Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

Suixin He; Shuangping Tao; Suixin He; Shuangping Tao

doi:10.3934/math.2022060

AIMS Mathematics

2022, Volume 7, Issue 1: 1000-1014. doi: 10.3934/math.2022060

Previous Article Next Article

Research article

Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

Suixin He ,
Shuangping Tao ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 09 July 2021 Accepted: 11 October 2021 Published: 19 October 2021
MSC : 26A33, 42B20, 42B35

The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ over non-homogeneous spaces, where $ G\subset $ $ \mathbb{R}^{n} $ is a bounded domain. Under assumption that functions $ \varphi $ and $ \phi $ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $ \theta $-type Calderón-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $. Moreover, the boundedness of commutator $ [b, T^{G}_{\theta}] $ which is generated by $ \theta $-type Calderón-Zygmund operator $ T_{\theta} $ and $ b\in\mathrm{RBMO}(\mu) $ on spaces $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ is also established.
- non-doubling measure,
- maximal operator,
- fractional integral operator,
- $ \theta $-type Calderón-Zygmund operator,
- grand generalized Morrey space
Citation: Suixin He, Shuangping Tao. Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces[J]. AIMS Mathematics, 2022, 7(1): 1000-1014. doi: 10.3934/math.2022060

Related Papers:

Abstract

The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ over non-homogeneous spaces, where $ G\subset $ $ \mathbb{R}^{n} $ is a bounded domain. Under assumption that functions $ \varphi $ and $ \phi $ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $ \theta $-type Calderón-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $. Moreover, the boundedness of commutator $ [b, T^{G}_{\theta}] $ which is generated by $ \theta $-type Calderón-Zygmund operator $ T_{\theta} $ and $ b\in\mathrm{RBMO}(\mu) $ on spaces $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ is also established.

References

[1]	B. Bilalov, S. R. Sadigova, On solvability in the small of higher order elliptic equations in grand-Sobolev spaces, Complex Var. Elliptic, 2020. doi: 10.1080/17476933.2020.1807965. doi: 10.1080/17476933.2020.1807965
[2]	L. D'Onofrio, C. Sbordone, R. Schiattarella, Grand Sobolev spaces and their applications in geometric function theory and PDEs, J. Fixed Point Theory A., 13 (2013), 309–340. doi: 10.1007/s11784-013-0140-5. doi: 10.1007/s11784-013-0140-5
[3]	A. Fiorenza, M. Formica, J. Rakototson, Pointwise estimate for $G\Gamma$-functions and applications, Diff. Int. Equ., 30 (2017), 809–824.
[4]	A. Fiorenza, M. R. Formica, A. Gogatishvili, On Grand and small Lebesgue and Sobolev spaces and some applications to PDEs, Differ. Equ. Appl., 10 (2018), 21–46. doi: 10.7153/dea-2018-10-03. doi: 10.7153/dea-2018-10-03
[5]	X. Fu, G. Hu, D. Yang, A remark on the boundedness of Calderón-Zygmund operators in non-homogeneous spaces, Acta Math. Sin., 23 (2007), 449–456. doi: 10.1007/s10114-005-0723-1. doi: 10.1007/s10114-005-0723-1
[6]	N. Fusco, P. Lions, C. Sbordone, Sobolev imbedding theorems in borderline case, Proc. Amer. Math. Soc., 124 (1996), 561–565. doi: 10.1090/S0002-9939-96-03136-X. doi: 10.1090/S0002-9939-96-03136-X
[7]	S. He, Multi-Morrey spaces for non-doubling measures, Czech. Math. J., 69 (2019), 1039–1052. doi: 10.21136/CMJ.2019.0031-18. doi: 10.21136/CMJ.2019.0031-18
[8]	D. I. Hakim, M. Izuki, Y. Sawano, Complex interpolation of grand Lebesgue spaces, Monatsh. Math., 184 (2017), 245–272. doi: 10.1007/s00605-017-1022-5. doi: 10.1007/s00605-017-1022-5
[9]	T. Iwaniec, C. Sbordone, On the integrability of Jacobian under minimal hypotheses, Arch. Ration. Mech. Anal., 119 (1992), 129–143. doi: 10.1007/BF00375119. doi: 10.1007/BF00375119
[10]	L. Greco, T. Iwaniec, C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math., 92 (1997), 249–258. doi: 10.1007/BF02678192. doi: 10.1007/BF02678192
[11]	V. Kokilashvili, Boundedness criteria for singular integrals in weighted grand Lebesgue space, J. Math. Sci. (N.Y.), 170 (2010), 20–33. doi: 10.1007/s10958-010-0076-x. doi: 10.1007/s10958-010-0076-x
[12]	V. Kokilashvili, A. Meskhi, Boundedness of integral operators in generalized weighted grand Lebesgue space with non-doubling measures, Mediterr. J. Math., 18 (2021), 1–16. doi: 10.1007/s00009-020-01694-1. doi: 10.1007/s00009-020-01694-1
[13]	V. Kokilashvili, A. Meskhi, Fractional integrals with measure in grand Lebesgue and Morrey spaces, Integr. Transf. Spec. F., Published online, 2020. doi: 10.1080/10652469.2020.1833003.
[14]	V. Kokilashvili, A. Meskhi, Weighted Sobolev inequality in grand mixed norm Lebesgue spaces, Positivity, 25 (2021), 273–288. doi: 10.1007/s11117-020-00764-8. doi: 10.1007/s11117-020-00764-8
[15]	V. Kokilashvili, A. Meskhi, M. Ragusa, Weighted extrapolation in grand Morrey spaces and applications to partial differential equations, Rend. Lincei-Mat. Appl., 30 (2019), 67–92. doi: 10.4171/RLM/836. doi: 10.4171/RLM/836
[16]	V. Kokilashvili, A. Meskhi, M. Ragusa, Commutators of sublinear operators in grand morrey spaces, Stud. Sci. Math. Hung., 56 (2019), 211–232. doi: 10.1556/012.2019.56.2.1425. doi: 10.1556/012.2019.56.2.1425
[17]	G. Lu, Parameter marcinkiewicz integral on non-homogeneous morrey space with variable exponent, U. Politeh. Buch. Ser. A, 83 (2021), 89–98.
[18]	G. Lu, Commutators of bilinear $\theta$-type Calderón-Zygmund operators on Morrey spaces over non-homogeneous spaces, Anal. Math., 46 (2020), 97–118. doi: 10.1007/s10476-020-0020-3. doi: 10.1007/s10476-020-0020-3
[19]	G. Lu, L. Rui, $\theta$-Type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces, AIMS Math., 6 (2021), 9619–9632. doi: 10.3934/math.20210560. doi: 10.3934/math.20210560
[20]	A. Meskhi, Y. Sawano, Density, duality and preduality in grand variable exponent Lebesgue and Morrey spaces, Mediterr. J. Math., 15 (2018), 1–15. doi: 10.1007/s00009-018-1145-5. doi: 10.1007/s00009-018-1145-5
[21]	Y. Sawano, Generalized Morrey spaces for non-doubling measures, NODEA-Nonlinear Diff., 15 (2008), 413–425. doi: 10.1007/s00030-008-6032-5. doi: 10.1007/s00030-008-6032-5
[22]	X. Tolsa, BMO, $H^{1}$, and Calderón-Zygmund operators for non doubling measures, Math. Ann., 319 (2001), 89–149. doi: 10.1007/pl00004432. doi: 10.1007/pl00004432
[23]	X. Tolsa, A $T(1)$ theorem for non-doubling measures with atoms, P. Lond. Math. Soc., 82 (2001), 195–228. doi: 10.1112/S0024611500012703. doi: 10.1112/S0024611500012703
[24]	X. Tolsa, Painlevé's problem and the semiadditivity of analytic capacity, Acta Math., 190 (2003), 105–149. doi: 10.1007/BF02393237. doi: 10.1007/BF02393237
[25]	M. Wang, S. Ma, G. Lu, Littlewood-Paley $g^{\ast}_{\lambda, \mu}$-function and its commutator on non-homogeneous generalized Morrey spaces, Tokyo J. Math., 41 (2018), 617–626. doi: 10.3836/tjm/1502179247. doi: 10.3836/tjm/1502179247

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)