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

TL;DR

The paper develops Riesz-transform characterizations of the endpoint spaces $H^1$ and ${\rm BMO}$ on Ahlfors regular sets with small oscillations, generalizing Fefferman–Stein type results from flat Euclidean space to geometric contexts with UR boundaries. By employing distributional and principal-value Riesz transforms, along with the Clifford-algebra/Cauchy–Clifford framework, the authors prove that on infinitesimally flat AR domains (and their δ-AR extensions) the Hardy space on the boundary satisfies $H^1(\partial\Omega,\sigma)=\{f\in L^1: R_j f \in L^1, \ 1\le j \le n\}$ and ${\rm BMO}(\partial\Omega,\sigma)$ admits a Fefferman–Stein type representation $f_0+\sum_j R_j f_j$ with $f_0,f_j\in L^{\infty}$. They treat both domains with compact boundaries and those with unbounded boundaries, building Green functions, non-tangential boundary traces, and Dirichlet problem solvability in $L^p$-settings, and they extend the results to the δ-AR regime (small oscillation of the normal) and to UR unbounded boundaries. The work provides a natural maximal/minimality interpretation for $H^1$ and ${\rm BMO}$ in this rough setting and yields a robust framework for boundary value problems on rough, low-regularity interfaces.

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.

Theorems & Definitions (47)

• Theorem 1.1
• proof
• Theorem 1.2
• proof
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Definition 2.2