Geodesic
Extension by zero of functions of several variables
Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 351-365

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We obtain sufficient conditions on a domain $G\subset\mathbb R^n$ for functions defined on $G$ to be extendable by zero to the entire space $\mathbb R^n$ with smoothness preserved in an integral norm. 
@article{MZM_1998_64_3_a2,
     author = {O. V. Besov},
     title = {Extension by zero of functions of several variables},
     journal = {Matemati\v{c}eskie zametki},
     pages = {351--365},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a2/}
}
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O. V. Besov. Extension by zero of functions of several variables. Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 351-365. http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a2/