On Rough maximal inequalities: an extension of
Fefferman–Stein results

D. Salim; W. S. Budhi; Y. Soeharyadi

doi:10.30970/ms.52.2.185-194

We prove some vector-valued inequalities for a rough maximal operator on Lebesgue spaces. These results are an extension of Fefferman--Stein (1971) and Sawano (2006) since the rough maximal operator is a generalization of the Hardy--Littlewood maximal operator and also a fractional maximal operator, respectively. We also establish some vector-valued inequalities for a rough maximal operator on Morrey spaces.

1. Chanillo S., Watson D., Wheeden R., Some integral and maximal operators related to starlike sets, Studia Math., 3 (1993), №107, 223-255.

2. Chiarenza F., Frasca M., Morrey spaces and Hardy.Littlewood maximal function, Rend. Mat., 7 (1987), 273-279.

3. Ding Y., Lai X., Weak type (1,1) behavior for the maximal operator with $L^1$-Dini kernel, Potential Anal., 47 (2017), №2, 169-187.

4. Duoandikoetxea J., Fourier analysis. American Mathematical Soc V.29, 2001.

5. Fefferman C., Stein E.M., Some maximal inequalities, Am. J. Math., 93 (1971), №1, 107-115.

6. Grafakos L., Classical fourier analysis, V.2, Springer, New York, 2008.

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

8. Iida T., Weighted inequalities on Morrey spaces for linear and multilinear fractional integrals with homogeneous kernels, Taiwan J. Math., 18 (2014), №1, 147-185.

9. Morrey C.B., Functions of several variables and absolute continuity, Duke Math. J., 6 (1940), №1, 187-215.

10. Muckenhoupt B., Wheeden R.L., Weighted norm inequalities for singular and fractional integrals, T. Am. Math. Soc., 161 (1971), 249-258.

11. Salim D., Soeharyadi Y., Budhi W.S., Rough fractional integral operators and beyond Adams inequalities, Math. Inequal. Appl., 22 (2019), №2, 747-760.

12. Sawano Y., .q-valued extension of the fractional maximal operators for non-doubling measures via potential operators, Int. J. Pure Appl. Math., 26 (2006), №4, 505-522.

13. Stein E.M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, V.43, Princenton, University Press, Princeton, NJ, 1993.

	On Rough maximal inequalities: an extension of Fefferman–Stein results
Author	D. Salim¹, W. S. Budhi², Y. Soeharyadi³ daniel.salim@sci.ui.ac.id¹, wono@math.itb.ac.id², yudish@math.itb.ac.id³ Analysis and Geometry Research Division, Bandung Institute of Technology, Indonesia
Abstract	We prove some vector-valued inequalities for a rough maximal operator on Lebesgue spaces. These results are an extension of Fefferman--Stein (1971) and Sawano (2006) since the rough maximal operator is a generalization of the Hardy--Littlewood maximal operator and also a fractional maximal operator, respectively. We also establish some vector-valued inequalities for a rough maximal operator on Morrey spaces.
Keywords	Rough maximal operator; vector-valued inequality; Fefferman–Stein maximal inequality; Morrey space
DOI	doi:10.30970/ms.52.2.185-194
Reference	1. Chanillo S., Watson D., Wheeden R., Some integral and maximal operators related to starlike sets, Studia Math., 3 (1993), №107, 223-255. 2. Chiarenza F., Frasca M., Morrey spaces and Hardy.Littlewood maximal function, Rend. Mat., 7 (1987), 273-279. 3. Ding Y., Lai X., Weak type (1,1) behavior for the maximal operator with $L^1$-Dini kernel, Potential Anal., 47 (2017), №2, 169-187. 4. Duoandikoetxea J., Fourier analysis. American Mathematical Soc V.29, 2001. 5. Fefferman C., Stein E.M., Some maximal inequalities, Am. J. Math., 93 (1971), №1, 107-115. 6. Grafakos L., Classical fourier analysis, V.2, Springer, New York, 2008. 7. Guliyev, Vagif S., Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N. Y.), 193 (2013), №2, 211-227. 8. Iida T., Weighted inequalities on Morrey spaces for linear and multilinear fractional integrals with homogeneous kernels, Taiwan J. Math., 18 (2014), №1, 147-185. 9. Morrey C.B., Functions of several variables and absolute continuity, Duke Math. J., 6 (1940), №1, 187-215. 10. Muckenhoupt B., Wheeden R.L., Weighted norm inequalities for singular and fractional integrals, T. Am. Math. Soc., 161 (1971), 249-258. 11. Salim D., Soeharyadi Y., Budhi W.S., Rough fractional integral operators and beyond Adams inequalities, Math. Inequal. Appl., 22 (2019), №2, 747-760. 12. Sawano Y., .q-valued extension of the fractional maximal operators for non-doubling measures via potential operators, Int. J. Pure Appl. Math., 26 (2006), №4, 505-522. 13. Stein E.M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, V.43, Princenton, University Press, Princeton, NJ, 1993.
Pages	185-194
Volume	52
Issue	2
Year	2019
Journal	Matematychni Studii
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Table of content of issue	html

On Rough maximal inequalities: an extension of Fefferman–Stein results