Introduction

We present our recently developed concept of envelopes in function spaces – a relatively simple tool for the study of classical, and also rather complicated spaces, say, of Besov or Triebel-Lizorkin type, respectively, in so-called “limiting” situations. This subject is still very new, but in our opinion it has grown to such a degree of maturity that it is now worth a coherent account. The topic is studied in two steps: on a more general level, in Part I, where we do not assume any knowledge of the scales of function spaces mentioned above, and subsequently, in Part II, the results are essentially related to spaces of type Bsp,q and F

s p,q. This also explains the main structure of this book.