## Tokyo Journal of Mathematics

### Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function

#### Abstract

In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),\lambda}$, $0 < \lambda < n$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\lambda}$ for radially decreasing functions. The boundedness of the iterated maximal operator $M^2$ from $\mathcal{M}_{L(\log L),\lambda}$ to weak Zygmund-Morrey space $\mathcal{W \! M}_{L(\log L),\lambda}$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(\log L),\lambda}$ to $\mathcal{W \! M}_{L(\log L),\lambda}$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b \in \text{BMO}(\mathbb{R}^n)$ such that $b^- \in L_{\infty}(\mathbb{R}^n)$ is bounded from $\mathcal{M}_{L(\log L),\lambda}$ to $\mathcal{W \! M}_{L(\log L),\lambda}$. New pointwise characterizations of $M_{\alpha} M$ by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.

#### Article information

Source
Tokyo J. Math., Volume 41, Number 1 (2018), 193-218.

Dates
First available in Project Euclid: 18 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1513566018

Mathematical Reviews number (MathSciNet)
MR3830814

Zentralblatt MATH identifier
06966864

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis

#### Citation

GOGATISHVILI, Amiran; MUSTAFAYEV, Rza; AǦCAYAZI, Müjdat. Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function. Tokyo J. Math. 41 (2018), no. 1, 193--218. https://projecteuclid.org/euclid.tjm/1513566018

#### References

• M. Agcayazi, A. Gogatishvili, K. Koca and R. Mustafayev, A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Japan. 67 (2015), no. 2, 581–593.
• J. Bastero, M. Milman and F. J. Ruiz, Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3329–3334 (electronic). MR1777580 (2001i:42027)
• C. Bennett and R. Sharpley, Weak-type inequalities for (H^p) and BMO, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 201–229. MR545259 (80j:46044)
• C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. xiv+469. 0-12-088730-4, MR928802 (89e:46001)
• A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in BMO and $H^1$, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405–1439. MR2364134 (2009d:42054)
• M. Carozza and A. Passarelli Di Napoli, Composition of maximal operators, Publ. Mat. 40 (1996), no. 2, 397–409. MR1425627 (98f:42013)
• F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), no. 3–4, 273–279 (1988). MR985999 (90f:42017)
• J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104.
• J. Garcia-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397–1420. MR1142721 (93f:42031)
• D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983.
• A. Gogatishvili and R. Ch. Mustafayev, A note on boundedness of the Hardy-Littlewood maximal operator on Morrey spaces, Mediterr. J. Math. 13 (2016), no. 4, 1885–1891.
• L. Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR2445437 (2011c:42001)
• L. Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR2463316 (2011d:42001)
• M. de Guzmán, Differentiation of integrals in (R^n), Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón.
• G. Hu, H. Lin and D. Yang, Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal. (2008), Art. ID 237937, 21. MR2393116 (2009d:42046)
• G. Hu and 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), no. 1, 249–262. MR2510436 (2010c:43018)
• T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR1859913
• F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR0131498 (24 #A1348)
• H. Kita, On maximal functions in Orlicz spaces, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3019–3025. MR1376993 (97b:42031)
• M. A. Leckband and C. J. Neugebauer, A general maximal operator and the (A_p)-condition, Trans. Amer. Math. Soc. 275 (1983), no. 2, 821–831, DOI 10.2307/1999056. MR682735 (84c:42029)
• M. A. Leckband, A note on maximal operators and reversible weak type inequalities, Proc. Amer. Math. Soc. 92 (1984), no. 1, 19–26, DOI 10.2307/2045145. MR749882 (86i:42009)
• D. Li, G. Hu and X. Shi, Weighted norm inequalities for the maximal commutators of singular integral operators, J. Math. Anal. Appl. 319 (2006), no. 2, 509–521. MR2227920 (2007a:42041)
• M. Milman and T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc. 110 (1990), no. 4, 961–969. MR1075187 (91k:46088)
• C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166, DOI 10.2307/1989904. MR1501936
• C. Perez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163–185. MR1317714 (95j:42011)
• M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. MR1113700 (92e:46059)
• Y. Sawano, S. Sugano and H. Tanaka, Orlicz-Morrey spaces and fractional operators, Potential Anal. 36 (2012), no. 4, 517–556, DOI 10.1007/s11118-011-9239-8. MR2904632
• C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat. 35 (1991), no. 1, 209–235. Conference on Mathematical Analysis (El Escorial, 1989). MR1103616 (93f:42035)
• C. Segovia and J. L. Torrea, Higher order commutators for vector-valued Calderón-Zygmund operators, Trans. Amer. Math. Soc. 336 (1993), no. 2, 537–556. MR1074151 (93f:42036)
• E. M. Stein, Note on the class $L$ log, Studia Math. 32 (1969), 305–310. MR0247534 (40 #799)
• E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR0290095 (44 #7280)
• E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192 (95c:42002)
• A. Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR869816 (88e:42001)
• C. P. Xie, Some estimates of commutators, Real Anal. Exchange 36 (2010/11), no. 2, 405–415. MR3016724