Two versions of the Nikodym maximal function on the Heisenberg group (Q837066)

The Nikodym maximal function on the Euclidean plane \(\mathbb R^2\) is defined as the supremum over averages over rectangles of eccentricity \(N\), and its operator norm in \(L^2(\mathbb R^2)\) is known to be \(O(\log N)\). The author considers two variants, one on the standard Heisenberg group \(\mathbb H^1\) and the other on the polarized Heisenberg group \(\mathbb H_p^1\). The group law of \(\mathbb H_p^1\) is given by \(x\cdot_p y=(x_1+y_1,x_2+y_2, x_3+y_3+x_1y_2)\). Associated with the two cases of Heisenberg groups, the author considers two maximal averages over all rectangles \(R\) of eccentricity \(N\) supported on the hyperplane \(\Pi=\{(x_1,x_2,0); x_1,x_2\in\mathbb R\}\): on \(\mathbb H^1\), \(\mathcal M_Nf(x)=\sup_{R}|R|^{-1}\int_R|f(x\cdot(y_1,y_2,0))|dy_1dy_2\), and on \(\mathbb H_p^1\), \(\mathcal M_N^pf(x)=\sup_{R}|R|^{-1}\int_R|f(x\cdot_p(y_1,y_2,0))| dy_1dy_2\). His main results are: (1) There are \(c,C>0\) such that \(c\log N\leq \|\mathcal M_N^p\|_{L^2(\mathbb H_p^1)\to L^2(\mathbb H_p^1)} \leq C(\log N)^{3/2}\). (2) There are \(c,C>0\) such that \(cN^{1/4}\leq \|\mathcal M_N\|_{L^2(\mathbb H^1)\to L^2(\mathbb H^1)} \leq CN^{1/4}(\log N)^{2}\).