Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space

Let $\mathfrak{R}=\mathrm{\nabla }(-{\mathrm{\Delta }}_D{)}^{-1\mathrm{/}2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space ${\mathbb{R}}_{+}^n$, and let $H_r^1({\mathbb{R}}_{+}^n)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak{R}$ on $H_r^1({\mathbb{R}}_{+}^n)$. For every $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}_{+}^n)$, the commutator $[b,\mathfrak{R}]$ extends boundedly from $H_r^1({\mathbb{R}}_{+}^n)$ to $L^{1,\mathrm{\infty }}({\mathbb{R}}_{+}^n)$. We then prove the sharp strong-endpoint characterization $$[b,\mathfrak{R}]:H_r^1({\mathbb{R}}_{+}^n)\to H_r^1({\mathbb{R}}_{+}^n)⟺b\in {\mathrm{B}\mathrm{M}\mathrm{O}}^{\log }({\mathbb{R}}_{+}^n)\mathrm{.}$$ Moreover, the ${\mathrm{B}\mathrm{M}\mathrm{O}}^{\log }$ norm is quantitatively equivalent to the sum of the usual $\mathrm{B}\mathrm{M}\mathrm{O}$ norm and the $H_r^1$ operator norms of the component commutators $[b,{\mathfrak{R}}_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.