Adams' inequality and limiting Sobolev embeddings into Zygmund spaces (Q1930870)

The author exhibits sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second-order Sobolev spaces into Lorentz spaces. One of the excellent results reads as follows. Let \(n> 2\), \(q> 1\), and \(\Omega\) be a bounded domain in \(\mathbb{R}^n\). Then, the following embedding holds: \[ \begin{gathered} W^2 L^{n/2,q}(\Omega)\cap W^1_0 L^{n/2,q}(\Omega)\to Z^{q-1/q}(\Omega),\text{ namely,}\\ \| u\|_{Z^{q-1/q}}\leq \gamma_{n,2}\|\Delta u\|_{n/2, q}\end{gathered}\tag{1} \] for any \(u\in W^2 L^{n/2,q}(\Omega)\cap W^1_0 L^{n/2,q}(\Omega)\), where \(\gamma_{n,2}= \omega^{{-2\over n}}_n[n(n-2)]^{-1}\) and \(\omega_n= \pi^{{n\over 2}}/\Gamma({n\over 2}+ 1)\) is the volume of the unit ball in \(\mathbb{R}^n\). Moreover, the constant appearing in equation (1) is sharp. As an easy consequence of this result, the author then gives a new proof of Adams' inequality, which holds under the weaker hypothesis of homogeneous Navier boundary conditions.