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PreprintarXiv licencearXiv

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

  • The Anh Bui
  • Xuan Thinh Duong
  • Xuejing Huo
  • Luong Dang Ky
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Abstract

Let R=∇(−ΔD)−1/2 be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space ℝn+, and let H1r(ℝn+) denote the Hardy space of restrictions. We establish an endpoint theory for commutators with R on H1r(ℝn+). For every b∈ BMO(ℝn+), the commutator [b,R] extends boundedly from H1r(ℝn+) to L1,∞(ℝn+). We then prove the sharp strong-endpoint characterization [b,R]:H1r(ℝn+)→ H1r(ℝn+) Longleftrightarrow b∈ BMOlog(ℝn+). Moreover, the BMOlog norm is quantitatively equivalent to the sum of the usual BMO norm and the H1r operator norms of the component commutators [b,Rj]. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.

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Cite

BibTeXRIS
BibTeX
@misc{bui2026endpoint,
  title = {{Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space}},
  author = {The Anh Bui and Xuan Thinh Duong and Xuejing Huo and Luong Dang Ky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23672},
  eprint = {2609.23672},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23672},
}
RIS
TY  - UNPB
TI  - Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space
AU  - The Anh Bui
AU  - Xuan Thinh Duong
AU  - Xuejing Huo
AU  - Luong Dang Ky
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23672
UR  - https://arxiv.org/abs/2609.23672
AB  - Let R=∇(−Δ_D)^−1/2 be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space ℝ^n_+, and let H^1_r(ℝ^n_+) denote the Hardy space of restrictions. We establish an endpoint theory for commutators with R on H^1_r(ℝ^n_+). For every b∈ BMO(ℝ^n_+), the commutator [b,R] extends boundedly from H^1_r(ℝ^n_+) to L^1,∞(ℝ^n_+). We then prove the sharp strong-endpoint characterization [b,R]:H^1_r(ℝ^n_+)→ H^1_r(ℝ^n_+) Longleftrightarrow b∈ BMO^log(ℝ^n_+). Moreover, the BMO^log norm is quantitatively equivalent to the sum of the usual BMO norm and the H^1_r operator norms of the component commutators [b,R_j]. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.
ER  -