×

zbMATH — the first resource for mathematics

Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces. (English) Zbl 07229051
Summary: The boundedness of the weighted iterated Hardy-type operators \(T_{u,b}\) and \(T_{u,b}^*\) involving suprema from weighted Lebesgue space \(L_p(v)\) into weighted Cesàro function spaces \({\operatorname{Ces}}_q(w,a)\) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator \(R_u\) from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator \(P_{u,b }\) from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. Under additional condition on \(u\) and \(b\), we are able to characterize the boundedness of weighted iterated Hardy-type operator \(T_{u,b}\) involving suprema from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function \(M_{\gamma}\) from \(\Lambda^p(v)\) into \(\Gamma^q(w)\).
MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
PDF BibTeX Cite
Full Text: DOI Euclid
References:
[1] R. Askey and R. P. Boas, Jr.,Some integrability theorems for power series with positive coefficients, Mathematical Essays Dedicated to A.J. Macintyre, 23-32. Ohio Univ. Press, Athens, Ohio, 1970.
[2] S. V. Astashkin and L. Maligranda,Structure of Ces‘aro function spaces, Indag. Math.,20(3)(2009), 329-279. · Zbl 1200.46027
[3] S. V. Astashkin and L. Maligranda,Structure of Ces‘aro function spaces: a survey, Banach Center Publ.102, (2014), 13-40. · Zbl 1327.46028
[4] G. Bennett,Factorizing the classical inequalities, Mem. Amer. Math. Soc., 120(576), Providence, 1996. · Zbl 0857.26009
[5] G. Bennett and K. -G. Grosse-Erdmann,Weighted Hardy inequalities for decreasing sequences and functions, Math. Ann.,334(3), (2006). 489-531. · Zbl 1119.26018
[6] R. P. Boas, Jr.,Integrability theorems for trigonometric transforms, Ergebnisse der Mathematik und ihrer Grenzgebiete,38, Springer-Verlag New York Inc., New York, 1967. · Zbl 0145.06804
[7] R. P. Boas, Jr.,Some integral inequalities related to Hardy’s inequality, J. Analyse Math.,23, (1970), 53-63. · Zbl 0206.06803
[8] M. Carro, A. Gogatishvili, J. Martin and L. Pick,Weighted inequalities involving two Hardy operators with applications to embeddings of function spaces, J. Operator Theory,59(2), (2008), 309-332. · Zbl 1150.26001
[9] M. Carro, L. Pick, J. Soria and V. D. Stepanov,On embeddings between classical Lorentz spaces, Math. Inequal. Appl.,4(3), (2001), 397-428. · Zbl 0996.46013
[10] A. Cianchi, R. Kerman, B. Opic and L. Pick,A sharp rearrangement inequality for the fractional maximal operator, Studia Math.,138(3), (2000), 277-284. · Zbl 0968.42014
[11] S. Chen, Y. Cui, H. Hudzik and B. Sims,Geometric properties related to fixed point theory in some Banach function lattices, Handbook of metric fixed point theory, 339-389. Kluwer Acad. Publ., Dordrecht, 2001. · Zbl 1013.46015
[12] Y. Cui and R. Płuciennik,Local uniform nonsquareness in Ces‘aro sequence spaces, Comment. Math. (Prace Mat.),37, (1997), 47-58. · Zbl 0898.46006
[13] Y. Cui and H. Hudzik,Some geometric properties related to fixed point theory in Ces‘aro spaces, Collect. Math.,50(3), (1999), 277-288. · Zbl 0955.46007
[14] Y. Cui and H. Hudzik,Packing constant for Cesaro sequence spaces, Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000). Nonlinear Anal.,47(4), (2001), 2695-2702. · Zbl 1042.46505
[15] Y. Cui, H. Hudzik and Y. Li,On the Garcia-Falset coefficient in some Banach sequence spaces, (Function spaces, Pozna´n, 1998), 141-148. Lecture Notes in Pure and Appl. Math.,213, Dekker, New York, 2000. · Zbl 0962.46011
[16] M. Cwikel and E. Pustylnik,Weak type interpolation near “endpoint” spaces, J. Funct. Anal.,171(2), (2000), 235-277. · Zbl 0978.46008
[17] R.Ya.Doktorskii,Reiterativerelationsoftherealinterpolation method,Russian, Dokl. Akad. Nauk SSSR,321(2), (1991), 241-245. Soviet Math. Dokl.,44(3), (1992), 665-669.
[18] W. D. Evans and B. Opic,Real interpolation with logarithmic functors and reiteration, Canad. J. Math.,52(5), (2000), 920-960. · Zbl 0981.46058
[19] A. Gogatishvili, M. Johansson, C. A. Okpoti and L.-E. Persson,Characterisation of embeddings in Lorentz spaces, Bull. Austral. Math. Soc.,76(1), (2007), 69-92. · Zbl 1128.26012
[20] A. Gogatishvili and R. Ch. Mustafayev,Weighted iterated Hardy-type inequalities, Math. Inequal. Appl.,20(3), (2017), 683-728. · Zbl 1375.26032
[21] A. Gogatishvili and R. Ch. Mustafayev,Iterated Hardy-type inequalities involving suprema, Math. Inequal. Appl.,20(4), (2017), 901-927. · Zbl 1379.26018
[22] A. Gogatishvili, R. Mustafayev and T. ¨Unver,Embeddings between weighted Copson and Ces‘aro function spaces, Czechoslovak Math. J.,67(142)(4), (2017), 1105-1132. · Zbl 06819576
[23] A. Gogatishvili, B. Opic and L. Pick,Weighted inequalities for Hardy-type operators involving suprema, Collect. Math.,57(3), (2006), 227-255. · Zbl 1116.47041
[24] A. Gogatishvili and L. Pick,Embeddings and duality theorems for weak classical Lorentz spaces, Canad. Math. Bull.,49(1), (2006), 82-95. · Zbl 1106.26018
[25] A. Gogatishvili and L. Pick,A reduction theorem for supremum operators, J. Comput. Appl. Math.,208(1), (2007), 270-279. · Zbl 1127.26012
[26] A. Gogatishvili and V. D. Stepanov,Reduction theorems for weighted integral inequalities on the cone of monotone functions, Russian, with Russian summary, Uspekhi Mat. Nauk,68(4(412)), (2013), 3-68. English transl., Russian Math. Surveys,68(4), (2013), 597-664. · Zbl 1288.26018
[27] K.-G. Grosse-Erdmann,The blocking technique, weighted mean operators and Hardy’s inequality, Lecture Notes in Mathematics,1679, Springer-Verlag, Berlin, 1998. · Zbl 0888.26014
[28] A. A. Jagers,A note on Ces‘aro sequence spaces, Nieuw Arch. Wisk. (3),22, (1974), 113-124. · Zbl 0286.46017
[29] A. Kami´nska and D. Kubiak,On the dual of Ces‘aro function space, Nonlinear Anal.,75(5), (2012), 2760-2773. · Zbl 1245.46024
[30] R. Kerman and L. Pick,Optimal Sobolev imbeddings, Forum Math.,18(4), (2006), 535-570. · Zbl 1120.46018
[31] M. Kˇrepela,Integral conditions for Hardy-type operators involving suprema, Collect. Math.,68(1), (2017), 21-50.
[32] G. M. Leibowitz,A note on the Ces‘aro sequence spaces, Tamkang J. Math.,2, (1971), 151-157. · Zbl 0236.46012
[33] V. G. Maz’ja,Sobolev spaces, Springer Series in Soviet Mathematics, Translated from the Russian by T.O. Shaposhnikova, Springer-Verlag, Berlin, 1985.
[34] R.Ch. Mustafayev and N. Bilgic¸li,Generalized fractional maximal functions in Lorentz spacesΛ, J. Math. Inequal.,12(3), (2018), 827-851. · Zbl 1403.42021
[35] R. Oinarov,Two-sided estimates for the norm of some classes of integral operators, Russian, Trudy Mat. Inst. Steklov.,204, (1993), Issled. po Teor. Differ. Funktsii Mnogikh Peremen. i ee Prilozh.16, 240-250. English transl., Proc. Steklov Inst. Math.,3(204), (1994), 205-214. · Zbl 0883.47048
[36] B. Opic,On boundedness of fractional maximal operators between classical Lorentz spaces, Function spaces, differential operators and nonlinear analysis, Pudasj¨arvi, 1999, 187-196, Acad. Sci. Czech Repub., Prague, 2000.
[37] B. Opic and A. Kufner,Hardy-type inequalities, Pitman Research Notes in Mathematics Series,219, Longman Scientific & Technical, Harlow, 1990. · Zbl 0698.26007
[38] L. Pick,Supremum operators and optimal Sobolev inequalities, Function spaces, differential operators and nonlinear analysis, Pudasj¨arvi, 1999, 207- 219, Acad. Sci. Czech Repub., Prague, 2000.
[39] L. Pick,Optimal Sobolev embeddings—old and new, Function spaces, interpolation theory and related topics (Lund, 2000), 403-411, de Gruyter, Berlin, 2002. · Zbl 1022.46019
[40] Programma van Jaarlijkse Prijsvragen (Annual Problem Section), Nieuw Arch. Wiskd.,16, (1968), 47-51.
[41] D. V. Prokhorov and V. D. Stepanov,On weighted Hardy inequalities in mixed norms, Proc. Steklov Inst. Math.,283, (2013), 149-164. · Zbl 1296.46029
[42] D. V. Prokhorov and V. D. Stepanov,Weighted estimates for a class of sublinear operators, Russian, Dokl. Akad. Nauk,453(5), (2013), 486-488. English transl., Dokl. Math.,88(3), (2013), 721-723. · Zbl 1310.47066
[43] D. V. Prokhorov and V. D. Stepanov,Estimates for a class of sublinear integral operators, Russian, Dokl. Akad. Nauk,456(6), (2014), 645-649. English transl., Dokl. Math.,89(3), (2014), 372-377. · Zbl 1303.45008
[44] D. V. Prokhorov,On the boundedness of a class of sublinear integral operators, Russian, Dokl. Akad. Nauk,92(2), (2015), 602-605. · Zbl 1330.47062
[45] E. Pustylnik,Optimal interpolation in spaces of Lorentz-Zygmund type, J. Anal. Math.,79, (1999), 113-157. · Zbl 0987.46035
[46] G. ‘E. Shambilova,Weighted inequalities for a class of quasilinear integral operators on the cone of monotone functions, Russian, with Russian summary, Sibirsk. Mat. Zh.,55(4), (2014), 912-936. English transl., Sib. Math. J.,55(4), (2014), 745-767. · Zbl 1318.26047
[47] J. -S. Shiue,On the Ces‘aro sequence spaces, Tamkang J. Math.,1(1), (1970), 19-25. · Zbl 0215.19504
[48] J. -S. Shiue,A note on Ces‘aro function space, Tamkang J. Math.,1(2), (1970), 91-95. · Zbl 0215.19601
[49] V. D. Stepanov,The weighted Hardy’s inequality for nonincreasing functions, Trans. Amer. Math. Soc.,338(1), (1993), 173-186. · Zbl 0786.26015
[50] V. D. Stepanov and G. ‘E. Shambilova,Weight boundedness of a class of quasilinear operators on the cone of monotone functions, Dokl. Math.,90(2), (2014), 569-572. · Zbl 1311.47013
[51] P. W. Sy, W. Y. Zhang and P. Y. Lee,The dual of Ces‘aro function spaces, English, with Serbo-Croatian summary, Glas. Mat. Ser. III,22(
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.