Canceling effects in higher-order Hardy-Sobolev inequalities

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 [CW], and Castro, D'avila and Wang [CDW1, CDW2], where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in $L^1$.