Maximal operators associated with some singular submanifolds
HTML articles powered by AMS MathViewer
- by Yaryong Heo, Sunggeum Hong and Chan Woo Yang PDF
- Trans. Amer. Math. Soc. 369 (2017), 4597-4629 Request permission
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 \begin{equation*} \langle \mu , f \rangle =\int _{\mathbb {R}^d} f(y, \gamma (y)) \psi (y) \chi _{\mathrm {U}\setminus \Omega }(y)\; dy, \end{equation*} where $\psi$ is a smooth function supported in $\mathrm {U}$. In this paper we give some conditions under which the Fourier decay estimates $|\widehat {\mu }(\xi )| \le C (1+|\xi |)^{-\epsilon }$ 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., \[ \mathrm {M}_Sf(x)=\sup _{r>0} r^{-d}\int _{|y|<r} \big {|}f\big {(}x-(y,\gamma (y))\big {)}\big {|} \chi _{\mathbb {R}^d \setminus \Omega _{\text {sym}}} dy,\] 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., \[ \mathrm {M}_Hf(x)=\sup _{r>0}\left |\int _{\mathbb {R}^d} f\big {(}x-(y,\gamma (y))\big {)} r^{-d} \psi (r^{-1}y) dy \right |,\] 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 $m>0$.References
- Alberto-P. Calderón, An atomic decomposition of distributions in parabolic $H^{p}$ spaces, Advances in Math. 25 (1977), no. 3, 216–225. MR 448066, DOI 10.1016/0001-8708(77)90074-3
- Michael Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. MR 951506, DOI 10.2307/1971461
- Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
- Ya Ryong Heo, An endpoint estimate for some maximal operators associated to submanifolds of low codimension, Pacific J. Math. 201 (2001), no. 2, 323–338. MR 1875897, DOI 10.2140/pjm.2001.201.323
- Ya Ryong Heo, Weak type estimates for some maximal operators on Hardy spaces, Math. Nachr. 280 (2007), no. 3, 281–289. MR 2292150, DOI 10.1002/mana.200410481
- Yaryong Heo, Endpoint estimates for some maximal operators associated to the circular conical surface, J. Math. Anal. Appl. 351 (2009), no. 1, 152–162. MR 2472929, DOI 10.1016/j.jmaa.2008.10.011
- Robert H. Latter and Akihito Uchiyama, The atomic decomposition for parabolic $H^{p}$ spaces, Trans. Amer. Math. Soc. 253 (1979), 391–398. MR 536954, DOI 10.1090/S0002-9947-1979-0536954-6
- Daniel M. Oberlin, An endpoint estimate for some maximal operators, Rev. Mat. Iberoamericana 12 (1996), no. 3, 641–652. MR 1435478, DOI 10.4171/RMI/209
- F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 637–670 (English, with English and French summaries). MR 1182643, DOI 10.5802/aif.1304
- Andreas Seeger and Terence Tao, Sharp Lorentz space estimates for rough operators, Math. Ann. 320 (2001), no. 2, 381–415. MR 1839769, DOI 10.1007/PL00004479
- Andreas Seeger, Terence Tao, and James Wright, Singular maximal functions and Radon transforms near $L^1$, Amer. J. Math. 126 (2004), no. 3, 607–647. MR 2058385, DOI 10.1353/ajm.2004.0020
- Andreas Seeger and James Wright, Problems on averages and lacunary maximal functions, Marcinkiewicz centenary volume, Banach Center Publ., vol. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 235–250. MR 2918096, DOI 10.4064/bc95-0-11
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Additional Information
- Yaryong Heo
- Affiliation: Department of Mathematics, Korea University, Seoul 136-701, South Korea
- MR Author ID: 688436
- Email: yaryong@korea.ac.kr
- Sunggeum Hong
- Affiliation: Department of Mathematics, Chosun University, Gwangju 501-759, South Korea
- MR Author ID: 648474
- 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
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4597-4629
- MSC (2010): Primary 42B20; Secondary 42B15
- DOI: https://doi.org/10.1090/tran/6785
- MathSciNet review: 3632544