Maximal functions with polynomial densities in lacunary directions

Abstract:

Given a real polynomial $p(t)$ in one variable such that $p(0)=0$, we consider the maximal operator in $\mathbb {R}^{2}$, \begin{equation*}M_{p}f(x_{1},x_{2})=\sup _{h>0 , i,j\in \mathbb {Z}}\frac {1}{h}\int _{0}^{h} \big |f\big (x_{1}-2^{i}p(t),x_{2}-2^{j}p(t)\big )\big | dt \ . \end{equation*} We prove that $M_{p}$ is bounded on $L^{q}(\mathbb {R}^{2})$ for $q>1$ with bounds that only depend on the degree of $p$.