On functions of bounded β-dimensional mean oscillation

In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q₀ ⊂ ℝ^d → ℝ which are integrable on β-dimensional subsets of the cube Q₀: 1 u_BMOβ_(Q0) := _Qsup _{⊂Q0 c}inf _∈ℝ _l(Q_)β ∫ |u − c| dH_∞^β , 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 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(Q₀), Q ⊂ Q₀, 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.