# Two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type

## Keywords:

Maximal operator, commutator, generalized weighted Morrey space, spaces of homogeneous type## Abstract

In this paper we give a characterization of two-weighted inequalities for maximal commutators in generalized weighted Morrey spaces on spaces of homogeneous type $\mathcal{M}_{\omega }^{p,\varphi }(X)$. We prove the boundedness of maximal commutators $[M,b]$ from the spaces $\mathcal{M}_{\omega _{1}^{\delta }}^{p,\varphi _{1}}(X)$ to the spaces $\mathcal{M}_{\omega _{2}^{\delta }}^{p,\varphi _{2}}(X)$, where $1<p<\infty $, $0<\delta <1$ and $(\omega _{1},\omega _{2})\in \widetilde{A}_{p}(X)$.

## 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.

M. Agcayazi, A. Gogatishvili and R. Mustafayev, Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function, Tokyo J. Math. 41 (2018), no. 1, 193-218.

C. Avcar, C. Aykol, J.J. Hasanov and A.M. Musayev, Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces $mathcal {GM}_omega^p,theta,varphi(mathbb R^n)$, Advanced Studies: Euro-Tbilisi Math. J. 16 (2023), no. 1, 33-50.

A. Aydogdu and C. Aykol, Two-weighted inequalities for maximal operator in generalized weighted Morrey spaces on spaces of homogeneous type, Baku Math. J. 2 (2023), no. 1, 113-121.

C. Aykol, H. Armutccu and M.N. Omarova, Maximal commutator and commutator of maximal function on modified Morrey spaces, Trans. Natl.

Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 36 (2016), no. 1, 29-35.

C. Aykol, X.A. Badalov and J.J. Hasanov, Maximal and singular operators in the local "complementary'' generalized variable

exponent Morrey spaces on unbounded sets, Quaest. Math. 43 (2020), no. 10, 1487-1512.

C. Aykol, J.J. Hasanov and Z.V. Safarov, Two weighted inequalities for Riesz potential and its commutator in generalized weighted Morrey spaces, Mat. Vesnik 75 (2023), no. 1, 37-49.

C. Aykol, J.J. Hasanov and Z.V. Safarov, A characterization of two-weighted inequalities for maximal, singular operators and their commutators in generalized weighteed Morrey spaces, Funct. Approx. Comment. Math. 67 (2022), no. 2, 145-167.

V. Burenkov, A. Gogatishvili, V.S. Guliyev and R. Mustafayev, Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 739-758.

R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certain espaces homogenes, in Lecture Notes in Math., No. 242, Springer-Verlag, Berlin, (1971).

D. Cruz-Uribe, New proofs of Two-weight norm inequalities for the maximal operator, Georgian Math. J. (2000), no. 7, 33-42.

D. Cruz-Uribe, Two weight norm inequalities for fractional integral operators and commutators, Advanced Courses of Mathematical Analysis VI, World Sci. Publ., Hackensack, NJ, (2017), 25-85.

G. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Bollettino U.M.I. 7 5-A (1991), 323-332.

D.E. Edmunds and V.M. Kokilashvili, Two weighted inequalities for singular integrals, Canad. Math. Bull. 38 (1995), no. 3, 295-303.

V.S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in $mathbb G$, (Russian) Doctor's degree dissertation, Moscow, Mat. Inst. Steklov, 1-329 (1994).

V.S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups, Some applications. (Russian) Baku, 1-332 (1999).

V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. Art. 1 (2009).

V.S. Guliyev, Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J. 3 (2012), no. 3, 33-61.

V.S. Guliyev, Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N. Y.) 193 (2013), no. 2, 211-227.

V.S. Guliyev, Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math. 3 (2013), no. 2, 79-94.

V.S. Guliyev, T. Karaman, R.Ch. Mustafayev and A. Serbetci, Commutators of sublinear operators generated by Caldero n-Zygmund operator on generalized weighted Morrey spaces, Czechoslovak Math. J. 60 (2014), no. 1, 365-386.

D.D. Haroske and L. Skrzypczak, Embeddings of weighted Morrey spaces, Math. Nachr. 290 (2017), no. 7, 1066-1086.

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 (2013), no. 3, 693--712.

K.P. Ho, Singular integral operators, John-Nirenberg inequalities and Tribel-Lizorkin type spaces on weighted

Lebesgue spaces with variable exponents, Revista De La Union Matematica Argentina 57 (2016), no. 1, 85-101.

T. Karaman, V.S. Guliyev and A. Serbetci, Boundedness of sublinear operators generated by Calderon-Zygmund operators on

generalized weighted Morrey spaces, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N. S.) 60 (2014),no. 1, 227-244.

V. Kokilashvili and A. Meskhi, Two-weight inequalities for fractional maximal functions and singular integrals in $L^p(cdot )$ spaces, J. Math. Sci. (N.Y.) 173 (2011), no. 6, 656-673.

Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2009), no. 2, 219-231.

M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure, Anal. PDE 5 (2012), no. 1, 1-60.

T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90

Satellite Proceedings, Springer - Verlag, Tokyo 183-189 (1991).

C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165(1972), 207-226.

E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95-103.

S. Nakamura, Y. Sawano, and H. Tanaka, The fractional operators on weighted Morrey spaces, J. Geom. Anal. 28 (2018), no. 2, 1502-1524.

C.J. Neugebauer, Inserting Ap-weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 644-648.

J. Pan and W. Sun, Two weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces, Math. Nachr. 293 (2020), no. 5, 970-982.

M.A. Ragusa, Regularity of solutions of divergence form elliptic equations, Proc. Amer. Math. Soc. 128 (2000), no. 1, 533-540.

N. Samko, Weighted Hardy and singular operators in Morrey spaces, J. Math. Anal. Appl. 350 (2009), no. 1, 56-72.

Y. Sawano, A thought on generalized Morrey spaces, J. Indonesian Math. Soc. 25 (2019), no. 3, 210-281 .

H. Tanaka, Two-weight norm inequalities on Morrey spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 773-791.

L.W. Wang, The commutators of multilinear maximal and fractional-type operators on central Morrey spaces with variable exponent, J. Funct. Spaces 2022, art.n. 4875460, (2022).

T.L. Yee, K.L. Cheung , K.P. Ho, Integral operators on local Orlicz-Morrey spaces, Filomat 36 (2022), no. 4, 1231-1243.

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*Electron. J. Appl. Math.*, vol. 1, no. 2, pp. 18–28, Sep. 2023.

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Copyright (c) 2023 Ayşenur Aydoğdu; Canay Aykol (Corresponding Author); Javanshir J. Hasanov

This work is licensed under a Creative Commons Attribution 4.0 International License.

Accepted 2023 Aug 10

Published 2023 Sep 09