Abstract
We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface $M$ and holomorphic convexity of compact sets in $M$, or bounded in part by $M$. Applications include extendability of Cauchy-Riemann functions, solvability of the $\overline {\partial }_b$-equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the $\overline {\partial }$-Neumann operator.
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