CONTINUITY OF THE MULTILINEAR MAXIMAL COMMUTATORS IN SOBOLEV SPACES

Xixi Jiang; Feng Liu

doi:10.11948/20230334

2024 Volume 14 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(3): 1674-1697

Next Article Previous Article

Xixi Jiang, Feng Liu. CONTINUITY OF THE MULTILINEAR MAXIMAL COMMUTATORS IN SOBOLEV SPACES[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1674-1697. doi: 10.11948/20230334

Citation:

Xixi Jiang, Feng Liu. CONTINUITY OF THE MULTILINEAR MAXIMAL COMMUTATORS IN SOBOLEV SPACES[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1674-1697. doi: 10.11948/20230334

CONTINUITY OF THE MULTILINEAR MAXIMAL COMMUTATORS IN SOBOLEV SPACES

Xixi Jiang^1,,
Feng Liu^1, ,

1.
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

Author Bio: Email: jxx15166021527@163.com(X. Jiang)
Corresponding author: Email: FLiu@sdust.edu.cn(F. Liu)
Fund Project: This work was supported partly by the National Natural Science Foundation of China (Grant No. 12326371) and the Natural Science Foundation of Shandong Province (Grant No. ZR2023MA022)

Abstract

Abstract

In the present paper we study the Sobolev continuity of the multilinear maximal commutators and their fractional variants with Lipschitz symbols. More precisely, let $\mathfrak{M}_{\alpha,\vec{b}}$ be the multilinear fractional maximal commutators, where $0\leq\alpha<mn$ and $\vec{b}=(b_{1},\ldots,b_{m})$ with each $b_i\in \rm Lip(\mathbb{R}^n)$. We establish the continuity of $\mathfrak{M}_{\alpha,\vec{b}}:W^{1,p_1} (\mathbb{R}^n)\times\cdots\times W^{1,p_m}(\mathbb{R}^n)\rightarrow W^{1,q}(\mathbb{R}^n)$, provided that $1<p_1,\ldots,p_m<\infty$, $1/q=\sum_{i=1}^m1/p_i-\alpha/n$ and $1\leq q<\infty$. The main result we obtain answers a question originally posed by Chen and Liu in 2022. Our main result is new, even in the linear case m=1.
- Multilinear maximal commutator /
- fractional variants /
- Sobolev space /
- continuity
MSC: 42B25, 46E35

FullText(HTML)

References

[1]	M. Agcayazi, A. Gogatishvili, K. Koca and R. Mustafayev, A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Jpn., 2015, 67(2), 581–593. Google Scholar
[2]	J. M. Aldaz and J. Pérez Lázaro, 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. Google Scholar
[3]	A. M. Alphonse, An end point estimate for maximal commutators, J. Fourier Anal. Appl., 2000, 6, 449–456. doi: 10.1007/BF02510149 CrossRef Google Scholar
[4]	D. Beltran and J. Madrid, Regularity of the centered fractional maximal function on radial functions, J. Funct. Anal., 2020, 279, 108686. doi: 10.1016/j.jfa.2020.108686 CrossRef Google Scholar
[5]	R. Bu, Z. W. Fu and Y. D. Zhang, Weighted estimates for bilinear square function with non-smooth kernels and commutators, Front. Math. China, 2020, 15, 1–20. doi: 10.1007/s11464-020-0822-4 CrossRef Google Scholar
[6]	E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc., 2008, 136(12), 4395–4404. doi: 10.1090/S0002-9939-08-09515-4 CrossRef Google Scholar
[7]	P. Chen, X. T. Duong, J. Li and Q. Y. Wu, Compactness of Riesz transform commutator on stratified Lie groups, J. Funct. Anal., 2019, 277, 1639–1676. doi: 10.1016/j.jfa.2019.05.008 CrossRef Google Scholar
[8]	T. Chen and F. Liu, Regularity of commutators of multilinear maximal operators with Lipschitz symbols, Math. Inequal. Appl., 2022, 25(1), 109–134. Google Scholar
[9]	F. Deringoz, V. S. Guliyev and S. G. Hasanov, Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, Positivity, 2018, 22, 141–158. doi: 10.1007/s11117-017-0504-y CrossRef Google Scholar
[10]	F. Deringoz, V. S. Guliyev and S. Samko, Vanishing generalized Orlicz–Morrey spaces and fractional maximal operator, Publ. Math. Debrecen, 2017, 90(1–2), 125–147. doi: 10.5486/PMD.2017.7494 CrossRef Google Scholar
[11]	X. T. Duong, M. Lacey, J. Li, B. D. Wick and Q. Y. Wu, Commutators of Cauchy-Szego type integrals for domains in Cn with minimal smoothness, Indiana Univ. Math. J., 2021, 70, 1505–1541. doi: 10.1512/iumj.2021.70.8573 CrossRef Google Scholar
[12]	H. Federer and W. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J., 1972, 22, 139–158. doi: 10.1512/iumj.1973.22.22013 CrossRef Google Scholar
[13]	Z. W. Fu, R. M. Gong, E. Pozzi and Q. Y. Wu, Cauchy-Szegö commutators on weighted Morrey spaces, Math. Nachr., 2023, 296(5), 1859–1885. doi: 10.1002/mana.202000139 CrossRef Google Scholar
[14]	Z. W. Fu, S. L. Gong, S. Z. Lu and W. Yuan, Weighted multilinear Hardy operators and commutators, Forum Math., 2015, 27, 2825–2852. doi: 10.1515/forum-2013-0064 CrossRef Google Scholar
[15]	Z. W. Fu, L. Grafakos, Y. Lin, Y. Wu and S. H. Yang, Riesz transform associated with the fractional Fourier transform and applications in image edge detection, Appl. Comput. Harmon. Anal., 2023, 66, 211–235. doi: 10.1016/j.acha.2023.05.003 CrossRef Google Scholar
[16]	Z. W. Fu, X. M. Hou, M. Y. Lee and J. Li, A study of one-sided singular integral and function space via reproducing formula, J. Geom. Anal., 2023, 33. DOI: 10.1007/s12220-023-01340-8. CrossRef Google Scholar
[17]	J. García-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J., 1991, 40, 1397–1420. doi: 10.1512/iumj.1991.40.40063 CrossRef Google Scholar
[18]	R. M. Gong, M. N. Vempati, Q. Y. Wu and P. Z. Xie, Boundedness and compactness of Cauchy-type integral commutator on weighted Morrey spaces, J. Aust. Math. Soc., 2022, 113(1), 36–56. doi: 10.1017/S1446788722000015 CrossRef Google Scholar
[19]	C. González-Riquelme, Continuity for the one-dimensional centered Hardy–Littlewood maximal opeator at the derivative level, J. Funct. Anal., 2023, 285(9), 110097. doi: 10.1016/j.jfa.2023.110097 CrossRef Google Scholar
[20]	V. S. Guliyev and F. Deringoz, Some characterizations of Lipschitz spaces via commutators on generalized Orlicz–Morrey spaces, Mediterr. J. Math., 2018, 15(180), 1–19. Google Scholar
[21]	P. Hajłasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 2004, 29(1), 167–176. Google Scholar
[22]	J. Kinnunen, The Hardy–Littlewood maximal function of a Sobolev function, Israel J. Math., 1997, 100, 117–124. doi: 10.1007/BF02773636 CrossRef Google Scholar
[23]	J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine. Angew. Math., 1998, 503, 161–167. Google Scholar
[24]	J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc., 2003, 35(4), 529–535. doi: 10.1112/S0024609303002017 CrossRef Google Scholar
[25]	A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory, Adv. Math., 2009, 220, 1222–1264. doi: 10.1016/j.aim.2008.10.014 CrossRef Google Scholar
[26]	F. Liu, Continuity and approximate differentiability of multisublinear fractional maximal functions, Math. Inequal. Appl., 2018, 21(1), 25–40. Google Scholar
[27]	F. Liu and G. Wang, Regularity of commutators of maximal operators with Lipschitz symbols, Taiwan. J. Math., 2021, 25(5), 1007–1039. Google Scholar
[28]	F. Liu and H. Wu, On the regularity of the multisublinear maximal functions, Canad. Math. Bull., 2015, 58(4), 808–817. doi: 10.4153/CMB-2014-070-7 CrossRef Google Scholar
[29]	F. Liu and S. Xi, Sobolev regularity for commutators of the fractional maximal functions, Banach. J. Math. Anal., 2021, 15(5), 1–36. Google Scholar
[30]	F. Liu, Q. Xue and K. Yabuta, Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function, Annales Fennici Mathematici, 2022, 47, 203–235. Google Scholar
[31]	F. Liu, Q. Xue and P. Zhang, Regularity and continuity of commutators of the Hardy–Littlewood maximal function, Math. Nachr., 2020, 293(3), 491–509. doi: 10.1002/mana.201900013 CrossRef Google Scholar
[32]	H. Luiro, Continuity of the maixmal operator in Sobolev spaces, Proc. Amer. Math. Soc., 2007, 135(1), 243–251. Google Scholar
[33]	H. Luiro, On the regularity of the Hardy–Littlewood maximal operator on subdomains of $\mathbb{R}^n$, Proc. Edinburgh Math. Soc., 2010, 53(1), 211–237. Google Scholar
[34]	C. Pérez and R. H. Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math., 2003, 320, 323–331. Google Scholar
[35]	J. M. Ruan, Q. Y. Wu and D. S. Fan, Weighted Moes for Hausdorff operator and its commutator on the Heisenberg group, Math. Inequal. Appl., 2019, 22(1), 307–329. Google Scholar
[36]	C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat., 1991, 35, 209–235. doi: 10.5565/PUBLMAT_35191_09 CrossRef Google Scholar
[37]	C. Segovia and J. L. Torrea, Higher order commutators for vector-valued Calderón–Zygmund operators, Trans. Amer. Math. Soc., 1993, 336, 537–556. Google Scholar
[38]	S. G. Shi, Z. W. Fu and S. Z. Lu, On the compactness of commutators of Hardy operators, Pacific J. Math., 2020, 307, 239–256. doi: 10.2140/pjm.2020.307.239 CrossRef Google Scholar
[39]	S. G. Shi and S. Z. Lu, A characterization of Campanato space via commutator of fractional integral, J. Math. Anal. Appl., 2014, 419, 123–137. doi: 10.1016/j.jmaa.2014.04.040 CrossRef Google Scholar
[40]	Q. Y. Wu and Z. W. Fu, Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces, Bull. Malays. Math. Sci. Soc., 2017, 40, 635–654. doi: 10.1007/s40840-017-0444-5 CrossRef Google Scholar
[41]	M. H. Yang, Z. W. Fu and J. Y. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov–Morrey spaces, J. Differ. Equations, 2019, 266, 5867–5894. doi: 10.1016/j.jde.2018.10.050 CrossRef Google Scholar
[42]	M. H. Yang, Z. W. Fu and J. Y. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 2017, 60, 1837–1856. doi: 10.1007/s11425-016-0490-y CrossRef Google Scholar
[43]	P. Zhang, Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function, C. R. Acad. Sci. Paris Ser. I, 2017, 355(3), 336–344. doi: 10.1016/j.crma.2017.01.022 CrossRef Google Scholar
[44]	P. Zhang, Characterization of boundedness of some commutators of maximal functions in terms of Lipschitz spaces, Anal. Math. Phys., 2019, 9(3), 1411–1427. doi: 10.1007/s13324-018-0245-5 CrossRef Google Scholar
[45]	P. Zhang, Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. Engl. Ser., 2015, 31(6), 973–994. doi: 10.1007/s10114-015-4293-6 CrossRef Google Scholar