ABSTRACT
We present a few applications of our envelope results. Though our original intention to introduce envelopes of function spaces was certainly of a different flavour, concentrating essentially on some better understanding of rather complicated function spaces, our studies admit quite a number of interesting results: rather direct there is a link to Hardy-type inequalities and limiting embeddings, presented first. With the help of some rather astonishing “lift arguments” for envelopes, as described in Section 11.2, we can finally tackle problems of compactness, that is, asymptotic estimates for entropy and approximation numbers of compact embeddings, with surprisingly sharp results. In this section we present some general results, but otherwise exemplify the possible applications with a few cases only.
