Geodesic
Extension of operators defined on reflexive subspaces of~$L^1$ and $L^1/H^1$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 103-123

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Inperpolation theory is used to develop a general pattern for proving extension theorems mentioned in the title. In the case where the range space $G$ is a $w^*$-closed subspace of $L^\infty$ or $H^\infty$ with reflexive annihilator $F$, a necessary and sufficient condition on $G$ is found for such an extension to be always possible. Specifically, $F$ must be Hilbertian and become complemented in $L^p$ $(1$ after a suitable change of density. 
@article{ZNSL_2000_270_a5,
     author = {S. V. Kislyakov},
     title = {Extension of operators defined on reflexive subspaces of~$L^1$ and $L^1/H^1$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {103--123},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a5/}
}
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S. V. Kislyakov. Extension of operators defined on reflexive subspaces of~$L^1$ and $L^1/H^1$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 103-123. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a5/