Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 1105-1132, 2017
Embeddings between weighted Copson and Cesàro function spaces
Amiran Gogatishvili, Rza Mustafayev, Tuğçe Ünver
Received August 9, 2016. First published October 11, 2017.
Abstract: In this paper, characterizations of the embeddings between weighted Copson function spaces ${\rm Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\rm Ces}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $ \align\biggl( \int_0^{\infty} &\biggl( \int_0^t f(\tau)^{p_2}v_2(\tau)\dd\tau\biggr)^{\!\!\frc{q_2}{p_2}} u_2(t)\dd t\biggr)^{\!\!\frc1{q_2}} $ \ $\le c \biggl( \int_0^{\infty} \biggl( \int_t^{\infty} f(\tau)^{p_1} v_1(\tau)\dd\tau\biggr)^{\!\!\frc{q_1}{p_1}} u_1(t)\dd t\biggr)^{\!\!\frc1{q_1}}, $ where $p_1,p_2,q_1,q_2 \in(0,\infty)$, $p_2 \le q_2$ and $u_1,u_2,v_1,v_2$ are weights on $(0,\infty)$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.
Keywords: Cesàro and Copson function spaces; embedding; iterated Hardy inequalities
References: [1] R. Askey, R. P. Boas, Jr.: Some integrability theorems for power series with positive coefficients. Math. Essays Dedicated to A. J. Macintyre. Ohio Univ. Press, Athens (1970), 23-32. MR 0277956 | Zbl 0212.41401 [2] S. V. Astashkin: On the geometric properties of Cesàro spaces. Sb. Math. 203 (2012), 514-533. (In English. Russian original.); translation from Mat. Sb. 203 (2012), 61-80. DOI 10.1070/SM2012v203n04ABEH004232 | MR 2976287 | Zbl 1253.46022 [3] S. V. Astashkin, L. Maligranda: Cesàro function spaces fail the fixed point property. Proc. Am. Math. Soc. 136 (2008), 4289-4294. DOI 10.1090/S0002-9939-08-09599-3 | MR 2431042 | Zbl 1168.46014 [4] S. V. Astashkin, L. Maligranda: Structure of Cesàro function spaces. Indag. Math., New Ser. 20 (2009), 329-379. DOI 10.1016/S0019-3577(10)00002-9 | MR 2639977 | Zbl 1200.46027 [5] S. V. Astashkin, L. Maligranda: Rademacher functions in Cesàro type spaces. Stud. Math. 198 (2010), 235-247. DOI 10.4064/sm198-3-3 | MR 2650988 | Zbl 1202.46031 [6] S. V. Astashkin, L. Maligranda: Geometry of Cesàro function spaces. Funct. Anal. Appl. 45 (2011), 64-68. (In English. Russian original.); translation from Funkts. Anal. Prilozh. 45 (2011), 79-83. DOI 10.1007/s10688-011-0007-8 | MR 2848742 | Zbl 1271.46027 [7] S. V. Astashkin, L. Maligranda: A short proof of some recent results related to Cesàro function spaces. Indag. Math., New Ser. 24 (2013), 589-592. DOI 10.1016/j.indag.2013.03.001 | MR 3064562 | Zbl 1292.46008 [8] S. V. Astashkin, L. Maligranda: Interpolation of Cesàro sequence and function spaces. Stud. Math. 215 (2013), 39-69. DOI 10.4064/sm215-1-4 | MR 3071806 | Zbl 06172508 [9] S. V. Astashkin, L. Maligranda: Interpolation of Cesàro and Copson spaces. Proc. Int. Symp. on Banach and Function Spaces IV (ISBFS 2012), Kitakyushu 2012 (M. Kato et al., eds.). Yokohama Publishers, Yokohama (2014), 123-133. MR 3289767 | Zbl 1338.46034 [10] S. V. Astashkin, L. Maligranda: Structure of Cesàro function spaces: a survey. Function Spaces X (H. Hudzik et al., eds.). Proc. Int. Conf., Poznań 2012, Polish Academy of Sciences, Institute of Mathematics, Warszawa; Banach Center Publications 102 (2014), 13-40. DOI 10.4064/bc102-0-1 | MR 3330604 | Zbl 1327.46028 [11] E. S. Belinskii, E. R. Liflyand, R. M. Trigub: The Banach algebra $A^*$ and its properties. J. Fourier Anal. Appl. 3 (1997), 103-129. DOI 10.1007/BF02649131 | MR 1438893 | Zbl 0882.42002 [12] G. Bennett: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120 (1996), 130 pages. DOI 10.1090/memo/0576 | MR 1317938 | Zbl 0857.26009 [13] R. P. Boas, Jr.: Integrability Theorems for Trigonometric Transforms. Ergebnisse der Mathematik und ihrer Grenzgebiete 38, Springer, New York (1967). DOI 10.1007/978-3-642-87108-5 | MR 0219973 | Zbl 0145.06804 [14] R. P. Boas, Jr.: Some integral inequalities related to Hardy's inequality. J. Anal. Math. 23 (1970), 53-63. DOI 10.1007/BF02795488 | MR 0274685 | Zbl 0206.06803 [15] M. Carro, A. Gogatishvili, J. Martín, L. Pick: Weighted inequalities involving two Hardy operators with applications to embeddings of function spaces. J. Oper. Theory 59 (2008), 309-332. MR 2411048 | Zbl 1150.26001 [16] S. Chen,Y. Cui, H. Hudzik, B. Sims: Geometric properties related to fixed point theory in some Banach function lattices. Handbook of Metric Fixed Point Theory (W. Kirk et al., eds.). Kluwer Academic Publishers, Dordrecht (2001), 339-389. DOI 10.1007/978-94-017-1748-9_12 | MR 1904283 | Zbl 1013.46015 [17] Y. Cui, H. Hudzik: Some geometric properties related to fixed point theory in Cesàro spaces. Collect. Math. 50 (1999), 277-288. MR 1744077 | Zbl 0955.46007 [18] Y. Cui, H. Hudzik: Packing constant for Cesaro sequence spaces. Nonlinear Anal., Theory Methods Appl. 47 (2001), 2695-2702. DOI 10.1016/S0362-546X(01)00389-3 | MR 1972393 | Zbl 1042.46505 [19] Y. Cui, H. Hudzik, Y. Li: On the García-Falset coefficient in some Banach sequence spaces. Function Spaces. Proc. Int. Conf. (Poznań 1998) (H. Hudzik et al., eds.). Lect. Notes Pure Appl. Math. 213, Marcel Dekker, New York (2000), 141-148. MR 1772119 | Zbl 0962.46011 [20] Y. Cui, C.-H. Meng, R. Płuciennik: Banach-Saks property and property $(\beta)$ in Cesàro sequence spaces. Southeast Asian Bull. Math. 24 (2000), 201-210. DOI 10.1007/s100120070003 | MR 1810056 | Zbl 0956.46003 [21] Y. Cui, R. Płuciennik: Local uniform nonsquareness in Cesàro sequence spaces. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 37 (1997), 47-58. MR 1608225 | Zbl 0898.46006 [22] W. D. Evans, A. Gogatishvili, B. Opic: The reverse Hardy inequality with measures. Math. Inequal. Appl. 11 (2008), 43-74. DOI 10.7153/mia-11-03 | MR 2376257 | Zbl 1136.26004 [23] J. E. Gilbert: Interpolation between weighted $L^p$-spaces. Ark. Mat. 10 (1972), 235-249. DOI 10.1007/BF02384812 | MR 0324393 | Zbl 0264.46027 [24] A. Gogatishvili, R. Ch. Mustafayev: Iterated Hardy-type inequalities involving suprema. Math. Inequal. Appl. 20 (2017), 901-927. [25] A. Gogatishvili, R. Ch. Mustafayev: Weighted iterated Hardy-type inequalities. Math. Inequal. Appl. 20 (2017) 683-728 . DOI dx.doi.org/10.7153/mia-20-45 | MR 3653914 [26] A. Gogatishvili, R. Ch. Mustafayev, L.-E. Persson: Some new iterated Hardy-type inequalities. J. Funct. Spaces Appl. 2012 (2012), Article ID 734194, 30 pages. DOI 10.1155/2012/734194 | MR 3000818 | Zbl 1260.26023 [27] A. Gogatishvili, B. Opic, L. Pick: Weighted inequalities for Hardy-type operators involving suprema. Collect. Math. 57 (2006), 227-255. MR 2264321 | Zbl 1116.47041 [28] A. Gogatishvili, L.-E. Persson, V. D. Stepanov, P. Wall: Some scales of equivalent conditions to characterize the Stieltjes inequality: the case $q< p$. Math. Nachr. 287 (2014), 242-253. DOI 10.1002/mana.201200118 | MR 3163577 | Zbl 1298.26052 [29] K.-G. Grosse-Erdmann: The Blocking Technique, Weighted Mean Operators and Hardy's Inequality. Lecture Notes in Mathematics 1679, Springer, Berlin (1998). DOI 10.1007/BFb0093486 | MR 1611898 | Zbl 0888.26014 [30] B. D. Hassard, D. A. Hussein: On Cesàro function spaces. Tamkang J. Math. 4 (1973), 19-25. MR 0333700 | Zbl 0284.46023 [31] A. A. Jagers: A note on Cesàro sequence spaces. Nieuw Arch. Wiskd., III. Ser. 22 (1974), 113-124. MR 0348444 | Zbl 0286.46017 [32] R. Johnson: Lipschitz spaces, Littlewood-Paley spaces, and convoluteurs. Proc. Lond. Math. Soc., III. Ser. 29 (1974), 127-141. DOI 10.1112/plms/s3-29.1.127 | MR 0355578 | Zbl 0295.46051 [33] A. Kamińska, D. Kubiak: On the dual of Cesàro function space. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 2760-2773. DOI 10.1016/j.na.2011.11.019 | MR 2878472 | Zbl 1245.46024 [34] A. Kufner, L. Maligranda, L.-E. Persson: The Hardy Inequality. About Its History and Some Related Results. Vydavatelský Servis, Plzeň (2007). MR 2351524 | Zbl 1213.42001 [35] A. Kufner, L.-E. Persson: Weighted Inequalities of Hardy Type. World Scientific Publishing, Singapore (2003). DOI 10.1142/5129 | MR 1982932 | Zbl 1065.26018 [36] R. Mustafayev, T. Ünver: Reverse Hardy-type inequalities for supremal operators with measures. Math. Inequal. Appl. 18 (2015), 1295-1311. DOI 10.7153/mia-18-101 | MR 3414598 | Zbl 1331.26032 [37] B. Opic, A. Kufner: Hardy-Type Inequalities. Pitman Research Notes in Mathematics 219, Longman Scientific & Technical, New York (1990). MR 1069756 | Zbl 0698.26007 [38] Programma van Jaarlijkse Prijsvragen (Annual Problem Section). Nieuw Arch. Wiskd. 16 (1968), 47-51. [39] J.-S. Shiue: A note on Cesàro function space. Tamkang J. Math. 1 (1970), 91-95. MR 0276751 | Zbl 0215.19601 [40] P. W. Sy, W. Y. Zhang, P. Y. Lee: The dual of Cesàro function spaces. Glas. Mat., III Ser. 22(42) (1987), 103-112. MR 0940098 | Zbl 0647.46033
Affiliations: Amiran Gogatishvili, Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, e-mail: gogatish@math.cas.cz; Rza Mustafayev, Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, B. Vahabzade St. 9, Baku, AZ 1141, Azerbaijan, and Department of Mathematics Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, e-mail: rzamustafayev@gmail.com; Tuğçe Ünver, Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey, e-mail: tugceunver@gmail.com
