Necessary and sufficient conditions for boundedness of commutators of maximal function on the $ p $-adic vector spaces

Qianjun He; Xiang Li; Qianjun He; Xiang Li

doi:10.3934/math.2023719

AIMS Mathematics

2023, Volume 8, Issue 6: 14064-14085. doi: 10.3934/math.2023719

Previous Article Next Article

Research article

Necessary and sufficient conditions for boundedness of commutators of maximal function on the $ p $-adic vector spaces

Qianjun He ¹,
Xiang Li ^{2
,
,}

1.
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
2.
School of Science, Shandong Jianzhu University, Jinan 250000, China

Received: 10 December 2022 Revised: 23 February 2023 Accepted: 28 February 2023 Published: 14 April 2023
MSC : 11E95, 11K70, 11S80, 42B25, 42B35

In this paper, we first show that the $ p $-adic version of maximal function $ \mathcal{M}_{L\log L}^{p} $ is equivalent to the maximal function $ \mathcal{M}^{p}(\mathcal{M}^{p}) $ and that the class of functions for which the maximal commutators and the commutator with the $ p $-adic version of maximal function or the maximal sharp function are bounded on the $ p $-adic vector spaces are characterized and proved to be the same. Moreover, new pointwise estimates for these operators are proved.
- $ p $-adic vector space,
- maximal function,
- commutator,
- BMO space,
- Morrey spaces
Citation: Qianjun He, Xiang Li. Necessary and sufficient conditions for boundedness of commutators of maximal function on the $ p $-adic vector spaces[J]. AIMS Mathematics, 2023, 8(6): 14064-14085. doi: 10.3934/math.2023719

Related Papers:

Abstract

In this paper, we first show that the $ p $-adic version of maximal function $ \mathcal{M}_{L\log L}^{p} $ is equivalent to the maximal function $ \mathcal{M}^{p}(\mathcal{M}^{p}) $ and that the class of functions for which the maximal commutators and the commutator with the $ p $-adic version of maximal function or the maximal sharp function are bounded on the $ p $-adic vector spaces are characterized and proved to be the same. Moreover, new pointwise estimates for these operators are proved.

References

[1]	D. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765–778. http://dx.doi.org/10.1215/S0012-7094-75-04265-9
[2]	M. Agcayazi, A. Gogatishvili, K Koca, R. Mustafayev, A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Japan, 67 (2015), 581–593. http://dx.doi.org/10.2969/jmsj/06720581 doi: 10.2969/jmsj/06720581
[3]	M. Agcayazi, A. Gogatishvili, R. Mustafayev, Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function, Tokyo J. Math., 41 (2018), 193–218. http://dx.doi.org/10.3836/tjm/1502179258 doi: 10.3836/tjm/1502179258
[4]	J. Bastero, M. Milman, F. Ruiz, Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc., 128 (2000), 3329–3334.
[5]	M. Bramanti, M. Cerutti, Commutators of singular integrals and fractional integrals on homogeneous spaces, In: Harmonic analysis and operator theory, Provence: American Mathematical Society, 1995, 81–94.
[6]	M. Bramanti, M. Cerutti, M. Manfredini, $L^{p}$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl., 200 (1996), 332–354. http://dx.doi.org/10.1006/jmaa.1996.0209 doi: 10.1006/jmaa.1996.0209
[7]	A. Bonami, T. Iwaniec, P. Jones, M. Zinsmeister, On the product of functions in $BMO$ and $H^{1}$, Ann. Inst. Fourier, 57 (2007), 1405–1439.
[8]	L. Caffarelli, Elliptic second order equations, Seminario Mat. e. Fis. di Milano, 58 (1988), 253–284. http://dx.doi.org/10.1007/BF02925245 doi: 10.1007/BF02925245
[9]	F. Chiarenza, M. Frasca, P. Longo, Interior $W^{2, p}$ estimates for non divergence elliptic equations with discontinuous coefficients, Ric. Mat., 40 (1991), 149–168.
[10]	F. Chiarenza, M. Frasca, P. Longo, $W^{2, p}$-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients, Trans. Am. Math. Soc., 336 (1993), 841–853.
[11]	R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103 (1976), 611–635. http://dx.doi.org/10.2307/1970954 doi: 10.2307/1970954
[12]	F. Deringoz, K. Dorak and V. Guliyev, Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz-Morrey spacces on spaces of homogeneous type, Anal. Math. Phys., 11 (2021), 63. http://dx.doi.org/10.1007/s13324-021-00497-1 doi: 10.1007/s13324-021-00497-1
[13]	G. Difazio, M. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241–256. http://dx.doi.org/10.1006/jfan.1993.1032 doi: 10.1006/jfan.1993.1032
[14]	D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer-Verlag, 2001. http://dx.doi.org/10.1007/978-3-642-61798-0
[15]	L. Grafakos, Modern Fourier analysis, New York: Springer-Verlag, 2014. http://dx.doi.org/10.1007/978-1-4939-1230-8
[16]	I. Greco, T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincare, 11 (1994), 17–35. http://dx.doi.org/10.1016/S0294-1449(16)30194-9 doi: 10.1016/S0294-1449(16)30194-9
[17]	V. Guliyev, Y. Mannadov, F. Muslumova, Characterization of fractional maximal operator and its commutators on Orlicz spaces in the Dunkl setting, J. Pseudo-Differ. Oper. Appl., 11 (2020), 1699–1717. http://dx.doi.org/10.1007/s11868-020-00364-w doi: 10.1007/s11868-020-00364-w
[18]	Q. He, M. Wei, D. Yan, Characterizations of $p$-adic central Campanato spaces via commutator of $p$-adic Hardy type operators, J. Korean Math. Soc., 56 (2019), 767–787. http://dx.doi.org/10.4134/JKMS.j180390 doi: 10.4134/JKMS.j180390
[19]	Q. He, X. Li, Characterization of Lipschitz spaces via commutators of maximal function on the $p$-adic vector space, J. Math., 2022 (2022), 7430272. http://dx.doi.org/10.1155/2022/7430272 doi: 10.1155/2022/7430272
[20]	G. Hu, H. Lin, D. Yang, Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal., 2008 (2008), 237937. http://dx.doi.org/10.1155/2008/237937 doi: 10.1155/2008/237937
[21]	G. Hu, D. Yang, Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type, J. Math. Anal. Appl., 354 (2009), 249–262. http://dx.doi.org/10.1016/j.jmaa.2008.12.066 doi: 10.1016/j.jmaa.2008.12.066
[22]	S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16 (1978), 263–270. http://dx.doi.org/10.1007/BF02386000 doi: 10.1007/BF02386000
[23]	T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat., 22 (1992), 127–155. http://dx.doi.org/10.1007/BF01232939 doi: 10.1007/BF01232939
[24]	Y. Kim, Carleson measures and the BMO space on the $p$-adic vector space, Math. Nachr., 282 (2009), 1278–1304. http://dx.doi.org/10.1002/mana.200610806 doi: 10.1002/mana.200610806
[25]	Y. Kim, $L^{q}$-estimates of maximal operators on the $p$-adic vector space, Commun. Korean Math. Soc., 24 (2009), 367–379. http://dx.doi.org/10.4134/CKMS.2009.24.3.367 doi: 10.4134/CKMS.2009.24.3.367
[26]	R. Liu, J. Zhou, Weighted multilinear $p$-adic Hardy operators and commutators, Open Math., 15 (2017), 1623–1634. http://dx.doi.org/10.1515/math-2017-0139 doi: 10.1515/math-2017-0139
[27]	M. Milman, T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc., 110 (1990), 961–969.
[28]	C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126–166. http://dx.doi.org/10.2307/1989904 doi: 10.2307/1989904
[29]	C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163–185. http://dx.doi.org/10.1006/jfan.1995.1027 doi: 10.1006/jfan.1995.1027
[30]	C. Pérez, R. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal., 181 (2001), 146–188. http://dx.doi.org/10.1006/jfan.2000.3711 doi: 10.1006/jfan.2000.3711
[31]	M. Rao, Z. Ren, Theory of Orlicz spaces, New York: Marcel Dekker Inc., 1991.
[32]	C. Rios, The $L^{p}$ Dirichlet problem and nondivergence harmonic measure, Trans. Amer. Math. Soc., 355 (2003), 665–687. http://dx.doi.org/10.1090/S0002-9947-02-03145-8 doi: 10.1090/S0002-9947-02-03145-8
[33]	K. Rim, J. Lee, Estimate of weighted Hard-Littlewood averages on the $p$-adic vector space, J. Math. Anal. Appl., 324 (2006), 1470–1477. http://dx.doi.org/10.1016/j.jmaa.2006.01.038 doi: 10.1016/j.jmaa.2006.01.038
[34]	K. Rogers, A van der Corput lemma for the $p$-adic numbers, Proc. Amer. Math. Soc., 133 (2005), 3525–3534.
[35]	K. Rogers, Maximal averages along curves over the $p$-adic numbers, Bull. Austral. Math. Soc., 70 (2004), 357–375. http://dx.doi.org/10.1017/S0004972700034602 doi: 10.1017/S0004972700034602
[36]	C. Segovia, J. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat., 35 (1991), 209–235.
[37]	C. Segovia, J. Torrea, Higher order commutators for vector-valued Calderón-Zygmund operators, Trans. Amer. Math. Soc., 36 (1993), 537–556.
[38]	Z. Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains, Am. J. Math., 125 (2003), 1079–1115.
[39]	E. Stein, Note on the class $L\log L$, Stud. Math., 32 (1969), 305–310.
[40]	V. Vladimirov, I. Volovich, E. Zelenov, p-adic Analysis and Mathematical Physics, Singapore: World Scientific, 1992. http://dx.doi.org/10.1142/1581
[41]	D. Wang, J. Zhou, Necessary and sufficient conditions for boundedness of commutators of bilinear Hardy-Littlewood maximal function, arXiv: 1708.09549.
[42]	R. Wheeden, J. Wilson, Weighted norm estimates for gradients of half-space extensions, Indiana Univ. Math. J., 44 (1995), 917–969.
[43]	P. Zhang, Multiple weighted estimates for commutators of multilinear maximal function, Acta. Math. Sin.-English Ser., 31 (2015), 973–994. http://dx.doi.org/10.1007/s10114-015-4293-6 doi: 10.1007/s10114-015-4293-6
[44]	P. Zhang, Characterization of Lipschitz spaces via commutators of Hardy-Littlewood maximal function, CR Math., 355 (2017), 336–344. http://dx.doi.org/10.1016/j.crma.2017.01.022 doi: 10.1016/j.crma.2017.01.022
[45]	P. Zhang, Characterization of boundedness of some commutators of maximal functions in terms of Lipschitz spaces, Anal. Math. Phys., 9 (2019), 1411–1427. http://dx.doi.org/10.1007/s13324-018-0245-5 doi: 10.1007/s13324-018-0245-5

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)