Geodesic
Conditions of smoothness of density for $L_p$-norm distribution of Gaussian random vector
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 161-166

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In this paper sufficient conditions are given for the smoothness indicated in the title. The restrictions on the Gaussian measure are connected only with the dimension of its topological support. 
@article{ZNSL_1987_158_a16,
     author = {N. V. Smorodina},
     title = {Conditions of smoothness of density for $L_p$-norm distribution of {Gaussian} random vector},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {161--166},
     publisher = {mathdoc},
     volume = {158},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a16/}
}
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N. V. Smorodina. Conditions of smoothness of density for $L_p$-norm distribution of Gaussian random vector. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 161-166. http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a16/