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.
|Translated title of the contribution||An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation (Memoirs of the American Mathematical Society Vol 188 Issue 882)|
|Publisher||American Mathematical Society|
|Number of pages||97|
|Publication status||Published - 2007|
Bibliographical noteOther identifier: 0821839837
Other: Series ISSN: 0065-9266