Compact embeddings for weighted Orlicz-Sobolov spaces on \({\mathbb R}^ N\)
We prove the existence of compact embeddings between certain weighted Orlicz-Sobolev spaces on \({\mathbb{R}}^ N\); we also prove the existence of compact trace embeddings between those spaces on \({\mathbb{R}}^ N\) and certain weighted Orlicz-Sobolev spaces defined on lower k-dimensional hyperplanes \({\mathbb{R}}^ k\). The method of proof is similar in both cases: invoking the Donaldson-Trudinger theory, we first devise an approximation scheme to construct a sequence of compact operators \((I_ n)^{\infty}_{n=1}\) between the definition space and the target space of the concerned embedding; we then control the limit lim \(I_ n\) as \(n\to \infty\) in some appropriate operator topology with suitable asymptotic conditions on the weights at infinity. We finally prove that lim \(I_ n\) indeed coincides with the desired embedding. The basic difficulty of this problem is that those asymptotic conditions on the weights needed to control lim \(I_ n\) also interfere with the growth properties of the convex functions which appear in the definition- and in the target space of the embeddings. Accordingly, the standard conditions traditionally used to describe growth properties of convex functions, such as the \(\Delta _ 2\)- and \(\Delta _ 3\)-conditions, ply an important role in our considerations. We discuss several examples in detail.
existence of compact embeddings between certain weighted Orlicz-Sobolev spaces
scientific article; zbMATH DE number 3987987
scientific article