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The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces. (English) Zbl 1222.47045
Czech. Math. J. 60, No. 2, 327-337 (2010); corrigendum ibid. 63, No. 4, 1149-1152 (2013).
Summary: The main purpose of this paper is to prove the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application, we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space.

MSC:
47B38 Linear operators on function spaces (general)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:
[1] M.M. Abbasova and R.A. Bandaliev: On the boundedness of Hardy operator in the weighted variable exponent spaces. Proc. of Nat. Acad. of Sci. of Azerbaijan. Embedding theorems. Harmonic Analysis. 13 (2007), 5–17.
[2] E. Acerbi and G. Mingione: Gradient estimates for a class of parabolic systems. Duke Math. J. 136 (2007), 285–320. · Zbl 1113.35105
[3] R.A. Bandaliev: On an inequality in Lebesgue space with mixed norm and with variable summability exponent. Mat. Zametki 84 (2008), 323–333. (In Russian.)
[4] J. Bradley: Hardy inequalities with mixed norms. Canadian Mathematical Bull. 21 (1978), 405–408. · Zbl 0402.26006
[5] D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer: The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238. · Zbl 1037.42023
[6] L. Diening: Maximal function on generalized Lebesgue spaces L p({\(\cdot\)}). Math. Inequal. Appl. 7 (2004), 245–253. · Zbl 1071.42014
[7] L. Diening and S. Samko: Hardy inequality in variable exponent Lebesgue spaces. Frac. Calc. and Appl. Anal. 10 (2007), 1–18. · Zbl 1132.26341
[8] D.E. Edmunds and V. Kokilashvili: Two-weighted inequalities for singular integrals. Canadian Math. Bull. 38 (1995), 295–303. · Zbl 0839.47022
[9] D.E. Edmunds, V. Kokilashvili and A. Meskhi: On the boundedness and compactness of weighted Hardy operators in spaces L p(x). Georgian Math. J. 12 (2005), 27–44. · Zbl 1099.47030
[10] D.E. Edmunds, V. Kokilashvili and A. Meskhi: Bounded and compact integral operators. Math. and Its Applications. 543, Kluwer Acad.Publish., Dordrecht, 2002. · Zbl 1023.42001
[11] D.E. Edmunds, V. Kokilashvili and A. Meskhi: Two-weight estimates in L p(x) spaces for classical integral operators and applications to the norm summability of Fourier trigonometric series. Proc. A. Razmadze Math. Inst. 142 (2006), 123–128. · Zbl 1176.42016
[12] A.D. Gadjiev and I.A. Aliev: Weighted estimates for multidimensional singular integrals generated by a generalized shift operator. Mat. Sbornik 183 (1992), 45–66. (In Russian.)
[13] V. S. Guliev: Two-weight inequalities for integral operators in L p-spaces and their applications. Trudy Mat. Inst. Steklov. 204 (1993), 97–116.
[14] V. S. Guliev: Integral operators on function spaces defined on homogeneous groups and domains in R n. Doctor’s dissertation. Mat. Inst. Steklov. (1994), 1–329.
[15] P. Harjulehto, P. Hästö and M. Koskenoja: Hardy’s inequality in a variable exponent Sobolev space. Georgian Math. J. 12 (2005), 431–442. · Zbl 1096.46017
[16] V. Kokilashvili and A. Meskhi: Two-weight inequalities for singular integrals defined on homogeneous groups. Proc. A. Razmadze Math. Inst. 112 (1997), 57–90. · Zbl 0983.43501
[17] V. Kokilashvili and S.G. Samko: Singular integrals in weighted Lebesgue space with variable exponent. Georgian Math. J. 10 (2003), 145–156. · Zbl 1046.42006
[18] V. Kokilashvili and S.G. Samko: The maximal operator in weighted variable spaces on metric measure space. Proc. A. Razmadze Math. Inst. 144 (2007), 137–144. · Zbl 1284.42058
[19] T. S. Kopaliani: On some structural properties of Banach function spaces and boundedness of certain integral operators. Czech. Math. J. 54 (2004), 791–805. · Zbl 1080.47040
[20] O. Kováčik and J. Rákosník: On spaces L p(x) and W k,p(x). Czech. Math. J. 41 (1991), 592–618. · Zbl 0784.46029
[21] M. Krbec, B. Opic, L. Pick and J. Rákosník: Some recent results on Hardy type operators in weighted function spaces and related topics. Function spaces, differential operators and nonlinear analysis (Frie4ichroda, 1992). 158–184, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993. · Zbl 0803.46036
[22] G. Lu, Sh. Lu and D. Yang: Singular integrals and commutators on homogeneous groups. Analysis Math. 28 (2002), 103–134. · Zbl 1026.43007
[23] R.A. Mashiyev, B. Çekiç, F. I.M amedov and S. Ogras: Hardy’s inequality in power-type weighted L p({\(\cdot\)})(0, spaces. J. Math. Anal. Appl. 334 (2007), 289–298. · Zbl 1120.26020
[24] V.G. Maz’ya: Sobolev Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1985.
[25] B. Muckenhoupt: Hardy’s inequality with weights. Studia Math. 44 (1972), 31–38. · Zbl 0236.26015
[26] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Springer-Verlag, Berlin-Heidelberg-New York, 1983. · Zbl 0557.46020
[27] J. Musielak and W. Orlicz: On modular spaces. Studia Math. 18 (1959), 49–65. · Zbl 0086.08901
[28] H. Nakano: Modulared semi-ordered linear spaces. Maruzen. Co., Ltd., Tokyo, 1950. · Zbl 0041.23401
[29] A. Nekvinda: Hardy-Littlewood maximal operator in L p(x)(R n). Math. Ineq. Appl. 7 (2004), 255–265. · Zbl 1059.42016
[30] B. Opic and A. Kufner: Hardy-Type Inequalities. Pitman Research Notes in Math. ser., 219. Longman sci. and tech., Harlow, 1990. · Zbl 0698.26007
[31] W. Orlicz: Über konjugierte Exponentenfolgen. Studia Math. 3 (1931), 200–212. · Zbl 0003.25203
[32] M. Ružčka: Electrorheological Fluids: Modeling and Mathematical theory. Lecture Notes in Math. 1748. Springer-Verlag, Berlin, 2000. · Zbl 0962.76001
[33] S.G. Samko: Differentiation and integration of variable order and the spaces L p(x). Proc.Inter.Conf. ”Operator theory for Complex and Hypercomplex analysis”. 1994, 203–219. Contemp. Math., 212, AMS, Providence, RI, 1998.
[34] S.G. Samko: Convolution type operators in L p(x). Integ. Trans. and Special Funct. 7 (1998), 123–144. · Zbl 0934.46032
[35] I. I. Sharapudinov: On a topology of the space L p(t)([0, 1]). Mat. Zametki 26 (1979), 613–637. (In Russian.) · Zbl 0437.46024
[36] I. I. Sharapudinov: The basis property of the Haar system in the space L p(t)([0, 1]) and the principle of localization in the mean. Mat. Sbornik 130 (1986), 275–283. (In Russian.)
[37] F. Soria and G. Weiss: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43 (1994), 187–204. · Zbl 0803.42004
[38] Y. Zeren and V. S. Guliyev: Two-weight norm inequalities for some anisotropic sublinear operators. Turkish Math. J. 30 (2006), 329–350. · Zbl 1179.42015
[39] V.V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk Russian 50 (1986), 675–710. (In Russian.)
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