Tube structures, Hardy spaces and extension of CR distributions

Abstract:

We consider rough tubes $X+i\mathbb {R}^m\subset \mathbb {C}^m$, where $X\subset \mathbb {R}^m$ is a measurable set, and extend the notion of $CR$ function to the space $L^\infty (X,h^p(\mathbb {R}^m))$, where $h^p(\mathbb {R}^m)$, $0<p<\infty$, is Goldberg’s semilocal Hardy space. We show that if $X$ is the image of some connected manifold by some $C^1$ map, then all such $CR$ functions can be extended to the convex hull of the tube as $CR$ functions $\in L^\infty (\mathrm {ch}(X),h^p(\mathbb {R}^m))$. This extends previous work of Boggess.