1.1. DEFINITIONS AND BASIC PROPERTIES 9

DEFINITION

1.1.6. Suppose that e+, e- e R, r 0, and let E e S(e+,e_,r).

The space Y(E) consists of all distributions / G S', which have a representation

(1.1.7), converging in 5', satisfying (1.1.8) and

(1.1.10) supp^/o C 5(0,2),

(1.1.11) suppTji C 5(0,2

i + 1

) \ 5(0,2

i _ 1

), i e N.

EXAMPLE.

If £ =

IQ(LP)

with A e R, 0 p,0 oo, or E = Lp(lg) with

0 p o o , 0 # o o , then Definition 1.1.6 is a classical definition of the Besov

spaces Bpe and Lizorkin-Triebel spaces F£e, respectively. See e.g. the books by

J. Peetre [42] and H. Triebel [53], [54]. The spaces obtained in Definition 1.1.5

are in this case denoted BL^e and FL^e. They have been less studied (in the

cases when the two definitions do not give the same spaces; see Proposition 1.1.12

below), but see Netrusov [33], [34].

It is easily seen that if ||/||yz,(£;) is defined by

\\f\\YHE)=inf\\{fi}Zo\\B,

where the infimum is taken over all representations of / as in Definition 1.1.5,

and

||/||Y(# )

is defined analogously, the spaces YL(E) and Y(E) are quasi-normed

spaces, and normed spaces HE is.

We shall prove that the spaces YL(E) and Y(E) are complete, i.e., quasi-

Banach or Banach spaces.

For the proof we need the following lemma of so called Polya-Plancherel type,

which plays an important role for the whole theory. It is a slight modification of

Theorem 1.3.1 in Triebel [53].

LEMMA 1.1.7. Let f e Sf, suppTf c B(0,R), r 0, d 0, and M e N

0

.

Then there is a constant C, independent of R, such that for all multi-indices a with

\a\ M,

R-M\D°f(x + z)\ Y J _ f \f(x + y)\r N

1/r

S?" {l + R\z\)»/*+* -CZPo\a»JB{0,a)(l + R\y\y*C

PROOF. For the reader's convenience we give the proof, essentially following

Triebel [53]. The idea of the proof is due to Peetre [41]. For r 1 a simpler proof

using Holder's inequality is possible.

We first prove that for any multi-index a and any A 0 there is C so that

n i 1

^

R-^\Daf(x

+ z)\ ^ \f(x + z)\

(1.1.13) SUP —! ' ... C SUP -y- „. , ' .

We can assume that R = 1, i.e., that suppTf C 5(0,1); otherwise replace f{x) by

R~N

f(x/R). It is also no restriction to assume x = 0.

Let (pbea function in S such that J-^{i) = 1 on 5(0,1). Then / = / * y, and

Daf

= / * D . For any A 0 there is C such that

PV2/)IC

,

(1 + M)-

A

-"-

1

foraJlj/,