Tauberian conditions for geometric maximal operators

Abstract:

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 $M_{\mathcal {B}}f(x) = \sup _{x \in R \in \mathcal {B}}\frac {1}{|R|}\int _{R}|f|$. 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 $|\{x : M_{\mathcal {B}} \chi _{E}(x) > \alpha \}| \leq C|E|$ 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.