On functions of bounded β-dimensional mean oscillation

You Wei Chen, Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q0 ⊂ ℝd → ℝ which are integrable on β-dimensional subsets of the cube Q0: 1 uBMOβ(Q0) := Qsup ⊂Q0 cinf l(Q ∫ |u − c| dHβ , Q where the supremum is taken over all finite subcubes Q parallel to Q0, l(Q) is the length of the side of the cube Q, and Hβ 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 Hβ ({x ∈ Q : |u(x) − cQ| > t}) ≤ Cl(Q)β exp(− uBMOctβ(Q0) ) for every t > 0, u ∈ BMOβ(Q0), Q ⊂ Q0, and suitable cQ ∈ ℝ. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.

Original languageEnglish
JournalAdvances in Calculus of Variations
Publication statusPublished - 2023


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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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