allowed that is V = ∅. Our statement has also a local variant and, moreover, applies to functions that are defined, when yj = 0 for any j, only on one side of V . There is a great amount of work that has been done on the problem of joint analyticity of separately holomorphic functions based on the method of the “pluripotential theory” whose use was initiated by Siciak. In absence of singularities, that is for V = ∅, we quote among others [5,16–19]; in case of V = ∅ analytic, we refer to [10,14]. Also, the above extension principle, in its formulation with a set of singularities V , has interesting applications to the range characterization of the exponential Radon transform (cf. [1, 8, 12]).
2009
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/2381777
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