Geodesic
Hardy-Sobolev Inequalities for Hessian Integrals
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 951-967

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Zbl MR
Using appropriate symmetrization arguments, we prove the Hardy-Sobolev type inequalities for Hessian Integrals which extend the classical results, well known for Sobolev functions. For such inequalities the value of the best constant is given. Finally we give an improvement of these inequalities by adding a second term that, involves another singular weight which is a suitable negative power of $\log (|x|)$. 
Gavitone, Nunzia. Hardy-Sobolev Inequalities for Hessian Integrals. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 951-967. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a34/
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     author = {Gavitone, Nunzia},
     title = {Hardy-Sobolev {Inequalities} for {Hessian} {Integrals}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {951--967},
     year = {2007},
     volume = {Ser. 8, 10B},
     number = {3},
     zbl = {1184.35010},
     mrnumber = {2507908},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a34/}
}
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