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Maximal functions with polynomial densities in lacunary directions

Author(s): Kathryn Hare; Fulvio Ricci
Journal: Trans. Amer. Math. Soc. 355 (2003), 1135-1144.
MSC (2000): Primary 42B25
Posted: October 25, 2002
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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 $p$.

References:

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A. Carbery, Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem, Ann. Inst. Fourier (Grenoble) 38 (1988), 157-168. MR 89h:42026

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A. Carbery, F. Ricci, and J. Wright, Maximal functions and Hilbert transforms associated to polynomials, Rev. Mat. Iberoam. 14 (1998), 117-144. MR 99k:42014

[CRW2]
A. Carbery, F. Ricci, and J. Wright, Maximal functions and singular integrals associated to polynomial mappings of $\mathbb{R}^{n}$, preprint.

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J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. MR 87f:42046

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A. Nagel, E. M. Stein and S. Wainger, Differentiation in lacunary directions, Proc. Natl. Acad. Sci. U.S.A. 75 (1978), 1060-1062. MR 57:6349

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F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), 637-670. MR 94d:42020

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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: 10.1090/S0002-9947-02-03129-X
PII: S 0002-9947(02)03129-X
Received by editor(s): May 27, 2002
Posted: October 25, 2002
Additional Notes: The research of the first author was supported in part by NSERC and the Swedish Natural Sciences Research Council
Copyright of article: Copyright 2002, American Mathematical Society

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