Abstract
A classical first-order Hardy–Sobolev inequality in Euclidean domains, involving weighted norms depending on powers of the distance function from their boundary, is known to hold for every, but one, value of the power. We show that, by contrast, the missing power is admissible in a suitable counterpart for higher-order Sobolev norms. Our result complements and extends contributions by Castro and Wang (Calc Var 39(3–4):525–531, 2010), and Castro et al. (Comptes Rendus Math Acad Sci Paris 349:765–767, 2011; J Eur Math Soc 15:145–155, 2013), where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in \(L^1\).
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Acknowledgements
The authors are grateful to Yoshinori Yamasaki for some helpful discussions. This research was initiated during a visit of the second named author at the Department of Mathematics and Informatics “U.Dini” of the University of Florence, in the fall-winter semester 2013–2014. He wishes to thank the members of the Department for their kind hospitality. This work was partly funded by: Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2012 “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications” (Grant Number 2012TC7588); GNAMPA of the Italian INdAM (National Institute of High Mathematics); JSPS KAKENHI (Grant Number 25220702).
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Communicated by H. Brezis.