Geodesic
Estimates for Restrictions of Monotone Operators on the Cone of Decreasing Functions in Orlicz Space
Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 30-46

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

The restriction of a monotone operator $P$ to the cone $\Omega$ of nonnegative decreasing functions from a weighted Orlicz space $L_{\varphi,v}$ without additional a priori assumptions on the properties of the Orlicz function $\varphi$ and the weight function $v$ is considered. An order-sharp two-sided estimate of the norm of this restriction is established by using a specially constructed discretization procedure. Similar estimates are also obtained for monotone operators over the corresponding Orlicz–Lorentz spaces $\Lambda_{\varphi,v}$. As applications, descriptions of associated spaces for the cone $\Omega$ and the Orlicz–Lorentz space are obtained. These new results are of current interest in the theory of such spaces.
Keywords: monotone operator, weighted Orlicz space, cone of decreasing functions, associated norm, Orlicz–Lorentz class, discretization method. 
M. L. Gol'dman. Estimates for Restrictions of Monotone Operators on the Cone of Decreasing Functions in Orlicz Space. Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 30-46. http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a2/
@article{MZM_2016_100_1_a2,
     author = {M. L. Gol'dman},
     title = {Estimates for {Restrictions} of {Monotone} {Operators} on the {Cone} of {Decreasing} {Functions} in {Orlicz} {Space}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {30--46},
     year = {2016},
     volume = {100},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a2/}
}
TY  - JOUR
AU  - M. L. Gol'dman
TI  - Estimates for Restrictions of Monotone Operators on the Cone of Decreasing Functions in Orlicz Space
JO  - Matematičeskie zametki
PY  - 2016
SP  - 30
EP  - 46
VL  - 100
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a2/
LA  - ru
ID  - MZM_2016_100_1_a2
ER  - 
%0 Journal Article
%A M. L. Gol'dman
%T Estimates for Restrictions of Monotone Operators on the Cone of Decreasing Functions in Orlicz Space
%J Matematičeskie zametki
%D 2016
%P 30-46
%V 100
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a2/
%G ru
%F MZM_2016_100_1_a2

[1] M. A. Krasnoselskii, Ya. B. Rutitskii, Vypuklye funktsii i prostranstva Orlicha, Sovremennye problemy matematiki, Fizmatgiz, M., 1958 | MR | Zbl

[2] L. Maligranda, Orlicz Spaces and Interpolation, Seminários de Matematica, 5, Univ. Estadual de Campinas, Departamento de Matemática, Campinas, 1989 | MR

[3] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR | Zbl

[4] H. Hudzik, A. Kaminska, M. Mastylo, “On the dual of Orlicz–Lorentz space”, Proc. Amer. Math. Soc., 130:6 (2002), 1645–1654 | MR | Zbl

[5] V. I. Ovchinnikov, “Interpolyatsiya v kvazibanakhovykh prostranstvakh Orlicha”, Funkts. analiz i ego pril., 16:3 (1982), 78–79 | MR | Zbl

[6] P. P. Zabreiko, “Ob odnoi interpolyatsionnoi teoreme dlya lineinykh operatorov”, Matem. zametki, 2:6 (1967), 593–598 | MR | Zbl

[7] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, Nauka, M., 1984 | MR | Zbl

[8] G. Ya. Lozanovskii, “O nekotorykh banakhovykh reshetkakh”, Sib. matem. zhurn., 10 (1969), 584–599 | MR | Zbl

[9] G. Ya. Lozanovskii, “O nekotorykh banakhovykh reshetkakh. II”, Sib. matem. zhurn., 12 (1971), 562–567 | MR | Zbl

[10] V. I. Ovchinnikov, “The method of orbits in interpolation theory”, Math. Rep., 1:2 (1984), 349–515 | MR

[11] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., 129, Acad. Press, Boston, MA, 1988 | MR

[12] M. L. Goldman, R. Kerman, “On the principal of duality in Orlicz–Lorentz spaces”, Function Spaces. Differential Operators. Problems of Mathematical Education, Proc. Intern. Conf. Dedicated to 75-th Birthday of Prof. L. D. Kudrjavtsev, Moscow, 1998, 179–183

[13] H. Heinig, A. Kufner, “Hardy operators on monotone functions and sequences in Orlicz Spaces”, J. London Math. Soc. (2), 53:2 (1996), 256–270 | DOI | MR | Zbl

[14] A. Kaminska, L. Maligranda, “Order convexity and concavity in Lorentz spaces $\Lambda_{p,w}$, $0

\infty$”, Studia Math., 160:3 (2004), 267–286 | DOI | MR | Zbl

[15] A. Kaminska, M. Mastylo, “Abstract duality Sawyer formula and its applications”, Monatsh. Math., 151:3 (2007), 223–245 | DOI | MR | Zbl

[16] A. Kaminska, Y. Raynaud, “New formula for decreasing rearrangements and a class of Orlicz–Lorentz spaces”, Rev. Mat. Complut., 27:2 (2014), 587–621 | DOI | MR | Zbl

[17] E. Sawyer, “Boundedness of classical operators on classical Lorentz spaces”, Studia Math., 96:2 (1990), 145–158 | MR | Zbl