We establish new improvements of the optimal Hardy inequality in the half-space. We ﬁrst add all possible linear combinations of Hardy type terms, thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufﬁcient conditions for the validity of Hardy–Sobolev–Maz’ya type inequalities.
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Stathis Filippas, Achilles Tertikas, Jesper Tidblom, On the structure of Hardy–Sobolev–Maz'ya inequalities. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1165–1185DOI 10.4171/JEMS/178