On functions of bounded \(\beta\)-dimensional mean oscillation (Q6566020)

We recall that the space \(BMO(Q_0)\) introduced by \N\textit{F.~John} and \textit{L.~Nirenberg} in [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)]\N is defined as the space of functions \(u\) having bounded mean oscillation, i.e. \[BMO(Q_0) := \{u \in L^1(Q_0) : \Vert u \Vert_{BMO(Q_0)} < \infty \}\] where \[\Vert u \Vert_{BMO(Q_0)} := \sup_{Q \subset Q_0} \frac 1{\vert Q\vert} \int_Q \vert u - u_Q \vert \, dx\] (here \(Q_0\) is a cube of \(\mathbb{R}^d\) and \(Q \subset Q_0\) any cube parallel to \(Q_0\)).\N\NIt is well known that functions in BMO are characterized by the exponential decay of their level sets according to the fundamental John and Nirenberg inequality: There exist constants \(C, c\) independent of \(u\) such that \N\[ \vert \{x \in Q : \vert u(x) - u_Q \vert > t \}\vert \leq C \vert Q \vert \exp(-ct) \tag{1}\] \Nfor every \(t > 0\) and every subcube \(Q\).\N\NIn this paper the authors introduce a generalization of this space based on the notion of \(\beta\)-dimensional mean oscillation, where the Lebesgue measure is replaced by the spherical Hausdorff content. We recall here the definition: \[\mathcal{H}_\infty^\beta (E):=\inf \left\{\sum_{i=1}^\infty \omega_\beta r_i^\beta : E \subset \bigcup_{i=1}^\infty B(x_i, r_i) \right\}\] More precisely, assuming \(\beta \in (0, d]\), the authors introduce the space \(BMO^\beta(Q_0)\) of functions with \(\beta\)-dimensional mean oscillation defined as follows: \N\[\NBMO^\beta(Q_0) := \{u \in L^1(Q_0; \mathcal{H}_\infty^\beta) : \Vert u \Vert_{BMO^\beta(Q_0)} < \infty \}\] where \[\Vert u \Vert_{BMO^\beta(Q_0)} := \sup_{Q \subset Q_0} \inf_{c\in \mathbb{R}} \frac 1{l(Q)^\beta} \int_Q \vert u - c \vert \, d\mathcal{H}_\infty^\beta.\N\]\N The essential achievement of their construction is that functions in this new space enjoy a characterization equivalent to the John and Nirenberg inequality given by the following exponential decay of their level sets: \N\[ \N\mathcal{H}_\infty^\beta(\{x \in Q : \vert u(x) - c_Q \vert > t \})\leq C'_\beta l(Q)^\beta \exp(-c'_\beta t) \tag{2}\N\] \NIt is not difficult to verify that for \(\beta = d\) the definition of \(BMO^\beta(Q_0)\) gives back the original \(BMO(Q_0)\) space, and the authors prove in this new space \(BMO^\beta(Q_0)\) some of the classical results known in \(BMO(Q_0)\) (e.g., the closure under composition with Lipschitz functions, the exponential integrability, etc.).\N\NMethods and techniques to obtain for the Hausdorff content the analogues of the classical tools known for the Lebesgue measure provide an elegant theory of independent interest.\N\NMotivations and the interesting results are clearly presented and the paper is very well written.