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)

Abstract

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
DOIs
Publication statusPublished - 2023

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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