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Weak type estimates of genuine Calderón–Zygmund operators on the local Morrey spaces associated with ball quasi-Banach function spaces

  • Mingwei Shi , Jiang Zhou EMAIL logo and Songbai Wang
Published/Copyright: March 26, 2024
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Abstract

Weak type estimates for genuine Calderón–Zygmund operators are established on the local Morrey spaces associated with ball quasi-Banach function spaces by two different methods. Above all, we obtain weak type estimates for the operator on the local weak Morrey spaces with variable exponents.

Award Identifier / Grant number: 12061069

Funding statement: The research was supported by National Natural Science Foundation of China (12061069) and the Natural Science Foundation Project of Chongqing, China (Grant No. cstc2021jcyj-msxmX0705).

Acknowledgements

The authors would like to thank the reviewers for their valuable advice on a previous version of the article.

References

[1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. Search in Google Scholar

[2] V. I. Burenkov and H. V. Guliyev, Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia Math. 163 (2004), no. 2, 157–176. 10.4064/sm163-2-4Search in Google Scholar

[3] V. I. Burenkov, H. V. Guliyev and V. S. Guliyev, Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces, J. Comput. Appl. Math. 208 (2007), no. 1, 280–301. 10.1016/j.cam.2006.10.085Search in Google Scholar

[4] V. I. Burenkov and V. S. Guliyev, Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 30 (2009), no. 3, 211–249. 10.1007/s11118-008-9113-5Search in Google Scholar

[5] V. I. Burenkov, V. S. Guliyev, A. Serbetci and T. V. Tararykova, Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces, Eurasian Math. J. 1 (2010), no. 1, 32–53. Search in Google Scholar

[6] B. Cekic, R. Mashiyev and G. T. Alisoy, On the Sobolev-type inequality for Lebesgue spaces with a variable exponent, Int. Math. Forum 1 (2006), no. 25–28, 1313–1323. 10.12988/imf.2006.06108Search in Google Scholar

[7] D. Cruz-Uribe, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable L p spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264. Search in Google Scholar

[8] A. Gogatishvili and R. Mustafayev, Dual spaces of local Morrey-type spaces, Czechoslovak Math. J. 61(136) (2011), no. 3, 609–622. 10.1007/s10587-011-0034-xSearch in Google Scholar

[9] C.-x. Miao and B.-q. Yuan, Weak Morrey spaces and strong solutions to the Navier–Stokes equations, Sci. China Ser. A 50 (2007), no. 10, 1401–1417. 10.1007/s11425-007-0101-9Search in Google Scholar

[10] Y. Sawano and S. R. El-Shabrawy, Weak Morrey spaces with applications, Math. Nachr. 291 (2018), no. 1, 178–186. 10.1002/mana.201700001Search in Google Scholar

[11] Y. Sawano, K.-P. Ho, D. Yang and S. Yang, Hardy spaces for ball quasi-Banach function spaces, Diss. Math. 525 (2017), Paper No. 102. 10.4064/dm750-9-2016Search in Google Scholar

[12] M. Shi and J. Zhou, The local Morrey-type space associated with ball quasi-banach function spaces and application, preprint (2022), https://arxiv.org/abs/2209.03861. Search in Google Scholar

[13] M. Shi and J. Zhou, Some estimates of operators on Local mixed Morrey-type spaces, Indian J. Pure Appl. Math. (2023), 10.1007/s13226-023-00432-z. 10.1007/s13226-023-00432-zSearch in Google Scholar

[14] J. Sun, D. Yang and W. Yuan, Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations, Forum Math. 34 (2022), no. 6, 1539–1589. 10.1515/forum-2022-0074Search in Google Scholar

[15] J. Sun, D. Yang and W. Yuan, Weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: decompositions, real interpolation, and Calderón–Zygmund operators, J. Geom. Anal. 32 (2022), no. 7, Paper No. 191. 10.1007/s12220-022-00927-xSearch in Google Scholar

[16] S. Wang, D. Yang, W. Yuan and Y. Zhang, Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood–Paley characterizations and real interpolation, J. Geom. Anal. 31 (2021), no. 1, 631–696. 10.1007/s12220-019-00293-1Search in Google Scholar

[17] S. Wang and J. Zhou, Another proof of the boundedness of Calderón–Zygmund singular integrals on generalized Orlicz spaces, Bull. Sci. Math. 179 (2022), Article ID 103176. 10.1016/j.bulsci.2022.103176Search in Google Scholar

[18] M. Wei, Linear operators and their commutators generated by Calderón–Zygmund operators on generalized Morrey spaces associated with ball Banach function spaces, Positivity 26 (2022), no. 5, Paper No. 84. 10.1007/s11117-022-00949-3Search in Google Scholar

[19] M. Wei and D. Yan, Operators on Herz-type spaces associated with ball quasi-Banach function spaces, preprint (2022), https://arxiv.org/abs/2209.04323. Search in Google Scholar

[20] X. J. Yan, Z. Y. He, D. C. Yang and W. Yuan, Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Littlewood–Paley characterizations with applications to boundedness of Calderón–Zygmund operators, Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 7, 1133–1184. 10.1007/s10114-022-1573-9Search in Google Scholar

[21] Y. Zhang, D. Yang, W. Yuan and S. Wang, Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón–Zygmund operators, Sci. China Math. 64 (2021), no. 9, 2007–2064. 10.1007/s11425-019-1645-1Search in Google Scholar

Received: 2023-05-27
Revised: 2023-12-04
Accepted: 2023-12-06
Published Online: 2024-03-26
Published in Print: 2024-12-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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