Introduction. Notation
The Sobolev spaces W^n(RiV) of functions, whose derivatives (understood in
the weak or distribution sense) of order not exceeding m belong to the Lebesgue
space Lp, 1 p oo, are an indispensable tool in analysis. They have been
generalized to Bessel potential spaces (fractional order Sobolev spaces) L*, where
s can be any real number, Besov spaces (or B-spaces)
Q, and Lizorkin-Triebel
spaces (or F-spaces) e, and the theory of the B- and F-spaces has been extended
to allow all p 0 and 6 0. The 5-spaces have many applications, and appear
naturally for example as restrictions of Sobolev spaces. The main importance of
the F-spaces is perhaps that they include the spaces
1 p oo, and the Hardy
spaces Hpi 0 p 1, in the same scale of spaces. The first chapter of H. Triebel's
book [54] gives a good survey of the development of the theory.
Usually results for 5-spaces and F-spaces are given separate proofs. The pur-
pose of this work is first (Chapter 1) to give a unified treatment of the scales of
^-spaces and F-spaces by including them in a larger class of function (or distri-
bution) spaces which is defined by simple axioms, and then (Chapters 2 and 3) to
extend some theorems of analysis to this general setting.
In Section 1.1 we first define certain classes of sequence spaces F, which will
be used throughout the paper. The unifying link, which makes the theory possi-
ble, is the assumption that the spaces E satisfy a vector-valued maximal function
inequality of Fefferman-Stein type.
Given F we define two spaces of functions or distributions, denoted YL(E) and
F(F), by means of representations as sums J2 fi °f entire functions of exponential
type with {fi} G F . After proving some of the basic properties of these spaces we
formulate two of the main theorems of the paper, Theorems 1.1.14, and 1.1.15. In
these theorems, which generalize known results for B- and F-spaces, we give several
equivalent definitions of the spaces YL(E) and F(F), including characterizations
by means of local approximations, higher differences, and atomic representations.
The proofs of Theorem 1.1.14, and of the lemmas required, occupy Sections 1.2
and 1.3, and Theorem 1.1.15 is proved in Sections 1.4 and 1.5.
Sections 1.6 - 1.8, which are not necessary for reading the following sections,
are devoted to analogous results for so called homogeneous spaces, corresponding
to the homogeneous Besov and Lizorkin-Triebel spaces which are often referred to
as B- and F-spaces.
In Chapter 2 we apply the results of Chapter 1 to an approximation property
known as spectral synthesis, and to an extension of Whitney's ideal theorem. In
Section 2.1 we define capacities associated to our spaces and prove a theorem about
pointwise Lp-differentiability outside exceptional sets of zero capacity. In Sections
2.2 - 2.4 we give extensions to the present general setting of the results given in
Chapter 10 of the book [4].
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