Riesz Transform Characterizations of $H^1$ and {\rm BMO} on Ahlfors Regular Sets with Small Oscillations

Abstract: We employ the Riesz transform as a means for describing geometric properties of sets in ${\mathbb{R}}^n$, and study the extent to which they can be used to characterize function spaces defined on said sets. In particular, characterizations of the end-point spaces on the Lebesgue scale $L^p$ with $1<p<\infty$, namely the Hardy space $H^1$ and the John-Nirenberg space {\rm BMO}, are produced in terms of the Riesz transforms on Ahlfors regular sets in ${\mathbb{R}}^n$ with small oscillations (quantified in terms of the {\rm BMO} nature of the outward unit normal). These generalize the celebrated results of C.~Fefferman and E.~Stein in the flat Euclidean setting.