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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Maximal functions with polynomial densities in lacunary directions


Authors: Kathryn Hare and Fulvio Ricci
Journal: Trans. Amer. Math. Soc. 355 (2003), 1135-1144
MSC (2000): Primary 42B25
Published electronically: October 25, 2002
MathSciNet review: 1938749
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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{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 $q>1$ with bounds that only depend on the degree of $p$.


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Additional Information

Kathryn Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Email: kehare@math.uwaterloo.ca

Fulvio Ricci
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: fricci@sns.it

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03129-X
PII: S 0002-9947(02)03129-X
Received by editor(s): May 27, 2002
Published electronically: October 25, 2002
Additional Notes: The research of the first author was supported in part by NSERC and the Swedish Natural Sciences Research Council
Article copyright: © Copyright 2002 American Mathematical Society