Geodesic
Fatou theorem on nontangential limits and questions of extension on the ideal boundary
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 101-109

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The positive answer is given to a question of S. Axler and A. Shields: it is possible to extend continuously on the Shilov boundary $M(L^\infty)$ of the algebra $H^\infty$ arbitrary continuous boundary function in the unit disc having almost everywhere non-tangential limits. So, a description is obtained, in terms of continuous extension on the part of boundary $M(H^\infty)$; of the maximum class of continuous functions satisfying the Fatou theorem on non-tangential limits. 
@article{ZNSL_1991_190_a4,
     author = {O. V. Ivanov},
     title = {Fatou theorem on nontangential limits and questions of extension on the ideal boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {101--109},
     publisher = {mathdoc},
     volume = {190},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a4/}
}
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O. V. Ivanov. Fatou theorem on nontangential limits and questions of extension on the ideal boundary. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 101-109. http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a4/