\(L^p(\mathbb{R}^2)\) bounds for geometric maximal operators associated with homothecy invariant convex bases (Q6631928)

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\N\[\NM_{\mathcal{B}}f(x)=\sup_{x\in R\in \mathcal{B}}\frac 1R \int_{R}|f(y)|dy,\N\]\Nwhere the supremum is over all members \(R\) in \(\mathcal{B}\) containing \(x\), which can be regarded as the generalizations of the classical Hardy-Littlewood maximal operator, the strong maximal operator, the lacunary maximal operator, and the Kakeya-Nikodym maximal operator, etc. This paper shows 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, the authors also obtain that any density basis that is a homothety invariant collection of convex sets in \(\mathbb{R}^2\) must differentiate \(L^p(\mathbb{R}^2)\) for every \(1<p\le \infty\). The corresponding results for the classical Hardy-Littlewood maximal operator, the strong maximal operator, the lacunary maximal operator, and the Kakeya-Nikodym maximal operator, etc can be found in [\textit{M. Bateman}, Duke Math. J. 147, No. 1, 55--77 (2009; Zbl 1165.42005)], [\textit{M. Bateman} and \textit{N. H. Katz}, Math. Res. Lett. 15, No. 1, 73--81 (2008; Zbl 1160.42010)], [\textit{E. Kroc} and \textit{M. Pramanik}, J. Fourier Anal. Appl. 22, No. 3, 623--674 (2016; Zbl 1355.42016)], [\textit{J. Parcet} and \textit{K. M. Rogers}, Am. J. Math. 137, No. 6, 1535--1557 (2015; Zbl 1337.42020)], [\textit{A. M. Stokolos}, Sib. Math. J. 36, No. 6, 1210--1216 (1995; Zbl 0857.42005); translation from Sib. Mat. Zh. 36, No. 6, 1392--1398 (1995)], etc..