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ap, then
there exists a constant c such that IIFMILpII`cJIM IHZII holds for all f EH2. Let
(8)
Ko={xIIxI-1}, Kj={xI2j-1=IxI=2j} for
j=1,2,3,...
(9)
Then we have (10)
IIFMI LLIIp=2] II(FM)(x)Ipdx j=o Kj
2 'n
2p 2
j=o
(f I(FM) (x)I2dx)2 Kj
11p
=c
r I2'ap(KjII(FM)(x)12dx)2J j=oL 2
<=c'
j=o
22'8 f I (FM) (x)12 dx Kj
I H2IIp
This proves (8). Step 3. Let MEHZ with swap and supp Mc 521, cf. (5). If p=min (1, p), then (8)
yields F-1MEL, . Now (4) follows from Proposition 1.5.1 and (8). The proof is complete. Remark 1. In Step 2 we proved a bit more than stated : If 0
0. If I ELp then we have f(b-'x) ELP, where w=B1, cf. (1.3.2/6) (which holds for any 9 and the argument given there. Let M(x) E H2 with s >op, where ap has the same meaning as in the theorem. Iteration of (1.3.2./6) (extended to S'(R,,)) yields (F-1MFf) (x) ={F-1[M(b .)
(Ff(b-1 .)) ('))} (bx) .
1. Spaces of Entire Analytic Functions
28
Now, by (4) with to instead of 0 and
instead of /we have
n
IIF-iMFI I LPII =b P IIF-'[M(b ') (Ff(b-1
I LPII
n
cb
(13)
IIM(b') I H2 II -II/(b-1') I LPII=cIIM(b') I H2 II Ii/ I LPII
where c is independent of b and f ELP. 1.5.3.
Convolution Algebras
If I EL, and gEL1 then the convolution f*g is defined by (f*g) (x) = f / (x - y) g(y) dy
(1)
Rn
If c is an appropriate positive number, then (1) can be rewritten as f*g= cF-1[Ff Fg], and one can define f*g by the last formula for more general distributions f and g if F-1 [Ff Fg] makes sense. Proposition. Let Q be a compact subset of Rn and let 0
Remark 2. Let 0=Ba={y I IyI mob} with b>0. If /ELP and 9ELP, then we apply (2) to f(b-1x) and g(b-lx) instead off and g, respectively, cf. Remark 1.5.2/3. Elementary transformations of coordinates, cf. also (1), show that there exists a constant c such that IIf *g I LPII
scbn ("
1) III I LpII IIg I LPII
(3)
holds for all b -0, all f ELP and all gELP .
1.5.4.
Further Multiplier Assertions
Improvement of Theorem 1.5.2. In 2.3.1. we define the spaces BsU,v=Bu,v(Rn). where sER1, 0<uso,, and 0
where opf has the same meaning as in 2.3.1., and it is not hard to see that the right-hand /1 1 ide of (1.5.2/11) is an equivalent quasi-norm in B2,PP
2
. Hence (1.5.2/11) can be re-formu-
ated as follows. If 0
-+ , then there 2 min (p, q)
exists a constant c such that
III''-hMkF/k I Lp(lq)II -_ c sup IIMi(da') I II2II
(1)
' II/k I Lp(lq)II
holds for all systems {/k(x)}k_oELp(lq) and all sequences Proof. First we remark that F/k] is well-defined: By nk (1.5.2/8) we have F-'MkEL1. Furthermore, /kELp, and hence by (1.4.1/3), /kELm. Remark 1.5.1/1 shows that F-'MkF/k makes sense. Next we want to prove that F 'MkFlx (n - z) c sup Ilk (x - z) I II Mk(dk-) sup F-'MkF1k=F-1[Mk
I II?
1+Idkzlr
zERa
(2)
zERn1+Idkzlr
holds where 0 < r < min (p, q) and x Z + r . Here c is independent of x and k (and
the functions Mk and /k). By (1.5.1/2) and the above remarks we have I(F-1MkFfk) (x - z)I t c J I (F- 'Mk) (x-z-y) II lk(y)I dy
(3)
Rn
c sup u E Rn.
But
ilk(u)I
n
(I(F-1Mk) (x-z-y) I (1+ I dk (x-y) r) dy
J 1+Idk (x-u)I 'n
n
n
n
1+ I dk(x-y)Ir
Substituting this estimate into (3) and dividing both sides by 1 + Idkzlr , we find that sup zERn
n
I(F-'MkF/k) (x - z)I n
1+Idkzlr
-=csuP I lk (x - z)In f I(F-'Mk) (y)I (1+Idkyl r)dy zERn1+IdkzIrRn
(4)
32
1. Spaces of Entire Analytic Functions
Using (F-1Mk(dk )) (y)=d; (F-1Mk) (dk ly) and x r +2 , we then obtain n
n
f I (F-'Mk) (y)I (1+Idkyl') dy=Rnf I(F-1Mk(dk')) (y)I (1+IyI') dy Rn sc [ f (1 +IyIx)2I (F-1Mk(dk')) (y)12dy12 `c'IIMk(dk-) I H2II Rn
Putting this estimate into (4), we obtain (2). Now (1) is an immediate consequence of (2), Theorem 1.6.2 and the trivial estimate I(F-1MkF/k) (x)Issup I(F-1MkFfk) (x-z)I ZERn
1+Idkzl'
Remark. The scalar case of (1) is related to (1.5.2/13). However (1.5.2/13) holds also for p=oo and the restrictions for s=x are more natural, cf. also the following subsection.
1.6.4.
Further Multiplier Assertions
The number op in Theorem 1.5.2 and in (1.5.4/2) is natural. The vector-valued counterpart is 1
(mm n
(1, p, q)
l
2/
(1)
This number is smaller than the critical number n
1 + in Theorem 1.6.3. 1 2 min (p, q) The reason for this awkward difference is the use of the technique of maximal functions:
it spoils some constants and this has the above effect, of. the proof of Theorem 1.6.3. However,
a combination of Theorem 1.6.3, a Fourier multiplier theorem of Michlin-Hormander type
for Lp(lq) with 1-p<-, 1- q-oo, and complex interpolation yields the desired result: the assertion of Theorem 1.6.3 is valid for all x>-up,,, cf. 2.4.9.
2.
Function Spaces on R.
2.1.
Introduction
As we mentioned in the preface, this chapter is the heart of the book. We give a thorough description of the two scales and of spaces on R,,. On this basis we switch over later on from R,2 to smooth bounded domains Q in R,,, study the spaces B7,,q(Q) and F,,q(Q) (Chapter 3), and apply the results to regular elliptic boundary value problems (Chapter 4). The spaces B',, and F'P q contain many well-known spaces as special cases: Besov-Lipschitz, Holder, Zygmund, Sobolev, Bessel-potential, Lebesgue and Hardy spaces, and also spaces of functions of bounded mean oscillation. All these spaces have their own defini-
tions, but we give for them (and their generalizations Bp,q and F q) a unified approach via decomposition methods in the framework of Fourier analysis. The advantage is clear: one deals with all these apparently rather diversified spaces from a common point of view. The disadvantage is that the formal definition of B' q and FSP,q looks somewhat complicated at first glance. In order to provide a better understanding of what happens we give in Section 2.2. some (more or less heuristic) motivations. We also add a few historical comments and list the above-mentioned spaces. The rigorous definition of B' ,,(R.,,) and F' is given in 2.3., where the chapter starts in earnest. The rest of the chapter (which
is the main part) deals with properties of these spaces. 2.2.
The Historical Background, Motivations, and Principles
2.2.1.
On the History of Function Spaces
In this Section 2.2. "function space" always means a normed or quasi-normed space of functions and distributions on R,,, which is (with the usual technical terms)
unweighted, isotropic and non-homogeneous. The historical remarks below and the list of spaces given in 2.2.2. deal mostly with spaces of such type. There is hardly any doubt that this type of space was and is the backbone (in the widest sense of the word) of the theory of function spaces from the very beginning up to our time. Corresponding spaces, defined on bounded smooth domains in R,,, will be considered in Chapter 3. Also, in Part II, we deal briefly with their homogeneous, periodic and weighted counterparts. A lot of other spaces, which have attracted much attention in recent years, remain outside the scope of this book, e. g. Lorentz spaces, Campanato-Morrey spaces, Orlicz spaces, Orlicz-Sobolev spaces etc. An excellent survey of the realm of function spaces in this wider sense has been given by A. Kufner, O. John, S. Fucik [1]. Cf. also the books mentioned in the Preface.
The history of function spaces (in the above restricted sense) can be divided into three periods. The first (basic) period starts at the beginning of our century and ends in the mid-thirties. During that time, the classical basic spaces Lp of p-inte-
grable functions and Cm of m times differentiable functions were thoroughly investigated (here m = 0, 1, 2.... ; we set C° = C; definitions of all the spaces
mentioned here may be found in 2.2.2.) The Holder spaces C' with 0-<s*
integer and the Hardy spaces H 7 , with 0- p-- belong to that period and antici pate the second (constructive) period. Originally, the Hardy spaces were introduced 3
Triebel, Function engi.
34
2. Function Spaces on Iin 67-75
systematic period
constructive period
basic period P 1P
Fig. 2.2.1 (45 means the
(complex)
year 1945 etc.)
in order to characterize the boundary values of analytic functions in a disk in the complex plane. The close connection between the Hardy spaces and the methods of Fourier analysis was quite clear from the beginning. The second (constructive) period starts with the papers by S. L. Sobolev [1-3] in 1935-1938, where the spaces that are nowadays called the Sobolev spaces WP`, with m = 0, 1, 2. ..., were introduced. A new too], the theory of distributions, was discovered and new techniques and results (e. g. embedding theorems) were used successfully in order to investigate
partial differential equations. The second period is characterized by more or less speculative constructions of many new spaces with the help of explicit norms (and quasi-norms) on the basis of the above classical spaces. This period culminated at the end of the fifties and in the sixties, and flourishes also in our time. First of all we must note the direct descendants of the above-mentioned classical spaces (which include the Holder spaces, Hardy spaces and Sobolev spaces) : (i) The Zygmund spaces (or classes) Cps. These are extensions of the Holder spaces
C' to values s=1,2,3.... The beginning is marked by A. Zygmund [1], who
discovered the remarkable fact that sometimes second differences 4hf of functions f (cf. 2.2.2. for the definition) are much more useful than first differences 41f. (ii) The Slobodeckij spaces W,, with 0-<s* integer. These constitute an attempt to fill the gaps between the Sobolev spaces LWi,, WP, ... in the spirit of the Holder spaces: use of the first differences 4hf. A second, and even more convincing, reason for the introduction of the spaces W. with fractional s is the fact that these spaces are necessary for the description of the exact traces on hyper-planes in R,, of functions belonging to Sobolev spaces. These spaces were introduced simultaneously by N. Aronszajn [1], L. N. Slobodeckij [1] and L. Gagliardo [1].
(iii) The Besov (or Lipschitz) spaces PA. The combination of the above ideas (Zygnrund's proposal to use second differences 4,/ instead of first differences .,'f, and the replacement of the sup-norm in the Holder spaces by the L9-norm .1
in the Slobodeckij spaces) yields essentially the spaces 4p,q, defined by 0. V. Besov [1, 2]. However two points should be emphasized. Firstly, there was from the very beginning a close connection with problems in approximation theory. We shall return to that point in 2.2.5. Secondly, the spaces AP,,, (sometimes called Nikol'skij spaces) had been investigated in the early fifties by S. M. Nikol'skij [1]. (iv) The Bessel-potential (or Lebesgue or Liouville) spaces HP, where s is real. This
is another attempt to fill the gaps between the Sobolev spaces L5, Wp, WP, .... because HP = W'n if no = 1, 2, 3.... Instead of derivatives and (first and higher) differences (as in the above spaces) the spaces ' are defined via Fourier transforms. This seemed to be reasonable after L. Schwartz discovered in the late forties the im-
portance of the Fourier transform in the framework of the theory of distributions.
2.2. The Historical Background, Motivations, and Principles
The spaces
35
' were introduced by N. Aronszajn, K. T. Smith [1] and A. P.
Calderon [1].
(v) Finally we mention the space BMO of functions of bounded mean oscillation,
cf. F. John, L. Nirenberg [1], and the n-dimensional real version of the Hardy spaces Hr, due to E. M. Stein, G. Weiss [1]. Besides these spaces, many extensions, generalizations, and modifications have
been treated extensively during the last 20 years: anisotropic spaces, weighted spaces, spaces with dominating derivatives, Lorentz spaces, Campanato-Morrey spaces, Orlicz spaces, Orlicz-Sobolev spaces etc. The third (systematic) period begins in the sixties and overlaps heavily with the constructive period. This inflation of the number of spaces demands simple and far-reaching methods which will enable us to deal with function spaces (either in the above restricted or in a wider sense)
from the point of view of a few general principles. The first remarkable success was achieved in the framework of the abstract interpolation theory of Banach spaces and its applications to function spaces (E. Gagliardo, J. L. Lions, E. Magenes,
J. Peetre, A. P. Calderon, 1960-1964). A thorough description of the theory of function spaces from the standpoint of interpolation theory and detailed references
have been given in [I], in particular in Chapter 2. Starting with simple spaces (e. g. L, Cm, WP) one obtains on the basis of general abstract procedures (interpolation methods) a lot of new spaces (e. g. A,q and H,) in a systematic way. The disadvantage of this method is that one needs "endpoint" spaces. At the end of the sixties and the beginning of the seventies, new methods (and essentially improved old methods) of Fourier analysis and of the theory of the so-called maximal inequalities gave a new impetus in the theory of function spaces (J. Peetre,
E. M. Stein, C. Fefferman in 1967-1975, detailed references will be given later on). With the help of these new far-reaching powerful techniques one can deal with all the above-mentioned function spaces in (i)-(v) (cf. also Fig. 2.2.1) from unified point of view and on the basis of very few general principles. All these spaces
are contained in the two scales BP q and FP, q, which are defined in the framework of methods of Fourier analysis. (In the case of the Hardy spaces H, and BMO one needs the so-called "local" versions, h1, and bnm, respectively. H and BMO are special homogeneous spaces of the type Fpq, of. Chapter 5.) Remark. References concerning the spaces B8 q and Fp,q will be given in 2.3.5. Of course, the above historical remarks about the "constructive" function spaces are very rough. Definitions of these spaces are given in the following subsection. In 2.3.5. we describe how the "basic" and "constructive" spaces are related with Bp q and Fy q. More detailed historical references concerning the "basic" and "constructive" spaces may be found in the note sections of S. M. Nikol'skij [3], A. Kufner, 0. John, S. Fu6ik [1, in particular Chapter 8] and [I, in particular Chapter 2]. The theory of the "constructive" spaces (and also of the "basic" spaces) has been treated extensively from several points of view in the books by R. A. Adams [1], 0. V. Besov, V. P. Il'in, S. M. Nikol'skij [1], A. Kufner, 0. John, S. Fucik [1], S. M. Nikol'skij [3], E. M. Stein [1], E. M. Stein, G. Weiss [2] and [I]. Systematic treatments in the framework of interpolation theory or Fourier analysis may be found in J. Peetre [8], [I], [F] and [S].
The Constructive Spaces We summarize the spaces from (i)-(v) of the preceding subsection as constructive
2.2.2.
spaces: We use the notations introduced in 1.2.1. and 1.2.2., in particular R,,, L1,(Rn) with If and O- p=oo (for the sake of clarity we add now the suffix Furthermore, as in those subsections, D" stands for derivative, F and F-1 denote the n-dimensional Fourier transform and its inverse on I
respectively. 3.
36
2. Function Spaces on B.
First we introduce C(R,,) as the set of all complex-valued bounded and uniformly continuous functions on R. If m = 1, 2, 3, ..., we define Cm(Rn)={f I DafEC(R,,) for all
IaI - m}.
(1)
Let C°(Rn) = C(R,,). Here Da are classical derivatives. Of course, Cm(Rn.) endowed with the norm 111
1 Cm(Rn)II = 2] IIDaf I L_(Rn)Il 1a1sm
is a Banach space. Concerning the space C(R,,) (and o'onsequently also for the spaces Cm(R,,,)) one has at least two other candidates: (i) the space of all bounded and con-
tinuous (but not necessarily uniformly continuous) functions on Rn and (ii) the completion of S(Rn) in C(Rn). The first space is larger and the second one is smaller
than C(R,,). There are some reasons (which will not be explained in detail at this moment) that C(R,,) is the best choice. After some "basic" spaces in the sense of 2.2.1. have been defined, we come to the constructive spaces and the Holder spaces. First we describe the spaces and the (in Remark 1, below) we give the necessary explanations. (i) The Holder spaces C8(Rn). If s is a real number, then (2) s = [s] + {s} with [s] integer and 0s {s} < 1 .
If s>-0 is not an integer, then C8(Rn)={f /EC[8](Rn) 111
1
(3)
,
C8(Rn)II =IIf I C[8](Rn)II + 2] lal=[8]
IDaf(x) -Daf(y)I
sup x$y
Ix-yi{8}
xER,y,YERn
(ii) The Zygmuftd spaces @8(Rn). If s is a real number, then
s=[s]-+{s}+ with [s]- integer and 0<{s}+- 1.
(4)
Furthermore, if I (x) is an arbitrary function on Rn and h E R, then (4U)(x) =f(x+h) -f(x), (d/)(x) = dn(di, 1f)(x) where 1 = 2, 3, ...and x E R,,. If s >- 0 then C8(Rn)
(5)
=U Y I IEC[8]-(R.), if I @8(Rn)II=IIf I C[8] (Rn)II sup hI -{8}+ II dhDaf I C(Rn)II <-) .
+I
(6)
O+hER,,
(iii' The Sobolev spaces W '(R,,). (R,,). If 1
L from (18) is larger than Isi+3 number c such that 11211
sup
0
n in (2) and if the natural number min (p, q)
min (p, q)
(ryk*f)(.) ! LL(Rn, lq)I!
+ra+2, then there exists a positive
c sup C(qr)!I f I FP.q(Rn)I)
(20)
0-cr<1
holds for all systems ry={ryk(x)}k-oEO(Rf), all families of systems
ryr={wk(x)}k=oEAL(Rn) with 0
if f E FP.q(Rn) and a>
lq)!!=c1!2k8F-1rykFf
min (p, q)
I Lp(Rn, lq)f
(21)
in (2), where c is independent of ry E O(Rn). Re-
call that the right-hand side of (21) coincides with !!f I Fy q(Rn)l!', cf. (2.3.1/6). However, (21) is an immediate consequence of Theorem 1.6.2. with dk=2k+2 (cf. Definition 2.3.1/1), a=q- and fk=F-1rykFf. r
Step 2. We prove (i). By (5) it is sufficient to show that (22)
lwkf I LP(Rn)i!=c!!F-1rykFf I LP(Rn)II it
if f E BP,q(Rn) and a - in (2), where c is independent of q? E O(Ra) and k = 0, 1, 2, .. . P
If p
then (22) is trivial :
!!rykf 1 Lm(Rn)1!=IIF-1rykFf I L-(Rn)II
If 0
then it is easy to see that ry can be approximated in the norm In particular,
C(ry, L)=li+01 C(TE, L),
C(ryE-rys, L)---0 if
e40 and 8)0
(and similarly, if we start with a family ryrof systems such that sup C (ryr, L+1)Now 0
we assume that ryEI(Rn) is as in the statement of thetheorem and that {99r}ocrc1 is an arbitrary family with sup C (ryr, L+ 1) < . We apply the above approximation procedure to (14) 0
with {ryE,k}x.oE&tL(Rf). This shows that the pointwise limit of (ryF;ki) (x) exists and hence so
2.3. Definition and Fundamental Properties
57
does the pointwise limit of (F-1Te,kF/)(x) (which by modulus is less than or equal to (qie k/) (x)).
By (3), this convergence is also valid in a weighted L. space (the weight is an appropriate polynomial). Hence, (F-1q,E*kF/)(x) converges also in if sl0 and we take the limit as the definition of F ' F/. Then (14) holds for any family of systems rpT={epk(x)}k_o with
sup C (99r, L+1)
2.3.7.
A Fourier Multiplier Theorem
Our next aim is to prove that the spaces B-,,J(R,,) and FP q(Rn) from Definition 2.3.1/2 (and also from Definition 2.3.4) satisfy Criterion 2.2.3, of. also the program which has been described in 2.3.5. We recall Convention 2.3.3, which will be used consistently in the sequel.
Definition. Let m(x) be a complex-valued infinitely differentiable function on R.
Let -o<s<- and 0
with 0
Then m(x) is said to be a Fourier multiplier for A if there exists a positive constant c such that IIF-1mF1 I All =cIIf I Al;
(1)
holds for all f EA. Remark 1. F-1mFf=F-'[mF/] makes sense for any /ES'(R,t) if 7n(x) and all its derivatives have at most polynomial growth. In order to avoid technical difficulties we assume for the present that m(x) alway has this property (later on we shall relax this assumption essentially via limiting procedures).
Remark 2. If i =FPo,q(R,,) with 1 and a Fourier multiplier for the spaces from Definition 2.3.4.
then (1) gives the definition of
Let N be a natural number. For the sake of brevity we introduce IirnlIv= sup sup (1+IxJ2)2 ;Damn(x)l . ]a] sN XERn
(2)
Theorem. Let --<S<- and 0
exists a positive number c such that IIF-1niFf I Alltz- cIImllNll/ I All
(3)
holds for all infinitely differentiable functions m(x) and all /EA. Proo/. Let A=Fp5q(Rn). We use (2.3.6/20) with 92={9?k(x)}7^eE0(R,,). We have
F-1gkF[F-1mF/]=F-1TkmF/ . Now (2.3.6/20) with q:k=mTk. and the obvious fact that I (F kF/)(x)I n y ields (3) with N- si+3 + n +2 . Simil arl y if A = Bp,q( Rn).
(4)
1
min (p, q) Remark 3. The estimates for V are very rough:
V,js1+3+n+2 NHJsj+3
mm
( 7>> 4)
for B8,s(R,),
+n+2 for
Fn . q(Rn)
(5)
(6)
58
2. Function Spaces on R.
Later on we obtain substantial improvement on the basis of Theorem 1.5.2 and (1.5.4/2) for the spaces BP,q(Rn), and on the basis of Theorem 1.6.3 (and the interpolation theory which will be developed in Section 2.4.) for the space F°p,q(Rn). Roughly speaking, instead of the unnatural right-hand sides of (5) and (6) we shall deal with the numbers a from Theorem 1.5.2 (for BP,q(Rn)) and op,q from (1.6.4/1) (for FP,q(Rn)). For details we refer to Subsection 2.4.8. In particular, the parameter 181 in (5) and (6) can be cancelled. The latter fact is also an easy
consequence of the simple lifting property from 2.3.8. by which it follows that the set of all Fourier multipliers for FP q(Rn) is independent of s (similarly for BP,q(Rn)). On the other hand,
the above simple proof of the theorem makes it clear that sufficiently general and strong Fourier multiplier assertions can be obtained via maximal inequalities. The right-hand sides of (5) and (6) (and also the numbers aP and op,q) tend to infinity if py0. This effect (in connection with Fourier multipliers) is known, cf. J. Peetre [6] and (for more general spaces) [F, p. 96], cf.
also C. Fefferman, E. M. Stein [2, p. 191] as far as the Hardy spaces are concerned, which by (2.3.5/4) and (2.3.5/6) are closely related to spaces of FP,q type.
and 1
Next we use (Fq q(Rn))'=Fm q(Rn) with
(7)
+j;=1, cf. Proposition 2.3.4/2. By Theorem
2.3.3, the set S(Rn) is dense in Fi,q (Rn). Then it follows easily from (7) that any Fourier multiplier for Fj,q.(Rn) is also a Fourier multiplier for F",q(Rn).
Corollary. The spaces BP,q(Rn) with FP,q(Rn) with F.,q(Rn) with satisfy Criterion 2.2.3.
0
and
<
Proof. This follows immediately from the above theorem and Remark 4. Remark 5. In [S, p. 32] we proved the following assertion: if a Banach space A has the weak Fourier multiplier property in the sense of Definition 2.2.3 (hence, it satisfies Criterion 2.2.3),
then there exists a natural number N and a positive number c, such that (3) holds for all m(x) and all tEA.
2.3.8.
Lifting Property and Related Equivalent Quasi-Norms
In order to show the usefulness of Theorem 2.3.6 and Theorem 2.3.7 we prove the so-called lifting property and some related assertions. If a is a real number, then the operator IQ is defined by Q
(1) Ia/=F-1 (I+ X11)' Ff, fES'(Rn) . It is well-known that Ia is a one-to-one mapping from S'(Rn) onto itself. Furthermore, the restriction of IQ to S(Rn) is a one-to-one mapping from S(Rn) onto itself. Obviously, IQI, =I,,+,,. We shall say in the sequel that "IQ maps A0 onto A1" instead of "the restriction of I. to A0 maps A0 onto Al".
Theorem. Let -oo<s
59
2.3. Definition and Fundamental Properties
is an equivalent quasi-norm on B8,q(Rn). Furthermore,
jllaf I BP-q (Rn)jj
and
2: I!Dal I Bn,q (Rn)II
II/
Ialsm
aml
BP,q
7=i axj
BP,q (Rn)
(2)
are equivalent quasi-norms on BP,q(Rn). (ii) Let 0-
lalsm
and
IIDa/ I FP,q (Rn)II
lit
I FP.q (Rn)Ii+
aml j=1
I
axj
FP,9 (R,,)
(3)
are equivalent quasi-norms on F1,q(Rn). Proof. Step 1. First we prove the mapping properties of I. If 9' _ {Tk(x) )k=o E q(R,,), then e = {v!k(x).)k=o Eo2L(Rn), where L is an arbitrary natural number and yk(x) _ a
= 2-ka (1 + IxI2) 2 99k(x), cf. Definitions 2.3.1/1 and 2.3.6/1. If /E Fs1,,q(Rn), then Theorem 2.3.6 and the obvious estimate I(F-1ykFf)(x)I --'Ipk f(x) yield a IIIL/
F'n,q (Rn)II =II2(8--)kF-1(1 + 1.12) 2 91kF1 I LP(Rn, lq)II
I
(4)
=II28kF-1VkFf I LP(Rn, lq)Ij =cl!/ I F,,q(Rn)ll
This proves that Ia maps F'',,q(Rn) into FP-Q (Rn). If gEFP,q (Rn), then we have by the same argument f =I_a qEF' q(Rn) and ll/ I FP,q(Rn)II `C119 I FP-'q"q (Rn)Il =CIIIaf I FP-1q"e (Rn)ll
(5)
Because, Ia is a one-to-one mapping in S'(Rn), it is also a one-to-one mapping from F'r,q(Rn) onto FP-Q (Rn). By (4) and (5), IIIQf
1
F'r-Q (Rn)II is an equivalent quasi-
norm in F ,,q(R.). This proves the first part of (ii). The proof of the first part of (i) is the same. Step 2. We prove that the quasi-norms in (3) are equivalent on F'p,5(Rn). If x= (x1, ...,
xn) E Rn, we put xa = if xj 1, where a = (Oct, ... , an) is a multi-index. It is an easy j=1
m
consequence of Theorem 2.3.7 that xa(1+Ix12) 2 is a Fourier multiplier for any space FP,q(Rn), provided that Ial Vim. Consequently, if f EF, q(Rn), then we have lalsm
IIF-1x2F1
IIDa/ I FP Q (Ba)ll =
lalsm
m
m
lal?m
I Fr,4 (Rn)II
IIF-1x2(1+Ixj2) 2FF-1 (1+IxI2)2 Ff I Fv,q (Rn)!I
(6)
eIIImf I F'p- (Rn)li `c'll/ ! FP,q(Rn)II . am
Next we assume that f E F'- (Rn) and axf E Fp,v (Rn) for j =1, ... , n. We wish to 1
show that f belongs to F',,q(Rn). In the next step we shall prove that there exist Fourier multipliers ei(x), ... , e,,(x) for FP.Q (Rn) (in the sense of Theorem 2.3.7) and a positive number c such that 1
n
j=1
!!t
ej(x) xrc (1+Ix12)2
for all xERn
(7)
(if m is even, then ej(x) __ 1 has the desired properties ; if m is odd, then the situation is a bit more complicated). Again by Theorem 2.3.7, (1 + 1x12) 2
r1 + L
1
j=1
ej(x)xr l
= M(x)
2. Function Spaces on R.
60
is also a Fourier multiplier for FP-Q (Rn). Consequently, FP,q(Rn)Il <=c - C'
JIF-1
F-1M(x) FF-1
m (1+xI2)2Ff I F8p,q
(Rn)I(
[1+I ej(x) xrl Ff I F'p.Q (Rn)
7-1 c"Ilf I F'p- (Rn)II + c" 1n IIF-1ei(x) xjLFf I FP,q (Rn)Ii
j=1
We have xrFf =cF
m
Hence, the multiplier property of ej(x) yields
If I FP,q(Rn)II __ C(f I Fv,q (Rn)II +c
j=1
Gm; 8xi FP,g (Rn)
I
(8)
Now (6) and (8) prove that the quasi-norms in (3) are equivalent in FP,q(Rn). In the same way one proves that the quasi-norms in (2) are equivalent in B',q(R,,). Step 3. We prove that there exist Fourier multipliers ei(x), ... , en(x) with the
property (7). If m is even, we choose ej(x) -1. Let m be odd and n = 2. Let St ={xER9 I x1>-0, x2I=tx1},
S9 ={xER, I x1<0, 1X21 -tIx,I},
t =O. Let Ti(x) be an infinitely differentiable function in R,,-{O} with 99j(x)_-O,
\x
O(x)=l if xESi and supp qcS,, where j=1, 2. Let y,(x) be a non-
negative infinitely differentiable function in R2 with e(x) = 1 if fix; ? 1 and 0 ( supp V.
Let of (x) = a(x) [9'1(x) - w2(x)]
In a similar way one constructs a function P2(x). It is not hard to see that e,(x) and e2(x) have the desired properties. This method can be extended to n = 3, 4, 5, .. . If h E Rn then the translation operator Th is defined by (Thf)(x) = f (x+h)
for any regular distribution. Th can be extended in the usual way to S'(Rn): f Recall that if f ES'(Rn) and T ES(R,L), then we put FThf=e"LxFf
and F-1(e;1:x/)=ThF-1/,
(9)
cf. (1.2.1/2) and (1.2.1/3) and the usual extension procedures from S(R,,) to S'(R,,). In particular, if T = {Tj(x) }T=o E O(Rn), then
(F-1pfFThf)(x)=Th(F-1gjFf)(x)=(F-1q'1Fl)(x+h)
.
(10)
Proposition. Let --<s<-and0
Remark 2. Both the above theorem and the proposition can be extended to FW,q(R,L) with In the case of the proposition this follows simply by duality, cf. Proposition 2.3.4/2. As for the theorem, a proof can be based on Remark 2.3.7/4.
2.3. Definition and Fundamental Properties
2.3.9.
61
Diversity of the Spaces BP4q(Rn.) and
The diversity of the spaces A ,q(Rn) and H,(R,t) from 2.2.2. attracted much attention in the history of these spaces, cf. 0. V. Besov [2], K. K. Golovkin [2] and M. H. Taibleson [1, I, Lemma 2]. Cf. also S. M. Nikol'skij [3, 9.7] and [I, 2.12] . Thus it is'natural to ask the same question in our more general framework. Theorem. Let 0
and 0
B"
(ii) Let 0
if and only if so=si, qo=q1 and po=p1 (iii) Let 0<po<= and 0
and po=qo=pl=q1<=. Proof. Step 1. The "if"-part of the theorem is obvious because FP,1,(Rn) =B°r,v(B,,). We prove
the "only if"-part. Step 2. Let V(x) ES(Rn) be an appropriate function with supp Fy, c {x I i s Ixl 52}. If Vj(x) V(21x), then supp F?pj c {x 121 s I xl 52i+1}, cf. (1.3.2/6). Then it follows from the Definitions 2.3.1/1 and 2.3.1/2 (and appropriate choices of V and {q1}EO(Rn)) that in
I LP(Rn)p=2 p I!w I LP(Rn)I!
BP,q(Rn)II =IYj I
liy,j
(1)
if j =0, -1, - 2, ... and 2
IlVi I Br,q(Rn)II 1Vj I F8,q(Rn)( =28' II,v(2'.) I LP(Rn)II =28 p )II=v I LP(Rn)II
(2)
if j =1, 2, 3, . . . Now we assume that two spaces, on the one hand BPo,go(Rn) or FPo,go(Rn) and on the other hand Bp,,q,(Rn) or FPl,gl(Rn) coincide. Then (1) yields po=p1 and (2) (afterwards) so=si.
Step 3. Let xc = (1, 0, 0, ... , 0) and let V(x) ES(Rn) be an appropriate function with supp 1 P c {x
Ix -xol s 1}. Let x' _ (x2, ... , xn). If f is defined by N
(Ff)(x) = Z ajp (x1- 21, x') , j=1
aj complex numbers,
then we have 1
q
IIf I FP,q(Rn)II = IIF-1VI LP(Rn)II
2: 28101 ajl q,
j=1 (modification if q==), of. also formula (3) below. Consequently, qo=qi if FP,go(Rn) =Fg1 g1(Rn).
This assertion and Step 2 prove (ii).
Step 4. In order to prove (i) and (iii) we may assume that so=si=0 and po=pi=p. This follows. from Step 2 and the lifting property from Theorem 2.3.8. Let V be the same function as in the previous step. Let N
(F/) (x) _ Y_ e22x ip (xi-2j, x') j=1
If
is an appropriate system, then we have (F-14,1Ff)(x)
=F-1(e 2'v11V(y1-21, y'))(x)
=ei21(x1+2j)(F-1,)(x1+21,x')
,
(3)
of. (1.2.1/3). Consequently, 1
Ill
Bp,q(Rn)II ".I]F-1V I LP(Rn)II Nq
IIf
FP,q(Rn)II ^'IIF-1y, I Ly(R1.)II Nq
(4)
and 1
(5)
62
2. Function Spaces on R.
Now (i) follows from (4). Finally, if then (4) and (5) yield q1=p. Hence Fp,p(R.)=Bp,p(R.) =Fp,q,()?,i). Then we obtain from (ii) that q0=p. This proves (iii).
Remark. The Besov spaces A4,q(R,) and the Bessel-potential spaces from 2.2.2. are always different (with the exception of p = q = 2). This classical result, which has been treated in the above-cited papers, is a special case of the theorem, cf. (2.3.5/2) and (2.3.5/3). Furthermore, in [I, 2.12] we proved that the spaces Ap,q(R.) and H, with p $2 are not only different from a set-theoretical point of view, but they differ also in their isomorphic structure as Banach spaces. For details we refer to [1, 2.12]. A corresponding result for the more general spaces Bp,q(R.n) and Fps,q(R,a) would be desirable. Finally we mention that the above theorem with - m s - and 1
2.4.
Interpolation
2.4.1.
Preliminaries
Interpolation methods are of some interest for our later considerations. But the central ideas of this chapter are independent of the results of this section. On the other hand, we shall occasionally use some interpolation formulas which are proved in this section. A reader who is not interested in interpolation theory may skip this section, provided he is ready to consult the corresponding results later on, when they will be needed from time to time. Furthermore, we shall not develop the abstract interpolation theory (with the exception of the few remarks below about the real method in quasi-Banach spaces). We prefer to give direct proofs for the main assertions of this section which are not based on general interpolation assertions.
Interpolation methods. As far as concrete interpolation procedures are concerned there exist essentially two methods: The real interpolation method by J.-L. Lions and J. Peetre, and the complex interpolation method by J.-L. Lions, A. P. Calder6n and S. G. Krejn. An extensive treatment of these abstract methods (and many references) has been given in [I], cf. also J. Bergh, J. Lofstrom [1]. Originally, both methods were developed in the framework of Banach space theory. However it is not difficult to see (and well-known nowadays) that the real method (or at least some of its crucial assertions) can be extended immediately to quasi-Banach spaces (concerning the notion of a quasiBanach space of. 1.2.2.). Below we give a brief description of the real method for quasi-Banach spaces and we prove the so-called interpolation property. In contrast to the real method, the complex method cannot be extended immediately from Banach spaces to quasi-Banach spaces. Recently, A. P. Calder6n, A. Torchinsky [1] discovered a modification of the usual complex method, which can be applied
to some concrete quasi-Banach spaces in the framework of Fourier analysis, in particular to the above spaces B,,g(Rn) and FP q(Rn). This modified concrete complex method (and not the usual abstract complex method) will be described in 2.4.4.-2.4.7. (including a weak interpolation property). The real method. The general background of an interpolation method is the following: let 'fC be a linear complex Hausdorff space, and let AD and Al be two complex quasi-Banach spaces such that AD c X and Al c X Recall that " c " indicates a linear and continuous embedding. In particular, the algebraic operations
(addition of elements and multiplication by complex numbers) are the same in
2.4. Interpolation
63
A0, A1, and `l( (as far as they make sense). Let Ao+A1 be the set of all elements a E'C which can be represented as a = ao + a1 with aoEAo and a1 E A1. If 0 < t < and aEAo + A1, then Peetre's celebrated K-functional is given by K(t, a)=K(t, a; A0, A1)=inf (Ilao I A0II +tIIa1 I A1II),
(1)
where the infimum is taken over all representations of a of the form a=ao+a1 with aoEAo and a1EA1.
Definition. Let 0 < 0 < 1. If 0 < q < oo then 1
(2)
(t-°K(t,a)]4dtl q
(A0,
A1)°,q={a I aEAo+A1, Ila I (A0, A1)°,gll=1 f o
/I
If q = oo, then (A0, A1)°,-={a I aEAo+A1, Ila (A0, AI)°,,.II= sup t-°K(t, a)
(3)
Remark 1. These are the well-known real interpolation spaces. One can.prove that (A0, A1)°,q is always a quasi-Banach space. In our concrete cases, which will be treated in the following subsections, this will be clear from the very beginning, because we shall only deal with the spaces and F,,q(Rn).
Remark 2. One can replace the complex spaces A0, Al and 2C by corresponding real spaces. This is immaterial for the real interpolation method.
Proposition (Interpolation property). X, A0 and Al have the above meaning. Let T be a linear operator mapping 'M into itself such that its restriction to Ak yields a linear and bounded mapping from Ak into itself, where k = 0 or 1. Then the restriction of T to (A0, A1) °,q (where 0 < 0 < 1 and 0 < q - oo) yields a linear and bounded mapping from (A0, A1)°,q into itself.
Proof. It is easy to see that (A0, Ai)°,g is a linear space. Hence, the restriction of T to (A0, A1)e,q is a linear operator. Let (4) II Tak I AkII `=IITIIk Ilak I Akli for all akEAk
if k=0 or 1. If a=ao+a1 with aoEAo and a1 EA1, then Ta= Tao+ Tat and (1) gives K(t, Ta)inf (11Tao I A0II+tIITa1 I A1II)
(5)
IIT110 inf (Ilao I A0II+tIITI11 IITII51 Ila1 I A1II) 11 T110 K(tIIT111 11 T110 1, a)
where the infimuun in (5) is taken over all representations a=ao+a1 (we assume that IITIIo>-0). By a simple transformation of coordinates we have IITa I (A0, A1)°,q II`IITII0'-°IIT111 °IIa I (A0,A1)°,gll .
(6)
In particular Ta E (Ao, A1)e,q. The proof is complete. Remark 3. We proved slightly more than we stated. If II Ta I (A0, A,)e,gll SIITII°,q Ila I (A0, A1)°,gI)
then (6) shows that IITII°,gSIlTllo-°IITII° . By a limiting argument this is also valid if IITII0=O.
(7)
Remark 4. The following elementary remark will be useful later on. Let 7C be the above linear complex Hausdorff space and let Ac c Bo c 2C and Al c B1 c JC, where A0, A1, B0 and B1
2. Function Spaces on R.
64
are quasi-Banach spaces. If 0<19< 1 and 0-
(8)
This follows easily from (1), (2) and (3).
2.4.2.
Real Interpolation for the Spaces
and
with Fixed p
The basis for the real interpolation theory for the spaces B,,q(Rn) and FP,q(Ra)
is Theorem 2.3.3, which shows that the Hausdorff space X from 2.4.1. can be identified with
Theorem. Let 0<0-<1, 0
-<so< -<s1< s0+s1 and s=(1-0)80+081. then
(i) If 0
BA,q1(Rn))Cc,q= B$,q(R,,)
(1)
(ii) If 0
si
s
>
Fp,gi(Rn) &,q=B.g(Rn)
(2)
Proof. Step 1. First we prove that (BP -(Rn), BD,m(Rn))©,4C B,,q(R,)
We may assume that
(3)
Let q<-. We have
(0,-)= U
2k(8o-8j))
and Jet-')ghq(t, f; B'-)
BA .,e(Rn))
dt t
0
e
k=0
(4)
2-egk(so-80Kq(2k(sn-s,), f; BP ..(R.), Bp`...(Rn))
c is an appropriate positive constant and f E B' (Rn) +BP,,,(R,,). Let f = fo+f1 with foEBBp,,,(Rn) and f1EBpm(Rn). If {q1j(x)}j=oEO(R,,) then (2.3.1/5) yields (5)
2180IIF-1gPkF1 I Lp(Rn)II c2k8oflF-1TkFf0 I Lp(Rn)II
+c2k(8o-81)2k81lI F-199kF11 I LP(Rn)II
`C Ilfo B ._(Rn)II The infimum with respect to all representations f =f0+f1 yields the K-functional in (4). Consequently, BS1;,..(Rn)II+c2x'(8°_81)Iif1
I
I
2] 2gk8IIF-1wkFf I Lp(R,)IIq
k=0
can be estimated from above by the left-hand side of (4). This proves (3). A small modification yields a corresponding result for q = oo.
Step 2. If 0--r-:q, we prove that B,,,(Rn)C(BP,r(Rn), BP,r(Rn)).9,q.
Again we assume that s0>s1. Because s=:-81, Proposition 2.3.2/2 gives K(t, f; BPr(R,,), Bs'r(Rn))Setllf I BBn,r(Rn)II Sc'tllf I B,,q(Rn)II
(6)
2.4. Interpolation
65
In particular, 1
(ft_eK(t,/; BP r(R0), BP,r(Rn)) dt
`cll/ I BP,q(Rn)II
(7)
I
0
(modification if q = 00). Furthermore, I - J t-°4K4(t, f; BP,r( L0), BP,r(Rn)l i
t
dt (8)
2-°Qk(so-80Kq(2k(so-ail,
2E; C
k=0
/; Bp r(Rn), Bp,r(R0))
If again q' = {(pj(x) }j--o E q(R0), then we put (for the sake of brevity) j =11 F-ipjF/ I
Lp(R0)II. We use the representation /=/0+/1 with k
lo=
2]F-1g5Ff
j=0
and f1=
F-igjF/ i=k+1
The properties of the system p and the scalar case of Theorem 1.6.3 (cf. also Remark 1.5.2/3) show that k+i
11/01 BP r(Rn)IIr j=0
2j8r;
11/11 BP
2'8fr>
j=k
If we use this special decomposition, then (8) yields 4
('k+i
1--c E 2gk8 12] 2sor(j
I
k=0
Let so
x0
s
r q
r + a =1 we have
q
k+1
20(8-so)
1
(Y=O
j
a k+1
2(so-xo)a
2
(=o q
20(1 -81)
(2(8i
kT0
_C
(9)
s1. By Holder's inequality with
k=o
+C
l
j=k
Lj=0
x1
c-c
2eir(j-k)
r
k=j-1
Ci G 274sf j=0
j=k
20(s-xo)+c
2xo4j jq j=0
2xigjpf)
Q
j=k
2xa i j=0
(10)
/I 7,2gk(s-x1)
k=0
1 BP,q(Rn)IIq .
(7), (8) and (10) prove (6) (if q = 00, then the above calculations must be modified in the usual way).
Step 3. In order to prove (1), we use Remark 2.4.1/4 and the embeddings from Proposition 2.3.2/2. If r-- qo and r _-q1, then (3) and (6) yield BP,q(Rn) C (BP r(Rn), B-';,(Rn))s,q C (BP go(Rn), BP q,(Rn))s,q C (BP,m(R.), BP,,o(R.)) s,q c BP,q(Rn)
This proves (1). Finally, (2) follows in the same way from (1) and (2.3.2/9). 5
Triebel, Function engl.
66
2. Function Spaces on R.
Remark 1. The proof is essentially a modification of a corresponding proof in [I, pp. 124/125].
Remark 2. If 1
up Q, then there exists a constant c such that IIF-1MkF/k I L,(]?,,,
sup!JM,(2i')I H2(Ba)II Iltk I Lp(Rn., l9)JI
(8)
holds for all systems {fk(x)}k=oELP(R,,, lq) and all sequences {Mk(x)}k=ocH2(Bn)
Remark 4. The proof of this theorem is essentially the same as the proof of Theorem 2.4.8.
It is based on Theorem I above. The restriction x>op, is more natural than the corresponding restriction in Theorem 1.6.3. Later on we shall use this improvement. In contrast to Theorem 1.6.3, the case q= - is excluded (but this depends probably only on our method).
2.5. 2.5.1.
Equivalent Quasi-Norms and Representations An Orientation
The spaces Bpq(Rn) and Fp,9(Rn) are constructed on the basis of Principle 2.2.4, in particular (2.2.4/13) and (2.2.4/14). We specialized the number a in (2.2.4/13) and (2.2.4/15) to a=2, cf. Definition 2.3.1/2. However on the basis of Theorem 1.6.3 (and its scalar counterpart) it is easy to see that any choice of a>-1 yields the same two scales B' ,,(R,,) and F' ,,(R.) (for all admissible parameters s, p, and q in the sense of Definition 2.3.1/2). Furthermore we proved that all these spaces satisfy Criterion 2.2.3, cf. Corollary 2.3.7. Hence our spaces are meaningful in the sense of the considerations in 2.2.3. and 2.2.4. What about the approximation procedures front 2.2.5., which we described as a further possibility in order to study smoothness properties of functions and distributions? In 2.5.2. and 2.5.3. we shall see that the corresponding spaces, cf. e.g. (2.2.5/7) and (2.2.5/9), coincide with the spaces BP,Q(R.) and (under some restrictions on the para-
meters). In 2.2.2. we gave a list of constructive spaces and claimed in Remark 2.2.2/3 and in 2.3.5. that these spaces coincide with special spaces BP Q(R,,) and This also will be done in this Section 2.5. The Bessel-potential spaces Hp(R,,) (which include the classical Sobolev spaces) will be treated in 2.5.6.
Subsection 2.5.8. deals with the local Hardy spaces h,(R.) (and also with
brrro(R,,)). Concerning the Besov spaces AP9(R,,,) and the Holder-Zygmund spaces Lx(R.n) we describe two rather different approaches. In 2.5.7. we discuss briefly
the spaces and (28(R,,) from the standpoint of interpolation theory. This approach is very convincing and comparatively easy. The disadvantage
is that one needs some statements of the abstract real interpolation method (and we promised in 2.4.1. to use interpolation methods only occasionally). A more direct, but also essentially more complicated approach (which covers also the cases p -I in contrast to the interpolation method in 2.5.7.) is contained in 2.5.12. The Subsections 2.5.9.-2.5.13. are crucial for subsequent considerations.
79
2.5. Equivalent Quasi-Norms and Representations
2.5.2.
Nikol'skij Representations
If 0.
