$L^p({\mathbb{R}}^2)$ 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 $$M_{\mathcal{B}}f(x)=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{|R|}{\int }_R|f|.$$ We show that either $M_{\mathcal{B}}$ is bounded on $L^p({\mathbb{R}}^2)$ for every $1<p\le \infty$ or that $M_{\mathcal{B}}$ is unbounded on $L^p({\mathbb{R}}^2)$ for every $1\le p<\infty$. 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 $1<p\le \infty$.

arXiv commentThe manuscript has been accepted for publication in the Indiana University Mathematics Journal