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NORM INEQUALITY FOR INTRINSIC SQUARE FUNCTION AND ITS COMMUTATOR IN A GENERALIZED HARDY SPACES WITH VARIABLE EXPONENT $\mathcal{H}^{p(\cdot), q}\left(\mathbb{R}^d\right)$
Keywords:
amalgam spaces, Hardy-amalgam spaces, variable exponents spaces, intrinsic square functions, commutators, BMO functionsDOI:
https://doi.org/10.17654/0972087126014Abstract
In this paper, we carry on with the study of the variable exponent Hardy-amalgam spaces introduced in [26]. Precisely, we investigate some boundedness properties of the intrinsic square function and its commutator generated by BMO functions in the above mentioned spaces via their atomic decomposition. Furthermore, our theorems extend some well-known existing results.
Received: July 22, 2025
Accepted: September 8, 2025
References
[1] A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226(5) (2011), 3912-3926.
[2] C. Fefferman and E. M. Stein, spaces of several variables, Acta Math. 129(3-4) (1972), 137-193.
[3] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.
[4] C. Zhuo, D. Yang and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bulletin of the Malaysian Mathematical Sciences Society 39 (2016), 1541-1577.
[5] M. A. Dakoury and J. Feuto, Norm inequality for intrinsic square functions in a generalized Hardy-Morrey space, Open Access Library Journal 9 (2022), e8463.
[6] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhauser, Basel, 2013.
[7] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces and Hyperbolic Systems, Birkhauser, 2014.
[8] D. Cruz-Uribe, A. Fiorenza, J. M. Martell and C. Perez, The boundedness of classical operators on variable spaces, Ann. Acad. Sci. Fenn. Math. 31(1) (2006), 239-264.
[9] D. Cruz-Uribe, J. M. Martell and C. Perez, Weights, Extrapolation and the Theory of Rubio de Francia, Birkhauser, Basel, 2011.
[10] D. Cruz-Uribe and L. A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63(2) (2014), 447-493.
[11] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), 3665-3748.
[12] F. Holland, Harmonic analysis on amalgams of and J. London Math. Soc. 2(10) (1975), 295-305.
[13] J. Huang and Y. Liu, Some characterizations of weighted Hardy spaces, J. Math. Anal. Appl. 363 (2010), 121-127.
[14] I. Aydin, On vector-valued classical and variable exponent amalgam spaces, Commun. Fac. Sci. Univ. Ank. Series A1 66(2) (2017), 100.
[15] I. Aydin and C. Unal, Amalgam spaces with variable exponent, Conference Proceeding of Science and Technology 2(1) (2019), 22-26.
[16] J. Garcia-Cuerva, Weighted spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63.
[17] K. P. Ho, Atomic decomposition of Hardy-Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), 31-62.
[18] K. P. Ho, Sublinear operators on weighted Hardy spaces with variable exponents, Forum Math. 31(3) (2019), 607-617.
[19] K. Saibi, Intrinsic square function characterizations of variable Hardy-Lorentz spaces, Journal of Function spaces (2020), Art. ID 2681719, 9 pp.
DOI :10.1155/2020/2681719.
[20] L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator theory 78(1) (2014), 115-150.
[21] L. Diening, Maximal function on generalized Lebesgue spaces Math. Inequal. Appl. 7(2) (2004), 245-253.
[22] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011.
[23] L. Grafakos, Modern Fourier Analysis, 2nd ed., Springer, New York, 2009 .
[24] L. Traore, Hardy-amalgam spaces with variable exponent and their duals, Universal Journal of Mathematics and Mathematical Sciences 16 (2022), 1-20.
[25] L. Traore and C. Unal, On the boundedness of the classical fractional integral operator in the variable exponent Hardy-amalgam spaces, preprint.
[26] L. Traore and C. Unal, Atomic decomposition of Hardy-amalgam spaces with variable exponents, preprint.
[27] L. Wang, Boundedness of the commutator of the intrinsic square function in variable exponent spaces, J. Korean Math. Soc. 55(4) (2018), 939-962.
[28] M. Izuki and T. Noi, An intrinsic square function on weighted Herz spaces with variable exponent, J. Math. Inequal. 11 (2017), 799-816.
[29] M. Wilson, The intrinsic square function, Rev. Mat. Ibreroam. 23(3) (2007), 771-791.
[30] M. Wilson, Weighted Littlewood-Paley Theory and Exponential-square Integrability, Vol. 1924, Lecture Notes in Math, Springer, New York, NY, 2007.
[31] N. Wiener, On the representation of functions by trigonometric integrals, Math. Z. 24 (1926), 575-616.
[32] Nouffou Diarra, Boundedness of some operators on variable exponent Fofana’s spaces and their preduals, Mathematical Analysis and its Contemporary Applications 5(3) (2023), 69-90.
[33] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47(4) (1980), 959-982.
[34] H. Wang and H. Liu, The intrinsic square function characterizations of weighted Hardy spaces, Illinois J. Math. 56 (2012), 367-381.
[35] W. Orlicz, Uber eine gewisse klasse Von Raumen Von Typus B, Bull. Int. Acad. Pol. Ser. A 8 (1932), 207-220.
[36] Y. Liang and D. Yang, Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Trans. Amer. Math. Soc. 367(5) (2015), 3225-3256.
DOI :10.1090/s0002-9947-2014-06180-1.
[37] Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Math. Z. 24 (1926), 575-616.
[38] Z. Birnbaum and W. Orlicz, Uber die Verallgemeinerung des Begriffes der Zueinander Konjugierten potenzen, Studia Math. 3 (1931), 1-67.
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