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 $H_r^1({\mathbb{R}}_{+}^n)$ 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 $b\in \mathrm{BMO}({\mathbb{R}}_{+}^n)$, the commutator $[b,\mathfrak{R}]$ extends boundedly from $H_r^1({\mathbb{R}}_{+}^n)$ 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 $H_r^1$ 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.

22 pages