Narrow Results

We study the so-called limiting Sobolev cases for embeddings of the spaces , where Ω ⊂ ℝⁿ is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler–Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ${ℝ}^n$ with respect to upper Ahlfors regular measures $\nu$, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $\nu$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

The fractional derivative operator is an extension of the usual derivative operator $D^n$ to an arbitrary (integer, rational, irrational or complex) values of $n$. In this paper, weighted fractional differentiation composition operators from mixed-norm spaces to Zygmund spaces are characterized in terms of boundedness and compactness.