A theorem on analytic representation on hypersurfaces with singularities (Q2765972)

Suppose that \(S\) is a smooth of class \(\mathcal C^1\) closed orientable hypersurface in \(D\subset\mathbb C^n\), dividing \(D\) into two open sets \(D^+\) and \(D^-\). The authors prove a theorem on analytic representation of integrable CR functions in the form \(f=h^+-h^-\) on \(S\) with singular points, where \(h^\pm\) are holomorphic functions in \(D^\pm\), respectively. The authors bring together two areas in which the problem of analytic representation can be studied. The first is complex analysis with its explicit integral formulas which enable one to treat also problems of piecewise smooth ``real'' geometry. The important point to note here is the nature of singularities which are purely ``real'', namely conical points, power-like cusps, etc. The second area is the analysis of pseudodifferential operators on manifolds with singular points. It introduces rather specific tools of real analysis in the complex problem, such as special weighted Sobolev spaces, asymptotic, ``regularizations'' of operators near singularities, etc. Using this approach the authors describe those locally integrable functions \(f\) on a hypersurface with singular points, which are still representable in the form \(f=h^+-h^-\) on \(S\). The asymptotic behavior of \(h^\pm(z)\) close to every singular point is specified.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00014].