We define axiomatically a large class of function (or distribution) spaces on
AT-dimensional Euclidean space. The crucial property postulated is the validity of
a vector-valued maximal inequality of Fefferman-Stein type. The scales of Besov
spaces (B-spaces) and Lizorkin-Triebel spaces (F-spaces), and as a consequence
also Sobolev spaces, and Bessel potential spaces, are included as special cases. The
main results of Chapter 1 characterize our spaces by means of local approximations,
higher differences, and atomic representations. In Chapters 2 and 3 these results
are applied to prove pointwise differentiability outside exceptional sets of zero ca-
pacity, an approximation property known as spectral synthesis, a generalization of
Whitney's ideal theorem, and approximation theorems of Luzin (Lusin) type.
Received by the editor May 15, 2004, in revised form November 16, 2004.
2000 Mathematics Subject Classification. Primary 46E35; Secondary 26B35, 31B15, 31C15,
The first author is grateful for a grant from the ESPRC (GR/N12985/01), and for the hos-
pitality of the University of Sussex.
The second author has been the holder of an ESPRC Advanced Fellowship (GR/A00249/01).
He is also grateful for several travel grants from the Swedish Natural Science Research Council
(NFR), and for the hospitality of Linkoping University.