Title: CR functions on subanalytic hypersurfaces

Authors: Debraj Chakrabarti and Rasul Shafikov

Issue: Volume 59 (2010), Issue 2, 459-494

Abstract: A general class of singular real hypersurfaces, called \textit{subanalytic}, is defined. For a subanalytic hypersurface $M$ in $\mathbb{C}^{n}$, Cauchy-Riemann (or simply CR) functions on $M$ are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point $p$ on a subanalytic hypersurface $M$ to admit a germ at $p$ of a smooth CR function $f$ that cannot be holomorphically extended to either side of $M$. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface $M$, which guarantees one-sided holomorphic extension of CR functions on $M$, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case.