Geodesic
An equivalent definition of $H^p$ spaces in the half-plane and some applications
Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 69-76 
A. M. Sedletskii. An equivalent definition of $H^p$ spaces in the half-plane and some applications. Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 69-76. http://geodesic.mathdoc.fr/item/SM_1975_25_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Classes of functions that are holomorphic for $\operatorname{Im}z>0$ and satisfy $$ \sup_{0<t<\pi}\int_0^\infty|f(re^{it})|^p\,dr<\infty,\qquad p\in(0,\infty), $$ are considered. It is proved that they coincide with the usual classes $H^p$ in the half-plane. This result is applied to an interpolation problem in $H^p$ in a strip and to the problem of basicity of exponential functions in the space $L^2$ on the line, with exponentially decreasing weight. Bibliography: 8 titles.

[1] V. I. Krylov, “O funktsiyakh, regulyarnykh v poluploskosti”, Matem. sb., 6 (48) (1939), 95–138

[2] P. L. Duren, Theory of $H^p$ spaces, New York, 1970 | MR

[3] K. Gofman, Banakhovy prostranstva analiticheskikh funktsii, IL, Moskva, 1963

[4] M. M. Dzhrbashyan, Integralnye preobrazovaniya i predstavleniya v kompleksnoi oblasti, izd-vo «Nauka», Moskva, 1966

[5] E. Titchmarsh, Vvedenie v teoriyu integralov Fure, Gostekhizdat, Moskva, 1948

[6] H. S. Shapiro, A. L. Shields, “On some interpolation problems for analytic functions”, Amer. J. Math., 83 (1961), 513–532 | DOI | MR | Zbl

[7] A. M. Sedletskii, “Interpolyatsiya v prostranstvakh $H^p$ v poluploskosti”, DAN SSSR, 208:6 (1973), 1293–1295

[8] M. Plancherel, G. Pólya, “Fonctions entières et integrales de Fourier multiples”, Comm. math. Helv., 10:2 (1938), 110–163 | MR