Startseite The L p -L q compactness of commutators of oscillatory singular integrals
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The L p -L q compactness of commutators of oscillatory singular integrals

  • Wenchang Sun und Shifen Wang EMAIL logo
Veröffentlicht/Copyright: 14. November 2024

Abstract

We study the compactness of commutator of a locally integrable function and an oscillatory singular integral operator defined by a real-valued polynomial and a kernel function satisfying the Hölder condition. We show that such a commutator is compact from L p to L q if and only if the locally integrable function is the sum of a constant and a function in certain Lebesgue space.

MSC 2020: 42B20; 42B35

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12171250

Award Identifier / Grant number: U21A20426

Award Identifier / Grant number: 12271267

Funding statement: This work was partially supported by the National Natural Science Foundation of China (12171250, U21A20426 and 12271267).

References

[1] E. Airta, T. Hytönen, K. Li, H. Martikainen and T. Oikari, Off-diagonal estimates for bi-commutators, Int. Math. Res. Not. IMRN 2022 (2022), no. 23, 18766–18832. 10.1093/imrn/rnab239Suche in Google Scholar

[2] H. Al-Qassem, L. Cheng and Y. Pan, Oscillatory singular integral operators with Hölder class kernels, J. Fourier Anal. Appl. 25 (2019), no. 4, 2141–2149. 10.1007/s00041-018-09660-ySuche in Google Scholar

[3] H. Al-Qassem, L. Cheng and Y. Pan, A van der Corput type lemma for oscillatory integrals with Hölder amplitudes and its applications, J. Korean Math. Soc. 58 (2021), no. 2, 487–499. Suche in Google Scholar

[4] F. Beatrous and S.-Y. Li, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (1993), no. 2, 350–379. 10.1006/jfan.1993.1017Suche in Google Scholar

[5] A. Bényi, W. Damián, K. Moen and R. H. Torres, Compactness properties of commutators of bilinear fractional integrals, Math. Z. 280 (2015), no. 1–2, 569–582. 10.1007/s00209-015-1437-4Suche in Google Scholar

[6] M. Cao, A. Olivo and K. Yabuta, Extrapolation for multilinear compact operators and applications, Trans. Amer. Math. Soc. 375 (2022), no. 7, 5011–5070. 10.1090/tran/8645Suche in Google Scholar

[7] P. Chen, X. T. Duong, J. Li and Q. Wu, Compactness of Riesz transform commutator on stratified Lie groups, J. Funct. Anal. 277 (2019), no. 6, 1639–1676. 10.1016/j.jfa.2019.05.008Suche in Google Scholar

[8] Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood-Paley operators, Kodai Math. J. 32 (2009), no. 2, 256–323. 10.2996/kmj/1245982907Suche in Google Scholar

[9] Y. Chen, Y. Ding and X. Wang, Compactness for commutators of Marcinkiewicz integrals in Morrey spaces, Taiwanese J. Math. 15 (2011), no. 2, 633–658. 10.11650/twjm/1500406226Suche in Google Scholar

[10] Y. P. Chen and Y. Ding, Compactness of commutators of singular integral operations with variable kernel, Chinese Ann. Math. Ser. A 30 (2009), no. 2, 201–212. Suche in Google Scholar

[11] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. 10.2307/1970954Suche in Google Scholar

[12] Y. Ding and S. Lu, Weighted L p -boundedness for higher order commutators of oscillatory singular integrals, Tohoku Math. J. (2) 48 (1996), no. 3, 437–449. 10.2748/tmj/1178225342Suche in Google Scholar

[13] Y. Ding, T. Mei and Q. Xue, Compactness of maximal commutators of bilinear Calderón–Zygmund singular integral operators, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville (2016), 163–175. Suche in Google Scholar

[14] Z. Fu, S. Lu, Y. Pan and S. Shi, Some one-sided estimates for oscillatory singular integrals, Nonlinear Anal. 108 (2014), 144–160. 10.1016/j.na.2014.05.016Suche in Google Scholar

[15] R. Gong, M. N. Vempati, Q. Wu and P. Xie, Boundedness and compactness of Cauchy-type integral commutator on weighted Morrey spaces, J. Aust. Math. Soc. 113 (2022), no. 1, 36–56. 10.1017/S1446788722000015Suche in Google Scholar

[16] T. Hytönen, The L p -to- L q boundedness of commutators with applications to the Jacobian operator, J. Math. Pures Appl. (9) 156 (2021), 351–391. 10.1016/j.matpur.2021.10.007Suche in Google Scholar

[17] T. Hytönen and S. Lappas, Extrapolation of compactness on weighted spaces, Rev. Mat. Iberoam. 39 (2023), no. 1, 91–122. 10.4171/rmi/1325Suche in Google Scholar

[18] T. Hytönen, K. Li, J. Tao and D. Yang, The L p -to- L q compactness of commutators with p > q , Studia Math. 271 (2023), no. 1, 85–105. 10.4064/sm220910-10-1Suche in Google Scholar

[19] T. Hytönen, T. Oikari and J. Sinko, Fractional Bloom boundedness and compactness of commutators, Forum Math. 35 (2023), no. 3, 809–830. 10.1515/forum-2022-0252Suche in Google Scholar

[20] M. Lacey and J. Li, Compactness of commutator of Riesz transforms in the two weight setting, J. Math. Anal. Appl. 508 (2022), no. 1, Paper No. 125869. 10.1016/j.jmaa.2021.125869Suche in Google Scholar

[21] A. K. Lerner and F. Nazarov, Intuitive dyadic calculus: The basics, Expo. Math. 37 (2019), no. 3, 225–265. 10.1016/j.exmath.2018.01.001Suche in Google Scholar

[22] F. Liu, S. Wang and Q. Xue, On oscillatory singular integrals and their commutators with non-convolutional Hölder class kernels, Banach J. Math. Anal. 15 (2021), no. 3, Paper No. 51. 10.1007/s43037-021-00138-6Suche in Google Scholar

[23] S. Lu and H. Wu, Oscillatory singular integrals and commutators with rough kernels, Ann. Sci. Math. Québec 27 (2003), no. 1, 47–66. Suche in Google Scholar

[24] S. Lu and Y. Zhang, Weighted norm inequality of a class of oscillatory integral operators, Chinese Sci. Bull. 37 (1992), 9–13. Suche in Google Scholar

[25] Y. Pan, Hardy spaces and oscillatory singular integrals, Rev. Mat. Iberoam. 7 (1991), no. 1, 55–64. 10.4171/rmi/105Suche in Google Scholar

[26] Y. Pan, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100 (1991), no. 1, 207–220. 10.1016/0022-1236(91)90108-HSuche in Google Scholar

[27] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), no. 1–2, 99–157. 10.1007/BF02392592Suche in Google Scholar

[28] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal. 73 (1987), no. 1, 179–194. 10.1016/0022-1236(87)90064-4Suche in Google Scholar

[29] S. Shi, Z. Fu and S. Lu, On the compactness of commutators of Hardy operators, Pacific J. Math. 307 (2020), no. 1, 239–256. 10.2140/pjm.2020.307.239Suche in Google Scholar

[30] S. Shi and L. Zhang, Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals, J. Inequal. Appl. 2016 (2016), Paper No. 88. 10.1186/s13660-016-1025-0Suche in Google Scholar

[31] J. Tao, Q. Xue, D. Yang and W. Yuan, XMO and weighted compact bilinear commutators, J. Fourier Anal. Appl. 27 (2021), no. 3, Paper No. 60. 10.1007/s00041-021-09854-xSuche in Google Scholar

[32] R. Torres and Q. Xue, On compactness of commutators of multiplication and bilinear pseudodifferential operators and a new subspace of BMO, Rev. Mat. Iberoam. 36 (2020), no. 3, 939–956. 10.4171/rmi/1156Suche in Google Scholar

[33] A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), no. 1, 163–171. 10.2748/tmj/1178230105Suche in Google Scholar

[34] S. Wang, Compactness of commutators of fractional integrals, Chinese Ann. Math. Ser. A 8 (1987), no. 4, 475–482. Suche in Google Scholar

[35] S. Wang and Q. Xue, On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators, Forum Math. 34 (2022), no. 2, 307–322. 10.1515/forum-2020-0357Suche in Google Scholar

[36] Q. Xue, K. Yabuta and J. Yan, Weighted Fréchet-Kolmogorov theorem and compactness of vector-valued multilinear operators, J. Geom. Anal. 31 (2021), no. 10, 9891–9914. 10.1007/s12220-021-00630-3Suche in Google Scholar

[37] K. Yosida, Functional Analysis, Class. Math., Springer, Berlin, 1995. 10.1007/978-3-642-61859-8Suche in Google Scholar

Received: 2024-07-07
Revised: 2024-10-04
Published Online: 2024-11-14
Published in Print: 2025-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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