Complements and Results on h-sets

Abstract

This is the celebrated enticing title of Chapter 1 of the monographTheory of function spaces IIby Hans Triebel ([Tri92]). On this occasion we could not withstand the temptation to quote such an impressive heading title. But instead of considering function spaces we shall take into consideration geometrical sets, i.e., smoothness is to be referred to the geometry of some sets in $\mathbb{R}^n$ To be honest, our efforts to describe smoothness (or beautiful badness, as it shall be clear) of some irregular sets in $\mathbb{R}^n$ are motivated by the desire to define function spaces of Besov and Triebel-Lizorkin typeon these sets. We shall not treat this aspect here and we rather refer to [Bri02] and to forthcoming papers for a complete discussion on this subject.