Some estimates of multilinear operators on tent spaces

Heng Yang; Jiang Zhou; Heng Yang; Jiang Zhou

doi:10.3934/cam.2024031

Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 700-716. doi: 10.3934/cam.2024031

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Some estimates of multilinear operators on tent spaces

Heng Yang ,
Jiang Zhou ^,

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 06 December 2023 Revised: 04 July 2024 Accepted: 04 July 2024 Published: 09 October 2024
42B20, 42B25, 42B35

Let $ 0 < \alpha < mn $ and $ 0 < r, q < \infty $. In this paper, we obtain the boundedness of some multilinear operators by establishing pointwise inequalities and applying extrapolation methods on tent spaces $ T_{r}^{q}(\mathbb{R}_{+}^{n+1}) $, where these multilinear operators include multilinear Hardy–Littlewood maximal operator $ \mathcal{M} $, multilinear fractional maximal operator $ \mathcal{M}_{\alpha} $, multilinear Calderón–Zygmund operator $ \mathcal{T} $, and multilinear fractional integral operator $ \mathcal{I}_{\alpha} $. Therefore, the results of Auscher and Prisuelos–Arribas [Math. Z. 286 (2017), 1575–1604] are extended to the general case.
- multilinear maximal operator,
- multilinear Calderón–Zygmund operator,
- multilinear fractional integral operator,
- tent space
Citation: Heng Yang, Jiang Zhou. Some estimates of multilinear operators on tent spaces[J]. Communications in Analysis and Mechanics, 2024, 16(4): 700-716. doi: 10.3934/cam.2024031

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Abstract

Let $ 0 < \alpha < mn $ and $ 0 < r, q < \infty $. In this paper, we obtain the boundedness of some multilinear operators by establishing pointwise inequalities and applying extrapolation methods on tent spaces $ T_{r}^{q}(\mathbb{R}_{+}^{n+1}) $, where these multilinear operators include multilinear Hardy–Littlewood maximal operator $ \mathcal{M} $, multilinear fractional maximal operator $ \mathcal{M}_{\alpha} $, multilinear Calderón–Zygmund operator $ \mathcal{T} $, and multilinear fractional integral operator $ \mathcal{I}_{\alpha} $. Therefore, the results of Auscher and Prisuelos–Arribas [Math. Z. 286 (2017), 1575–1604] are extended to the general case.

References

[1]	R. Coifman, Y. Meyer, E. M. Stein, Un nouveal espace fonctionnel adapte a l'etude des operateurs definis par des integrales singulieres, in Harmonic Analysis. Lecture Notes in Mathematics, (eds. G. Mauceri, F. Ricci, G. Weiss), Springer, Berlin, Heidelberg, (1983), 1–15. https://doi.org/10.1007/BFb0069149
[2]	S. Hofmann, S. Mayboroda, Hardy and BMO spaces to divergence form elliptic operators, Math. Ann., 344 (2009), 37–116. https://doi.org/10.1007/s00208-008-0295-3 doi: 10.1007/s00208-008-0295-3
[3]	R. Coifman, Y. Meyer, E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304–335. https://doi.org/10.1016/0022-1236(85)90007-2 doi: 10.1016/0022-1236(85)90007-2
[4]	P. Auscher, C. Kriegler, S. Monniaux, P. Portal, Singular integral operators on tent spaces, J. Evol. Equ., 12 (2012), 741–765. https://doi.org/10.1007/s00028-012-0152-4 doi: 10.1007/s00028-012-0152-4
[5]	P. Auscher, C. Prisuelos-Arribas, Tent space boundedness via extrapolation, Math. Z., 286 (2017), 1575–1604. https://doi.org/10.1007/s00209-016-1814-7 doi: 10.1007/s00209-016-1814-7
[6]	J. Cao, D. Chang, Z. Fu, D. Yang, Real interpolation of weighted tent spaces, Appl. Anal., 95 (2016), 2415–2443. https://doi.org/10.1080/00036811.2015.1091924 doi: 10.1080/00036811.2015.1091924
[7]	C. Cascante, J. M. Ortega, Imbedding potentials in tent spaces, J. Funct. Anal., 198 (2003), 106–141. https://doi.org/10.1016/s0022-1236(02)00087-3 doi: 10.1016/s0022-1236(02)00087-3
[8]	R. Coifman, Y. Meyer, On commutators of singular integral and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315–331. https://doi.org/10.1090/S0002-9947-1975-0380244-8 doi: 10.1090/S0002-9947-1975-0380244-8
[9]	R. Coifman, Y. Meyer, Commutateurs d'intégrales singuliéres et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble), 28 (1978), 177–202. https://doi.org/10.5802/aif.708 doi: 10.5802/aif.708
[10]	G. Lu, Multiple weighted estimates for bilinear Calderón-Zygmund operator and its commutator on non-homogeneous spaces, Bull. Sci. Math., 187 (2023), 103311. https://doi.org/10.1016/j.bulsci.2023.103311 doi: 10.1016/j.bulsci.2023.103311
[11]	S. He, J. Zhang, Endpoint estimates for multilinear fractional maximal operators, Bull. Korean Math. Soc., 57 (2020), 383–391. https://doi.org/10.4134/BKMS.b190269 doi: 10.4134/BKMS.b190269
[12]	A. Lerner, S. Ombrosi, C. Pérez, R. H. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math., 220 (2009), 1222–1264. https://doi.org/10.1016/j.aim.2008.10.014 doi: 10.1016/j.aim.2008.10.014
[13]	T. Iida, The boundedness of the Hardy-Littlewood maximal operator and multilinear maximal operator in weighted Morrey type spaces, J. Funct. Space. 2014 (2014), 648251. https://doi.org/10.1155/2014/648251 doi: 10.1155/2014/648251
[14]	L. Grafakos, R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 65 (2002), 124–164. https://doi.org/10.1006/aima.2001.2028 doi: 10.1006/aima.2001.2028
[15]	L. Grafakos, L. Liu, D. Maldonado, D. Yang, Multilinear analysis on metric spaces, Diss. Math., 497 (2014), 1–121. https://doi.org/10.4064/dm497-0-1 doi: 10.4064/dm497-0-1
[16]	L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49–56. https://doi.org/10.4064/sm-102-1-49-56 doi: 10.4064/sm-102-1-49-56
[17]	K. Li, K. Moen, W. Sun, Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals, Math. Nachr., 288 (2015), 619–632. https://doi.org/10.1002/mana.201300287 doi: 10.1002/mana.201300287
[18]	L. Grafakos, Classical Fourier Analysis, in Graduate Texts in Mathematics. 3 Eds., Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-1194-3
[19]	J. García-Cuerva, J. Rubio de Francia, Weighted norm inequalities and related topics, Elsevier, 1985.
[20]	L. Grafakos, Modern Fourier Analysis, in Graduate Texts in Mathematics. 3 Eds., Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-1230-8
[21]	J. M. Martell, C. Prisuelos-Arribas, Weighted Hardy spaces associated with elliptic operators Part: Ⅰ. Weighted norm inequalities for conical square functions, Trans. Am. Math. Soc., 369 (2017), 4193–4233. https://doi.org/10.1090/tran/6768 doi: 10.1090/tran/6768
[22]	K. Li, J. M. Martell, S. Ombrosi, Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform, Adv. Math., 373 (2020), 107286. https://doi.org/10.1016/j.aim.2020.107286 doi: 10.1016/j.aim.2020.107286
[23]	K. Moen, Weighted inequalities for multilinear fractional integral operators. Collect. Math., 60 (2009), 213–238. https://doi.org/10.1007/bf03191210 doi: 10.1007/bf03191210
[24]	D. Cruz-Uribe, J. M. Martell, Limited range multilinear extrapolation with applications to the bilinear Hilbert transform, Math. Ann., 371 (2018), 615–653. https://doi.org/10.1007/s00208-018-1640-9 doi: 10.1007/s00208-018-1640-9
[25]	M. Cao, A. Olivo, K. Yabuta, Extrapolation for multilinear compact operators and applications, Trans. Amer. Math. Soc., 375 (2022), 5011–5070. https://doi.org/10.1090/tran/8645 doi: 10.1090/tran/8645
[26]	P. Auscher, S. Hofmann, J. M. Martell, Vertical versus conical square functions, Trans. Am. Math. Soc., 364 (2012), 5469–5489. https://doi.org/10.1090/S0002-9947-2012-05668-6 doi: 10.1090/S0002-9947-2012-05668-6
[27]	A. Amenta, Interpolation and embeddings of weighted tent spaces, J. Fourier. Anal. Appl., 24 (2018), 108–140. https://doi.org/10.1007/s00041-017-9521-2 doi: 10.1007/s00041-017-9521-2
[28]	X. Chen, Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362 (2010), 355–373. https://doi.org/10.1016/j.jmaa.2009.08.022 doi: 10.1016/j.jmaa.2009.08.022

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