On functions of bounded β-dimensional mean oscillation

Abstract

Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ⁢ ( Q 0 ) := sup Q ⊂ Q 0 ⁡ inf c ∈ ℝ ⁡ 1 l ⁢ ( Q ) β ⁢ ∫ Q | u - c | ⁢ 𝑑 ℋ ∞ β , \displaystyle\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}\vcentcolon=\sup_{Q\subset Q_{% 0}}\inf_{c\in\mathbb{R}}\frac{1}{l(Q)^{\beta}}\int_{Q}|u-c|\,d\mathcal{H}^{% \beta}_{\infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ⁢ ( Q ) {l(Q)} is the length of the side of the cube Q, and ℋ ∞ β {\mathcal{H}^{\beta}_{\infty}} is the Hausdorff content. In the case β = d {\beta=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 ] {\beta\in(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 {c,C>0} such that ℋ ∞ β ⁢ ( { x ∈ Q : | u ⁢ ( x ) - c Q | > t } ) ≤ C ⁢ l ⁢ ( Q ) β ⁢ exp ⁡ ( - c ⁢ t ∥ u ∥ BMO β ⁢ ( Q 0 ) ) \displaystyle\mathcal{H}^{\beta}_{\infty}(\{x\in Q:|u(x)-c_{Q}|>t\})\leq Cl(Q)% ^{\beta}\exp\biggl{(}-\frac{ct}{\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}}\biggr{)} for every t > 0 {t>0} , u ∈ BMO β ⁢ ( Q 0 ) {u\in\mathrm{BMO}^{\beta}(Q_{0})} , Q ⊂ Q 0 {Q\subset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}\in\mathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.