Geodesic
Equivalent norms in spaces of entire functions
Sbornik. Mathematics, Tome 21 (1973) no. 1, pp. 33-55 
V. È. Katsnelson. Equivalent norms in spaces of entire functions. Sbornik. Mathematics, Tome 21 (1973) no. 1, pp. 33-55. http://geodesic.mathdoc.fr/item/SM_1973_21_1_a1/
@article{SM_1973_21_1_a1,
     author = {V. \`E. Katsnelson},
     title = {Equivalent norms in spaces of entire functions},
     journal = {Sbornik. Mathematics},
     pages = {33--55},
     year = {1973},
     volume = {21},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1973_21_1_a1/}
}
TY  - JOUR
AU  - V. È. Katsnelson
TI  - Equivalent norms in spaces of entire functions
JO  - Sbornik. Mathematics
PY  - 1973
SP  - 33
EP  - 55
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1973_21_1_a1/
LA  - en
ID  - SM_1973_21_1_a1
ER  - 
%0 Journal Article
%A V. È. Katsnelson
%T Equivalent norms in spaces of entire functions
%J Sbornik. Mathematics
%D 1973
%P 33-55
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1973_21_1_a1/
%G en
%F SM_1973_21_1_a1

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

It is shown that if $E\subset\mathbf R^n$ is relatively dense with respect to Lebesgue mesure and $p\in(0,\infty)$, then for any entire function $f(z)$ of $n$ complex variables and of exponential type not exceeding $\sigma$ the inequality $$ \int_E|f(x)|^p\,dx_1\dots dx_n\geqslant c\int_{\mathbf R^n}|f(x)|^p\,dx_1\dots dx_n $$ is satisfied, where $c$ is a constant depending only on $\sigma$, $L$, $\delta$ and $p$, but not on $f(z)$, and the integrals on both sides of the inequality converge or diverge simultaneously. Bibliography: 11 titles.

[1] B. P. Paneyakh, “O nekotorykh teoremakh tipa Peli-Vinera”, DAN SSSR, 138:1 (1961), 47–50 | Zbl

[2] B. P. Paneyakh, “O nekotorykh zadachakh garmonicheskogo analiza”, DAN SSSR, 142:5 (1962), 1026–1029 | Zbl

[3] B. P. Paneyakh, “Nekotorye neravenstva dlya funktsii eksponentsialnogo tipa i apriornye otsenki dlya obschikh differentsialnykh operatorov”, Uspekhi matem. nauk, XXI:3 (127) (1966), 75–114

[4] V. Ya. Lin, “Ob ekvivalentnykh normakh v prostranstvakh preobrazovanii Fure finitnykh funktsii”, DAN SSSR, 144:1 (1962), 40–43 | MR | Zbl

[5] V. Ya. Lin, “Ob ekvivalentnykh normakh v prostranstve summiruemykh s kvadratom tselykh funktsii eksponentsialnogo tipa”, Matem. sb., 67 (109) (1965), 586–608 | MR | Zbl

[6] P. Lax, “An inequality for functions of exponential type”, Comm. Pure Appl. Math., 16:2 (1963), 241–246 | DOI | MR | Zbl

[7] V. N. Logvinenko, Yu. F. Sereda, “Ekvivalentnye normy v prostranstve tselykh funktsii eksponentsialnogo tipa”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, Respublikanskii nauchnyi sbornik, no. 19, Kharkov, 1973 | Zbl

[8] B. Ya. Levin, “Subgarmonicheskie mazhoranty i ikh prilozheniya”, Tezisy dokladov Vsesoyuznoi konf. po teorii funktsii kompleksnogo peremennogo (tselye i meromorfnye funktsii i funktsii mnogikh kompleksnykh peremennykh), AN USSR, fiziko-tekhnicheskii in-t nizkikh temperatur, Kharkov, 1971

[9] L. I. Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, izd-vo «Nauka», Moskva, 1971 | MR

[10] N. S. Landkof, Osnovy sovremennoi teorii potentsiala, izd-vo «Nauka», Moskva, 1966 | MR

[11] A. A. Goldberg, I. V. Ostrovskii, Raspredelenie znachenii meromorfnykh funktsii, izd-vo «Nauka», Moskva, 1970 | MR