Versões do teorema tubular de Bochner

Abstract

The aim of this work is to study the classical Bochner Tubular Theorem from a more modern perspective. The classic Bochner Tubular Theorem is presented in the classical theory of holomorphic functions and provides a sufficient condition for the extension of holomorphic functions defined on tubular sets in C m, i.e., sets of the form U ˆ R m, where U is an open, connected, non-empty subset of R m. We present the ideas from the paper [21], which suggest an alternative proof for the Bochner Tubular Theorem. For a complete understanding of this proof, we need to establish some notions and results about convexity, analytic discs, and the Baouendi-Treves Approximation Formula. We only need the particular case of the Baouendi-Treves Formula for the structure generated by the Cauchy-Riemann operators. However, to make the text more comprehensive, we will first present it for arbitrary structures and then for the desired structure (emphasizing that the convergence of the approximate formula can be obtained in other topologies, but in this text it is sufficient to consider only the uniform convergence). We will also present a microlocal version of the Bochner Tubular Theorem. In this version, we will use a class of FBI transforms (introduced in [6]), the notion of analytic wavefront set, and a relation between these concepts. The ideas in the second part of this work are in the paper [5] written by S. Berhanu. It is important to observe that using this second version, we can extend holomorphic functions defined on sets more general than tubular sets in C m.