Abstract
In this paper, we study the boundedness of the Hardy averaging operator between weighted variable grand Lebesgue spaces for nonincreasing functions.
References
M. A. Arino and B. Muckenhoupt, “Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions,” Trans. Amer. Math. Soc. 320, 727–735 (1990).
C. J. Neugebauer, “Weighted norm inequalities for averaging operators of monotone functions,” Publ. Mat. 35, 429–447 (1991).
I. V. Tsenov, “Generalization of the problem of the best approximation of a function in the space \(\mathcal L^s\),” Uch. Zap. Dagestan. Gos. Univ. 7, 25–37 (1961).
O. Kováčik and J. Rákosník, “On \(L^{p(x)}\) and \(W^{k,p(x)}\) spaces,” Czech. Math. J. 41, 592–618 (1991).
I. I. Sharapudinov, “Topology of the space \(\mathscr L^{p(t)}([0,1])\),” Math. Notes 26 (4), 796–806 (1979).
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis (Birkhäuser, Heidelberg, 2013).
C. J. Neugebauer, “Weighted variable \(L^{p}\) integral inequalities for the maximal operator on non-increasing functions,” Studia Math. 192, 51–60 (2009).
A. Fiorenza, V. Kokilashvili, and A. Meskhi, “Hardy–Littlewood maximal operator in weighted grand variable exponent Lebesgue space,” Mediterr. J. Math. 14, Paper No. 118 (2017).
L. Greco, T. Iwaniec, and C. Sbordone, “Inverting the p-harmonic operator,” Manuscripta Math. 92, 249–258 (1997).
T. Iwaniec and C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses,” Arch. Ration. Mech. Anal. 119, 129–143 (1992).
A. Fiorenza, B. Gupta, and P. Jain, “The maximal theorem for weighted grand Lebesgue spaces,” Studia Math. 188, 123–133 (2008).
C. Capone and A. Fiorenza, “On small Lebesgue spaces,” J. Funct. Spaces Appl. 3, 73–89 (2005).
A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces,” Collect. Math. 51, 131–148 (2000).
A. Fiorenza and G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs,” Z. Anal. Anwendungen 23, 657–681 (2004).
A. Fiorenza and J. M. Rakotoson, “New properties of small Lebesgue spaces and their applications,” Math. Ann. 326, 543–561 (2003).
P. Jain and S. Kumari, “On grand Lorentz spaces and the maximal operator,” Georgian Math. J. 19, 235–246 (2012).
P. Jain, M. Singh, and A. P. Singh, “Hardy type operators on grand Lebesgue spaces for non-increasing functions,” Trans. A. Razmadze Math. Inst. 170, 34–46 (2016).
P. Jain, M. Singh, and A. P. Singh, “Integral operators on fully measurable weighted grand Lebesgue spaces,” Indag. Math. 28, 516–526 (2017).
P. Jain, M. Singh, and A. P. Singh, “Recent trends in grand Lebesgue spaces,” in Function Spaces and Inequalities, Springer Proc. Math. Stat., Ed. by P. Jain and H. J. Schmeisser (Springer, Singapore, 2017), Vol. 206, pp. 137–159.
V. M. Kokilashvili, “Boundedness criterion for singular integrals in weighted grand Lebesgue spaces,” J. Math. Sci. 170, 20–33 (2010).
V. Kokilashvili and A. Meskhi, “Maximal and Calderón–Zygmund operators in grand variable exponent Lebesgue spaces,” Georgian Math. J. 21, 447–461 (2014).
V. Kokilashvili and A. Meskhi, “Maximal and singular integral operators in weighted grand variable exponent Lebesgue spaces,” Ann. Funct. Anal. 12, 48 (2021).
A. N. Meskhi, “Weighted criteria for the Hardy transform under the \(B_p\) condition in grand Lebesgue spaces and some applications,” J. Math. Sci. 178, 622–636 (2011).
M. Singh, “Grand and small \(X^p\) spaces and generalized duality,” Positivity 25, 1469–1488 (2021).
A. P. Singh, M. Singh, P. Jain, and R. Panchal, “Rubio de Francia extrapolation theorem in variable Lebesgue spaces for \(B_p( \cdot )\) weights,” Ricerche Mat. (2021).
S. Boza and J. Soria, “Weighted Hardy modular inequalities in variable \(L_p\) spaces for decreasing functions,” J. Math. Anal. Appl. 348, 383–388 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 283–294 https://doi.org/10.4213/mzm13885.
Rights and permissions
About this article
Cite this article
Singh, M., Jain, P. Hardy Inequality in Variable Grand Lebesgue Spaces for Nonincreasing Functions. Math Notes 113, 282–291 (2023). https://doi.org/10.1134/S0001434623010315
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434623010315