## Abstract

In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q_{0} ⊂ ℝ^{d} → ℝ which are integrable on β-dimensional subsets of the cube Q_{0}: 1 u_{BMO}β_{(Q0)} := _{Q}sup _{⊂Q0 c}inf _{∈}ℝ _{l}_{(}Q_{)β} ∫ |u − c| dH_{∞}^{β} , Q where the supremum is taken over all finite subcubes Q parallel to Q_{0}, l(Q) is the length of the side of the cube Q, and H_{∞}^{β} is the Hausdorff content. In the case β = d we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ (0, d] one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c, C > 0 such that H_{∞}^{β} ({x ∈ Q : |u(x) − c_{Q}| > t}) ≤ Cl(Q)^{β} exp(− u_{BMO}^{ct}_{β}_{(Q0)} ) for every t > 0, u ∈ BMO^{β}(Q_{0}), Q ⊂ Q_{0}, and suitable c_{Q} ∈ ℝ. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.

Original language | English |
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Journal | Advances in Calculus of Variations |

DOIs | |

Publication status | Published - 2023 |

## Keywords

- Bounded mean oscillation
- Hausdorff content
- capacitary John–Nirenberg lemma

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics