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

For a function that is defined and continuous on $\R^n$ exceptfrom a $C^1$-hypersurface $V\subset \R^n$ and that extends as aholomorphic function separately in each complex direction$z_j=x_j+{\rm i}y_j$ to $y_j>0$, jointly continuous up to$\R^n\setminus V$, we prove simultaneous holomorphic extensionto the domain $\{z=x+{\rm i}y\in\C^n:\,y_j>0\ \T{ for any }j\}$provided that the conormal $v\,{=}\,v_x$ to $V$ at any $x\,{\in}\,V$satisfies $v_j>0$\break (or $v_j<0$) for any $j$. This is ageneralization of the Ajrapetyan–Henkin “edge of the wedgetheorem” where singularities are not allowed thatis $V=\emptyset$. Our statement has also a local variant and,moreover, applies to functions that are defined, when $y_j=0\\T{ for any }j$, only on one side of $V$. There is a greatamount of work that has been done on the problem of jointanalyticity of separately holomorphic functions based on themethod of the “pluripotential theory” whose use was initiatedby Siciak. In absence of singularities, that is for$V=\emptyset$, we quote among others\cite{Br22,Si70,Si81,Si82,Za76}; in case of $V\neq\emptyset$analytic, we refer to $\cite{JP01,PN04}$. Also, the aboveextension principle, in its formulation with a set ofsingularities $V$, has interesting applications to the rangecharacterization of the exponential Radon transform (cf.$\cite{A-E-K96,Eh00,O98}$).

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Published 1 January 2009