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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Maximal functions associated with curves and the Calderón-Zygmund method of rotations


Author: Shuichi Sato
Journal: Trans. Amer. Math. Soc. 293 (1986), 799-806
MSC: Primary 42B25
MathSciNet review: 816326
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Abstract: Let $ {\delta _t}(t > 0)$ be a dilation in $ {{\mathbf{R}}^n}(n \geqslant 2)$ defined by

$\displaystyle {\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),$

where $ {\alpha _i} > 0(i = 1,2, \ldots ,n)$ and $ {\alpha _i} \ne {\alpha _j}$ if $ i \ne j$. For $ \nu \in {{\mathbf{R}}^n}$ with $ \vert\nu \vert = 1$, let $ {\Gamma _\nu }:(0,\infty ) \to {{\mathbf{R}}^n}$ be a curve defined by $ {\Gamma _\nu }(t) = {\delta _t}\nu \,(0 < t < \infty )$. Using maximal functions associated with the curves $ {\Gamma _\nu }$, we define an operator $ M$ which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that $ M$ is a bounded operator on $ {L^p}({{\mathbf{R}}^n})$ for some $ p$ with $ 1 < p < 2$. As its application we prove the $ {L^p}$ boundedness of operators of the form $ {T^{\ast}}(f)(x) = {\sup _{\varepsilon > 0}}\vert{T_\varepsilon }(f)(x)\vert$, where $ {T_\varepsilon }$ is an integral operator associated with a variable kernel with mixed homogeneity.

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DOI: https://doi.org/10.1090/S0002-9947-1986-0816326-0
Article copyright: © Copyright 1986 American Mathematical Society