| Title:
|
The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces (English) |
| Author:
|
Bandaliev, Rovshan A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
327-337 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
variable Lebesgue space |
| Keyword:
|
weights |
| Keyword:
|
Hardy operator |
| Keyword:
|
boundedness |
| MSC:
|
42B20 |
| MSC:
|
46B50 |
| MSC:
|
47B38 |
| idZBL:
|
Zbl 1222.47045 |
| idMR:
|
MR2657952 |
| . |
| Date available:
|
2010-07-20T16:41:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140572 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/143621 |
| . |
| Reference:
|
[1] Abbasova, M. M., Bandaliev, R. A.: 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. MR 2719561 |
| Reference:
|
[2] Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems.Duke Math. J. 136 (2007), 285-320. Zbl 1113.35105, MR 2286632, 10.1215/S0012-7094-07-13623-8 |
| Reference:
|
[3] Bandaliev, R. A.: On an inequality in Lebesgue space with mixed norm and with variable summability exponent.Mat. Zametki 84 (2008), 323-333 Russian. Zbl 1171.46023, MR 2473750 |
| Reference:
|
[4] Bradley, J.: Hardy inequalities with mixed norms.Canadian Mathematical Bull. 21 (1978), 405-408. Zbl 0402.26006, MR 0523580, 10.4153/CMB-1978-071-7 |
| Reference:
|
[5] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J.: The maximal function on variable $L^p$ spaces.Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238. MR 1976842 |
| Reference:
|
[6] Diening, L.: Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$.Math. Inequal. Appl. 7 (2004), 245-253. MR 2057643 |
| Reference:
|
[7] Diening, L., Samko, S.: Hardy inequality in variable exponent Lebesgue spaces.Frac. Calc. and Appl. Anal. 10 (2007), 1-18. Zbl 1132.26341, MR 2348863 |
| Reference:
|
[8] Edmunds, D. E., Kokilashvili, V.: Two-weighted inequalities for singular integrals.Canadian Math. Bull. 38 (1995), 295-303. Zbl 0839.47022, MR 1347301, 10.4153/CMB-1995-043-5 |
| Reference:
|
[9] Edmunds, D. E., Kokilashvili, V., Meskhi, A.: On the boundedness and compactness of weighted Hardy operators in spaces $\smash{L^{p(x)}}$.Georgian Math. J. 12 (2005), 27-44. MR 2136721 |
| Reference:
|
[10] Edmunds, D. E., Kokilashvili, V., Meskhi, A.: Bounded and compact integral operators.Math. and Its Applications. 543, Kluwer Acad.Publish., Dordrecht (2002). Zbl 1023.42001, MR 1920969 |
| Reference:
|
[11] Edmunds, D. E., Kokilashvili, V., Meskhi, A.: 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. MR 2294574 |
| Reference:
|
[12] Gadjiev, A. D., Aliev, I. A.: Weighted estimates for multidimensional singular integrals generated by a generalized shift operator.Mat. Sbornik 183 (1992), 45-66 Russian. MR 1198834 |
| Reference:
|
[13] Guliev, V. S.: Two-weight inequalities for integral operators in $L_p$-spaces and their applications.Trudy Mat. Inst. Steklov. 204 (1993), 97-116. MR 1320021 |
| Reference:
|
[14] Guliev, V. S.: Integral operators on function spaces defined on homogeneous groups and domains in $R^n.$ Doctor's dissertation.Mat. Inst. Steklov. (1994), 1-329. |
| Reference:
|
[15] Harjulehto, P., Hästö, P., Koskenoja, M.: Hardy's inequality in a variable exponent Sobolev space.Georgian Math. J. 12 (2005), 431-442. Zbl 1096.46017, MR 2174945 |
| Reference:
|
[16] Kokilashvili, V., Meskhi, A.: Two-weight inequalities for singular integrals defined on homogeneous groups.Proc. A. Razmadze Math. Inst. 112 (1997), 57-90. Zbl 0983.43501, MR 1468962 |
| Reference:
|
[17] Kokilashvili, V., Samko, S. G.: Singular integrals in weighted Lebesgue space with variable exponent.Georgian Math. J. 10 (2003), 145-156. MR 1990694, 10.1515/GMJ.2003.145 |
| Reference:
|
[18] Kokilashvili, V., Samko, S. G.: The maximal operator in weighted variable spaces on metric measure space.Proc. A. Razmadze Math. Inst. 144 (2007), 137-144. MR 2387413 |
| Reference:
|
[19] 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:
|
[20] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k, p(x)}$.Czech. Math. J. 41 (1991), 592-618. MR 1134951 |
| Reference:
|
[21] Krbec, M., Opic, B., Pick, L., Rákosník, J.: 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). MR 1242582 |
| Reference:
|
[22] Lu, G., Lu, Sh., Yang, D.: Singular integrals and commutators on homogeneous groups.Analysis Math. 28 (2002), 103-134. Zbl 1026.43007, MR 1918254, 10.1023/A:1016568918973 |
| Reference:
|
[23] Mashiyev, R. A., Çekiç, B., Mamedov, F. I., Ogras, S.: Hardy's inequality in power-type weighted $L^{p(\cdot)}(0, \infty)$ spaces.J. Math. Anal. Appl. 334 (2007), 289-298. MR 2332556, 10.1016/j.jmaa.2006.12.043 |
| Reference:
|
[24] Maz'ya, V. G.: Sobolev Spaces.Springer-Verlag, Berlin-Heidelberg-New York (1985). Zbl 0727.46017, MR 0817985 |
| Reference:
|
[25] Muckenhoupt, B.: Hardy's inequality with weights.Studia Math. 44 (1972), 31-38. Zbl 0236.26015, MR 0311856, 10.4064/sm-44-1-31-38 |
| Reference:
|
[26] Musielak, J.: Orlicz Spaces and Modular Spaces.Lecture Notes in Math. 1034. Springer-Verlag, Berlin-Heidelberg-New York (1983). Zbl 0557.46020, MR 0724434 |
| Reference:
|
[27] Musielak, J., Orlicz, W.: On modular spaces.Studia Math. 18 (1959), 49-65. Zbl 0099.09202, MR 0101487, 10.4064/sm-18-1-49-65 |
| Reference:
|
[28] Nakano, H.: Modulared semi-ordered linear spaces.Maruzen. Co., Ltd., Tokyo (1950). Zbl 0041.23401, MR 0038565 |
| Reference:
|
[29] Nekvinda, A.: Hardy-Littlewood maximal operator in $L^{p(x)}(R^n)$.Math. Ineq. Appl. 7 (2004), 255-265. MR 2057644 |
| Reference:
|
[30] Opic, B., Kufner, A.: Hardy-Type Inequalities.Pitman Research Notes in Math. ser., 219. Longman sci. and tech., Harlow (1990). Zbl 0698.26007, MR 1069756 |
| Reference:
|
[31] Orlicz, W.: Über konjugierte Exponentenfolgen.Studia Math. 3 (1931), 200-212. Zbl 0003.25203, 10.4064/sm-3-1-200-211 |
| Reference:
|
[32] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical theory.Lecture Notes in Math. 1748. Springer-Verlag, Berlin (2000). MR 1810360 |
| Reference:
|
[33] Samko, S. G.: 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. MR 1486602 |
| Reference:
|
[34] Samko, S. G.: Convolution type operators in $L^{p(x)}$.Integ. Trans. and Special Funct. 7 (1998), 123-144. MR 1658190, 10.1080/10652469808819191 |
| Reference:
|
[35] Sharapudinov, I. I.: On a topology of the space $L^{p(t)}([0,1])$.Mat. Zametki 26 (1979), 613-637 Russian. MR 0552723 |
| Reference:
|
[36] Sharapudinov, I. I.: 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 Russian. MR 0854976 |
| Reference:
|
[37] Soria, F., Weiss, G.: A remark on singular integrals and power weights.Indiana Univ. Math. J. 43 (1994), 187-204. Zbl 0803.42004, MR 1275458, 10.1512/iumj.1994.43.43009 |
| Reference:
|
[38] Zeren, Y., Guliyev, V. S.: Two-weight norm inequalities for some anisotropic sublinear operators.Turkish Math. J. 30 (2006), 329-350. Zbl 1179.42015, MR 2248753 |
| Reference:
|
[39] Zhikov, V. V.: Averaging of functionals of the calculus of variations and elasticity theory.Izv. Akad. Nauk Russian 50 (1986), 675-710 Russian. Zbl 0599.49031, MR 0864171 |
| . |
