Abstract dyadic cubes, maximal operators and Hausdorff content (Q310074)

For a fixed collection \({\mathcal D}\) of dyadic cubes and \(0< \alpha\leq n\), consider the Hausdorff content \(H_{\mu}^{\alpha}\) of a subset \(E\subset \mathbb{R}^n\) with respect to the measure \(\mu\) defined by NEWLINE\[NEWLINE H_{\mu}^{\alpha}(E)=\inf \sum_j \mu(Q_j)^{\alpha/n}, NEWLINE\]NEWLINE where the infumum is taken over all coverings of \(E\) by countable families of dyadic cubes \(\{ Q_j\} \subset {\mathcal D}\). The Choquet integral of \(f\geq 0\) with respect to \(H_{\mu}^{\alpha}\) is defined by NEWLINE\[NEWLINE \int f \;d H_{\mu}^{\alpha}=\int_0^{\infty} H_{\mu}^{\alpha} (\{x \in \mathbb{R}^n: f(x)>t \} ) \;dt. NEWLINE\]NEWLINE The main result of the paper consists in the proof of boundedness theorems concerning the dyadic maximal operator \(M_ {\mathcal D}^{\mu}\) adapted to \({\mathcal D}\) and \(\mu\), defined by NEWLINE\[NEWLINE M_ {\mathcal D}^{\mu} f(x)= \sup_{Q\in {\mathcal D}} 1_Q(x) \frac{1}{\mu(Q)} \int_Q |f| d\mu.NEWLINE\]NEWLINE Let \(0<\alpha<n\) and \(\alpha/n<p<+\infty\). It is shown that NEWLINE\[NEWLINE \int (M_ {\mathcal D}^{\mu} f)^p d H_{\mu}^{\alpha} \leq \frac{2^{2p+2}}{\min(1,p)-(\alpha/n)} \int |f|^p d H_{\mu}^{\alpha}NEWLINE\]NEWLINE and, for \(0<\alpha\leq n\), the following weak type \((\alpha/n, \alpha/n)\) inequality holds: NEWLINE\[NEWLINE H_{\mu}^{\alpha} (\{x \in \mathbb{R}^n: M_ {\mathcal D}^{\mu} f(x)>t \} ) \leq 4 (\frac{n}{\alpha})^{\alpha/n}t^{-\alpha/n} \int |f|^{\alpha/n} d H_{\mu}^{\alpha}, \;t>0.NEWLINE\]