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An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation
Lars Inge Hedberg, Linköping University, Sweden, and Yuri Netrusov, University of Bristol, UK
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Memoirs of the American Mathematical Society
2007; 97 pp; softcover
Volume: 188
ISBN-10: 0-8218-3983-7
ISBN-13: 978-0-8218-3983-6
List Price: US$66 Individual Members: US$39.60
Institutional Members: US\$52.80
Order Code: MEMO/188/882

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.

• Appendix. Whitney's approximation theorem in $$L_p(\mathbf{R}^N), p>0$$