Geodesic
Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals
Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 77-89 
P. I. Lizorkin. Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals. Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 77-89. http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/
@article{SM_1970_10_1_a5,
     author = {P. I. Lizorkin},
     title = {Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals},
     journal = {Sbornik. Mathematics},
     pages = {77--89},
     year = {1970},
     volume = {10},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/}
}
TY  - JOUR
AU  - P. I. Lizorkin
TI  - Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 77
EP  - 89
VL  - 10
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/
LA  - en
ID  - SM_1970_10_1_a5
ER  - 
%0 Journal Article
%A P. I. Lizorkin
%T Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals
%J Sbornik. Mathematics
%D 1970
%P 77-89
%V 10
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/
%G en
%F SM_1970_10_1_a5

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

This article gives a necessary and sufficient condition for a $p$-integrable function to have partial derivatives of specified orders which are $p$th power integrable over $R^n$. This condition is expressed using integrals of differences which in general converge conditionally in the $L_p$-norm. We also prove a Fubini theorem for these function spaces. Bibliography: 7 titles.

[1] P. I. Lizorkin, “Obobschennoe liuvillevskoe differentsirovanie i funktsionalnye prostranstva $L_p^r(E_n)$”, Matem. sb., 60(102) (1963), 325–353 | MR | Zbl

[2] P. I. Lizorkin, “Neizotropnye besselevy potentsialy. Teoremy vlozheniya dlya prostranstv tipa Soboleva $L_p^{(r_1,\dots,r_n)}$”, DAN SSSR, 170:3 (1966), 508–511 | MR | Zbl

[3] E. M. Stein, “The characterization of functions arising ias potentials. I”, Bull. Amer. Math. Soc., 67 (1961), 102–104 | DOI | MR

[4] O. V. Besov, P. I. Lizorkin, “Singulyarnye integralnye operatory i posledovatelnosti svertok v prostranstvakh $L_p$”, Matem. sb., 73(115) (1967), 65–88 | MR | Zbl

[5] P. I. Lizorkin, “$(L_p,L_q)$-multiplikatory integralov Fure”, DAN SSSR, 152:4 (1963), 808–811 | MR | Zbl

[6] R. S. Strichartz, “Multipliers on fractional Sobolev spaces”, J. Math. and Mech., 16:9 (1967), 1031–1060 | MR | Zbl

[7] P. I. Lizorkin, “Obobschennoe liuvillevskoe differentsirovanie i metod multiplikatorov v teorii vlozhenii klassov funktsii”, Trudy matem. in-ta im. V. A. Steklova AN SSSR, 105 (1969), 89–167 | MR | Zbl