Article
| Title: | A boundedness criterion for the discrete Hardy operator on weighted Musielak-Orlicz sequence spaces (English) |
| Author: | Bandaliyev, Rovshan |
| Author: | Aliyev, Mehraly |
| Author: | Omarova, Konul |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1393-1410 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We establish a necessary and sufficient condition on weight functions for the boundedness of the discrete Hardy operator on weighted Musielak-Orlicz sequence spaces. In particular, we get similar results for the dual operator of the discrete Hardy operator. We give sufficient pointwise conditions on generalized $\Phi $-functions that guarantee continuous embeddings between weighted Musielak-Orlicz sequence spaces. The results are illustrated by a number of corollaries. (English) |
| Keyword: | discrete Hardy operator |
| Keyword: | dual operator |
| Keyword: | weight function |
| Keyword: | weighted Musielak-Orlicz sequence space |
| Keyword: | embedding theorem |
| Keyword: | boundedness |
| MSC: | 26D15 |
| MSC: | 46E30 |
| MSC: | 47G10 |
| DOI: | 10.21136/CMJ.2025.0181-25 |
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| Date available: | 2025-12-20T07:57:29Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153249 |
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| Reference: | [1] Andersen, K. F., Heinig, H. P.: Weighted norm inequalities for certain integral operators.SIAM J. Math. Anal. 14 (1983), 834-844. Zbl 0527.26010, MR 0704496, 10.1137/0514064 |
| Reference: | [2] Bandaliev, R. A.: On a two-weight criterion for Hardy type operator in the variable Lebesgue spaces with measures.Trans. Natl. Acad. Sci. Azerb., Ser. Phys.-Tech. Math. Sci. 30 (2010), 45-54. Zbl 1226.47028, MR 2780350 |
| Reference: | [3] Bandaliev, R. A.: The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces.Czech. Math. J. 60 (2010), 327-337 corrigendum ibid. 63 1149-1152 2013. Zbl 1222.47045, MR 2657952, 10.1007/s10587-010-0038-y |
| Reference: | [4] Bandaliev, R. A.: The boundedness of multidimensional Hardy operators in weighted variable Lebesgue spaces.Lith. Math. J. 50 (2010), 249-259. Zbl 1219.46026, MR 2719561, 10.1007/s10986-010-9083-3 |
| Reference: | [5] Bandaliev, R. A.: Embedding between variable exponent Lebesgue spaces with measures.Azerb. J. Math. 2 (2012), 119-126. Zbl 1278.46028, MR 2967280 |
| Reference: | [6] Bandaliev, R. A.: Applications of multidimensional Hardy operator and its connection with a certain nonlinear differential equation in weighted variable Lebesgue spaces.Ann. Funct. Anal. 4 (2013), 118-130. Zbl 1285.46025, MR 3034935, 10.15352/afa/1399899530 |
| Reference: | [7] Bandaliev, R. A.: On Hardy-type inequalities in weighted variable Lebesgue spaces $L_{p(x),\omega}$ for $0< p(x)< 1$.Eurasian Math. J. 4 (2013), 5-16. Zbl 1308.47042, MR 3382899 |
| Reference: | [8] Bandaliyev, R. A., Aliyeva, D. R.: On boundedness and compactness of discrete Hardy operator in discrete weighted variable Lebesgue spaces.J. Math. Inequal. 16 (2022), 1215-1228. Zbl 1516.46018, MR 4502066, 10.7153/jmi-2022-16-81 |
| Reference: | [9] Bandaliyev, R. A., Serbetci, A., Hasanov, S. G.: On Hardy inequality in variable Lebesgue spaces with mixed norm.Indian J. Pure Appl. Math. 49 (2018), 765-782. Zbl 1451.46028, MR 3882009, 10.1007/s13226-018-0300-9 |
| Reference: | [10] Bennett, G.: Some elementary inequalities.Q. J. Math., Oxf. II. Ser. 38 (1987), 401-425. Zbl 0649.26013, MR 0916225, 10.1093/qmath/38.4.401 |
| Reference: | [11] Bennett, G.: Some elementary inequalities. II.Q. J. Math., Oxf. II. Ser. 39 (1988), 385-400. Zbl 0687.26007, MR 0975904, 10.1093/qmath/39.4.385 |
| Reference: | [12] Bennett, G.: Some elementary inequalities. III.Q. J. Math., Oxf. II. Ser. 42 (1991), 149-174. Zbl 0751.26007, MR 1107279, 10.1093/qmath/42.1.149 |
| Reference: | [13] Braverman, M. S., Stepanov, V. D.: On the discrete Hardy inequality.Bull. Lond. Math. Soc. 26 (1994), 283-287. Zbl 0806.26011, MR 1289048, 10.1112/blms/26.3.283 |
| Reference: | [14] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2013). Zbl 1268.46002, MR 3026953, 10.1007/978-3-0348-0548-3 |
| Reference: | [15] Cruz-Uribe, D., Mamedov, F. I.: On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces.Rev. Mat. Complut. 25 (2012), 335-367. Zbl 1284.42053, MR 2931417, 10.1007/s13163-011-0076-5 |
| Reference: | [16] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces With Variable Exponents.Lecture Notes in Mathematics 2017. Springer, Berlin (2011). Zbl 1222.46002, MR 2790542, 10.1007/978-3-642-18363-8 |
| Reference: | [17] Diening, L., Samko, S.: Hardy inequality in variable exponent Lebesgue spaces.Fract. Calc. Appl. Anal. 10 (2007), 1-18. Zbl 1132.26341, MR 2348863 |
| Reference: | [18] Edmunds, D. E., Kokilashvili, V., Meskhi, A.: On the boundedness and compactness of weighted Hardy operators in spaces $L^{p(x)}$.Georgian Math. J. 12 (2005), 27-44. Zbl 1099.47030, MR 2136721, 10.1515/GMJ.2005.27 |
| Reference: | [19] Gogatishvili, A., Stepanov, V. D.: Reduction theorems for operators on the cones of monotone functions.J. Math. Anal. Appl. 405 (2013), 156-172. Zbl 1326.47063, MR 3053495, 10.1016/j.jmaa.2013.03.046 |
| Reference: | [20] Gogatishvili, A., Stepanov, V. D.: Reduction theorems for weighted integral inequalities on the cone of monotone functions.Russ. Math. Surv. 68 (2013), 597-664. Zbl 1288.26018, MR 3154814, 10.1070/RM2013v068n04ABEH004849 |
| Reference: | [21] Goldman, M. L.: Sharp estimates of the norms of Hardy-type operators on the cone of quasimonotone functions.Proc. Steklov Inst. Math. 232 (2001), 109-137. Zbl 1030.42013, MR 1851444 |
| Reference: | [22] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities.Cambridge University Press, Cambridge (1934). Zbl 0010.10703, MR 0197653 |
| Reference: | [23] Heinig, H. P.: Weighted norm inequalities for certain integral operators. II.Proc. Am. Math. Soc. 95 (1985), 387-395. Zbl 0592.26017, MR 0806076, 10.1090/S0002-9939-1985-0806076-3 |
| Reference: | [24] Kalybay, A., Oinarov, R., Temirkhanova, A.: Boundedness of $n$-multiple discrete Hardy operators with weights for $1< q< p< \infty$.J. Funct. Spaces Appl. 2013 (2013), Article ID 121767, 9 pages. Zbl 1383.42018, MR 3066263, 10.1155/2013/121767 |
| Reference: | [25] Kalybay, A., Persson, L.-E., Temirkhanova, A.: A new discrete Hardy-type inequality with kernels and monotone functions.J. Inequal. Appl. 2015 (2015), Article ID 321, 10 pages. Zbl 1336.26036, MR 3405707, 10.1186/s13660-015-0843-9 |
| Reference: | [26] Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-Standard Function Spaces. Volume 1. Variable Exponent Lebesgue and Amalgam Spaces.Operator Theory: Advances and Applications 248. Birkhäuser, Basel (2016). Zbl 1385.47001, MR 3559400, 10.1007/978-3-319-21015-5 |
| Reference: | [27] Kopaliani, T. S.: On some structural properties of Banach function spaces and boundedness of certain integral operators.Czech. Math. J. 54 (2004), 791-805. Zbl 1080.47040, MR 2086735, 10.1007/s10587-004-6427-3 |
| Reference: | [28] Kufner, A., Maligranda, L., Persson, L.-E.: The prehistory of the Hardy inequality.Am. Math. Mon. 113 (2006), 715-732. Zbl 1153.01015, MR 2256532, 10.2307/27642033 |
| Reference: | [29] Mamedov, F. I., Harman, A.: On a weighted inequality of Hardy type in spaces $L^{p(\cdot)}$.J. Math. Anal. Appl. 353 (2009), 521-530. Zbl 1172.26006, MR 2508955, 10.1016/j.jmaa.2008.12.029 |
| Reference: | [30] Mamedov, F., Mammadova, F. M., Aliyev, M.: Boundedness criterions for the Hardy operator in weighted $L^ {p( \cdot)}(0, l)$ space.J. Convex Anal. 22 (2015), 553-568. Zbl 1329.42018, MR 3346203 |
| Reference: | [31] Musielak, J.: Orlicz Spaces and Modular Spaces.Lecture Notes in Mathematics 1034. Springer, Berlin (1983). Zbl 0557.46020, MR 0724434, 10.1007/BFb0072210 |
| Reference: | [32] Musielak, J., Orlicz, W.: On modular spaces.Stud. Math. 18 (1959), 49-65. Zbl 0086.08901, MR 0101487, 10.4064/sm-18-1-49-65 |
| Reference: | [33] Nakano, H.: Modulared Semi-Ordered Linear Spaces.Maruzen, Tokyo (1950). Zbl 0041.23401, MR 0038565 |
| Reference: | [34] Nakano, H.: Topology of Topological Linear Spaces.Maruzen, Tokyo (1951). MR 0046560 |
| Reference: | [35] Nekvinda, A.: Embeddings between discrete weighted Lebesgue spaces with variable exponents.Math. Inequal. Appl. 10 (2007), 165-172. Zbl 1121.46028, MR 2286702, 10.7153/mia-10-14 |
| Reference: | [36] Okpoti, C. A., Persson, L.-E., Wedestig, A.: Scales of weight characterizations for discrete Hardy and Carleman inequalities.Function Spaces, Differential Operators and Nonlinear Analysis Mathematical Institute of the AS CR, Praha (2005), 236-258. |
| Reference: | [37] Okpoti, C. A., Persson, L.-E., Wedestig, A.: Weight characterizations for the discrete Hardy inequality with kernel.J. Inequal. Appl. 2006 (2006), Article ID 18030, 14. Zbl 1090.26018, MR 2221216, 10.1155/JIA/2006/18030 |
| Reference: | [38] Orlicz, W.: Über konjugierte Exponentenfolgen.Stud. Math. 3 (1931), 200-211 German. Zbl 0003.25203, 10.4064/sm-3-1-200-211 |
| Reference: | [39] Rafeiro, H., Samko, S.: Hardy type inequality in variable Lebesgue spaces.Ann. Acad. Sci. Fenn., Math. 34 (2009), 279-289 corrigendum ibid. 35 679-680 2010. Zbl 1177.47045, MR 2489027 |
| Reference: | [40] Wedestig, A.: Some new Hardy type inequalities and their limiting inequalities.JIPAM, J. Inequal. Pure Appl. Math. 4 (2003), Article ID 61, 15 pages. Zbl 1064.26023, MR 2044410 |
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