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Sharp Constants and Extremizers in Homogeneous Tree BMO

  • Mohammadreza Mohajer
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Abstract

We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered L1-BMO over homogeneous q-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.

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Cite

BibTeXRIS
BibTeX
@misc{mohajer2026sharp,
  title = {{Sharp Constants and Extremizers in Homogeneous Tree BMO}},
  author = {Mohammadreza Mohajer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14459},
  eprint = {2610.14459},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14459},
}
RIS
TY  - UNPB
TI  - Sharp Constants and Extremizers in Homogeneous Tree BMO
AU  - Mohammadreza Mohajer
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14459
UR  - https://arxiv.org/abs/2610.14459
AB  - We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered L^1-BMO over homogeneous q-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.
ER  -