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Maximal functions with polynomial densities in lacunary directions

Authors:	Kathryn Hare Fulvio Ricci

Institution:	Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 ; Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

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

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