Kakeya极大算子及其分数次情形的正则性

刘风, 吴玉荣

数学学报 ›› 2018, Vol. 61 ›› Issue (5) : 783-800.

PDF(609 KB)
PDF(609 KB)
数学学报 ›› 2018, Vol. 61 ›› Issue (5) : 783-800. DOI: 10.12386/A2018sxxb0073
论文

Kakeya极大算子及其分数次情形的正则性

    刘风1, 吴玉荣2
作者信息 +

Regularity of the Kakeya Maximal Operator and Its Fractional Variant

    Feng LIU1, Yu Rong WU2
Author information +
文章历史 +

摘要

研究中心Kakeya(Nikodym)极大算子KNN>2)及其分数次情形Kα,N(0 < α < d)的正则性.特别地,建立了中心分数次Kakeya极大算子Kα,N是从W1,pRdW1,pRd上的有界连续算子,其中1 < p < ∞,q=dp/(d-αp)和0 ≤ α < d/p.还证明了中心Kakeya极大算子KN是分数次Sobolev空间Ws,pRd,非齐次Triebel—Lizorkin空间Fsp,qRd以及非齐次Besov空间Bsp,qRd上的有界连续算子,其中0 < s < 1,1 < p,q < ∞.此外,也考虑分数次Kakeya极大函数的弱导数的两种点态估计以及其离散情形的正则性.

Abstract

In this article, the authors investigate the regularity properties of the centered Kakeya (Nikodym) maximal operator KN(with N>2) and its fractional variant Kα,N (with 0 < α < d). More precisely, the authors prove that, the operator Kα,N is bounded and continuous from W1,p(Rd to W1,p(Rd for 1 < p < ∞ and q=dp/(d -αp) with 0 ≤ α < d/p, and the operator KN is bounded and continuous on the fractional Sobolev spaces Ws,p(Rd, inhomogeneous Triebel-Lizorkin spaces Fsp,q(Rd and inhomogeneous Besov spaces Bsp,q(Rd for all 0 < s < 1 and 1 < p, q < ∞. In addition, two pointwise estimates for the derivatives of the fractional Kakeya maximal functions and the regularity properties for the discrete versions of these operators are also presented.

关键词

Kakeya极大算子 / 分数次Kakeya极大算子 / Sobolev空间 / Triebel-Lizorkin空间 / 连续性

Key words

Kakeya maximal operator / fractional Kakeya maximal operator / Sobolev space / Triebel-Lizorkin space / continuity

引用本文

导出引用
刘风, 吴玉荣. Kakeya极大算子及其分数次情形的正则性. 数学学报, 2018, 61(5): 783-800 https://doi.org/10.12386/A2018sxxb0073
Feng LIU, Yu Rong WU. Regularity of the Kakeya Maximal Operator and Its Fractional Variant. Acta Mathematica Sinica, Chinese Series, 2018, 61(5): 783-800 https://doi.org/10.12386/A2018sxxb0073

参考文献

[1] Aldaz J. M., Pérez Lázaro J., Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc., 2007, 359(5):2443-2461.
[2] Bober J., Carneiro E., Hughes K., Pierce L. B., On a discrete version of Tanaka's theorem for maximal functions, Proc. Amer. Math. Soc., 2012, 140(5):1669-1680.
[3] Brezis H., Lieb E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 1983, 88:486-490.
[4] Carneiro E., Finder R., Sousa M., On the variation of maximal operators of convolution type Ⅱ, Revista Mat. Iberoamericana (to appear).
[5] Carneiro E., Hughes K., On the endpoint regularity of discrete maximal operators, Math. Res. Lett., 2012, 19(6):1245-1262.
[6] Carneiro E., Mardid J., Derivative bounds for fractional maximal functions, Trans. Amer. Math. Soc., 2017, 369(60):4063-4092.
[7] Carneiro E., Moreira D., On the regularity of maximal operators, Proc. Amer. Math. Soc., 2008, 136(12):4395-4404.
[8] Carneiro, E., Svaiter B. F., On the variation of maximal operators of convolution type, J. Funct. Anal., 2013, 265:837-865.
[9] Córdoba A., The Kakeya maximal function and the spherical summation multiplier, Amer. J. Math., 1977, 99:1-22.
[10] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, 2nd edn., SpringerVerlag, Berlin, 1983.
[11] Grafakos L., Modern Fourier Analysis, Volume 250 of Graduate Texts in Mathematics, Springer, New York, 2nd Edition, 2008.
[12] Haj lasz P., Onninen J., On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 2004, 29(1):167-176.
[13] Kinnunen J., The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math., 1997, 100:117-124.
[14] Kinnunen J., Lindqvist P., The derivative of the maximal function, J. Reine Angew. Math., 1998, 503:161-167.
[15] Kinnunen J., Saksman E., Regularity of the fractional maximal function. Bull. London Math. Soc., 2003, 35(4):529-535.
[16] Korry S., Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces, Rev. Mat. Complut., 2002, 15(2):401-416.
[17] Korry S., A class of bounded operators on Sobolev spaces, Arch. Math., 2004, 82(1):40-50.
[18] Kurka O., On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math., 2015, 40:109-133.
[19] Liu F., A remark on the regularity of the discrete maximal operator, Bull. Austral. Math. Soc., 2017, 95:108-120.
[20] Liu F., Continuity and approximate differentiability of multisublinear fractional maximal functions, Math. Inequal. Appl., 2018, 21(1):25-40.
[21] Liu F., On the regularity of one-sided fractional maximal functions, Math. Slovaca (accepted).
[22] Liu F., Chen T., Wu H., A note on the endpoint regularity of the Hardy-Littlewood maximal functions, Bull. Austral. Math. Soc., 2016, 94:121-130.
[23] Liu F., Mao S., On the regularity of the one-sided Hardy-Littlewood maximal functions, Czech. Math. J., 2017, 67(142):219-234.
[24] Liu F., Wu H., On the regularity of the multisublinear maximal functions, Canad. Math. Bull., 2015, 58(4):808-817.
[25] Liu F., Wu H., Endpoint regularity of multisublinear fractional maximal functions, Canad. Math. Bull., 2017, 60(3):586-603.
[26] Liu F., Wu H., Regularity of discrete multisublinear fractional maximal functions, Sci. China Math., 2017, 60(8):1461-1476.
[27] Liu F., Wu H., On the regularity of maximal operators supported by submanifolds, J. Math. Anal. Appl., 2017, 453:144-158.
[28] Luiro H., Continuity of the maixmal operator in Sobolev spaces. Proc. Amer. Math. Soc., 2007, 135(1):243-251.
[29] Luiro H., On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rn, Proc. Edinburgh Math. Soc., 2010, 53(1):211-237.
[30] Madrid J., Sharp inequalities for the variation of the discrete maximal function, Bull. Austral. Math. Soc., 2017, 95:94-107.
[31] Pierce L. B., Discrete Analogues in Harmonic Analysis, Ph.D. Thesis, Princeton University, 2009.
[32] Strömberg J. O., Maximal functions associated to rectangles with uniformly distributed directions, Ann. Math., 1978, 107:399-402.
[33] Tanaka H., A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc., 2002, 65(2):253-258.
[34] Temur F., On regularity of the discrete Hardy-Littlewood maximal function, preprint (2015), arxiv.org/abs/1303.3993.
[35] Yabuta K., Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces, Appl. Math. J. Chinese Univ. Ser. B, 2015, 30(4):418-446.

基金

国家自然科学基金(11701333,11771395);山东科技大学人才引进科研启动基金(2015RCJJ053);山东科技大学数学与系统科学学院优秀青年科技拔尖人才支持计划项目(Sxy2016ko1)

PDF(609 KB)

361

Accesses

0

Citation

Detail

段落导航
相关文章

/