  In Chapter 4 of [25] Triebel proved two theorems concerning pointwise
multipliers and diffeomorphisms in function spaces of Besov and
Triebel-Lizorkin type. 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 [22]) and
one by Triebel and Winkelvoss (see [30]) 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 [25].
  References:
  [22] L. Skrzypczak. Atomic decompositions on manifolds with bounded geometry.
Forum Math., 10(1):19-38, 1998.
  [25] H. Triebel. Theory of Function Spaces II. Birkh\"auser, Basel, 1992.
  [30] H. Triebel and H. Winkelvo{\ss}. Intrinsic atomic characterizations of
function spaces on domains. Math. Z., 221(1):647-673, 1996.
