The authors define axiomatically a large class of function (or distribution) spaces on \(N\)-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 capacity, an approximation property known as spectral synthesis, a generalization of Whitney's ideal theorem, and approximation theorems of Luzin (Lusin) type.

Table of Contents

• Introduction. Notation
• A class of function spaces
• Differentiability and spectral synthesis
• Luzin type theorems
• Appendix. Whitney's approximation theorem in \(L_p(\mathbf{R}^N), p>0\)
• Bibliography