Weak-L∞ and BMO with Hausdorff Content
Abstract
The space of weak-L∞ on measure space was introduced by Bennett et al. (Ann. of Math., 113(1981), 601-611) and proved to contain the space of L∞ and possess the appropriate interpolation property. In this paper, we are aimed to continue this line of research on the Euclidean space ℝn endowed with Hausdorff content, which is a low-dimensional outer measure on ℝn, and fails to be additive or absolutely continuous with respect to Lebesgue measures. But low-dimensional subsets, such as fractals, of ℝn are Hausdorff capacitable. Precisely, in the setting of Hausdorff content, an analogue of weak-L∞ is defined and proved to possess an improved Marcinkiewicz interpolation theorem. Moreover, we show that the space of functions with bounded mean oscillation associated to the Hausdorff content is embedded into this new weak-L∞ space by establishing a key estimate for the distribution via a packing condition. As a by-product, the boundedness of the Hardy-Littlewood operator on the space of weak-L∞ is established.
Related
- A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
- Reverse Hölder Class and Two-Parameter Good-λ Inequality over Hausdorff Content
- Hausdorff-Choquet Angular Spaces and Weak-Type (1,1) Bounds for Rough Maximal Operators
- Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
- Valeur exacte de la mesure de Hausdorff d'un sous-ensemble fractal de dimension log 3 / log 4 du triangle de Sierpinski
BibTeX
@misc{guo2026weakl,
title = {{Weak-$L^\infty$ and BMO with Hausdorff Content}},
author = {Hao Guo and Long Huang and Yangzhi Zhang and Ciqiang Zhuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32231},
eprint = {2610.32231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32231},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Weak-L^∞ and BMO with Hausdorff Content
AU - Hao Guo
AU - Long Huang
AU - Yangzhi Zhang
AU - Ciqiang Zhuo
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32231
UR - https://arxiv.org/abs/2610.32231
AB - The space of weak-L^∞ on measure space was introduced by Bennett et al. (Ann. of Math., 113(1981), 601-611) and proved to contain the space of L^∞ and possess the appropriate interpolation property. In this paper, we are aimed to continue this line of research on the Euclidean space ℝ^n endowed with Hausdorff content, which is a low-dimensional outer measure on ℝ^n, and fails to be additive or absolutely continuous with respect to Lebesgue measures. But low-dimensional subsets, such as fractals, of ℝ^n are Hausdorff capacitable. Precisely, in the setting of Hausdorff content, an analogue of weak-L^∞ is defined and proved to possess an improved Marcinkiewicz interpolation theorem. Moreover, we show that the space of functions with bounded mean oscillation associated to the Hausdorff content is embedded into this new weak-L^∞ space by establishing a key estimate for the distribution via a packing condition. As a by-product, the boundedness of the Hardy-Littlewood operator on the space of weak-L^∞ is established.
ER -