Geodesic
Three weight Hardy inequality on measure topological spaces
Eurasian mathematical journal, Tome 14 (2023) no. 2, pp. 58-78.

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For the Hardy inequality to hold on a Hausdorff topological space, we obtain necessary and sufficient conditions on the weights and measures. As in the recent paper by G. Sinnamon (2022), we assume total orderedness of the family of sets that generate the Hardy operator. Sinnamon’s method consists in the reduction of the problem to an equivalent one-dimensional problem. We provide a different, direct proof which develops the approach suggested by D. Prokhorov (2006) in the one-dimensional case. 
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K. T. Mynbaev. Three weight Hardy inequality on measure topological spaces. Eurasian mathematical journal, Tome 14 (2023) no. 2, pp. 58-78. http://geodesic.mathdoc.fr/item/EMJ_2023_14_2_a3/

[1] R. A. Bandaliev, “Application of multidimensional Hardy operator and its connection with a certain nonlinear differential equation in weighted variable Lebesgue spaces”, Ann. Funct. Anal., 4:2 (2013), 118–130 | DOI | MR | Zbl

[2] G. A. Bliss, “An integral inequality”, J. London Math. Soc., 5:1 (1930), 40–46 | DOI | MR

[3] P. Drábek, H. P. Heinig, A. Kufner, “Higher-dimensional Hardy inequality”, General inequalities, 7 (Oberwolfach, 1995), 1997, 3–16 | MR | Zbl

[4] D. E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and compact integral operators, Mathematics and its Applications, 543, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[5] A. Kalybay, R. Oinarov, “Boundedness of Riemann-Liouville operator from weighted Sobolev space to weighted Lebesgue space”, Eurasian Math. J., 12:1 (2021), 39–48 | DOI | MR | Zbl

[6] A. Kufner, L. Maligranda, L. E. Persson, The Hardy inequality. About its history and some related results, Vydavatelský Servis, Plzěn, 2007 | MR | Zbl

[7] A. Kufner, B. Opic, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific Technical, Harlow, 1990 | MR | Zbl

[8] A. Kufner, L. E. Persson, N. Samko, Weighted inequalities of Hardy type, Second edition, World Scientific Pub lishing Co. Pte. Ltd., Hackensack, NJ, 2017 | MR | Zbl

[9] K. Kuliev, “On estimates for norms of some integral operators with Oinarov's kernel”, Eurasian Math. J., 13:3 (2022), 67–81 | DOI | MR | Zbl

[10] V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, Fundamental Principles of Mathematical Sciences, 342, Springer, Heidelberg, 2011 | MR | Zbl

[11] R. Oinarov, “Two-sided estimates for the norm of some classes of integral operators”, Proc. Steklov Inst. Math., 204:3 (1994), 205–214 | MR

[12] D. V. Prokhorov, “Hardy's inequality with three measures”, Proc. Steklov Inst. Math., 255 (2006), 223–233 | DOI | MR

[13] D. V. Prokhorov, V. D. Stepanov, “Weighted estimates for the Riemann-Liouville operators and applications”, Proc. Steklov Inst. Math., 243 (2003), 278–301 | MR | Zbl

[14] D. V. Prokhorov, V. D. Stepanov, E. P. Ushakova, “Hardy-Steklov integral operators: Part I”, Proc. Steklov Inst. Math., 300:2 (2018), 1–112 | MR

[15] W. Rudin, Real and complex analysis, Third edition, Co. McGraw-Hill Book, New York, 1987 | MR | Zbl

[16] M. Ruzhansky, D. Verma, “Hardy inequalities on metric measure spaces”, Proc. R. Soc. A, 475:2223 (2019), 20180310, 15 pp. | DOI | MR | Zbl

[17] M. Ruzhansky, D. Verma, “Hardy inequalities on metric measure spaces, II: the case $p > q$”, Proc. R. Soc. A, 477:2250 (2021), 20210136, 16 pp. | DOI | MR

[18] M. Ruzhansky, N. Yessirkegenov, Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups, 21 Feb 2019, arXiv: 1810.08845v2 [math.FA] | MR

[19] E. Sawyer, “Weighted inequalities for the two-dimensional Hardy operator”, Studia Math., 82:1 (1985), 1–16 | DOI | MR | Zbl

[20] E. Sawyer, “Boundedness of classical operators on classical Lorentz spaces”, Studia Math., 96:2 (1990), 145–158 | DOI | MR | Zbl

[21] G. Sinnamon, “One-dimensional Hardy-type inequalities in many dimensions”, Proc. Roy. Soc. Edinburgh Sect. A, 128:4 (1998), 833–848 | DOI | MR | Zbl

[22] G. Sinnamon, “Hardy inequalities in normal form”, Trans. Amer. Math. Soc., 375:2 (2022), 961–995 | MR | Zbl

[23] V. D. Stepanov, G. E. Shambilova, “On two-dimensional bilinear inequalities with rectangular Hardy operators in weighted Lebesgue spaces”, Proc. Steklov Inst. Math., 312 (2021), 241–248 | DOI | MR | Zbl

[24] A. M. Temirkhanova, A. T. Beszhanova, “Boundedness and compactness of a certain class of matrix operators with variable limits of summation”, Eurasian Math. J., 11:4 (2020), 66–75 | DOI | MR | Zbl