Tauberian conditions for geometric maximal operators

Let $\mathcal{B}$ be a collection of measurable sets in $\mathbb{R}^{n}$. The associated geometric maximal operator $M_{\mathcal{B}}$ is defined on $L^{1}(\mathbb{R}^n)$ by $M_{\mathcal{B}}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{\vert R\vert}\int_{R}\vert f\vert$. If $\alpha > 0$, $M_\mathcal{B}$ is said to satisfy a Tauberian condition with respect to $\alpha$ if there exists a finite constant $C$ such that for all measurable sets $E \subset \mathbb{R}^n$ the inequality $\vert\{x : M_{\mathcal{B}} \chi_{E}(x) > \alpha\}\vert \leq C\vert E\vert$ holds. It is shown that if $\mathcal{B}$ is a homothecy invariant collection of convex sets in $\mathbb{R}^{n}$ and the associated maximal operator $M_{\mathcal{B}}$ satisfies a Tauberian condition with respect to some $0 < \alpha < 1$, then $M_\mathcal{B}$ must satisfy a Tauberian condition with respect to $\gamma$ for all $\gamma > 0$ and moreover $M_{\mathcal{B}}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for sufficiently large $p$. As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^{n}$ must differentiate $L^{p}(\mathbb{R}^{n})$ for sufficiently large $p$.