Bilinear Calderón-Zygmund operator and its commutator on some variable exponent spaces of homogeneous type

Abstract

Let $(X,d,\mu)$ be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that a bilinear Calderon-Zygmund operator is bounded from the product of variable exponent Lebesgue spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is bounded from the product of variable exponent generalized Morrey spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $\mathcal{L}^{p(\cdot),\varphi}(X)$, where the Lebesgue measure functions $\varphi(\cdot,\cdot), \varphi_{1}(\cdot,\cdot)$ and $\varphi_{2}(\cdot,\cdot)$ satisfy $\varphi_{1}\times\varphi_{2}=\varphi$, and $\frac{1}{p(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Furthermore, by establishing sharp maximal estimate for the commutator $[b_{1},b_{2},BT]$ generated by $b_{1}, b_{2}\in\mathrm{BMO}(X)$ and $BT$, the author shows that the $[b_{1},b_{2},BT]$ is bounded from the product of spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is also bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $L^{p(\cdot),\varphi}(X)$.

Keywords

• space of homogeneous type
• bilinear Calderón-Zygmund operator
• commutator
• space BMO(X)
• variable exponent generalized Morrey space

References

1. [1] R.R. Coifman and G.Weiss, Analyse Harmonique Non-commutative sur certain Espaces Homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin-New York, 1971.
2. [2] R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (2), 611-635, 1976.
3. [3] D. Cruz-Uribe and J. Cummings, Weighted norm inequalities for the maximal operator on $L^{p(\cdot)}$ over spaces of homogeneous type, Ann. Fenn. Math. 47 (1), 457-488, 2022.
4. [4] F. Deringoz, K. Dorak and V.S. Guliyev, Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz-Morrey spaces on spaces of homogeneous type, Anal. Math. Phys. 11 (2), 1-30, 2021.
5. [5] I.A. Fernandes and S.A. Tozoni, Weighted norm inequality for a maximal operator on homogeneous space, Z. Anal. Anwend. 27 (1), 67-78, 2008.
6. [6] J. García-Cuerva and A. Gatto, Lipschitz spaces and Calderón-Zygmund operators associated to nondoubling measures, Publ. Mat. 49 (2), 285-296, 2005.
7. [7] R. Gong, J. Li, E. Pozzi and M.N. Vempati, Commutators on weighted Morrey spaces on spaces of homogeneous type, Anal. Geom. Metr. Spaces 8 (1), 305-334, 2020.
8. [8] L. Grafakos, G. Liu, D. Maldonado and D. Yang, Multilinear analysis on metric spaces, Dissertationes Math., 497, 1-121, 2014.

9. [9] L. Grafakos and R.H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (1), 124-164, 2002.
10. [10] V.S. Guliyev and S.G. Samko, Maximal operator in variable exponent generalized Morrey spaces on quasi-metric measure space, Mediterr. J. Math. 13 (3), 1151-1165, 2- 016.
11. [11] M.G. Hajibayov and S. Samko, Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces, Math. Nachr. 284 (1), 53-66, 2011.
12. [12] T. Heikkinen, J. Kinnunen, J. Nuutinen and H. Tuominen, Mapping properties of the discrete fractional maximal operator in metric measure spaces, Kyoto J. Math. 53 (3 ), 693-712, 2013.
13. [13] G. Hu, Y. Meng and D. Yang, Multilinear commutators of singular integrals with non doubling measures, Integral Equations Operator Theory 51 (2), 235-255, 2005.
14. [14] G. Hu, Y. Meng and D. Yang, Weighted norm inequalities for multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces, Forum Math. 26 (5), 1289-1322, 2014.
15. [15] G. Hu, X. Shi and Q. Zhang, Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type, J. Math. Anal. Appl. 336 (1), 1-17, 2007.
16. [16] G. Hu, D. Yang and Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math. 13 (1), 91-135, 2009.
17. [17] A. Huang and J. Xu, Multilinear singular integrals and commutators in variable exponent Lebesgue spaces, Appl. Math. J. Chinese Univ. Ser. B 25 (1), 69-77, 2010.
18. [18] T. Hytönen, S. Liu, Da. Yang and Do. Yang, Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces, Canad. J. Math. 64 (4), 892-923, 2012.
19. [19] M. Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend. Circ. Mat. Palermo 59 (2), 199-213, 2010.
20. [20] M. Kahbazi, The maximal operator in spaces of homogeneous type, Proc. A. Razmadze Math. Inst., 138: 17-25, 2005.
21. [21] V. Kokilashvili and A. Meskhi, Maximal and singualr integral operators in weighted grand variable exponent Lebesgue spaces, Ann. Funct. Anal. 12 (3), 1-29, 2021.
22. [22] H. Li, Multilinear Calderón-Zygmund operator on space of homogeneous type, Master’s Degree Dissertion, Mathematical Institute, Shijiazhuang, Hebei, 2002.
23. [23] W. Li, Q. Xue and K. Yabuta, Multilinear Calderón-Zygmund operators on weighted Hardy spaces, Studia Math. 199 (1), 1-16, 2010.
24. [24] G. Lu, Bilinear $\theta$-type Calderón-Zygmund operator and its commutator on nonhommogeneous weighted Morrey spaces, RACSAM, 115 (1), 1-15, Paper No. 16, 2021.
25. [25] G. Lu and S. Tao, Estimate for bilinear Calderón-Zygmund operator and its commutator on product of variable exponent spaces, Bull. Korean Math. Soc. 59 (6), 1471-1493, 2022.
26. [26] Y. Meyer and R. Coifman, Wavelets: Calderón-Zygmund and multilinear operators. Translated from the 1990 and 1991 French originals by David Salinger. Cambridge Studies in Advanced Math., 48. Cambridge University Press, Cambridge, 1997. xx+315 pp.
27. [27] M. Mirek and C. Thiele, A local $T(b)$ theorem for perfect multilinear Calderón-Zygmund operators, Proc. Lond. Math. Soc. 114 (1), 35-59, 2017.
28. [28] H. Mo, Boundedness of the multilinear Calderón-Zygmund singular integral maximal operator on spaces of homogeneous type, Beijing Shifan Daxue Xuebao 41 (1), 14-17, 2005.
29. [29] E. Nakai, On genealized fractioanl integral in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn. 54 (3), 473-487, 2001.
30. [30] G. Pradolini and O. Salinas, Commutators of singular integrals on spaces of homogeneous type, Czechoslovak Math. J. 57 (1), 75-93, 2007.
31. [31] C. Pérez and R.H. Torres, Sharp maximal function estimates for multilinear singular integrals, Harmonic Analysis at Mount Holyoke, Contemp Math, 320, 323-331, 2003.
32. [32] M. Rosenthal and H. Triebel, Calderón-Zygmund operators in Morrey spaces, Rev. Mat. Complut., 27 (1): 1-11, 2014.
33. [33] J. Tan, Bilinear Calderón-Zygmund operators on products of variable Hardy spaces, Forum Math. 31 (1), 187-198, 2019.
34. [34] L.Wang and L. Shu, Multilinear commutators of singular integral operators in variable exponent Herz-type spaces, Bull. Malays. Math. Sci. Soc. 42 (4), 1413-1432, 2019.
35. [35] W. Wang and J. Xu, Multilinear Calderón-Zygmund operators and their commutators with $\mathrm{BMO}$ functions in variable exponent Morrey spaces, Front. Math. China 12 (5), 1235-1246, 2017.
36. [36] P. Zhang and J. Sun, Commutators of multilinear Calderón-Zygmund operators with kernels of Dini’s type and applications, J. Math. Inequal. 13 (4), 1071-1093, 2019.