$L^p(\mathbb{R}^2)$ bounds for geometric maximal operators associated to homothecy invariant convex bases

Abstract: Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that $M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^2$ must differentiate $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$.