Geodesic
The diagonal mapping in general Hardy spaces in the polidisk
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 218-222

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For $p=(p_1,p_2)$, $1$, $i=1,2$, let $H^{p_1,p_2}$ be the Hardy class on the bidisk with mixed norm. We give a complete discription of all holomorphic function $\varphi$ in the unit disk that are representable in the form $\varphi(z)=f(z,z)$, $|z|1$, with $f\in H^{p_1,p_2}$. 
@article{ZNSL_2003_303_a11,
     author = {N. A. Chasova and F. A. Shamoyan},
     title = {The diagonal mapping in general {Hardy} spaces in the polidisk},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {218--222},
     publisher = {mathdoc},
     volume = {303},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a11/}
}
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N. A. Chasova; F. A. Shamoyan. The diagonal mapping in general Hardy spaces in the polidisk. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 218-222. http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a11/