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

Abstract

Let $\mathfrak R=\nabla(-\Delta_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\mathbb R^n_+$, and let $H^1_r(\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak R$ on $H^1_r(\mathbb R^n_+)$. For every $b\in \mathrm{BMO}(\mathbb R^n_+)$, the commutator $[b,\mathfrak R]$ extends boundedly from $H^1_r(\mathbb R^n_+)$ to $L^{1,\infty}(\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \[ [b,\mathfrak R]:H^1_r(\mathbb R^n_+)\to H^1_r(\mathbb R^n_+) \quad\Longleftrightarrow\quad b\in \mathrm{BMO}^{\log}(\mathbb R^n_+). \] Moreover, the $\mathrm{BMO}^{\log}$ norm is quantitatively equivalent to the sum of the usual $\mathrm{BMO}$ norm and the $H^1_r$ 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.