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0Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 163213,47 pages doi:10.1155/2012/163213
Research Article
Continuous Characterizations of Besov-Lizorkin-Triebel Spaces and New Interpretations as Coorbits
Tino Ullrich
Hausdorff Center for Mathematics, Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Correspondence should be addressed to Tino Ullrich, tino.ullrich@hcm.uni-bonn.de Received 7 April 2010; Accepted 20 July 2010 Academic Editor: Hans Triebel
Copyright © 2012 Tino Ullrich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Grochenig (H. G Feichtinger and K. Grochenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.
1. Introduction
This paper deals with Besov-Lizorkin-Triebel spaces Bpq(Rd) and Fp,q(Rd) on the Euclidean space Rd and their interpretation as coorbits. For this purpose we prove a number of characterizations for homogeneous and inhomogeneous spaces for the full range of parameters. Classically introduced in Triebel's monograph [1, Section 2.3.1] by means of a dyadic decomposition of unity, we use more general building blocks and provide in addition continuous characterizations in terms of Lusin and maximal functions. Equivalent (quasi-) normings of this kind were first given by Triebel in [2]. His proofs use in an essential way the fact that the function under consideration belongs to the respective space. Therefore, the obtained equivalent (quasi-)norms could not yet be considered as a definition or characterization of the space. Later on, Triebel was able to solve this problem partly in his monograph [3, Sections 2.4.2, 2.5.1] by restricting to the Banach space case.
Afterwards, Rychkov [4] completed the picture by simplifying a method due to Bui et al. [5, 6]. However, [4] contains some problematic arguments. One aim of the present paper is to provide a complete and self-contained reference for general characterizations of discrete and continuous type by avoiding these arguments. We use a variant of a method from Rychkov's subsequent papers [7, 8], which is originally due to Stromberg and Torchinsky developed in their monograph [9, Chapter 5].
In a different language, the results can be interpreted in terms of the continuous wavelet transform (see Appendix A.1) belonging to a function space on the ax + b-group G. Spaces on G considered here are mixed norm spaces like tent spaces [10] and Peetre-type spaces. The latter are indeed new and received their name from the fact that quantities related to the classical Peetre maximal function are involved. This leads to the main intention of the paper. We use the established characterizations for the homogeneous spaces in order to embed them in the abstract framework of coorbit space theory originally due to Feichtinger and Grochenig [11-15] in the 1980s. This connection was already observed by them in [11, 14, 15]. They worked with Triebel's equivalent continuous normings from [2] and the results on tent spaces, which were introduced more or less at the same time by Coifman et al. [10] to interpret Lizorkin-Triebel spaces as coorbits. On the one hand the present paper gives a late justification, and on the other hand, we observe that Peetre-type spaces on G are a much better choice for this issue. Their two-sided translation invariance is immediate and much more transparent as we will show in Section 4.1. Furthermore, generalizations in different directions are now possible. In a forthcoming paper, we will show how to apply a generalized coorbit space theory due to Fornasier and Rauhut [16] in order to recover generalized inhomogeneous spaces based on the characterizations given here. Moreover, the extension of the results to quasi-Banach spaces using a theory developed by Rauhut in [17,18] is possible.
Once we have interpreted classical homogeneous Besov-Lizorkin-Triebel spaces as certain coorbits, we are able to benefit from the achievements of the abstract theory in [1115]. The main feature is a powerful discretization machinery which leads in an abstract universal way to atomic decompositions. We are now able to apply this method, which results in atomic decompositions and wavelet bases for homogeneous spaces. More precisely, sufficient conditions in terms of vanishing moments, decay, and smoothness properties of the respective wavelet function are given. Compact support of the used atoms does not play any role here. In particular, we specify the order of a suitable orthonormal spline wavelet system depending on the parameters of the respective space.
The paper is organized as follows. After giving some preliminaries, we start in Section 2 with the definition of classical Besov-Lizorkin-Triebel spaces and their characterization via continuous local means. In Section 3, we give a brief introduction to abstract coorbit space theory, which is applied in Section 4 on the ax + b-group G. We recover the homogeneous spaces from Section 2 as coorbits of certain spaces on G. Finally, several discretization results in terms of atomic decompositions and wavelet isomorphisms are established. The underlying decay result of the continuous wavelet transform and some basic facts about orthonormal wavelet bases are shifted to the appendix.
1.1. Notation
Let us first introduce some basic notation. The symbols r, c, n, n0, and z denote the real numbers, complex numbers, natural numbers, natural numbers including 0, and the integers. The dimension of the underlying Euclidean space for function spaces is denoted by d, its
elements will be denoted by x,y,z,..., and | x | is used for the Euclidean norm. We will use | k11 for the ¿d-norm of a vector k. For a multi-index a and x e rd, we write
and define the differential operators Da and A by
ö|a|i
öx? ■■■dxa/'
d d2 A = V—.
k=1 uxk
If X is a (quasi-)Banach space and f e X, we use \\f | X\\ or simply \\f || for its (quasi-)norm. Operator norms of linear mappings A : X ^ Y are denoted by \\A : X ^ Y\\ or simply by \\A\\. As usual, the letter c denotes a constant, which may vary from line to line but is always independent of f, unless the opposite is explicitly stated. We also use the notation a < b if there exists a constant c > 0 (independent of the context-dependent relevant parameters) such that a< c b. If a < b and b < a, we will write a x b.
2. Function Spaces on Rd 2.1. Vector-Valued Lebesgue Spaces
The space Lp(rd), 0 <p <to, denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm
ll^lLp(Rd)ll = (Id f (x)|Pdx)^ (Z1)
with the usual modification if p = to. The Hilbert space L2(rd) plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions {fk }keI on rd, where I is a countable index set, we put
/ \ 1// ||fk|<Lp(rd))|| = (|||fk|L^r^||^ ,
||fk | Lp(eq,rd)|| =
(Xlfk(x)l^ Vp(rd) ,
where we modify appropriately in the case / = to.
2.2. Maximal Functions
For a locally integrable function f, we denote by Mf (x) the Hardy-Littlewood maximal function defined by
(Mf)(x) = sup^i f (y)\dy, x e rd, (2.3)
xeQ |Q| J Q
where the supremum is taken over all cubes centered at x with sides parallel to the coordinate axes. The following theorem is due to Fefferman and Stein [19].
Theorem 2.1. For 1 <p < to and 1 < q <to, there exists a constant c > 0, such that
\Mfk | Lv(eq/Rd) || < c\\fk | Lv{eq) || (2.4)
holds for all sequences {fk }keZ of locally Lebesgue-integrable functions on rd.
Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function {¥k}keN c S(rd), a tempered distribution f e S'(rd) and a positive number a> 0, we define the system of maximal functions
|(¥k * f )(x + y) d
(^kña(x) = sup" Jk yn, x eRd,k e z. (2.5)
(1 + 2k|y|)
Since (¥fc * f )(y) makes sense pointwise (see the following paragraph), everything is well-defined. However, the value "to" is also possible for (^*kf)a(x). This was the reason for the problematic arguments in [4] mentioned in the introduction. We will often use dilates ¥k (x) = 2kd¥(2kx) of a fixed function ¥ e S(rd), where ¥0(x) might be given by a separate function. Also continuous dilates are needed. Let the operator , t > 0, generate the p-normalized dilates of a function ¥ given by Dtp¥ := t-d/p¥(t-1-). If p = 1, we omit the super index and use additionally ¥t := Dt¥ := D^1 ¥. We define (¥*f )a(x) by
(¥*f) a(x) = sup
\(¥¿ * f)(x + y)\
(i + \y\/t)a
We will refer to this construction later on. It turned out that this maximal function construction can be used to interpret classical smoothness spaces as coorbits of certain function spaces on the group.
2.3. Tempered Distributions and Fourier Transform
As usual, S(rd) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on rd, where the topology is generated by the family of seminorms
||y||fc/ = sup |D>(x)|(1 + |x|)k, y eSe no. (2.7)
xeS,d,\a\i<e
The space S'(rd) is called the set of all tempered distributions on rd and defined as the topological dual of S(rd). Indeed, a linear mapping f : S(rd) ^ c belongs to S'(rd) if and only if there exist numbers k,£ e no and a constant c = Cf such that
If (y)|< Cf sup |Day(x)|(1 + |x|)k (2.8)
xeRd,\a\1<¿
for all y e S(rd). S'(rd) is equipped with the weak*-topology.
The convolution y * f of two integrable functions y,f is defined via the integral
(y * f) (x) = y(x - y)f (y)dy. (2.9)
If y,f e S(rd), then (2.9) still belongs to S(rd). The convolution can be generalized to S(rd) x S'(rd) via (y * f )(x) = f (y(x - ■)), makes sense pointwise, and is a C^-function in rd of at most polynomial growth.
As usual, the Fourier transform defined on both S(rd) and S'(rd) is given by (Ff )(y) := f (Fy), where f eS(rd),y eS(rd), and
Fy(D := (2n)-d/2 ( e-ixiy(x)dx. (2.10)
The mapping F is a bijection (in both cases) and its inverse is given by F-1y = Fy(-■).
In order to deal with homogeneous spaces, we need to define the subset S0(rd) c S(rd). Following [1, Chapter 5], we put
So(rd) = {y e^r^ : Da (Fy)(0) = 0 for every multi-index a e nd}. (2.11)
The set SO(rd) denotes the topological dual of So(rd). If f e S'(rd), the restriction of f to S0(rd) clearly belongs to SO (rd). Furthermore, if P(x) is an arbitrary polynomial in rd, we have (f + P( ■ ))(y) = f (y) for every y e So(rd). Conversely, if f e SO(rd), then f can be extended from S0(rd) to S(rd), that is, to an element of S'(rd). However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify SO(rd) with the factor space S'(rd)/P(rd), since two different extensions differ by a polynomial.
2.4. Besov-Lizorkin-Triebel Spaces
Let us first introduce the concept of a dyadic decomposition of unity, see also [1, Section 2.3.1].
Definition 2.2. (a) Let b(rd) be the collection of all systems {yj (x))-eNo c S(rd) with the following properties:
(i) yj (x) = y(2-jx), j e n,
(ii) supp yo c{x e rd : |x| < 2), supp y c{x e rd : 1/2 <|x|< 2),
(iii) E;c=0 yj (x) = 1 for every x e Rd.
(b) Moreover, the system ((rd) denotes the collection of all systems {y- (x))jeZ c S(rd) with the following properties:
(i) y- (x) = y(2-jx), j e z,
(ii) supp y = {x e Rd : 1/2 <|x|< 2),
(iii) yj = 1 for every x e Rd \{0). Remark 2.3. If we take y0 e S(rd) satisfying
and define y(x) = y0(x)-y0(2x), then the system {yj(x) )jeNo belongs to b(rd) and the system {yj (x))jeZ with y0 := y belongs to ((rd).
Now we are ready for the definition of the Besov and Lizorkin-Triebel spaces. See for instance [1, Section 2.3.1] for details and further properties.
Definition 2.4. Let {yj (x))^0 e b(rd) and (j = F-1yj, j e no. Let further -to < s < to and 0 < q < to.
(i) If 0 <p < to, then
( 1: |x| < 1 |0: |x| > 2
(2.12)
► < TO.
(2.13)
(ii) If 0 <p < go, then
rd) = j f eS'(rd) : ||f | F^rd) ||
(2.14)
¿2jsq|(0; *f)(x)|q ) | Lp(Rd) ;=0 /
In case q = go, we replace the sum by a supremum in both cases.
The homogeneous counterparts are defined as follows. For details, further properties and how to deal with ocurring technicalities we refer to [1, Chapter 5].
Definition 2.5. Let (yj (x)}jeZ e O(rd) and Oj = F-Vj. Let further -g < s < g and 0 < q < g.
(i) If 0 <p < g, then
(2.15)
(ii) If 0 <p < g, then
Fm(r^ = 1 f eS0(r^ : ||f I Frd)||
(2.16)
In case q = g, we replace the sum by a supremum in both cases.
2.5. Inhomogeneous Spaces
Essential for the sequel are functions ®q/ o e s(rd) satisfying
|(FOq)(x)| > 0 on {jxj < 2e}, |(?O)(x)j> 0 on{|< jxj < 2eJ,
(2.17)
for some e > 0, and
Da(FO)(0)= 0 Vjäj< R.
(2.18)
We will call the functions ®0 and ® kernels for local means. Recall that Ok = 2kdO(2k■), k e n, and ¥i = Dt¥.
Theorem 2.6. Let s e r, 0 < p < to, 0 < q < to, a > d/min{p,q} and R + 1 > s. Let further ®0/ ® e S(rd) be given by (2.17) and (2.18). Then the space Fs/q(Rd) can be characterized by
= {/ e S'(rd) : |/ | F^) | < to}, i = 1.....5, (2.19)
= №0 *
= OQ *
/ I Lp(. /1 Lp (
rsq * /) (x)\qj) I Lp(Rd)
■ \(O * /) (x + z)\ 2d (i + jzi/i)a
qdtx 1/q
Oq * / I r
N<i K O * /) (x + z)rdr)"lLp( rd)
^ 2 "''kqj
\(Ok */)(x + z)\ (1 + 2k |z|)a
q\ 1/q
£2skq\(®k */)(x)\M I Lp(rd) k=0 /
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
with the usual modification in case q = œ. Furthermore, all quantities \\f | Fpq(rd)\i i = 1,...,5, are equivalent (quasi-)norms in Fpq(rd ).
For the inhomogeneous Besov spaces, we have the following characterizations.
Theorem 2.7. Let s e r, 0 <p,q <œ, a> d/p, and R + 1 > s. Let further O0, O e S(rd) be given by (2.17) and (2.18). Then the space Bp,q(Rd) can be characterized by
Rd) = {f eS'(Rd) : ||f | ||i< to}, i = 1.....4, (2.25)
||f | Bp,q111 = ||Oo * f | Lp(rd)|| + ^£ f-sq||(®i * f )(x) | Lp(r'
| BP,q|
Oo . f|Lp( rd) | + ( JO rd)
l(Ok * f)(X + :)l|L,(rd) " ""
= I y2skq
qdt ^ 1/q
(1 + 2k |z|)
a | Lp\
| BP,q\\4
£2sfcq||(Ok * f (x) | LpÇrd) |
(2.26)
with the usual modification if q = œ. Furthermore, all quantities \\f|Bp,q(rd)\i, i = 1,...,4, are equivalent quasinorms in Bpq(Rd).
2.6. Homogeneous Spaces
The homogeneous spaces can be characterized in a similar way. Here we do not have a separate function O0 anymore. We put O0 = O.
Theorem 2.8. Let s e r, 0 < p < to, 0 < q < to, a > d/ min(p,q}, and ^ + 1 > s. Let further O e S(rd) be given by (2.17) and (2.18). Then the space ip q(rd) can be characterized by
Fp,q(rd) = {f e SO(rd) : |f | FPqÇrd) |i < œ}, i = 1.....5, (2.27)
(J" t-sq|(O * f )(x)\qd) 1/q|Lp(Kd)
' KOt * f) (x + z)| sup (i+1 zi /t)a
t) iLp (rd )
r dt \ 1/q
0 r*J |(0, . f)(x + |L„(ed)
KOk * f (x + z)| (1 + 2k | z | )a
q\ 1/q
|Lp( rd)
Ë 2skq |(®k * f (x)|M | Lp(rd)
(2.28)
with the usual modification if q = œ. Furthermore, all quantities \\f jf® (R^)^, i = 1,...,5, are
equivalent quasinorms in Fp, q
For thing characterizations.
Theorem 2.9. Let s e r, 0 <p,q <to, a> d/p, and R + 1 > s. Let further ® e S(rd) be given by (2.17) and (2.18). Then the space Bp,q(Rd) can be characterized by
BP,q(Rd) = {f eS>(:
&p,q(Rd) ||.< , i = 1.....4,
(2.29)
B pq 11
([t-q ||(Ot * f )(x)|Lp( Rd) ff)"',
|( Ot * f) (x + z)| (1 + |z|/t)
a ^ J-p
| Lp rt
qdt \ 1/q
BBP,q 13 = ( E 2skq
sup K Ok fk(X. +az)| | Lp (rd )
(1 + 2k |z|)
q\ 1/q
Bil = [ %2skq |(Ok * f (x) | Lp(Rd
(2.30)
with the usual modification if q = œ. Furthermore, all quantities \\f j Bp/q(Rd )\i, i = 1,...,4, are equivalent quasinorms in B;q(
Remark 2.10. Observe, that the (quasi-)norms ||- | ip/q(Rd)||3 and ||- | Fsq(Rd)||3 are characterizations via Lusin functions, see [3, Section 2.4.5] and [1, Section 2.12.1] and the references given there. We will return to it later when defining tent spaces, see Definition 4.1 and (4.3).
2.7. Particular Kernels
For more details concerning particular choices for the kernels O0 and O, we refer mainly to Triebel [3, Section 3.3].
The most prominent nontrivial examples (besides the one given in Remark 2.3) of functions O0 and O satisfying (2.17) and (2.18) are the classical local means. The name comes from the compact support of O0, O, which is admitted in the following statement.
Corollary 2.11. Let p, q, s as in Theorem 2.6. Let further k0, k0 e S(rd) such that
Fko(0), Fk0(0) = 0 (2.31)
and define
O0 = k0, O = ANk0, (2.32)
where N e n such that 2N > s. Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize Fspq
Corollary 2.12. Let p,q,s as in Theorem 2.6. Let further O0 e S(rd) be a radial function such that FO0 is non-increasing and atisfying
FO0(0) = 0, D"FO0(0) = 0 (2.33)
for 1 <|a|! < R, where R + 1 > s. Define
O(x) = O0(x) - 2dO)(2)• (2.34)
Then (2.23), and (2.24) characterize Fp,q(rd).
2.8. Proofs
We give the proof for Theorem 2.6 in full detail. The proof of Theorem 2.8 is more or less the same, even a bit simpler. We refer to the next paragraph for the necessary modifications. The proofs in the Besov scale are analogous, so we omit them completely. The strategy is a modification of Rychkov [4], where he proved the discrete case, that is, that (2.23) and (2.24) characterize Fp,q(rd). However, Hansen [21, Remark 3.2.4] recently observed that the arguments used for proving (23) and (23') in [4] are somehow problematic. The finiteness
of the Peetre maximal function is assumed. But this is not true in general under the stated assumptions. Consider for instance in dimension d = 1 the functions
¥0(0 = ¥1(0 = (2.35)
and, if a > 0 is given, the tempered distribution f (t) = |t|n with a < n e n. Then (¥f )a(x) is infinite in every point x e r. The mentioned incorrect argument was inherited to some subsequent papers dealing with similar topics, for instance [22-24]. Anyhow, the stated results hold true. An alternative strategy, in order to avoid the problematic Lemma 3 in [4], is given in Rychkov [7] as well as [8]. A variant of this method, which is originally due to Stromberg/Torchinsky [9, Chapter V], is also used in our proof below.
We start with a convolution-type inequality which will be often needed below. The following lemma is essentially [4, Lemma 2].
Lemma 2.13. Let 0 < p,q < to and, 6 > 0. Let {gk }keN0 be a sequence of nonnegative measurable functions on rd and put
Ge(x) = ^ 2-|k^|6gk(x), x e rd, t e (2.36)
Then there is some constant C = C(p, q, 6), such that
|| {G}e | ^(Lp(Rd))|| < C||{gk}k | ¿q(Lp(rd))||,
(2.37)
||{Gt}e | Lp(tq,rd)|| < C||{gk}k | Lp(tq,rd)||
hold true.
Proof of Theorem 2.6. The strategy of the proof is as follows. First, we prove the equivalence of the "continuous" characterizations (2.20) and (2.21). The next step is to build the bridge between the "continuous" (2.21) and the "discrete" characterization (2.23) and to change from the system (O0, O) to a system (¥0, ¥). The equivalence of (2.23) and (2.24) goes parallel to (2.20) and (2.21). This was the original proof by Rychkov in [4]. So, up to this point, we have that (2.20), (2.21), (2.23), and (2.24) generate the same space for every chosen functions (O0,O) satisfying (2.17) and (2.18), namely, F^R). Indeed, Definition 2.4 can be seen as a special case of (2.24).
Step 1. We are going to prove the following inequalities:
||f|Fp,q||2 ^ 11f | Fp,q111 ^||f|Fp,q||2 (2.38)
for every f eS'(rd).
Substep 1.1. Put yo = FOq and ye = (F®)(2-e■) if e > 1. Because of (2.17), it is possible to find functions fo,f e S(rd) withsupp fo c{ e rd : ||| < 2e},supp f c{ e rd : e/2 < |£| < 2e} and fe(x) = f (2-ex) such that
X ye(I) ■ fe(|) = 1. (2.39)
We need a bit more. Fix a 1 < t < 2. Clearly, we have also
X ye (t£) ■ fe (t£) = 1 (2.40)
for all | e rd. With ¥0 = F-1f0 and ¥ = F-1f, we obtain then
g = E (¥m)t * (®m)t * g.
(2.41)
The 2e dilation gives then
ge = X (¥m)t2-e * (Om)t2-e * ge (2.42)
for every g e S'(rd), where ge(n) = g(n-e), n e S(rd). Obviously, we can rewrite (2.42) to obtain
g = £ (^m)tl-e * (®m)t2-e * g (2.43)
for all g e S'(rd). Let us now choose g = (Oe)t * f. This gives the final version of the convolution identity
(®e)t * f = X (®e)t * (^m)t2-e * (®m)t2-e * f. (2.44)
For m,e e no we define
f 2ed®o(2ex): m = 0, Am/ (x) = ^ x e rd. (2.45)
^ ®e(x): m> 0,
Clearly, we have
(Oe)t * (Om)t2-e = (Am,e)t * (®m+e)t.
(2.46)
Plugging this into (2.44), we end up with the pointwise representation
((^)t * f )(y) = E ((^2-^ * (Am,e)t * (®m+e)t * f )(y)
= E [Cm)2^t * (Am/)t] * ((®m+e)t * f )(y) (2 47)
meN0 ^ ' '
= E I [(¥m)2-^t * (Am/)t](y - Z) ■ ((®m+e)t * f ) (z)dz
meNo -J Ki
for all y e rd.
Substep 1.2. Let us prove the following important inequality first. For every r > 0 and every N e n0, we have
|((®e)t * f)(x)\r < c V 2-fcNr2(k+e)d f K* f)(y)Jd (2.48)
•'Rd (1 + 2ex-vV
where c is independent of f e S(rd), x e rd, and t e n0.
The representation (2.47) will be the starting point to prove (2.48). Namely, we have for y e rd
|((G>/)t * f)(y)|< V f I((¥m)t2-/ * (Am/)t)(y - Z)|-|((®m+/)t * f)(z)|d.
meNo ^ Rd
. V S f K^m+e)t * f )(Z)| dZ
meNo J Rd (1 + 2%- z|)N
(2.49)
Sm,/,t = sup| [(¥m)2-et * (Am/)t] (x) | • (1 + 2e\x\) . (2.50)
Elementary properties of the convolution yield (compare with (2.84))
Sm,e,t = -d- sup
' xeRd
(¥m * (Am/)2e) ( —
(1 + 2e\x\) N
= —sup | ( fm * nm/) (x) | ■ (1 + \tx\)N
(2.51)
f®(x): e > 0, m> 0, Um,e (x) = } (2.52)
[®o(x): otherwise.
Lemma A.3 yields
Sm,e,t < 2ed2-mN, (2.53)
which we put into (2.49) to obtain
| ((®e)t * f )(y) | < Cn Z 2-m^ 2 V ^)t *fN ' dz. (2.54)
meNo Jrd (1 + 2e |y - z|)
We prefer the strategy used by Rychkov in [7, Theorem 3.2] and [8, Lemma 2.9], which is a variant of the Stromberg/Torchinsky technique introduced in [9, Chapter V].
Let us continue by replacing e by k + e in (2.54) and multiply on both sides with 2-kN. Then we can estimate
2-kN\ ((Ok+e)t * /)(y) \ < CN x 2-kN2-mN f 2-
mlNo JRd (1 + 2k+e|y - Z|)
2(m+k+e)d \((Om+k+e)t * /) (z) \
< Cn x 2-<-«n f 2"-+MI «°m+k.»)t /(z)l dz (2.55) Ä, Jr." (1 + 2<|y - z |)N
= Cn x W 2'-e'"K(°m+'). *N<z)\dz,
ict+ N JR
mek+ No JR" (1 + 2e | y - z|)
2(m+e)" \((Om+e). * /) (z)\ meNo JR" (1 + 2e\y - z\)N
< C^ 2-mN | ,.K ,.N ldz. (2.56)
Next, we apply the elementary inequalities
(1 + 2e\y - z\) ■ (1 + 2e\x - y\) > (1 + 2e|x - zj),
/ \ N(1-r)
\((Om+e)t * /)(z)\ < \((Om+e)t * /)(z)\r( 1 + 2e|x - zj) (2.57)
\((Om+e). * /)(y)\1-r
N(1-r)
(1 + 2e|x - y|) where 0 <r < 1. We define the maximal function
MerN(x, t) = sup sup 2-kN K(°k+e)t * fKy)N 1, x e rd, (2.58)
keNo yeRd (1 + 2e|x - y|)
and estimate
(x,t) < Cn 2 2-™~f 2'......1 * Ndz (2.59)
mlNo V (1 + 2^|x - z|)N
< Cn 2 2-mN^2-mNsup ^^)t * f WIN ^
meNo V yeRd (1 + 2e\x - y\) /
N ^ , - , „,
(2.60)
2(m+t)d |((Om+t)t * f (2)|r -r;-d2-
(1 + 2t|x - 2^Nr
Observe that we can estimate the term (•••)1-r in the right-hand side of (2.60) by Mt,N(x,t)1-r. Hence, if Mt,N(x,t) < to we obtain from (2.60)
2(m+e)d \((0m+£) * f) (Z)\'
meNo jRd (i + 2^\x - z|)N
M£rN(x,t)r < Cn 2 2-mN^ 2 Zm+eH dz, (2.61)
m n Jrj
where CN is independent of x, f, t, and t e [1,2]. We claim that there exists Nf e n0 such that Mt,N(x, t) < to for all N > Nf. Indeed, we use that f e S'(rd), that is, there is an M e n0 and cf > 0 such that
| ((Ok+t)t * f )(y)|< Cf sup sup|D^Ok+e(2)| ■ (1 + |y - 2|)M (2.62)
|a|1 <M 2eRd
Assuming N > M, we estimate as follows:
|((Ot)t * f )(x)| < Me,N (x,t)
< c sup sup 2-kN I ((Ok+t). * -OWI
fceNo yeRd + \x - y\2)
\Dajk+£(z)\ ■ (1 + \y - z\)
< c sup sup 2-kN 2(k+^)(M+d)sup sup 1 y\ N
keNo yeRd zeRd |a|1<M (1 + |x - y|)
< c 2^(M+d)sup sup sup \ Dajk+e(z) \ (1 + |x - z|)N,
keNo zeRd |a|1<M
(2.63)
where we again used the inequality (compare with (2.57))
1 + |y - 2| < (1 + |x - y|)(1 + |x - 2|) (2.64)
and have set
®o(f) : e = 0,
Ye (t)
(2.65)
O(t) : e> 0.
Hence jk+£ gives us only two different functions from S(rd). This implies the boundedness of M£N(x,t) for x e rd if N > M = Nf. Therefore, (2.61) together with (2.63) yield (2.48) with c = CN, independent of x, f, and £, for all N > Nf. But this is not yet what we want. Observe that the right-hand side of (2.48) decreases as N increases. Therefore, we have (2.48) for all N e no but with c = c(f) = CNf depending on f. This is still not yet what we want. Now we argue as follows: starting with (2.48) where c = c(f) and N e n0 arbitrary, we apply the same arguments as used from (2.55) to (2.56), switch to the maximal function (2.58) with the help of (2.57), and finish with (2.61) instead of (2.59) but with a constant that depends on f. But this does not matter now. It is Important, that a finite right-hand side of (2.61) (which is the same as rhs(2.48)) implies M£,N(x,t) < to.
We assume rhs (2.48) < to. Otherwise, there is nothing to prove in (2.48). Returning to (2.60) and having in mind that now M£,N(x, t) < to, we end up with (2.61) for all N and CN independent of f. Finally, from (2.61) we obtain (2.48) and are done in case 0 < r < 1.
Of course, (2.48) also holds true for r > 1 with a much simpler proof. In that case, we use (2.54) with N+1 instead of N and apply Holder's inequality with respect to 1 / r+1/r' = 1 first for integrals and then for sums.
Substep 1.3. The inequality (2.48) implies immediately a stronger version of itself. Using (2.57) again, we obtain for a < N and £ e n
We proved that the inequality (2.66) holds for all t e [1,2] where c > 0 is independent of t. If we choose r < min{p, q}, we can apply the norm
(2.66)
(2.67)
on both sides and use Minkowski's inequality for integrals, which yields
dy. (2.68)
If ar > d, then we have
ge(y) =
and we observe
/2 )a(-)Td^
(1 + 2e|y|)a
e li(rd),
(2.69)
< c ^ 2-kNr 2kd2esr
ge *(/2 \2es((Ok+e)t * / )(-)|q^
(2.70)
Now we use the majorant property of the Hardy-Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]) and continue estimating
f|2es(®2-et/)a(x)r d) r/ < c X 2ers2k(-Nr+d)M ( f|((®k+e)t * /)0|q ^
•H ' W keNo V1 t
An index shift on the right-hand side gives
/2 l^-et/)a(x)ff
= c ^ 2ers2(k-e)(-Nr+d)m kee+ N0
= C ^ 2(e-k)(Nr-d+rs)2krsM kee+ N0
|il((®k)t * /)(-)|qy !2i((®k )t * / )(-)iq^
Choose now 1/a <r < min{p, q}, N > max{0, -s} + a and put
(x). (2.71)
(x) (2.72)
6 = N + s - - > 0. r
We obtain for e e N
(2.73)
f|2es(®*-et/)a(x)|qd|^N) /q < cX 2-6re-k2krsM ( f|((Ok)t */)(-)lqdt
Jl t / keN V-'1 t
(x). (2.74)
Now we apply Lemma 2.13 in Lp/r (£q/r, Rd), which yields
r2 dt \r/q
j |2ks((O) * f )(■)
(2.75)
\ Lp/r(ßq/r)
The Fefferman-Stein inequality (see Section 2.2 /Theorem 2.1, having in mind that p/r,q/r > 1) gives
/2 )Mqd
\ Lp/r{ß-q/r)
f2 dt \r/q J \2fcS((Ok) * f (■) \ W(V)
(2.76)
,2 dt \ 1/q
jj2ks( (Ok )t * f (Off)
Hence, we obtain
|VsqKf )»r
\LP( kd)
£( |2"s(O2-,tf)a(x)|qd^ ) \Lp
j2 |2^s(O2-,tf)a(x)|q^y \L^^q)
ri dX \1/q
J ^(Ox * f (x)|qdX) \L^Rd)
(2.77)
This proves ||f \ Fp,q(Rd)||2 ^ |If \ Fp,q(Rd)||1. Since the reverse inequality is trivial, this finishes Step 1.
Step 2. Let ¥0, ¥ e S'(rd) be functions satisfying (2.18). In fact, the condition (2.17) for ¥0, ¥ is not necessary for what follows.
Substep 2.1. We are going to prove
l|/\fM(Rd)IL Rd)L (2.78)
for all / e S'(rd). We decompose / similar as in Step 1. Exploiting the property (2.17) for the system O, we find S(rd)-functions ¿0,1 e S(rd) such that supp X0 c{ e rd : \£\ < 2e} and supp 1 c { e rd : e/2 < \£\ < 2e} and
^Xe(tx) ■ ye(tx) = 1 (2.79)
for x e rd and t e [1,2] fix. Putting A0 = FX0 and A = FA, we obtain the decomposition
g = X (Ae)t * (Oe)t * g (2.80)
for every g e S'(rd). We put g = ¥e * / and see
¥e * / =£ ^e * (Ak)t * (Ok)t * /. (2.81)
Now, we estimate as follows:
|((¥e * (Ak)t) * ((Ok)t * /))(y)|
< f \(¥e * (Ak )t)(z)\-|((Ok )t * / )(y - z)|dz
< (®Ut/)a(y)i \(Oe * (Ak)t)(z)\ ■ (1 + 2k\z\Ydz
J Rd v '
< (O!-kt/)a(y)Je,k,
(2.82)
J£,k =f |(¥£ * (Afc),)(z)|(1 + 2k|z|)adz. (2.83)
J Rd X 7
We first observe that for x e rd and functions ¡,n e S(rd), the following identity holds true for u,v > 0
1 /Z\ 1 /z\
(pu * nv) (z) = Ud^ * nv/u\ (J = Vd [^u/v * n](v). (2.84)
This yields in case £ > k (with a minor change if k = o or £ = o)
J£,k = f \[(^£-k)1/t * A](z)\(1 + | tz|)a
< sup\ [(We-k)1/t * A(z)\(1 + |z|)a+d+1 (2.85)
< 2(k-£)(R+1),
where we used Lemma A.3 for the last estimate.
If k > l, we change the roles of ¥ and A to obtain again with Lemma A.3
Je,k = f |[¥ * (Ak-£)](z)|( 1 + \2k-£z\)adx
< 2(k-£)asup|[¥ * (Ak-£)t](z)|(1 + |z|)a+d+1 (2.86)
< 2(£-k)(L+1-a)
where L can be chosen arbitrary large since A satisfies (ML) for every L e n according to its construction. Let us further use the estimate
(¥*kf) a (y) < (¥*kf) a (x) (1 + 2k|x - y|)a
(2.87)
£ (¥*kf) a(x) (1 + 2e|x - y|) a max{ 1,2(k-t)a } •
Consequently,
sup2*' |(¥ f ))(y)' < 2k- ^^ m»{1.
(2.88)
r 2(£-k)(L+1-a+s) . k>l, 2(k-£)(R+1-s) . £>k.
< ^®2-kt/)a(x)-
Plugging this into (2.81) and choosing L > a + |s| and 6 = min{1,^ + 1 - s}, we obtain the inequality
2£s(¥*/)a(x) < £ 2-|k-£|fi2ks(®2_ktf)a(x) (2.89)
keNo a
for all x e rd and all t e [1,2]. Suppose first that q > 1. Then we take on both sides (j! \ ■ \qdt/t)1/q, which gives
(r2 dt \ 1/q
£2-'k-^2ks \(®*2-kJ)a(x)\qdt
tcN J 1
Applying Lemma 2.13 yields
)a(x)Lp(eq, rd)|| < ( 12ksqj2 \ (°2-kf/u^fj) ' i Lp(rd)
(2.90)
(2.91)
which gives the desired result.
In case q < 1, we argue as follows. The quantity (J^ | ■ |qdt/t)1/q is not longer a norm, but a q-norm. This gives
2£s(f(x)q < X 2-'fc-£'5q2fcsq [2|(®2-kif)a(x)|q^. (2.92)
keN0 J1 1
Notice that the right-hand side is nothing more than a convolution (y * f3)£ of the sequences
Yk = 2-|k|5q, ßk = 2ksqJ \(&2/a(x)
2 \ \qdt
(2.93)
Now we apply the ^-norm to both sides and get for all x e rd ||2'S(^2/)a(x) I ^ <||Y I *|| 'llßI ¿l||
œ r2 I
<X 2kSq\ \( ®2-kt/a(x)
k=1 J 1
(2.94)
We take both sides to the power (• ■ ■ )1/q and apply the Lp(rd)-norm. This gives (2.78).
Substep 2.2. With similar arguments and obvious modifications of Substep 2.1, we obtain for all f eS'(rd )
f^MÎ S \\f\F„(l" (2.95)
Step 3. Choosing t = 1 in Step 1 and omitting the integration over t, we see immediately
|f \ FPq (Rd)l5 S |f \ FPq(Rd)\\4 S |f \ Flq(Rd ) I5- 02.%)
Step 4. What remains is to show that (2.22) is equivalent to the rest.
Substep 4.1. Let us prove
l|f|fp,q(Rd)||2 < |f|fp,q(Rd)L- (2.97)
We return to (2.66) in Substep 1.3. If |2| < 2-(e+k)t, formula (2.66) implies by shift in the integral the following:
(O*-,.f)a(x)r < C^ 2-kNr2<k+»d i K(Ok+e2','f)(y +a2' '' dy
JRd 1 + 2e x - y
(1 + 2e|x - y\)'
(2.98)
Indeed, we have
1 + 2e |x - y| < 1 + 2e(|x - (y + 2) | + |2|)
< 1 + 2e(|x- (y + 2)|) + 2-k (2.99)
< 1 + 2e(|x- (y + 2)|),
where the last estimate follows from the fact that k e no in the sum. Instead of the integral (/2 | ■ |q/rdt/t)r/q (see Step 3), we take now on both sides of (2.98) the norm
f2 f dt \ r/q
, L'-"^) - (2100)
The integration over 2 does not influence the left-hand side. Instead of (2.68), we obtain
a j.\ r/q
2 q dt
/J« Ol-„f)aM|"f
< c j 2-kNs2lk.„^ (.Six | <(Ok+'). * f}y WW**'> )""
Jr^ (1 + 2e|x - y|)ar
tJ'aK~'\ t
(2.101)
We continue with analogous arguments as after (2.68) and end up with (2.97).
Substep 4.2. We prove \\f | Fpq(Rd) ||3 < \\f | Fsq(Rd) H2. Indeed, it is easy to see, that we have for all t> 0
1 f ,, , (l(®t * f )(x + z)|
- K* n(x+z)idz < su^1 y+f;
t J\z\<t \z\<t (1 + 1/t|z|) (2.102)
(®*f)a(x),
and we are done. The proof is complete. □
Proof of Theorem 2.8. The proof of Theorem 2.8 works almost analogously to the previous one. It is even a bit simpler, since we do not have to deal with a separate function <0, which causes some technical difficulties. However, there are still some technical obstacles which have to be discussed. Although we are in the homogeneous world, we use the same decomposition as used in (2.43), even with the inhomogeneity <0. In the definition of Am,£(x) in (2.45), we have to add <(x) if £ = 0 and m > 0. The consequence is (2.47) for every £ e z, where <0 = Hence, the inhomogeneity is buried in Am,£. This yields (2.66) for all £ e z, where k still runs trough n0. We need this for the argument in Substep 4.1. In contrast to the previous decomposition, we use (2.80) now for £ e z, where <0 = A0 = A, and k e z. This works since we assume g e S0 (rd). The rest is clear. □
Proof of Corollaries 2.11 and 2.12. Corollary 2.11 is more or less clear. We know that AN gives (2d=i Hkl2)N as factor on the Fourier side. This gives (2.18) immediately, and together with (2.31) we have (2.17) for e> 0 small enough.
In the case of Corollary 2.12, the situation is a bit delicate. Clearly, Condition (2.18) holds true. But the problem here is, that (2.17) may be violated for all e > 0. However, we argue as follows. In Step 2 in the proof above we have seen, that we do not need (2.17) for the system (¥0, ¥). Hence, we can estimate (2.20), (2.21), (2.22), (2.23), and (2.24) from above by a different characterization of F®,q(rd). For the remaining estimates we apply Theorem 2.6 with the system (®0, ®), where
<5 = ®0(x) - ^®0(|:) (2.103)
and k e n is chosen in such a way that (2.17) is satisfied. What remains is a simple consequence of the fact that
< = < + <-1 + ••• + <-(k-1) (2.104)
and the triangle inequality. This type of argument is due to Triebel [3, Section 3.3.3]. □
3. Classical Coorbit Space Theory
In [11-13,15], a general theory of Banach spaces related to integrable group representations was developed. The ingredients are a locally compact group G with identity e, a Hilbert space H, and an irreducible, unitary, and continuous representation n on H, which is at least
integrable. Then one can associate a Banach space CoY to any solid, translation-invariant Banach space Y of functions on the group G. The main achievement of this abstract theory is a powerful discretization machinery for CoY, that is, a universal approach to atomic decompositions and Banach frames. It allows to transfer certain questions concerning Banach space or interpolation theory from the function space to the associated sequence space level, see [12, 13, 26]. In connection with smoothness spaces of Besov-Lizorkin-Triebel type, the philosophy of this approach is to measure smoothness of a function in decay properties of the continuous wavelet transform Wgf, see the appendix for details. Indeed, homogeneous Besov and Lizorkin-Triebel-type spaces turn out to be coorbits of properly chosen spaces Y on the ax + b-group G.
The are many more examples according to this abstract theory. One main class of examples refers to the Heisenberg group h and the short-time Fourier transform and leads to the well-known modulation spaces as coorbits of weighted Lp (h) spaces, see [11, Section 7.1] and also [27].
3.1. Function Spaces on G
Integration on G will always be with respect to the left Haar measure d^(x). The Haar module on G is denoted by A. We define further LxF(y) = F(x-1y) and RxF(y) = F(yx), x,y e G, the left and right translation operators. A Banach function space Y on the group G is supposed to have the following properties:
(i) Y is continuously embedded in L11oc(G),
(ii) Y is invariant under left and right translation Ly and Ry, which represent continuous operators on Y,
(iii) Y is solid, that is, H e Y and | F(x)| < | H(x) | a.e. imply F e Y and ||F | Y|| < ||H |Y ||.
The continuous weight w is called submultiplicative if w(xy) < w(x)w(y) for all x,y e G. Further, another weight m is called w-moderate if m(xy2) < w(x)m(y)w(2), x,y,2 e G. The space Lp' (G) of functions F on the group G is defined via the norm
IF | Lw(G)|| = ( I |F(x)w(x)|pd^(x) ) , (3.1)
where 1 < p <œ (modification if p = <x>). If w = 1, then we simply write Lp(G). These spaces provide left and right translation invariance, which is easy to show. Later, in Section 4.1, we are going to introduce certain mixed norm spaces where the translation invariance is not longer automatic.
3.2. Sequence Spaces
Definition 3.1. Let X = {xi}ieI be some discrete set of points in G and V a relatively compact neighborhood of e eG.
(i) X is called V-dense if G = U xV.
(ii) X is called relatively separated if for all compact sets K c G, there exists a constant CK such that
sup *{i e I: XiK n XjK / 0} < Ck-
(iii) X is called V-well-spread (or simply well-spread) if it is both relatively separated and V-dense for some V.
Definition 3.2. For a family X = (xt}ieI which is V-well-spread with respect to a relatively compact neighborhood V of e eG we define the sequence space Yb and associated to Y as
Yb = j(Ai)iei : |Ui}iei | Yb|| = Y* = i (Ai}iei : INiei I Y*| =
^|Ai||XiV|-1!xiV I y
< G >,
B^XxV | Y
Remark 3.3. For a well-spread family X, the spaces Yb and do not depend on the choice of V, that is, different sets V define equivalent norms on Yb and respectively. For more facts on these sequence spaces, confer [12].
3.3. Coorbit Spaces
The starting point is a Hilbert space H and an integrable, irreducible, unitary and continuous representation n of G on H. Then the general voice transform Vgf is a function on the group G given by
Vgf (x) = (n (x)g,f),
where the brackets denote the inner product in H.
The continuous wavelet transform Wgf (appendix) is a voice transform for the situation H = L2(rd) and the ax + b-group.
Definition 3.4. For a weight w(-) > 1 on G, we define the space Aw cH of admissible vectors by
= {g eH: Vgg e LWW (GH -If Aw ^ (0} and g e Aw, we define further
Hi(Rd) = {f eH : |f IHl| = ||Vgf | LW(G)|| <
Finally, we denote by (Hw)~ the canonical antidual of Hw, that is, the space of conjugate linear functionals on Ji}n.
We see immediately that Aw c Hlw c H. The voice transform (3.4) can now be extended to Hw x (Hxw)~ by the usual dual pairing. The space Hlw can be considered as the space of test functions, whereas the space (HW)~ serves as reservoir, that is, distributions. Let now Y be a space on G such that (i)-(iii) in Section 3.1 hold true. We define further
wy(x) = max(\\Lx\\, \\Lx-11|, \\Rx\\, a(x-1) \\Rx-1 \\}, x eG, (3.7)
where the operator norms are considered from Y to Y.
Definition 3.5. Let Y be a space on G satisfying (i)-(iii) and wY(x) be given by (3.7). Let further g e Aw. We define the space CoY, called coorbit space of Y, through
CoY = f e (h£,)~ : Vgf e y} with ||f | CoY|| = \\Vgf | Y||. (3.8)
The following result is of crucial importance. See [14] for details.
Lemma 3.6 (Correspondence principle). Let G = Vgg e LW(G) for a fixed analyzing vector g e Aw with proper normalization. Then a function F e Y is of the form Vgf for some f e CoY if and only if
F (x) = (F * G)(x) := | F(y)LyG(x)d^i(y), x eG. (3.9)
In other words, CoY is isometrically isomorphic to the closed subspace Y * G of Y.
The following basic properties are proved for instance, in [28, Theorem 4.5.13].
Theorem 3.7. (i) The space CoY is independent of the analyzing vector g e Aw.
(ii) The definition of the space CoY is independent of the reservoir in the following sense. Assume that S c Hw is a nontrivial locally convex vector space, which is invariant under n. Assume further that there exists a nonzero vector g e S n Aw for which the reproducing formula
Vgf = Vgg * Vgf
(3.10)
holds for all f e S~. Then we have
CoY = f e (Hi: Vgf e Y} = f e S~ : Vgf e Y}. (3.11)
3.4. Discretizations
This section collects briefly the basic facts concerning atomic (frame) decompositions in coorbit spaces. We are interested in atoms of type {n(xt)g}ieI, where (xt}iei c G represents a discrete subset, whereas g denotes a fixed admissible analyzing vector.
Definition 3.8. A family {gi}ieI in a Banach space B is called an atomic decomposition for B if there exists a family of bounded linear functionals {li}ieI c B' (not necessary unique) and a Banach sequence space = B#(I) such that
(a) {li(f )}ieI e for all f e B and there exists a constant Ci > 0 with
|{Mf)}feI | B«| < C1 ||f | B||. (3.12)
(b) For all f e B, we have
f = XMf)gi (3.13)
in some suitable topology.
(c) If {Xi}ieI e B#, then ^ieI Xigi e B and there exists a constant C2 > 0 such that
Z^gi I B
< C2||Ui}fei | bC
(3.14)
Definition 3.9. A family {hi}ieI c B' is called a Banach frame for B if there exists a Banach sequence space Bb = Bb(I) and a linear bounded reconstruction operator 0 : Bb ^ B such that
(a) {hi(f) }ieI e Bb for all f e B and there exist constants C1, C2 such that
Cl||f I B\\ < |{HDhel I B1 < C2|lf I Bl
(3.15)
(b) 0({hi(f)}ieI)= f.
Remark 3.10. This setting differs slightly from the understanding of Triebel in [3, 29].
The following abstract result for the atomic decomposition in CoY is due to Feichtinger and Grochenig (see [12, Theorem 6.1]).
Theorem 3.11. Let Y be a function space on the group G which satisfies the hypotheses (i)-(iii). Let further
wy(x) := max{ ||Lx||, ||L*-11|, \\Rxy, ^x"^\\Rx-11|}, (3.16)
where the operator norms are taken from Y toY, and g e Aw such that
( sup(n(y)g,g)\ )w(x,t)d^(x) < go. (3.17)
yexV /
Then there exists a neighborhood U of e e G and constants C0,C1 > 1 such that for every U-well-spread discrete set X = {x^^ cG, the following is true.
(i) (Analysis) Every f e CoY has a representation
f = (xi) g (3.18)
with coefficients (li}ieI depending linearly on f and satisfying the estimate
||{Ai}iei | Y*W < Co\\f | Y||. (3.19)
(ii) (Synthesis) Conversely, for any sequence {li}ieI e , the element f = £ieIhn(xi)g is in CoY and one has
\\f | CoY|| < C1||{Ai}feI | Y«||. (3.20)
In both cases, convergence takes place in the norm of CoY if the finite sequences are norm dense in Y^, and in the weak* -sense of (Hxw)~ otherwise.
Remark 3.12. According to Definition 3.8, the family {n(xi)g}ieI represents an atomic decomposition for CoY.
Theorem 3.13. Under the same conditions as in Theorem 3.11, the system {n(xi)g}ieI represents a Banach frame for CoY, that is,
||f | CoY|| x||«n(xi)g,f)} | Yb||. (3.21)
The following powerful result goes back to Grochenig [14] and was generalized by Rauhut [17].
Theorem 3.14. Suppose that the functions gr,jr, r = 1,...,n, satisfy (3.17). Let X = {xi}ieI be a well-spread set such that
f = X E(n(xi)Tr,f)n(xi)gr (3.22)
r=1 ieI
for all f e H. Then expansion (3.22) extends to all f e CoY. Moreover, f e (Hw)~ belongs to CoY if and only if {(n (xi)jr,f)}ieI belongs to for each r = 1,...,n. The convergence is considered in CoY if the finite sequences are dense in Y$. In general, we have weak*-convergence.
Proof. The proof of this result relies on the fact that there exists an atomic decomposition {n(yOg}tej by Theorem 3.11 with a certain g satisfying (3.17) and a corresponding sequence of points Z = {yt}iej. This has to be combined with Theorem 3.13 and Theorem 3.11/(ii) and we are done. See [14] for the details. □
4. Coorbit Spaces on the ax + b-Group
Let G = rd x r+ the d-dimensional ax + b-group. Its multiplication is given by
(x, t) (y, s) = (x + ty, st). (4.1)
The left Haar measure i on G is given by dt/td+1, the Haar module is A(x, t) = t-d. Giving a function F on G, the left and right translation Ly = L(y,r) and Ry = R(y,r) are given by
L(y,r)F(x,t) = F( (y,r)-1(x,t^ = F(X7l,tr), (4.2)
R(y,r)F (x,t) = F( (x,t)( y,r)) = F(x + ty,rt).
4.1. Peetre-Type Spaces on G
The present paragraph is devoted to the definition of certain mixed norm spaces on the group. Such spaces were considered in various papers, see [10,11,14,15]. Especially tent spaces have some nice applications in harmonic analysis. In particular, they were used by many authors in order to apply coorbit space theory to Lizorkin-Triebel spaces.
Here we use a different approach and define a new scale of function spaces on the group G. We call them Peetre-type spaces since the Peetre maximal function is involved in the definition. It turned out that they are very easy to handle in connection with translation invariance. Compared with the tent space approach, they are the more natural choice for considering Lizorkin-Triebel spaces as coorbit spaces. Additionally they seem to be the suitable choice for inhomogeneous spaces and more general situations like weighted spaces, which will be studied in a subsequent contribution to the subject.
Definition 4.1. Let s e r, 0 <p,q and a > 0. We define the spaces Lp,q(G), Tp,q(G), and Ppq(G) on the group G via the finiteness of the following (quasi-)norms:
\F\L Pq(G)|| = ( [ t-sq || F M)|LP( R«)
f \ Tm(G)II =
I|F \ OG)|| =
C dt \1/q
1 r'"i,m:F(x+fdz^) |L>-(r")
|F(x + y,t)\V "t \ 1/q
(1 + |y|/t)a
Si) \Lp(r")
using the usual modification in case q = œ.
Proposition 4.2. The spaces Lpq(G), Tpq(G) and Pp,rq(G), are left and right translation invariant. Precisely, we have
\L(z,r) : Lp,q(G)
\L(z,r) : Tp,q(G)
L Pq(G)|| = r"(1/p-1/q)-s,
RM : Lp,q(G) LPq(G)| = rs+d/q,
TTSq(G)| = rd/q-s,
R(z,r) : TTPq(G) TTp,q(G)| < Crd/q+s max{1, r"b(1 + |z|)b},
where b > 0 is a constant depending on d, p, and q, and
|L(z,r) : fp£(G) — = rd(1/p-1/q)-s,
||R(z,r) : Ppq(G) ^a(G)|| < rs+d/q max{1,r-a}(1 + |z|)a
Proof.
Step 1. The left and right translation invariance of Lpq(G) and Tpq(G) was shown in [28, Lemma. 4.7.10].
Step 2. Let us consider Ppq(G). Clearly, we have for F e P}p,',a(G)
\L(z,r)F | Ppq (G) || =
IF ( (x + y - z) /r, t/r) I
= rd/P
(1 + |y|/t)a |F(x + y,t/r)\
dt td+1
| L^rd)
= rd(1/p-1/q)-s
sup . .
ye Rd (1 + r |y|/t)
|F (x + y^"
d-) Mr")
(1 + |y|/t)a
| Lp(R" )
Hence, we obtain
L(z,r): ppq (g)
Pp,q (G)|| = rd(1/p-1/q)-s.
■ /.
The right translation invariance is obtained by
\R(Z,r)F I Pm(G)|I =
(i: (C
_ rs+d/q
|F(x + tz + y, tr) I
(i + \y\/t)a
q ^ \1/q
|F(x + y, tr) I
(1 + |y - tz|/t)
F (x + y,t)|
qdt_ ^ 1/q
I !?(rd)
(1 + I y - tz\r/t)\
I Lp(rd)
Observe that
\F(x + y,t)\
suP^—i-TTTT^ _ suP
(1 + |y - tz\r/t)
|F(x + y,t)\ (1 + \y\/t)a
(1 + \y\/t)a (1 + |y- tz\r/ty
(1 + \y\/t)a Ü + |y - tz|/t + |z0a_ (1 + |y - tz|/00(1 + |z|)a
(1 + |y - tz\r/t)a (1 + Iy - tz\r/ty
(1 + Iy - tz\r/ty
This yields
\F(x + y,t)\ , a \F(x + y,t)\
sup^V- , nsa < max{1,ra}(1 + |z|)asup-1 v * n
1 + y - tz r/t a
(1 + |y|/0'
(4.10)
and consequently
\\R(z,r) : ^(0)11 < rs+d/qmax{1,r-a}(1 + |z|)a. (4.11)
Remark 4.3. Note that we did neither use the translation invariance of the Lebesgue measure nor any change of variable in order to prove the right translation invariance of Ppq(G). This gives room for generalizations, that is, replacing the space Lp (rd ) by some weighted Lebesgue space Lp(rd,o>) in the definition.
4.2. New Old Coorbit Spaces
We start with H = L2(rd) and the representation
n(x, t) _ TxdL2, (4.12)
which is unitary continuous on H. This representation is not irreducible. However, if we restrict to radial functions g e L2(rd), then span {n(x,t)g : (x,t) e G} is dense in L2(rd).
Another possibility to overcome this obstacle is to extend the group by SO(d), which is somehow equivalent. See [11,12] for details.
The voice transform in this special situation is represented by the so-called continuous wavelet transform Wgf, see Appendix A.1 in the appendix. Recall the abstract definition of the space Hw and Aw from Definition 3.4. The following lemma is proved for instance in [28, Lemma 4.7.11] and is also a consequence of our Lemma A.3 on the decay of the continuous wavelet transform. It states under which conditions on the weight w the space Hw is nontrivial.
Lemma 4.4. If the weight function w(x,t) > 1 satisfies the condition
w(x,t) < (1 + |x|)r(ts + t-') (4.13)
for some r, s, s' > 0, then
So(Md)^Hw. (4.14)
This is a kind of minimal condition which we need in order to define coorbit spaces in a reasonable way. Instead of (Hw)~, one may use SO (rd) as reservoir and a radial g e S0(rd) as analyzing vector. Considering (3.7), we have to restrict to such function spaces Y on G satisfying (i),(ii),(iii) in Section 3.1 where additionally (recall the definition of wY in (3.16)) (iv)
wy(x, t) < (1 + |x|)r (ts + t-') (4.15)
for some r, s, s' > 0. The following theorem shows how the spaces of Besov-Lizorkin-Triebel type from Section 2 can be recovered as coorbit spaces.
Theorem 4.5. (i) For 1 < p,q <g, and s e r, we have Bpq(Rd) = CoLp+qd/2-d/q(G), (ii) for 1 < p < g, 1 < q <g and s e r we have
F^) = Coj;+d/1 (G) (4.16)
(iii) and if additionally a> d/ min{p,q}, we obtain
Fprq(Rd) = CoPp+d/2-d/q,a(G). (4.17)
Proof. Theorem 4.5 is a direct consequence of Proposition 4.2, Theorems 2.8, 2.9, and the abstract result in Theorem 3.7. □
Remark 4.6. (a) The assertions (i) and (ii) are not new. They appear for instance in [11, 14, 15]. They rely on the characterizations given by Triebel in [2] and [3, Sections 2.4, 2.5], see in particular, [3, Section 2.4.5] for the variant in terms of tent spaces, which were invented
in [10]. From the deep result in [10, Proposition 4], it follows that TS^G) are translation invariant Banach function spaces on G, which makes them feasible for coorbit space theory.
(b) Assertion (iii) is indeed new and makes the rather complicated tent spaces T^G) obsolete for this issue. We showed that Y = P£'r%(G) is a much better choice since the right translation invariance is immediate and gives more transparent estimates for its norm. Once we are interested in reasonable conditions for atomic decompositions, this is getting important, see Section 4.5.
4.3. Sequence Spaces
In the following, we consider a compact neighborhood of the identity element in G given by U = [0,a] x [j-1,1], where a > 0 and 1 < j. Furthermore, we consider the discrete set of points
[xjrk = (akß~j, ßj : j e z, k e zd}.
(4.18)
Then the family {x,kU}¿eZ,fceZd defines a partition of G. Indeed,
XjkkU = Qjk x [ß~(j+1),ß j,
(4.19)
Qjk = [akiß-j,a(ki + 1)ßj x ■■■ x \akdß-j,a(kd + 1)ßj.
(4.20)
Note that in this case the spaces and Yb coincide. We will further use the notation
Xj,k (x) =
1: x e Qj,k
0: otherwise.
(4.21)
Definition 4.7. Let Y be a function space on G satisfying Section 3/(i), (ii), (iii). We put
Y«(a^) = {{jj : lUj-kjk I Y«(a,ß)| < ,
(4.22)
|{lj,k}j,k|Y«(a,ß)|
Y,\Xj,k \lj,k (x)X[ßj,ßj+1] (t) 1 Y
(4.23)
Theorem 4.8. Let 1 < p,q <<x>, s e r, and a> d/ min{p,q}. Then
\\{j }irki(ps:a)\«,p)\\
E ^e(s+d/q)ql^klqxek(x)) I Lp(rd)
ieZ keZd
{Xhk}hk I (ip^W) - ( Vpi(s+d/q-d/p)q( £ |p
(4.24)
Proof. We prove the first statement. The proof for the second one is even simpler. Let
F(x,t) = E|^;,k|X;,k(x) ■ X[p-(i+1),p-i](t).
(4.25)
Discretizing the integral over t by t ~ P *, we obtain
I|F | Y y =
|F(x + w,t)| T (1 + |w|/t)a
^pj(s+d/q)q ieZ •'P-(i+1)
f J0-(
q dt \ 1/q , ,
dO |L^rd)
q j, \ Vq
|F(x + w,t)|
s^p (TT^MT
f) |L^rd)
(4.26)
With t e [p-(i+!\p-i], we observe
F (x,t) = E
(4.27)
and estimate
11F | Y ||<
V pe(s+d/q)q sup -1--
ieZ weRd i1 + |w|)
Yílx^klxe,k(x + w)
| Lp rd
. (4.28)
In order to include also the situation min{p, q} < 1, we use the following trick. Obviously, we can rewrite and estimate (4.28) with 0 <r < 1 in the following way:
11F | Y|| <
, ie Z
V^(s+d/q)r|^k|r sup + W.rr
.fc weRd (1 + P |w|)
| Lp rd
(4.29)
The next observation is the useful estimate
|l^,k(x + w)| . supJ-1 <
(1 + pi|w|)ar ~ (1 + - kp-e\Y
Xik ()
(1 + Pe H)a
(x). (4.30)
Indeed, the first estimate is obvious. Let us establish the second one
(1 + ß* |-|) a
-)(x)=[ * i| i,
yt-ktß-e|<ß-* ¿=1,
(1 + ß*\x - y\)' 1
|y|<cß-* (1 + ß*\x - kß-* - y\)a 1
|y|<Cß-* (1 + ß*\x-kß-*\+ ß*\y\)a
dy (4.31)
o (1 + ß*\x - kß-*\ + u)c
(1 + ß*|x - kß-*\)c
Note, that the functions
g*(x) =
(1+ß*H)a
(4.32)
belong to Li(Rd) with uniformly bounded norm where we need ar > d. Putting (4.31) and (4.30) into (4.29), we obtain
IF | Y If <
g* * £ ß*(s+d/q)r MUk) (x)
| lp/r(rd)
(4.33)
Now we are in a position to use the majorant property of the Hardy/Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]), which states that a convolution of a function f with an Li (rd)-function (having norm one) can be estimated from above by the Hardy/Littlewood maximal function of f. We choose r < {min{p,q} and apply Theorem 2.1 for the Lp/r (£q/r) situation. This gives
||F | Y||r
^ß*(s+d/q)r |l*,k|rx*,k(x)
| Lp/r(rd)
(4.34)
EEß*(S+d/q)q|1*,k|qX*,k(xn | L^Rd)
*eZ keZd
and finishes the upper estimate. Both conditions, ar > d and r < min{p,q}, are compatible if a> d/ min{p,q} is assumed at the beginning.
For the estimate from below, we go back to (4.26) and observe
if (x + w, t) | sup F{ >|f (xj)l
w (1 + ߣ |w|)
(4.35)
which results in
IF I YII
Vß^+^W |F(x,t)q— ) I lJRd)
h-(e+1) t / V '
(4.36)
A further use of (4.27) gives finally
11F | Y ||
E Xß'(s+d/q)q|wx^(x)) | L^rd)
ieZ keZd
(4.37)
The proof is complete.
4.4. Atomic Decompositions
The following theorem is a direct consequence of the abstract results in Theorems 3.11, and 3.13.
Theorem 4.9. Let 1 < p,q < to, a > d/ min{p,q}, and s e r. Let further g e S0(rd) be a radial function. Then there exist numbers a0 > 0 and ¡¡0 > 1 such that for all 0 < a < a0 and 1 < ¡5 < ¡¡0 the family
jeZ,keZd " {T«kßiDß"2g}
(4.38)
has the following properties.
(i) {gj,k }j ez/ke z forms a Banach frame for CoLspq(Q) and CoPp'a(G), that is, we have a dual frame {ej/k}jeZ,keZd c S0(rd) with f = XjeZ/keZd (gj,k, f)ej,k and the norm equivalences
| CoLPq(Q)
gk,f) | (^(G))\a,ß)
, f e CoLp,q(G)
(4.39)
as well as
\\f | CoP;:a(G) II X I (ghk,f) | (Pp:a(G))\a,ß) ||, f e CoP^G). (4.40)
(ii) [gj,k}jez,fcezd is an atomic decomposition, that is, for f e CoPp^(G), we have a (not necessary unique) decomposition ^jeZ/keZd Xjrk(f)gj,k such that
||{Aj,k(f)}j,k I (P^G))^! < llf I Co^qa(G)||. (4.41)
Conversely, if {j} ¡ez,kezd e (Pp'q(G)f(a,p), then f = £yeZ/keZd jgjk converges and belongs to CoPp'^G) and moreover,
||f | Co^^(G)|| < ||{A;-k},k I (^^(G))*(«^)|| (4.42)
(analogously for CoLp^G)). Convergence is considered in the strong topology if the finite sequences are dense in (Pp^G))^«,^) and in the weak*-topology otherwise.
Remark 4.10. (i) Since the analyzing function or atom g can be chosen arbitrarily, we allow more flexibility here than in the results given in Frazier/Jawerth [30] and Triebel [3, 29].
(ii) Instead of regular families of sampling points ) rather irregular families
of points in G are allowed as long as they are distributed sufficiently dense, see Theorem 3.11.
4.5. Wavelet Frames
In the sequel, we consider wavelet bases on rd in the sense of Lemma A.5 from Appendix A.2 in the appendices. We have given an orthonormal scaling function ¥° and the associated wavelet ¥1 on r and consider the tensor products ¥c, c e E. Our aim is to specify, that is, give sufficient conditions to ¥°, such that (A.19) represents an unconditional basis in Bpq(Rd) and Fpq(Rd). We will apply the abstract Theorem 3.14.
In order to do so we need to have (3.17) for all functions ¥c. We will state certain smoothness, decay, and moment conditions to ¥1 and ¥°, see Definition A.1 in the appendix, to ensure this. Let us fix the neighborhood V = [-1,1]d x (1/2,1] cG of e e G.
Proposition 4.11. Let L e n, K > 0, and ¥° be an orthogonal scaling function with associated wavelet ¥1 on r. The function ¥° is supposed to satisfy (D) and (SK ) and ¥1 is supposed to satisfy (D), (SK), and (ML-1). For r1, r2 e r, the weight w(x, t) is given by
w(x,t) = (1 + |x|)v(tr2 + Tr1 ), (x,t) eG. (4.43)
r1 < min{L,K} - 2, r2 < min{L,K} + d - v, (4.44)
then we have
C i*œ Ai
sup |(n(y,s)¥c,¥c)\w(x,t) — dx< œ. (4.45)
jRd Jo (yrs)e(x,t)V *
Proof. With Lemma A.3, we obtain for W» »1 the following estimates:
tmin{L,K} + 1/2 1
l(W» »0 M< d ti)-»,i,i + |S|/(1 + ■ («6)
And in addition
| (W»i(s, t) l <7-^-1-T7, i = 1,2. (4.47)
IV » J , 1 (t +1) (1 + |s|/(1 + t))N, ,
Hence, for any c e E, the tensor product structure gives (assume without restriction that Q = 1)
tmin{L,K} td/2 d 1
W»c»C(-')|< (TTmnmKm- (T^U(1 + M/(1+ ,))N■ (448)
The expression sup(y s)e(xt)V|W»c»c(y,s)| can be estimated similar sup |W»c »c (y, s)1 = sup |W»c »c (;y,s)|
(y,s)e(x,t)V yi-xi |<t
t/2<s<t
xmin{L,K} d/2 d 1 <_t_:_:__t_n_1__(4.49)
" ----------¡11/1. i.. i //1 . .wN
(1 + t)2 mm{L,K} (1 + t)d (1 + |Xi|/(1 + t))N tmin{L,K} td/2 1
(1 + t)2 min{LK} (1 + t)d (1 + |x|/(1 + t))N '
Fubini's theorem and a change of variable yields
II „ , Sup ^ »c> M^) Jdn dx < I (1 + t)2 min{L,K} ^
JRiJ 0 (y,s)e(x,t)V t J0 (1 + t) 1,1 (4.50)
■(tr2 + t-r1 )d1 ■ V ' td + 1
Finally it is easy to see that the latter is finite if the conditions in (4.44) are valid. This proves Proposition 4.11. □
Theorem 4.12. Let L e n, K > 0, and »0 be an orthogonal scaling function with associated wavelet »1 on r. The function »0 is supposed to satisfy (D) and (SK), and »1 is supposed to satisfy (D), (Sk) and (Ml-1).
(a) If 1 < p,q < œ and
-min{L,K} + -<s< min{L,K} - d( 1 - ^, (4.51)
P \ PZ
then (A.19) is a Banach frame for bp (rd) in the sense of (3.22).
(b) If 1 < p,q < œ and
-min{L,K| + 2dmax^ 1, 1 j < s< min{L,K| - d ma^i1,1, 1 - 1 (4.52)
p q) lp q p
then (A.19) is a Banach frame for Fpq(rd)in the sense of (3.22).
Proof. Let us prove (a). First of all, we apply Theorem 4.5/(i). Afterwards, we use Proposition 4.2 in order to estimate the weight wy(y, t) for Y = Lp+q/2 d/q(G). We obtain
wy (x,t) = max{td(1/P-1/2)-s,ts-d(1/P-1/2),ts+d/2,t-s+d/2}
Yr1; 0 <t< 1, (4.53)
^tr2: t > 1.
Let us distinguish the cases, s > 0 and s < 0. In the first case we can put r1 = max{s - d(1/p -1/2),-s + d(1/p - 1/2),s - d/2}, r2 = max{s + d/2,-s + d(1/p - 1/2)}, and v = 0. Now we apply first Proposition 4.11. This gives the condition
0 < s< min{L,K} - d(1 - 1/p). (4.54)
In the second case we put r1 = max{s - d(1/p - 1/2), -s + d(1/p - 1/2),-s - d/2}, r2 = max{-s + d/2,-s + d(1/p - 1/2)} and v = 0. With Proposition 4.11, we obtain the condition
-min{L,K| + d/p < s < 0. (4.55)
Finally (4.54), (4.55), and Theorem 3.14 yield (a).
Step 2. We prove (b). We apply Theorem 4.5/(iii) and afterwards Proposition 4.2 and obtain
for Y = Pp+d/2-d/q,a (G)
WY (x,t) = max{ fd(1/p-1/2)-s/fs-d(1/p-1/2),
ts+d/2 max{1,ra}(1 + |x|a)/i-s+d/2max{i-a/ia}(1 + |x|)a}
(4.56)
ft-r1: 0 <t< 1,
< (1 + |x|)a^
tr2: t 1.
First, we consider the case s > 0. We can put n = max{s + a - d/2,s + d/2 - d/p}, r2 = max{s + d/2, -s + d/2 + a}, and v = a. Proposition 4.11 gives the condition
0 < s < min{L,K} - ma^ja,d(^ 1 - ^ j (4.57)
which can be rewritten to
f 11 11
0 < s< min{L,K} - dma^ -,-, 1 - - I (4.58)
[p q p\
since a can be chosen arbitrarily greater than d max {1 /p, 1 /q}. This gives the upper bound in (b). Now we consider s < 0. We put n = max{-s + d/p - d/2,s + d/2 - d/p}, r2 = -s + d/2 + a. This yields
-min{L,K} + 2a< s < 0 (4.59)
and can be rewritten to
- min{L,K} + 2d max{1/p, 1/q} <s< 0. (4.60)
This yields the lower bound in (b) and we are done. □
The following corollary is a direct consequence of Theorem 4.12 and the facts in Appendix A.2.
(a) If 1 < p,q <œ and -m + 1 + d < s <m - 1 - d( 1 - ,
p \ pj
Corollary 4.13. Let m> 0 and (¥0, ¥1) = (ym,fm) the spline wavelet system of order m. Then
d -i J. 1'
— < s <m - 1 - d 1 — p \ p,
then (A.19) is a Banach frame for Bpq(K.d) in the sense of (3.22); (b) If 1 < p,q and
(111 f 11 11
-m + 1 + 2d maw \ < s <m - 1 - d max^ -,-, 1 — \, (4.61)
p q p q p
then (A.19) is a Banach frame for i^p,q(Kd) in the sense of (3.22).
Remark 4.14. The (optimal) smoothness conditions in [31] are slightly weaker than (a) in case d = 1. However, compared to the approach of Triebel [29, 32], we admit some more degree of freedom. The wavelet or atom does not have to be compactly supported. Additionally, in case d = 1, we do not need that f e Cu(R) where u> s. Indeed, the conditions in (a) and (b) are slightly weaker.
Remark 4.15. More examples can be obtained by using compactly supported Daubechies' wavelets of order N or Meyer's wavelets. Based on the underlying abstract result in Theorem 3.14 even biorthogonal wavelet systems which provide sufficiently high smoothness and vanishing moments are suitable for this issue.
Appendix A. Wavelets
A.1. The Continuous Wavelet Transform
The vector g is said to be the analyzing vector for a function f e L2(rd )• The continuous wavelet transform Wgf is then defined through
Wgf (x, t) = (txD>L2 g,f), x e rd, t> 0, (A.1)
where the bracket (■, •) denotes the inner product in L2(rd). We call g an admissible wavelet if
!?g(i)T
Cg := ' ' n dl < go. (A.2)
If this is the case, then the family {TxDt2g}t>0/XeRd represents a tight continuous frame in L2(r). In particular, this means that the above family is dense in L2(rd). For a proof we refer to Theorem 1.5.1 in [28].
Let us now specify the conditions (ML), (D), and (SK), which we intend to impose on functions ¥ e L2(rd) in order to obtain a good decay of the continuous wavelet transform |WT®(x,t)|.
Definition A.1. Let L+1 e n0, K > 0, and fix the conditions (D), (ML), and (SK) for a function ¥ e L2(rd).
(D) For every N e n, there exists a constant cN such that
(1 + |x|)J
|¥(x)| (A.3)
(ML) We have vanishing moments
D"F¥(0) = 0 (A.4)
for all |a|1 < L. (SK) The function
(1 + |^|)K |DaF¥(^)| (A.5)
belongs to Li(rd) for every multi-index a e n.
Remark A.2. If a function g e L2(rg) satisfies (SK) for some K > 0 then by well-known properties of the Fourier transform we have g e (rg).
The following lemma provides a useful decay result for the continuous wavelet transform under certain smoothness, decay, and moment conditions, see also [4, 30, 33] for similar results in a different language. It represents a continuation of [4, Lemma 1] where one deals with S(rg) -functions.
Lemma A.3. Let L e no, K> 0, and O, ¥, O0 e L2(rg).
(i) Let O satisfy (D), (ML-1), and let O0 satisfy (D), (SK). Then for every N e n there exists a constant CN such that the estimate
tmin{L,K}+d/2
|(WoOo)(x, t)| < Cn „ , ixN (A.6)
(1 + |x|)N
holds true for x e rg and 0 <t< 1.
(ii) Let O, ¥ satisfy (D), (ML-1) and (SK). For every N e n there exists a constant CN such that the estimate
tmin{L,K}+d/2 / xi \~N K^)^ < CN (1 + t)2m1n{L,K)+^1 + ^
holds true for x e rg and 0 <t< to.
Proof.
Step 1. Let us prove (i). We follow the proof of Lemma 1 in [4]. This reference deals with S(rg)-functions, which makes the situation much more easy. We argue as follows. Clearly,
|(W®O0)(x,t)| = tg/2|[(DtO) *O0](x)|. (A.8)
Fix 0 <t< 1. Obviously, the convolution (DtO) *O0 satisfies (D). By well-known properties of the Fourier transform, the derivative DaF((DtO) *O0))(£) exists for every multi-index a e n. For fixed a, we estimate by using Leibniz' formula
(ft tOh$0))(i) = Id;
< \Dß?fc(t£) ■ Da-ßF®o(ö |
< 4tL(1 + | £|)l£|d^F®0(ö|.
In the last step we used property (ML-i). Assuming K > L and exploiting (SK), we obtain that the left-hand side of (A.9) belongs to L1(rd) and
||D»F((Di®) * ®o))(£) | L1(rd)|| < catL. (A.10)
We proceed as follows:
max ||DaF((Dt®) * ®,))(£) | L^rd) || > max ||F-1 DaF((Dt®) * ®,)1 | L^rd |a|1<N+1ll N /11 |a|1<N+1ll L J N
> CN||(1 + |x|)N[(Dt®) * ®0](x) | L„(r'
(A.11)
This estimate together with (A.8) and (A.10) yields (A.6).
Let us finally assume K <L and return to (A.9). Clearly, the resulting inequality keeps valid if we replace the exponent L by L' e n0 with L' < L. It is even possible to extend (A.9) to every 0 < L" < L by the following argument. Let L" / n. We have on the one hand
LHS (A.9) < c'at[n (1 + |t|) [L"J G(t) (A.12)
and on the other hand
LHS (A.9) < c'atlL"J+1(1 + |^|)[L"J+1 G(t), (A.13)
where G(t) = |Da-^ F®0(t)|. Choosing 0 <9< 1 such that L" = (1 - 0) [L"J + d([L"J + 1), we obtain by a kind of interpolation argument
LHS(A.9) = LHS(A.9) 1-0 LHS(A.9)0 < c'atL"(1 + i^j)1"G(l).
(A.14)
In particular, we obtain instead of (A.9)
|DaF ((Di®) * ®0))(t)|< c'atK(1 + |t|)K£|D^F®0(t)|, t e rd. (A.15)
We exploit property (SK) for ®0 and proceed analogously as above. This proves (A.6). Step 2. The estimate in (ii) is an immediate consequence of (A.6) and the fact
(W®¥)(x,f) = (WT®)( -X' 1). (A.16)
This completes the proof. □
Corollary A.4. Let ¥ belong to the Schwartz space 50(rd )• By Lemma A3/(ii) for every L,N e n there is a constant CL/N > 0 such that
tL+d/2 / |„i
|(W®¥)(x/i)|< CLrN (1 + t)2L+^ 1 + f^J , x e Rd,t> 0. (A.17)
Additionally, we obtain for ® e S°(rd ) and e S(rd) such that
tL+d/2
|(W®®o)(x,f)| < Cln-TT, x e rd, 0 <t< 1. (A.18)
' (1 + |x|)N
A.2. Orthonormal Wavelet Bases
The following Lemma is proved in Wojtaszczyk [34, Section 5.1].
Lemma A.5. Suppose we have a multiresolution analysis in L2 (r) with scaling functions ¥° (t) and
associated wavelets ¥1(t). Let E = (0,1}d \{(0,...,0)}. For c = (ci,...,cd) e E, let ¥c = ®dWct.
Then the system
12jd/2¥c(2jx - k)}
ceE,jeZ,keZd
(A.19)
is an orthonormal basis in L2(rd).
Spline Wavelets
As a main example, we will consider the spline wavelet system. The normalized cardinal B-spline of order m + 1 is given by
Nm+i(x) := Nm*X(x), x e r, m e n,
(A.20)
beginning with N1 = X, the characteristic function of the interval (0,1). By
^m(x) : =
(Z£-J?Nm(£ + 2nk)|2)
(x), x e ^
(A.21)
we obtain an orthonormal scaling function, which is again a spline of order m. Finally, by
fm(x) := X (vm(2^j,Vm(t - k)^(-1)k <^m(2x + k + 1) (A.22)
the generator of an orthonormal wavelet system is defined. For m = 1, it is easily checked that -f1(x - 1) is the Haar wavelet. In general, these functions fm have the following properties:
(i) fm restricted to intervals [k/2, (k + 1)/2], k e z, is a polynomial of degree at most m - 1.
(ii) fm e Cm-2(r) if m > 2.
(iii) fiT-^ is uniformly Lipschitz continuous on r if m > 2.
(iv) The function fm satisfies a moment condition of order m - 1, that is,
xefm(x)dx = 0, e = 0,1,...,m - 1. (A.23)
In particular, fm satisfies (D), (ML) for 0 <L < m and (SK) for K <m - 1.
Acknowledgments
The author would like to thank Holger Rauhut, Martin Schafer, Benjamin Scharf, and Hans Triebel for valuable discussions, a critical reading of preliminary versions of this paper and for several hints how to improve it.
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