Geodesic
Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$
Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 1041-1054 
B. I. Golubov. Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$. Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 1041-1054. http://geodesic.mathdoc.fr/item/SM_1997_188_7_a3/
@article{SM_1997_188_7_a3,
     author = {B. I. Golubov},
     title = {Boundedness of {the~Hardy} and {the~Hardy{\textendash}Littlewood} operators in the~spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$},
     journal = {Sbornik. Mathematics},
     pages = {1041--1054},
     year = {1997},
     volume = {188},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_7_a3/}
}
TY  - JOUR
AU  - B. I. Golubov
TI  - Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$
JO  - Sbornik. Mathematics
PY  - 1997
SP  - 1041
EP  - 1054
VL  - 188
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_1997_188_7_a3/
LA  - en
ID  - SM_1997_188_7_a3
ER  - 
%0 Journal Article
%A B. I. Golubov
%T Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$
%J Sbornik. Mathematics
%D 1997
%P 1041-1054
%V 188
%N 7
%U http://geodesic.mathdoc.fr/item/SM_1997_188_7_a3/
%G en
%F SM_1997_188_7_a3

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

The boundedness of the Hardy operator $\mathscr H$ and the Hardy–Littlewood operator $\mathscr B$ are established, respectively, in $\operatorname {Re}H^1$ and the space $\text {\textrm {BMO}}$ of functions of bounded mean oscillation on the real axis $\mathbb R$. Here the space $\operatorname {Re}H^1$ is isomorphic to the Hardy space of single-valued analytic functions $F(z)$ in the upper half-plane satisfying condition (0.3), the Hardy–Littlewood operator $\mathscr B$ is defined in $\mathbb R$ by equality (0.2), and the Hardy operator $\mathscr H$ is defined in $\mathbb R_+$ by equality (0.1) and its value $\mathscr Hf$ is continued to $\mathbb R_-$ as an even (odd) function if the function $f$ is even (odd). For an arbitrary function $f$ one sets $\mathscr H(f)=\mathscr H(f_+)+\mathscr H(f_-)$, where $f_+$ is the even and $f_-$ is the odd component of $f$.

[1] Khardi G. G., Litlvud D. E., Polia G., Neravenstva, IL, M., 1948

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

[3] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[4] Garsia-Cuerva Y., Rubio de Francia J. L., Weighted norm inequalities and related topics, North-Holland, New York–Oxford, 1985 | MR

[5] Titchmarsh E., Vvedenie v teoriyu integralov Fure, Gostekhizdat, M.-L., 1948

[6] Bellman R., “A note on a theorem of Hardy on Fourier constants”, Bull. Amer. Math. Soc., 50 (1944), 741–744 | DOI | MR | Zbl

[7] Golubov B. I., “Ob odnoi teoreme Bellmana o koeffitsientakh Fure”, Matem. sb., 185:11 (1994), 31–40 | MR | Zbl

[8] Hardy G. H., “Notes on some points in the integral calculus”, Messenger of Math., 58 (1928), 50–52 | Zbl

[9] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[10] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[11] Torchinsky A., Real variable methods in harmonic analysis, Academic Press, New York–London–Toronto, 1986 | MR | Zbl

[12] Benett C., Devore R. A., Sharpley R., “Weak $L^\infty$ and BMO”, Ann. of Math., 113 (1981), 601–611 | DOI | MR