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Transactions of the American Mathematical Society

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

 
 

 

Maximal operators associated with some singular submanifolds


Authors: Yaryong Heo, Sunggeum Hong and Chan Woo Yang
Journal: Trans. Amer. Math. Soc. 369 (2017), 4597-4629
MSC (2010): Primary 42B20; Secondary 42B15
DOI: https://doi.org/10.1090/tran/6785
Published electronically: January 9, 2017
MathSciNet review: 3632544
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Abstract: Let $ \mathrm {U}$ be a bounded open subset of $ \mathbb{R}^d$ and let $ \Omega $ be a Lebesgue measurable subset of $ \mathrm {U}$. Let $ \gamma =(\gamma _1, \cdots , \gamma _n) : \mathrm {U}\setminus \Omega \rightarrow \mathbb{R}^n$ be a Lebesgue measurable function, and let $ \mu $ be a Borel measure on $ \mathbb{R}^{d+n}$ defined by

$\displaystyle \langle \mu , f \rangle =\int _{\mathbb{R}^d} f(y, \gamma (y)) \psi (y)\,\chi _{\mathrm {U}\setminus \Omega }(y)\; dy,$    

where $ \psi $ is a smooth function supported in $ \mathrm {U}$. In this paper we give some conditions under which the Fourier decay estimates $ \vert\widehat {\mu }(\xi )\vert \le C (1+\vert\xi \vert)^{-\epsilon }$ hold for some $ \epsilon >0$. As a corollary we obtain the $ L^p$-boundedness properties of the maximal operators $ \mathrm {M}_{S}$ associated with a certain class of possibly non-smooth $ n$-dimensional submanifolds of $ \mathbb{R}^{d+n}$, i.e.,

$\displaystyle \mathrm {M}_Sf(x)=\sup _{r>0}\, r^{-d}\int _{\vert y\vert<r} \big... ...ig {)}\big {\vert} \,\chi _{\mathbb{R}^d \setminus \Omega _{\text {sym}}} \,dy,$

where $ \Omega _{\text {sym}}$ is a radially symmetric Lebesgue measurable subset of $ \mathbb{R}^d$, $ \gamma (y)=(\gamma _1(y), \cdots , \gamma _n(y))$, $ \gamma _i(t y)=t^{a_i} \gamma _i(y)$ for each $ t>0$ where $ a_i \in \mathbb{R}$, and the function $ \gamma _i : \mathbb{R}^d \setminus \Omega _{\text {sym}} \rightarrow \mathbb{R}$ satisfies some singularity conditions over a certain subset of $ \mathbb{R}^d$. Also we investigate the endpoint $ (parabolic\; H^1, L^{1,\infty })$ mapping properties of the maximal operators $ \mathrm {M}_H$ associated with a certain class of possibly non-smooth hypersurfaces, i.e.,

$\displaystyle \mathrm {M}_Hf(x)=\sup _{r>0}\left \vert\int _{\mathbb{R}^d} f\big {(}x-(y,\gamma (y))\big {)} r^{-d} \psi (r^{-1}y)\,dy \right \vert,$

where the function $ \gamma : \mathbb{R}^d \rightarrow \mathbb{R}$ satisfies some singularity conditions over a certain subset of $ \mathbb{R}^d$ and $ \gamma (t y)=t^m \gamma (y)$ for each $ t>0$ where $ m>0$.

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Additional Information

Yaryong Heo
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, South Korea
Email: yaryong@korea.ac.kr

Sunggeum Hong
Affiliation: Department of Mathematics, Chosun University, Gwangju 501-759, South Korea
Email: skhong@chosun.ac.kr

Chan Woo Yang
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, South Korea
Email: cw_yang@korea.ac.kr

DOI: https://doi.org/10.1090/tran/6785
Keywords: Maximal operators
Received by editor(s): February 23, 2015
Received by editor(s) in revised form: March 3, 2015, and July 7, 2015
Published electronically: January 9, 2017
Additional Notes: This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology NRF-2015R1A1A1A05001304, NRF-2014R1A1A3049983, and NRF-2013R1A1A2013659.
Article copyright: © Copyright 2017 American Mathematical Society

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