• Atomic decompositions;
  • pointwise multipliers;
  • diffeomorphisms;
  • Sobolev spaces;
  • Besov and Lizorkin-Triebel spaces;
  • MSC (2010) 46E35


In Chapter 4 of 28 Triebel proved two theorems concerning pointwise multipliers and diffeomorphisms in function spaces equation image and equation image. In each case he presented two approaches, one via atoms and one via local means. While the approach via atoms was very satisfactory concerning the length and simplicity, only the rather technical approach via local means proved the theorems in full generality.

In this paper we generalize two extensions of these atomic decompositions, one by Skrzypczak (see 25) and one by Triebel and Winkelvoss (see 33) so that we are able to give a short proof using atomic representations getting an even more general result than in the two theorems in 28.