Geodesic
On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions
Eurasian mathematical journal, Tome 8 (2017) no. 2, pp. 47-73.

Voir la notice de l'article provenant de la source Math-Net.Ru

We solve the characterization problem of $L_v^p-L_{\rho}^r$ weighted inequalities on Lebesgue cones of monotone functions on the half-axis for quasilinear integral operators of iterated type with Oinarov's kernels. 
@article{EMJ_2017_8_2_a4,
     author = {V. D. Stepanov and G. E. Shambilova},
     title = {On the boundedness of quasilinear integral operators of iterated type with {Oinarov's} kernels on the cone of monotone functions},
     journal = {Eurasian mathematical journal},
     pages = {47--73},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2017_8_2_a4/}
}
TY  - JOUR
AU  - V. D. Stepanov
AU  - G. E. Shambilova
TI  - On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions
JO  - Eurasian mathematical journal
PY  - 2017
SP  - 47
EP  - 73
VL  - 8
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2017_8_2_a4/
LA  - en
ID  - EMJ_2017_8_2_a4
ER  - 
%0 Journal Article
%A V. D. Stepanov
%A G. E. Shambilova
%T On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions
%J Eurasian mathematical journal
%D 2017
%P 47-73
%V 8
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2017_8_2_a4/
%G en
%F EMJ_2017_8_2_a4
V. D. Stepanov; G. E. Shambilova. On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions. Eurasian mathematical journal, Tome 8 (2017) no. 2, pp. 47-73. http://geodesic.mathdoc.fr/item/EMJ_2017_8_2_a4/

[1] S. Bloom, R. Kerman, “Weighted norm inequalities for operators of Hardy type”, Proc. Amer. Math. Soc., 113 (1991), 135–141 | DOI | MR | Zbl

[2] L. S. Bradley, “Hardy inequalities with mixed norms”, Canad. Math. Bull., 21 (1978), 405–408 | DOI | MR | Zbl

[3] V. I. Burenkov, R. Oinarov, “Necessary and sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesgue space to a Morrey-type space”, Math. Inequal. Appl., 16 (2013), 1–19 | MR | Zbl

[4] P. J. Martín-Reyes, E. Sawyer, “Weighted inequalities for Rieman–Liouville fractional integrals of order one and greater”, Proc. Amer. Math. Soc., 106 (1989), 727–733 | DOI | MR | Zbl

[5] V. G. Maz'ja, Sobolev spaces, Springer, Berlin, 1985 | MR | Zbl

[6] B. Muckenhoupt, “Hardy inequalities with weights”, Studia. Math., 44 (1972), 31–38 | DOI | MR | Zbl

[7] M. Carro, L. Pick, J. Soria, V. D. Stepanov, “On embeddings between clasical Lorentz spaces”, Math. Inequal. Appl., 4:3 (2001), 397–428 | MR | Zbl

[8] A. Gogatishvili, L. Pick, “Discretization and antidiscretization of rearrangement-invariant norms”, Publ. Mat., 47 (2003), 311–358 | DOI | MR | Zbl

[9] A. Gogatishvili, V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions”, Russian Math. Surveys, 68:4 (2013), 597–664 | DOI | MR | Zbl

[10] A. Gogatishvili, R. Mustafayev, L.-E. Persson, “Some new iterated Hardy-type inequalities”, J. Function Spaces Appl., 2013, 734194, 30 pp. | MR

[11] A. Gogatishvili, R. Mustafayev, L.-E. Persson, “Some new iterated Hardy-type inequalities: the case $\theta = 1$”, J. Inequal. Appl., 2013 (2013), 515, 29 pp. | DOI | MR | Zbl

[12] M. L. Goldman, H. P. Heinig, V. D. Stepanov, “On the principle of duality in Lorentz spaces”, Canad. J. Math., 48:5 (1996), 959–979 | DOI | MR | Zbl

[13] A. Kufner, L.-E. Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003 | MR | Zbl

[14] A. Kufner, L. Maligranda, L.-E. Persson, The Hardy inequality-about its history and some related results, Vydavatelsky Servis Publishing House, Pilsen, 2007 | MR | Zbl

[15] R. Oinarov, “Two-sided estimates of the norm of some classes of integral operators”, Proc. Steklov Inst. Math., 204 (1994), 205–214 | MR

[16] R. Oinarov, “Weighted inequalities for a class of integral operators”, Soviet Math. Dokl., 44 (1992), 291–293 | MR

[17] R. Oinarov, A. Kalybay, “Three-parameter weighted Hardy type inequalities”, Banach J. Math. Anal., 2:2 (2008), 85–93 | DOI | MR | Zbl

[18] R. Oinarov, A. Kalybay, “Weighted inequalities for a class of semiadditive operators”, Ann. Funct. Anal., 6 (2015), 155–171 | DOI | MR | Zbl

[19] L.-E. Persson, G. E. Shambilova, V. D. Stepanov, “Hardy-type inequalities on the weighted cones of quasiconcave functions”, Banach J. Math. Anal., 9:2 (2015), 21–34 | DOI | MR | Zbl

[20] L.-E. Persson, G. E. Shambilova, V. D. Stepanov, “Weighted Hardy-type inequalities for supremum operators on the cones of monotone functions”, J. Inequal. Appl., 2016 (2016), 237, 18 pp. | DOI | MR | Zbl

[21] O. V. Popova, “Hardy-type inequalities on the cones of monotone functions”, Sib. Math. J., 53:1 (2012), 187–204 | DOI | MR | Zbl

[22] D. V. Prokhorov, “On weighted inequalities for a Hardy-type operator”, Proc. Steklov Inst. Math., 284 (2014), 208–215 | DOI | MR | Zbl

[23] D. V. Prokhorov, “On a class of weighted inequalities involving quasilinear operators”, Proc. Steklov Inst. Math., 293 (2016), 272–287 | DOI | MR | Zbl

[24] D. V. Prokhorov, V. D. Stepanov, “On weighted Hardy inequalities in mixed norms”, Proc. Steklov Inst. Math., 283 (2013), 149–164 | DOI | MR | Zbl

[25] D. V. Prokhorov, V. D. Stepanov, “Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces”, Sbornik: Mathematics, 207:8 (2016), 135–162 | DOI | MR | Zbl

[26] D. V. Prokhorov, V. D. Stepanov, A. P. Ushakova, “Hardy–Steklov integral operators”, Sovremennye Problemy Matematiki, 22, Steklov institute of Mathematics, 2016, 1–186 (In Rusian) | DOI

[27] E. Sawyer, “Boundedness of classical operators on classical Lorentz spaces”, Studia Math., 96 (1990), 145–158 | DOI | MR | Zbl

[28] G. E. Shambilova, “The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions”, Siberian Math. J., 55:4 (2014), 745–767 | DOI | MR | Zbl

[29] G. Sinnamon, “Embeddings of concave functions and duals of Lorentz spaces”, Publ. Mat., 46 (2002), 489–515 | DOI | MR | Zbl

[30] G. Sinnamon, “Weighted Hardy and Opial type inequalities”, J. Math. Anal. Appl., 160 (1991), 434–445 | DOI | MR | Zbl

[31] G. Sinnamon, V. D. Stepanov, “The weighted Hardy inequality: new proofs and the case $p=1$”, J. London Math. Soc., 54:2 (1996), 89–101 | DOI | MR | Zbl

[32] V. D. Stepanov, E. P. Ushakova, “Kernel operators with variable intervals of integration in Lebesgue spaces and applications”, Math. Inequal. Appl., 13:3 (2010), 449–510 | MR | Zbl

[33] V. D. Stepanov, G. E. Shambilova, “Boudedness of quasilinear integral operators on the cone of monotone functions”, Siberian Math. J., 57:5 (2016), 884–904 | DOI | MR | Zbl

[34] V. D. Stepanov, “Two-weighted estimates for the Riemann–Liouville integrals”, Math USSR-Izv., 36 (1991), 669–681 | DOI | MR | Zbl

[35] V. D. Stepanov, “Weighted inequalities for a class of Volterra convolution operators”, J. London Math. Soc., 45 (1992), 232–242 | DOI | MR | Zbl

[36] V. D. Stepanov, “Integral operators on the cone of monotone functions”, J. London Math. Soc., 48:3 (1993), 465–487 | DOI | MR | Zbl

[37] V. D. Stepanov, “Weighted norm inequalities of Hardy type for a class of integral operators”, J. London Math. Soc., 50:1 (1994), 105–120 | DOI | MR | Zbl

[38] G. Tomaselli, “A class of inequalities”, Boll. Un. Mat. Ital. A, 6 (1969), 622–631 | MR