The content of the article may be best described by a citation form the author's introduction: ``In the present work we study essentially the properties of domains of regularity of \(\mathbb{H}^ n\). Among other results, we prove a theorem of Cartan-Thullen type, which asserts that every domain of regularity of \(\mathbb{H}^ n\) is regularly convex, and we give a differential geometric condition on the boundary of a domain of regularity. \dots In the last section we study the problem of the local regular extendibility of functions defined on hypersurfaces of \(\mathbb{H}^ n\) and we prove a theorem of Hans Lewy type''.