Best constants for two families of higher order critical Sobolev embeddings

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into [Formula presented] and those that embed into slightly larger target spaces. Concerning the former, we show that for [Formula presented], [Formula presented] even, one has an optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] (the case [Formula presented] was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For [Formula presented], [Formula presented] and [Formula presented], there exists [Formula presented] and optimal constant [Formula presented] such that [Formula presented]for all [Formula presented] such that [Formula presented], where [Formula presented] is the traditional semi-norm on the space [Formula presented].