with Applications to Wavelet Coorbit Spaces
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Felix Voigtlaender, M.Sc. aus Aachen, Deutschland
Berichter:
Universitätsprofessor Dr. rer. nat. Hartmut Führ
ao. Univ.-Prof. i.R. tit. Univ.-Prof. Dr. Hans Georg Feichtinger Universitätsprofessor Dr. rer. nat. Holger Rauhut
Tag der mündlichen Prüfung:03.11.2015
with Applications to Wavelet Coorbit Spaces
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Natur-wissenschaften genehmigte Dissertation
vorgelegt von
Felix Voigtlaender, M.Sc. aus Aachen, Deutschland
Berichter:
Universitätsprofessor Dr. rer. nat. Hartmut Führ
ao. Univ.-Prof. i.R. tit. Univ.-Prof. Dr. Hans Georg Feichtinger Universitätsprofessor Dr. rer. nat. Holger Rauhut
Tag der mündlichen Prüfung:03.11.2015
The main topic of this thesis is the development of criteria for the (non)-existence ofembeddings between decomposition spaces.
A decomposition space is defined in terms of
• a coveringQ= (Qi)i∈I of (a subset) of the frequency spaceRd,
• an integrability exponent pand
• a certain discrete sequence spaceYon the index set I.
The decomposition space norm of a distribution f is then computed by decomposing the frequency content of f according to the covering
Q, using a suitable partition of unity. Each of the localized pieces is measured in the Lebesgue space Lp and the contributions of the
individual pieces are aggregated using the discrete sequence space normk·kY.
Given two decomposition spaces, it is of interest to know whether there is an embedding between these two spaces. Since both decom-position spaces are defined only in terms of the respective coverings, weights and discrete sequence spaces, it should be possible to decide the existence of the embedding only based on these quantities.
Our findings will show that this is not only possible, but that the resulting criteria only involve discrete combinatorial considerations. In particular, no knowledge of Fourier analysis is needed for the ap-plicationof these criteria. Finally, our resultscompletely characterizethe existence of the desired embedding under mild assumptions on the two coverings and sequence spaces.
We apply our findings to a large number of concrete examples. Among others, we consider embeddings between
• α-modulation spaces,
• homogeneous and inhomogeneous Besov spaces and • shearlet-type coorbit spaces.
In all cases, the known results for embeddings between these spaces turn out to be special cases of our criteria; in some cases, our new approach even yields stronger results than those previously known.
For the discussion of shearlet-type coorbit spaces, we employ the second main result of this thesis which shows that the Fourier trans-form induces a natural isomorphism between a large class of wavelet coorbit spaces and certain decomposition spaces. This further empha-sizes the scope of our embedding results for decomposition spaces.
Das Hauptthema dieser Arbeit ist die Entwicklung von Kriterien für die (nicht)-Existenz vonEinbettungen zwischen Dekompositionsräumen.
Ein Dekompositionsraum ist hierbei definiert über
• eine ÜberdeckungQ= (Qi)i∈I des FrequenzraumesRd,
• einen Integrabilitätsexponenten pund
• einen diskreten FolgenraumYauf der Indexmenge I.
Zur Berechnung der Dekompositionsraum-Norm einer Distribution f zerlegt man f auf der Fourierseitegemäß der ÜberdeckungQ (mittels einer zugehörigen Zerlegung der Eins). Die einzelnen Teile werden in Lp gemessen und die Gesamt-Norm ergibt sich durch Zusammen-fassen aller einzelnen Normen mittels des FolgenraumesY.
Falls zwei verschiedene Dekompositionsräume gegeben sind, stellt sich die Frage, ob eine Einbettung zwischen diesen existiert. Da bei-de Räume nur über die zugehörigen Überbei-deckungen, Gewichte und Integrabilitätsexponenten definiert sind, sollte es möglich sein, die Existenz der Einbettung nur anhand dieser Größen zu entscheiden.
Wir werden nicht nur sehen, dass dies tatsächlich möglich ist, son-dern auch, dassfür die Anwendungder sich ergebenden Kriterien nur diskrete, kombinatorische Überlegungen nötig sind; insbesondere be-nötigt man keinerlei Wissen über Fourieranalysis. Weiterhin bemer-ken wir, dass unsere Resultate – unter milden Annahmen an die Überdeckungen und Folgenräume – eine äquivalente Bedingung für die Existenz der jeweiligen Einbettung geben.
Als Anwendung diskutieren wir unter anderem die Existenz von Einbettungen zwischen
• α-Modulations Räumen,
• homogenen und inhomogenen Besovräumen und • Coorbit-Räumen vom Shearlet-Typ.
Wir werden sehen, dass in jedem dieser Fälle die existierenden Resul-tate Spezialfälle unserer neuen Kriterien sind. In vielen Fällen liefern die neuen Kriterien sogar stärkere Aussagen als die bisher bekannten.
Die Behandlung der Einbettungen für Coorbit-Räume vom Shearlet-Typ wird durch das zweite Hauptergebnis der Arbeit möglich. Kon-kret werden wir sehen, dass die Fouriertransformation einen Isomor-phismus zwischen einer großen Klasse von Wavelet Coorbit-Räumen und gewissen Dekompositionsräumen liefert. Damit sind unsere Kri-terien für Einbettungen zwischen Dekompositionsräumen auch auf Wavelet Coorbit-Räume anwendbar.
— Blaise Pascal
A C K N O W L E D G M E N T S
I would like to thank my supervisor Professor Hartmut Führ for giv-ing me the opportunity to write this thesis, for many inspirgiv-ing dis-cussions and for having patience whenever I gave him another 100 pages to read.
Furthermore, I would like to express my thanks to Professor H.G. Feichtinger for helpful discussions and references to related papers and for making it possible for me to stay at the CIRM in Marseille for several weeks, where I had the chance to visit a summer school on computational harmonic analysis, a conference on harmonic analysis and function spaces and a workshop on coorbit theory.
Last, but not least, I would like to thank my family and friends, in particular my wife, for supporting me in writing this thesis, although that meant that I often had less time for them than they deserved.
This research was funded by the Excellence Initiative of the Ger-man federal and state governments, and by the GerGer-man Research Foundation (DFG), under the contract FU402/5-1.
1 i n t r o d u c t i o n 1
1.1 Coorbit theory (for Quasi-Banach spaces) 2 1.1.1 Examples 2
1.1.2 Features of coorbit theory 4
1.1.3 Coorbit theory for Quasi-Banach spaces 5 1.1.4 New contributions 6
1.2 (Fourier-side) decomposition spaces 7 1.2.1 General decomposition spaces 7 1.2.2 Fourier-side decomposition spaces 8 1.2.3 Related work 9
1.2.4 New contributions 10
1.3 Wavelet coorbit spaces as decomposition spaces 10 1.3.1 Wavelet coorbit spaces 10
1.3.2 Isomorphism between wavelet coorbit spaces and certain decomposition spaces 11
1.3.3 Applications of the decomposition space view of coorbit spaces 12
1.3.4 New contributions 13
1.4 Embeddings between decomposition spaces 13 1.4.1 New contributions 16
1.5 Applications 16
2 c o o r b i t t h e o r y f o r q ua s i-b a na c h s pa c e s 21 2.1 A crash course on Quasi-Banach spaces 22
2.1.1 Properties of Quasi-Banach spaces 22 2.1.2 Pathological examples of Quasi-Banach spaces
∗ 32
2.2 Solid Quasi-Banach spaces of functions 34 2.2.1 Definition and basic properties 35
2.2.2 Pointwise convergence and convergence in solid function spaces 38
2.2.3 Completeness of function spaces and the Riesz-Fischer property 43
2.2.4 Weighted spaces and translation invariance 48 2.2.5 Completions of solid function spaces and the
Fatou property∗ 55
2.3 Wiener amalgam spaces for Quasi-Banach spaces 59 2.3.1 Definition and elementary properties of Wiener
amalgam spaces 60
2.3.2 A discrete characterization of Wiener amalgam spaces 68
2.3.3 Convolution relations for Wiener amalgam spaces 84 2.3.4 Further properties of Wiener amalgam spaces 106
2.4 Coorbit theory for Quasi-Banach spaces 110 2.4.1 Assumptions for coorbit theory 111
2.4.2 The reproducing formula for square integrable representations 113
2.4.3 Construction of the reservoirR = H1
v
¬ 119 2.4.4 Definition of coorbit spaces with respect to
Quasi-Banach spaces 130
2.4.5 Atomic decomposition results 138 3 (f o u r i e r s i d e) d e c o m p o s i t i o n s pa c e s 159
3.1 Convolution relations forLp(Rd), 0< p< 1 160 3.2 Different types of coverings and their properties 168 3.3 Relations between coverings 186
3.4 (Fourier side) decomposition spaces 194 3.5 Dilations of decomposition spaces 210
4 wav e l e t c o o r b i t s pa c e s a s d e c o m p o s i t i o n s pa c e s 217 4.1 Admissible dilation groups and the dual action 219 4.2 Applicability of coorbit theory for the spacesLmp,q 225 4.3 Definition of FR:Rv → D0(O) 246
4.4 The induced covering and a special BAPU 258 4.4.1 The induced coveringQ 258
4.4.2 A special BAPU 264 4.5 Continuity ofFR: Co(Lmp,q)→ D(Q,Lp,`qu) 271 4.6 Continuity ofF−1 R :D(Q,Lp,` q u)→Co(Lmp,q) 284 4.7 Dilation invariance of coorbit spaces 298
5 e m b e d d i n g s b e t w e e n d e c o m p o s i t i o n s pa c e s 309 5.1 Sufficient conditions 310
5.2 Simplification of the sufficient conditions 347 5.3 Necessary conditions 376
5.3.1 The general case 376
5.3.2 The relatively moderate case 404 5.4 Summary 421
5.5 Decomposition spaces as spaces of tempered distribu-tions 439
6 a p p l i c at i o n s 445
6.1 Embeddings betweenα-modulation spaces 446 6.2 Embeddings for Besov spaces 458
6.3 Embeddings for shearlet type coorbit spaces 487 6.3.1 Shearlet type coorbit spaces as decomposition
spaces 488
6.3.2 Embeddings between shearlet type coorbit spaces 496 6.3.3 Shearlet type coorbit spaces and Besov spaces 510 6.4 Embeddings for shearlet smoothness spaces 535
6.5 Dilations of decomposition spaces and coorbit spaces 559 6.5.1 Isotropic dilations of shearlet-type
decomposi-tion spaces 560
6.5.3 Isotropic dilations ofα-modulation spaces 582 7 c o n c l u s i o n a n d o u t l o o k 611
Here, we fix commonly used notations. More specific notations are introduced in the text and are not listed here.
• The lettersZ,Q,R,Chave their usual meanings. We define the natural numbers as N := {n∈Z|n≥1}and N0 := N∪ {0}.
For convenience, we also define C∗ := C\ {0}, R∗+ := (0,∞) andR+:= [0,∞).
• We write A ⊂ B if A is a subset of B, where A = B is also allowed. For a strict inclusion, we use the notation A(B. • IfA,Bare sets, thenBA={f : A→ B}is the set of all functions
from Ato B.
• The notation M b U means that U ⊂ Rd is open and that M ⊂ U is relatively compact, with M ⊂ U, where M denotes the closure of M.
• For a finite set M, we write |M| ∈ N0 for the number of
ele-ments of M. If Mis an infinite set, we write|M|= ∞.
• IfXis a vector space, we writeY≤ XifY ⊂Xis a subspace of X.
• IfXis a vector space andM ⊂X, thenhMidenotes the span of M.
• The open/closed Euclidean balls aroundx ∈Rdof radiusr≥0
are denoted by Br(x)andBr(x), respectively.
• We denote the d-dimensional Lebesgue measure of a (measur-able) set M ⊂ Rd by
λd(M)or simply by λ(M). Furthermore,
we define
vd :=λd(B1(0))
as the measure of the Euclidean unit ball.
• Forn∈N0, we setn:={1, . . . ,n}. In particular, 0=∅.
• For any setXand any element x∈X, we define the Dirac delta
δx :X→ {0, 1},y7→δx,ywithδx,y =1 forx= yandδx,y=0 for
x6= y.
• Forx ∈ Rd, we write |x|:=kxk2 for the usual Euclidean norm of x. We will also use the “chinese bracket”
hxi:=
q
1+|x|2.
• If f :X →Cis a function, we denote its supremum norm by
kfksup :=sup
x∈X|
f(x)| ∈[0,∞].
Further, we set kfksup,M := kf|Mksup for each subset M ⊂ X,
where f|M denotes the restriction of f to M.
• For a set X and any subset A ⊂ X, we define the indicator functionχAof Aby χA :X→ {0, 1},x7→ 1, ifx ∈ A, 0, ifx ∈/ A.
• If G is a group and f : G → X is a function, where X 6= ∅ is any (nonempty) set, we define the left- and right translations of
f by
Lxf :G→X,y7→ f(x−1y),
Rxf :G→X,y7→ f(yx).
Furthermore, we let
f∨ :G→X,x7→ f(x−1). For X=C, we also define
fO :G→C,x7→ f(x−1).
• For d ∈ N and f : Rd → C, we define the modulation of f by
ξ ∈Rd as Mξf :=eξ ·f, with eξ :R d →C,x7→ e2πihx,ξi. • Ford∈N, f :Rd →CandT ∈GL Rd , we define DTf :Rd →C,x7→ |det(T)|− 1 2 ·f(T−1x), ∆Tf :Rd →C,x7→ f(TTx),
whereTT denotes the transpose ofT.
• A dual pairing of the form h·,·iis always supposed to be bilin-ear, whereas a dual pairing of the formh· | ·iis linear in the first and antilinear in the second component.
Ff(ξ) = bf(ξ) =
Z
Rd f(x)·e
−2πihx,ξidx
for every f ∈ L1 Rd
. The inverse Fourier transform is likewise given by F−1f(x) = b f(−x) = Z Rd f(ξ)·e 2πihx,ξid ξ.
We note that F extends to a unitary automorphism ofL2 Rd . • If Xis a topological space, then Cc(X)denotes the space of all
continuous functions f : X→Cwith compact support. • We set C0(Rd):= f :Rd →C lim |x|→∞|f(x)|=0 .
• We write A.Bif there is a constantCwith A≤C·B. Usually, the (implied) constant C is only allowed to depend on certain quantities which are either implicitly clear or mentioned explic-itly. In many cases, we will write something like
A.r,s B,
to indicate that the implied constant is only allowed to depend on rands.
• Finally, we write ABif we have A. BandB.A. The same restrictions as above apply.
A C R O N Y M S
l c h g r o u p locally compact Hausdorff group b a p u bounded admissible partition of unity b u p u bounded uniform partition of unity
1
I N T R O D U C T I O NThe author discusses valueless measures in pointless spaces. — Paul R. Halmos [48] The purpose of this thesis is the study of two different classes of (function) spaces, the c o o r b i t s pa c e s and the d e c o m p o s i t i o n s pa c e s. We will discover deep and novel connections between these two classes of spaces, but also between pairs of spaces belonging to one of the two classes:
Connecting these two classes of spaces is the Fourier transform. More precisely, if Co Lpm,q
is a wav e l e t c o o r b i t s pa c e (see be-low for definitions), then one can naturally extend the usual Fourier transformF : L2 Rd → L2 Rd to anisomorphism F : Co Lmp,q → D Q,Lp,`qu
between this wavelet coorbit space and a certain decomposition space
D Q,Lp,`qu
(also defined below).
The connection between different spaces of thesameclass is given by e m b e d d i n g s. In view of the isomorphism between wavelet coor-bit spaces and decomposition spaces obtained above, formulating these embeddings for decomposition spaces is sufficient. We will provide readily verifiable criteria for the existence of an embedding
ι:D Q,Lp1,`qu1
→ D P,Lp2,`q2
v
between two different decomposition spaces. Our criteria will be based on the relation between the geometries of the two coverings. Under restrictive, but commonly satisfied assumptions on the two coverings Q,P, our criteria completely characterize the existence of such an embedding.
We now describe the spaces under consideration and our results in more detail. We remark that the structure of this introduction parallels the structure of the entire thesis, i.e., the different topics are considered in the same order and the sections in the introduction correspond to the different chapters of the thesis.
The first part of the thesis is concerned with the two types of spaces mentioned above. We start with motivation for and with an introduc-tion to the general theory of c o o r b i t s pa c e s. Subsequently, we introduce the (f o u r i e r-s i d e) d e c o m p o s i t i o n s pa c e s which we will consider in the remainder of the thesis.
In the second part, we develop the connection between these two classes of spaces. We first introduce the special class of wav e l e t c o o r b i t s pa c e s. We then go on to show that the Fourier transform induces an isomorphism between these spaces and certain Fourier-side decomposition spaces.
In the final part, we study embeddings between different decom-position spaces. We then apply these abstract results to prove (or disprove) the existence of embeddings for a large class of concrete examples. In particular, we consider embeddings between
• (α)-modulation spaces,
• Besov spaces,
• Coorbit spaces of shearlet-type groups,
• Shearlet smoothness spaces as defined in [53]. We now decsribe these different topics in greater detail. 1.1 c o o r b i t t h e o r y (f o r q ua s i-b a na c h s pa c e s)
Many function spaces on Rd can be constructed using the following procedure: One fixes a certain family of functions Γ = (γi)i∈I on
Rdand correlates each suitable function or (tempered) distribution f
with all of theγi to obtain atransformed versionof f, i.e.,
WΓf : I →C,i7→ hf|γii.
By imposing suitable conditions on these coefficients, one obtains the actual function space, for example by requiring WΓf ∈ Lp(µ) for a
certain measureµon the set I. The resulting (coorbit) space is thus
Co(Γ,Y):= {f ∈ R |WΓf ∈Y},
where R is a suitable r e s e r v o i r of functions or distributions, for example R = S0 Rd
, andY is a certain space of functions onI, for exampleY =Lp(µ).
In many cases, the index setIis a locally compact Hausdorff group (LCH group) I =Gand the familyΓis given byγx= π(x)g forx∈G
and some fixed function g (called the a na ly z i n g w i n d o w) and a suitable representation π of G. The measure µis then usually taken
to be the Haar measure onG. The spaces obtained from such a family can be considered as coorbit spaces in the strict sense. In this thesis, we will only consider coorbit spaces that arise in this way.
1.1.1 Examples
To make this abstract approach more concrete, we mention the fol-lowing two well-known examples:
a. In (one-dimensional) wav e l e t a na ly s i s, one considers fami-lies of the form Γ= (LxDag)x∈R,a∈R∗, where the dilation
opera-torDa is defined by
Dag(x) =|a|−1/2·g(x/a).
The intuition here is that the different values of the dilation parametera lead to an analysis of the signal/function f at dif-ferentscales.
In the background of this construction is the (non-connected) “ax+b”-groupG=R×R∗ with multiplication given by
(x,a)·(y,b) = (x+ay,ab).
The family Γ results from a fixed window g by applying all values of the q ua s i-r e g u l a r r e p r e s e n tat i o n
G→ U L2(R),(x,a)7→ LxDa.
One can show that, for suitable analyzing windows g, the re-sulting family of coorbit spaces Co Γ,Lpws(G)
, with weight given by ws(x,a) = |a|−s, yields precisely the class of h o m o -g e n e o u s b e s ov s pa c e s B˙rp,p, see [28, Section7.2].
Since the homogeneous Besov spaces are special decomposition spaces, the above characterization of the spaces Co Γ,Lwps(G)
as homogeneous Besov spaces is also a special case of – as well as motivation for – the isomorphism between coorbit spaces and decomposition spaces which is obtained later in this thesis. See Chapters4and6for more details.
b. In the context of t i m e f r e q u e n c y a na ly s i s, the class of m o d u l at i o n s pa c e s plays an important role. Here, the fam-ily Γ = (LxMωg)x,ω∈Rd consists of all time-frequency shifts of
a fixed analyzing window g. One also says that Γ is a g a b o r f a m i ly.
More group theoretically, the modulation spaces can be inter-preted as coorbit spaces of the s c h r ö d i n g e r r e p r e s e n ta -t i o n
π:Hr → U L2(Rd)
,(x,ω,e2πiτ)7→ e2πiτ·eπihx,ωi·LxMω
of the (reduced) h e i s e n b e r g g r o u p Hr = Rd×Rd×T, see We remark that the
group structure of
Hr=Rd×Rd×T is of coursenotthat of the direct product of the three factors.
[28, Section7.1]. An extensive introduction to modulation spaces is given in Gröchenig’s book [45], especially in chapters 11and 12.
More recently, the discovery of the group theoretical background of s h e a r l e t s led to an investigation of the associated coorbit spaces,
cf. [13,15]. Certain coorbit spaces of the Blaschke group that give rise to spaces of holomorphic functions are discussed in [31].
1.1.2 Features of coorbit theory
One question that arises in each of the above examples is whether the resulting spaces differ if different analyzing windows g are chosen. Thus, it would be preferable to have a general framework at hand which shows (among other things) this independence of the analyz-ing window g, at least for “reasonable” windows g.
Such a framework, which exhibits the similarities between the con-crete examples mentioned above, is provided by the theory of c o o r -b i t s pa c e s. This theory was originally developed by Feichtinger and Gröchenig in the80s in a series of papers [28,29,30]. They considered the setting of unitary, irreducible, (square) integrable representations
π : G → U(H)of a locally compact group G and of a solid Banach
function space Y on G. In this case, the (g e n e r a l i z e d) wav e l e t t r a n s f o r m of a function/distribution f is given by
Wgf : G→C,x7→ hf|π(x)gi
and the resulting coorbit space is defined as Co(Y):=Cog(Y):= f ∈ RWgf ∈Y with normkfkCo(Y):=Wgf
Y, whereY is a solid function space on
G.
The main building blocks of coorbit theory are the following: 1. Construction of a suitabler e s e r v o i rR= RYsuch that Cog(Y)
Because of the generality of the approach, one cannot simply use a space likeS0(Rd) as the reservoirR, since we need to define the wavelet transform Wgf(x) = hf|π(x)gifor elements f ∈ R. But the representationπ only acts onH, so thatRneeds to be related toHin some way. ButHis a general Hilbert space, not necessarily a space of functions onRd.
becomes a well-defined Banach space.
2. Identification of a class of (g o o d) a na ly z i n g w i n d o w s AY such that Cog(Y) =Coh(Y)holds for allg,h∈ AY.
Here, it is relevant that the spaces RY,AY actually do not depend on
all properties of the space Y, but only on the norms |||Lx|||Y→Y and |||Rx|||Y→Y of the left/right translation operators Lx,Rx. Thus, one
can often choose the same reservoir and the same class of analyzing vectors for a huge class of spacesY.
Among the advantages of identifying a certain class of Banach spaces as a class of coorbit spaces are the following:
1. Coorbit theory provides at o m i c d e c o m p o s i t i o n s, cf. [29, Theorem 6.1]. This means that one can fix a suitable family
(xi)i∈I in G, as well as (bounded) linear functionals (λi)i∈I on
Co(Y)such that each f ∈Co(Y)can be written as f =
∑
i∈I
with a norm equivalencekfkCo(Y) (λi(f)) i∈I Yd. Here,Yd is
a certain sequence space associated toY.
The differentπ(x)g are called at o m s, which gives rise to the name at o m i c d e c o m p o s i t i o n.
2. The question of inclusions or embeddings between coorbit spaces is greatly simplified. In fact, if Y,Z are solid Banach function spaces on a common group G, then CoY ,→ CoZ holds if and only if the embedding Yd ,→ Zd is true, cf. [30, Theorem8.4]. Even compactness of the embedding between the coorbit spaces is equivalent to compactness of the embedding between the dis-crete sequence spaces, cf. [30, Theorem9.4].
Here, the two coorbit spaces of course have to use the same representationπ: G→ U(H).
1.1.3 Coorbit theory for Quasi-Banach spaces
Several generalizations of the construction of coorbit spaces have been considered: In [18], the authors develop a form of coorbit spaces in cases where the representation is not necessarily irreducible or in-tegrable. A construction of coorbit spaces on homogeneous spaces is developed in [11]. Here, the assumption is that the representation is square integrable modulo a certain subgroup ofG.
Finally, in [58], the theory of coorbit spaces is extended to cover the
case of Quasi-Banach spaces instead of genuine Banach spaces. For a Quasi-Banach space, the usual triangle inequality is replaced by
kx+yk ≤ C·[kxk+kyk].
This generalization is important in connection with n o n l i n e a r a p p r o x i m at i o n. Here, one is given a (possibly redundant) d i c -t i o na r y (fi)
i∈I of elements fi ∈ Y and seeks for arbitrary f ∈Y an
approximation ˜f =∑i∈If αi· fi to f under the restriction
If
≤ K, i.e., using only a fixed number of dictionary elements. The term n o n l i n -e a r a p p r o x i m at i o n is used since the map f 7→ f˜ is not required to be linear.
In the context of “usual” norms like the`p-norms on CN and with the dictionary given by the standard basis (e1, . . . ,eN), the best
ap-proximation is obtained by ˜x= x·χIx, where the set Ix ⊂ Ncontains
the (indices of) theKlargest entries of x. A vectorx∈CN is calledK -s pa r -s e[35, Definition2.1], if at most Kentries of x are nonzero and x is called c o m p r e s s i b l e if x can be well approximated by sparse vectors. The minimal approximation error is denoted by
σK(x)p :=min n kx−zkp z∈C N isK-sparseo.
A rigorous measure of the compressibility of a vector is given by the decay ofσK(x)pfor largeK(but of courseK N). In [35, Proposition 2.3] it is shown that this error obeys the bound
σK(x)q≤K
1
q−1p · kxk
Thus, the decay is faster for large values of 1/p, i.e., one wants to control kxkp for small p > 0. But the `p- and Lp-norms are only
genuine norms for p ≥1 and q ua s i-n o r m s for 0< p <1. Thus, it is worthwhile to also consider q ua s i-b a na c h s pa c e s, because the above considerations show that these are natural spaces for nonlinear approximation.
In the setting of coorbit spaces, the dictionary can be taken as
(π(xi)g)i∈I, where the family (xi)i∈I is as in the discussion of the
atomic decomposition results for coorbit spaces above. These results show that in order to approximate f by finite sums∑i∈Jαiπ(xi)g, it
is sufficient to approximate the coefficient sequence (λi(f))i∈I by a
finitely supported sequence, with respect to the (quasi)-normYd. For
Y = Lq(G), one has Y
d = `q(I). Thus, the decay in equation (1.1.1)
implies that elements f ∈Co(Lp)forsmall p>0 can be well approx-imatedin the norm ofCo(Lq)forq p by finite linear combinations of the dictionary (π(xi)g)i∈I. Again, this shows that elements in
Co(Lp)with small p >0 are compressible as elements of Co(Lq)for q p, so that it is desirable to be able to consider the space Co(Lp)
for 0< p<1.
1.1.4 New contributions
For the reasons outlined above, we will develop coorbit theory for Quasi-Banach spaces, following [58]. This is done mostly to make the thesis more self-contained. But we will see that the construction in [58] is based on an incorrect convolution relation for Wiener amalgam spaces given in [59]. Thus, the new contribution of our treatment is the realization that this convolution relation indeed fails in general, as well as the development of an alternative convolution relation for which we then verify that a modified version of the construction in [58] indeed yields well-defined coorbit spaces.
As a byproduct, we slightly reduce the assumptions that the space Yhas to fulfill: In [58], the spaceW(L∞,Y)is required to be left- and right-invariant, whereas we only require right-invariance.
Furthermore, we give an introduction to the theory of solid quasi-normed function spaces. Most of the results here are well-known for Banach spaces, see e.g. [69]. For the case of quasi-normed spaces they are probably folklore, but I am not aware of a publication in which comparable results are discussed.
We finally remark that the chapter about coorbit theory for Quasi-Banach spaces contains two subsections marked with an asterisk (∗). These subsections are not needed for the remainder of the thesis, but contain instructive examples indicating possibly pathological be-haviour of Quasi-Banach spaces and solid function spaces.
1.2 (f o u r i e r-s i d e) d e c o m p o s i t i o n s pa c e s
Decomposition spaces were first introduced in 1985 by Feichtinger and Gröbner[27, 26] in a very general setting. The underlying idea (which should also be compared to the concept of general Wiener amalgam spaces[24, 23]) is to introduce a space D(Q,B,Y) of func-tions or distribufunc-tions, whose localproperties are measured in the Ba-nach spaceB– which is a Banach space of functions/distributions, for example B=LporB= FLp with p≥1 – and theglobalbehaviour of these local properties is measured using the discrete sequence space Y, e.g.Y =`qwith q≥1.
1.2.1 General decomposition spaces
More precisely, Q = (Qi)i∈I is required to be a (well-behaved)
cov-ering of the (typically locally compact) space X andΨ = (ψi)i∈I is a
suitable partition of unity subordinate to Q. The general idea is to define the norm on the decomposition spaceD(Q,B,Y)by
kfkD(Q,B,Y):=(kψi·fkB)
i∈I
Y.
Well-known special cases of this construction include the class of (ho-mogeneous and inho(ho-mogeneous) b e s ov-s pa c e s[57, 66], as well as the class of (α-)m o d u l at i o n s pa c e s[25,43,45]. In both cases, one uses certain weighted `q spaces as the global component Y and cer-tain FLp-spaces as the local component B. The main difference be-tween the different spaces lies in the chosen coveringQ. For the Besov spaces, one uses adyadiccovering of the frequency space, whereas the usual modulation spaces use a uniformcovering. The α-modulation
spaces are defined using a family of intermediate coverings[43] be-tween these two extreme cases.
For the definition above to make sense, one has to impose suitable assumptions on the covering Q, the spacesBandYand on the parti-tion of unityΨin order to ensure that this indeed yields awell-defined Banach space. Well-defined here means (among other things) that the resulting space is independent of the chosen partition of unityΨ, with equivalent norms for different choices.
To see that this can indeed pose a problem, we remark that for B = FL1 Rd
, one can not simply take a partition of unity (ψi)i∈I
consisting of characteristic functions, because this will lead to
kψi·fkB= F −1( ψi· f) L1 =∞,
even for very well behaved functions like a smooth bump function f, because the crude localization by ψi will make ψi· f discontinuous,
which impliesF−1(ψ
The assumption made in [27] is to take Bas a Banach module over a Banach algebra A and to assume that the partition of unity Ψ is uniformly bounded in A.
1.2.2 Fourier-side decomposition spaces
In this thesis, we will consider a more concrete class of decomposition spaces which differs slightly from the general setting introduced by Feichtinger and Gröbner. The main reason for this is that we allow the choice of the Quasi-Banach spaces B = FLp for 0 < p < 1 for the local component B. We will see later that these spaces are not Banach modules of the Banach algebraFL1 which is usually used as
the space of multipliers for the choice B = FLp with p ≥ 1. In a certain sense, one can show that FLpis a “Quasi-Banach module” of
FLp. More precisely, it turns out that an estimate of the form kF−1(f·g)kLp ≤C· kF−1fkLp · kF−1gkLp
is true, but only if both f and g are compactly supported. Further-more, the constantCdepends in an unexpected way on the supports of f,g, so that the Fourier multiplier f doesnotyield a bounded oper-ator onFLp, but only on FKLp for each compact set K ⊂Rd, where FKLp is the space of all distributions f ∈ S0 Rd
∩Lp Rd with supp bf ⊂K.
Thus, this situation is not covered by the abstract setting considered in [27, 26]. Therefore, instead of trying to generalize the conditions considered by Feichtinger and Gröbner to match our present setting, we will stick to the less abstract (but less general) approach of Borup and Nielsen[5], which we will modify slightly. Nevertheless, we will make use of many of the ideas and results from [27], which mostly remain valid in the present setting.
More precisely, a f o u r i e r-s i d e d e c o m p o s i t i o n s pa c e s is de-fined as follows: It forms a space of distributions on an open set
O ⊂Rd (thought of as a subset of thefrequency domainRd) for which
a covering Q = (Qi)i∈I, together with a partition of unity (ϕi)i∈I
is given (both have to satisfy certain admissibility conditions). The decomposition space norm is then defined by localising f ∈ D0(O) to ϕif, by taking the Lp-norm of the individual pieces on the space
side and by finally measuring the norms of the pieces in a suitable sequence spaceY, that is
kfkD(Q,Lp,Y)= (kF−1(ϕif)k Lp)i∈I Y.
We remark that this definition seems to differ significantly from the usual definition of these spaces, since these are defined on the space
side, i.e. for a given f ∈ S0 Rd , the quantity F−1(ϕif) Lp used above is replaced by kF−1(ϕibf)kLp
in the usual definition. But if one is willing to replace the reservoir
S0 Rd
by the space [F(C∞c (O))]0, then it is easy to see that the Fourier transform yields an isomorphism of the “classical” space-side decomposition spaces to the Fourier-side decomposition spaces as defined above. Since the spaceF(C∞c (O))and its dual space appear less natural than the spacesC∞c (O)andD0(O), we prefer to work on the Fourier-side directly.
In section Section5.5, we will give a natural criterion which ensures
D Q,Lp,`qu
,→ S0 Rd
, so thatS0 Rd
can be used as the reservoir for the “space-side” versions of the decomposition spaces.
In any case, it is crucial to note that the partition of unity(ϕi)i∈I in
both cases partitions the frequency domain. 1.2.3 Related work
Recently, (the space-side versions of) these decomposition spaces re-ceived increased attention. This began with the work of Borup and Nielsen [5], who introduced the class of s t r u c t u r e d a d m i s s i b l e c ov e r i n g s (but only in the case O = Rd). They showed that these coverings always admit a suitable partition of unity (ϕi)i∈I (called
a b o u n d e d a d m i s s i b l e pa r t i t i o n o f u n i t y) and constructed tight frames (ηn,T)(n,T)∈Zd×T for L2 Rd
which have the additional property that the decomposition space norm with respect to the struc-tured admissible covering Q = (TQ)T∈T of Rd can be characterized using the frame coefficients (cf. [5, Proposition3]), to wit
kfkD(Q,Lp,`q u) (hf,η (p) n,Ti)n∈Zd `p T∈T `qu with η(np,T) = |T| 1 2−1p ·η
n,T. Here, each T ∈ T is an affine map of the
formTx= ATx+bTand|T|:=|det(AT)|. The family(ηn,T)(n,T)∈Zd×T
even yields an atomic decomposition for the decomposition space (cf. [5, Theorem2]).
Further recent publications connected to the theory of decompo-sition spaces are the papers of Toft and Wahlberg[64] and by Han and Wang[49]. They considered α-m o d u l at i o n s pa c e s and their dilation- and embedding properties. Finally, Labate et al.[53] defined certain s h e a r l e t s m o o t h n e s s s pa c e s as special decomposition spaces. They also showed that these spaces coincide with the curvelet smoothness spaces (cf. [53, Proposition4.4]) and established embed-dings between shearlet smoothness spaces and Besov spaces (cf. [53, Proposition4.3]).
1.2.4 New contributions
Chapter 3 does not contain striking new results, but rather many small contributions, since it serves mainly to fix notation for the rest of the thesis.
Among these contributions are the following: We introduce a new class of coverings, thes e m i-s t r u c t u r e d a d m i s s i b l e c ov e r i n g s. Furthermore, we give a precise definition of the Fourier-side decom-position spaceD Q,Lp,`qu
and show its well-definedness and com-pleteness. In the main source[5] for this type of decomposition spaces, this is neglected by referring to the classical papers by Feichtinger and Gröbner[27,26], although these are strictly speaking not applicable in the setting of FLp for p < 1, or if the space S0 Rd
is used as the reservoir. This leads in particular to the problem that the decomposi-tion spaces as defined in [5, Definition3] arenotcomplete in general, as we will see.
As a final result in this section, we show that a decomposition space is invariant under dilation with a matrix g ∈ GL Rd
if and only if there is an embedding
D Q,Lp,`qu
→ D gQ,Lp,`qu
. (1.2.1) Although this is a rather simple observation, it seems to be new. Fur-thermore, we will see in Chapter 6 that this simple reformulation of dilation invariance using embeddings for decomposition spaces will allow us to asymptotically determine the operator norm of the isotropic dilation operators onα-modulation spaces and shearlet-type
coorbit spaces.
1.3 wav e l e t c o o r b i t s pa c e s a s d e c o m p o s i t i o n s pa c e s As indicated at the beginning of this introduction, the Fourier trans-form induces an isomorphism
F : Co Lmp,q
→ D Q,Lp,`qu
between certain coorbit spaces and suitable decomposition spaces. 1.3.1 Wavelet coorbit spaces
More precisely, the coorbit spaces for which we will prove this result are the wav e l e t c o o r b i t s pa c e s. These are formed as follows: We fix some closed matrix group H ≤ GL Rd
, which we call the d i l at i o n g r o u p. The groupGwhich is used to define the wavelet coorbit spaces is G=RdoH, which can be interpreted as the group of all affine mappings generated by arbitrary translations x∈Rd and by all dilationsh ∈ H.
On this group, we consider the q ua s i-r e g u l a r r e p r e s e n ta -t i o n, given as
π :G→ U L2(Rd),(x,h)7→ LxDh.
Here, Dhf(y) := |det(h)|−1/2· f h−1y
is the (L2-normalized)
dila-tion of f byh.
Finally, we only consider spacesYof the formY= Lpm,q(G), where
the (weighted) m i x e d l e b e s g u e s pa c e Lmp,q = Lmp,q(G) is defined for p,q∈(0,∞]as Lmp,q(G):= n f :G→C f measurable and kfkLmp,q <∞ o , with kfkLp,q m := Z Hk m(·,h)· f(·,h)kqLp(Rd) dh |det(h)| 1/q
forq∈(0,∞)and with
kfkLp,q
m :=ess sup
h∈H k
m(·,h)·f(·,h)kLp(Rd)
forq=∞. Here,m:G→(0,∞)is a (measurable) weight.
The results in [38,36] precisely characterize those dilation groupsH for which the quasi-regular representation is irreducible and square integrable; we call such groups a d m i s s i b l e. This characterization is expressed in terms of the d ua l a c t i o n
H×Rd→Rd,(h,
ξ)7→h−Tξ
of Hon the frequency space Rd. The dilation group H is admissible if and only if the dual action has a single open orbitO=HTξ0of full
measure, called the d ua l o r b i t, such that the isotropy group Hξ0 = n h∈ H h −T ξ0=ξ0 o is compact.
We will see that for any such group, all assumptions for the appli-cation of coorbit theory to the spaces Lmp,q(G) are satisfied, at least
under mild growth conditions on the weight m.
1.3.2 Isomorphism between wavelet coorbit spaces and certain decomposi-tion spaces
By virtue of the dual action, the dilation group H induces a covering of the frequency space Rd, or more precisely of the dual orbitO. To wit, if (hi)i∈I is a well-spread family in H, we will see that there
is always a set Q b O such that Q = hi−TQ
i∈I is a structured Here, QbOmeans that Q is compactly contained inO.
admissible covering ofO. We call each such coveringQa c ov e r i n g o f O i n d u c e d b y H.
We will see that if the weight m = m(h) only depends on the second factor, one can define a form of the f o u r i e r t r a n s f o r m which restricts to an isomorphism
F : Co Lmp,q
→ D Q,Lp,`qu between the wavelet coorbit space Co Lpm,q
and the decomposition space D Q,Lp,`qu
. Here, Q is an induced covering of O as above and the weight udepends on H,mand the exponentq.
Thus, wavelet coorbit theory becomes a branch of decomposition space theory.
Furthermore, this isomorphism opens the door to a Fourier-analytic understanding of wavelet coorbit spaces. More conceptually, the ex-istence of this isomorphism shows that a large part of the structure of the dilation groupHis actually not relevant for the approximation theoretic properties of the coorbit spaces Co Lmp,q
. More precisely, the only relevant property of H is the way in which the dual action h 7→ h−Tξ covers the dual orbitO. Once we have developed
embed-ding results for decomposition spaces, we will see in Chapter6how the geometry of the induced covering determines the existence of em-beddings of the coorbit space Co Lpm,q
into other coorbit spaces or into more classical smoothness spaces like Besov spaces. The next subsection discusses these and other applications in more detail. 1.3.3 Applications of the decomposition space view of coorbit spaces It turns out that many properties of the coorbit spaces Co Lmp,q
are more apparent from the decomposition space point of view than from the coorbit space point of view. One example is the property of d i l a -t i o n i n va r i a n c eof the space Co Lpm,q. Here, we ask whether for a given dilation γ∈ GL Rdand a function f ∈ L2 Rd∩Co Lmp,q
, it automatically follows that f ◦γ ∈ Co Lmp,q
, together with a norm estimate kf ◦γkCo(Lp,q
m ) . kfkCo(Lmp,q). As long as the space L
p,q m (G)
is left-invariant, it is not hard to see that this is the case for every dilation γ∈ H. But for dilations γ∈GL Rd\H, this is not clear at
all and not easy to answer from the point of view of coorbit theory. As an application of our results, we will show that for the simili-tude group H = R∗·O Rd
, the resulting coorbit spaces Co Lmp,q
are invariant under arbitrarydilations γ ∈GL Rd, while this is not
true for the shearlet group in dimension two, as we will see in Chap-ter6.
More generally, the benefit of the decomposition space point of view is that it becomes possible to compare coorbit spaces that are defined using differentdilation groups H≤ GL Rd
of view of coorbit spaces, it is not even clear what we mean by this: The coorbit space Co Lmp,q RdoH is a subspace of the reservoir
Rv, which depends heavily on the group RdoH, so that it is not clear how two coorbit spaces “living” on different groups Rd
oH1
andRdoH2 can be compared.
But in the decomposition space point of view, this is possible: The most transparent case is if the two dilation groups H1,H2 have the
same dual orbits O1 = O = O2. In this case, the decomposition
spaces D1 := D Q1,Lp1,`qu11
and D2 := D Q2,Lp2,`qu22
are both subspaces of the spaceD0(O)of distributions onOand one can thus ask whether an embedding D1 ,→ D2 (or vice versa) is true. By
employing the isomorphism ofD1 andD2with the respective coorbit
spaces, this leads to a natural form of embedding between the coorbit spaces Co Lp1,q1 m1 R d oH1 and Co Lp2,q2 m2 R d oH2 .
Thus, using the above isomorphism, the question of embeddings between coorbit spaces “living” on different groups can be reduced to the question of embeddings between decomposition spaces. The study of this question, and a multitude of applications, is taken on in Chapters 5and 6. We will see that one can obtain reasonable forms of embeddings, even if the two sets O1,O2do not coincide.
1.3.4 New contributions
The findings in this chapter are new contributions of this thesis, but some of the results (only Banach spaces, i.e., for p,q∈ [1,∞]) were al-ready submitted for publication in the paper [41], written by Hartmut Führ and the present author.
Furthermore, the applicability of coorbit theory (for p,q ∈ [1,∞]) for admissible dilation groups was already studied in [39,40].
1.4 e m b e d d i n g s b e t w e e n d e c o m p o s i t i o n s pa c e s
In the description of decomposition spaces above, we already men-tioned the papers of Toft and Wahlberg[64] and by Han and Wang[49] in which embeddings between α-m o d u l at i o n s pa c e s for differ-ent values of α are studied. Furthermore, Labate et al.[53] consider embeddings of the newly defined s h e a r l e t s m o o t h n e s s s pa c e s into the more classical Besov spaces (cf. [53, Proposition4.3]).
As these results show, there is great interest in establishing embed-dings between decomposition spaces that are defined using different coverings. Furthermore, an important application of the decompo-sition space view of wavelet coorbit spaces will be that embedding results for decomposition spaces will yield embeddings between coor-bit spaces that “live” on different groups.
In Chapter 5, we solve the problem of existence of embeddings between different decomposition spaces in a quite general setting. We
will see that all previously known embedding results as mentioned above are special cases of the results in this chapter. In a lot of cases, our approach even yields stronger results. In fact, our criteria are quite often sharp, i.e., they characterize the existence of the respective embedding completely.
The precise problem that we want to solve is the following: Given two coveringsQ= (Qi)i∈IandP = Pj
j∈J, we want to find sufficient,
as well as necessary criteria for the existence of an embedding
D Q,Lp1,`q1 u ,→ D P,Lp2,`q2 v . (1.4.1) Since these spaces are defined solely in terms of the coverings Q,P, the weights u,v and the exponents p1,p2,q1,q2, it should be
possi-ble to decide the existence of the embedding only in terms of these quantities.
We will show that this intuition is justified by proving that the existence of the embedding for decomposition spaces can be derived as a consequence of embeddings between certain nested sequence spaces. These sequence spaces encode the geometric relation between the two coverings Q,P.
For a more concrete description of our criteria, let us assume for simplicity that Q is subordinate to P; this means Qi ⊂ Pji for each
i∈ I and a suitableji ∈ J. In this case, the geometric relationship
be-tween the two coverings is predominantly determined by the family of i n t e r s e c t i o n s e t s
Ij :=
i∈ IQi∩Pj 6=∅ , for j∈ J.
Our results will show that the embedding (1.4.1) holds, if we have p1≤ p2and if we have an embedding
The exponent pO
2 used here is called
thel o w e r c o n j u g at e e x p o n e n tfor p2, defined by pO2 =min p2,p02 . (ci)i∈Ij `pO2 j∈J `q2 v . |det(Ti)| 1 p2−p11 c i i∈I `q1 u (1.4.2) of certain (nested) sequence spaces which are defined using the inter-section sets Ij. We observe that it does not suffice to only consider
the family of sets Ij; in case ofp16= p2 the geometry ofQalso enters
through the term |det(Ti)|, which corresponds to the measure of the
sets Qi = TiQ0i+bi, where we assume Q0i ⊂ BR(0) for all i ∈ I and
some fixed R>0.
The condition given above is chosen as to make the proof of our result more streamlined. If one actually wants to apply this crite-rion to concrete examples, checking estimate (1.4.2) for allsequences
(ci)i∈I (for which the right-hand side is finite) seems overly
considerably. In fact, we will show that validity of this embedding of nested sequence spaces is equivalent to finiteness of the nested norm
u−i 1|det(Ti)| 1 p1−p12 i∈Ij `pO2·(q1/pO2)0 j∈J `q2·(q1/q2) 0 v .
In other words, we have reduced validity of an infinite number of estimates to finiteness of the quasi-norm of a singlesequence.
We have thus found a readily verifiable sufficient criterion for the existence of an embedding for decomposition spaces. We remark that the resulting criterion only involves discrete combinatorial considera-tions and can thus be applied without having to know anything about Fourier analysis. Furthermore, it is worth noting that the geometry of the coveringsQ,P only enters through three factors:
• We assumed thatQis subordinate toP,
• the size/measure of the setsQienters through the term|det(Ti)|,
• the relative geometry of the coverings Q,P enters through the sets Ij.
In particular, the absolute position of the sets Qi,Pj is not relevant. Given these sufficient conditions for the existence of an embedding for decomposition spaces, it is natural to studysharpnessof these con-ditions. We will see that the condition p1 ≤ p2 is always necessary.
Furthermore, for the range p2 ∈ (0, 2] and under very mild
condi-tions on the coveringQ, we will be able to show that existence of the embedding (1.4.1) already implies validity of the embedding for dis-crete sequence spaces given in equation (1.4.2). Thus, for p ∈ (0, 2], the condition given above achieves a complete characterization of the existence of an embedding for decomposition spaces.
For the range p2 ∈ (2,∞], we will also show that the embedding
in equation (1.4.2) is necessary for the existence of an embedding between the decomposition spaces. But in this case, we will need to impose much stronger assumptions on the relation between the coverings Q,P. Precisely, we will assume that the covering Q and the weight u are r e l at i v e ly P-m o d e r at e; this means that the weights u and(|det(Ti)|)i∈I are almost constant on each of the sets
Ij. Thus, we also achieve a characterization of the existence of an
em-bedding for decomposition spaces for the range p2∈ (2,∞], but only
under more restrictive assumptions. Nevertheless, these assumptions are fulfilled for many concrete examples, especially if Q,P are both coverings corresponding to α-modulation spaces, even for different
values ofα.
Finally, we remark that we only assumed Q to be subordinate to
P for simplicity. In Chapter 5, we will consider the more general setting in whichQisalmostsubordinate toP. In this case, we do not
only study the embedding in equation (1.4.1), but also the “reverse” embedding. Even more generally, we will consider the case of two coveringsR,Sfor which there are certain setsA,Bso thatRis almost subordinate to S “near A” and S is almost subordinate to R “near B”.
1.4.1 New contributions
The embedding results derived in Chapter 5 are new results of this thesis. To my knowledge, similar results have only been established for the special case ofα-modulation spaces in [43], [64] and [49] and furthermore in [53], where the authors consider embeddings between Besov spaces and the newly defined class of shearlet smoothness spaces.
Our approach also seems to be new: The proof idea used in [49] is to establish certain special cases of the embedding and then to derive the general case by interpolation. In contrast to this, we only use interpolation to derive certain “local” results. Once these have been obtained, we use embedding results for nested sequence spaces (as in equation (1.4.2) above), to establish the global embedding.
Furthermore, it should be observed that the approach in this thesis is applicable in the following cases which have (to my knowledge) not been treated elsewhere:
1. The case of different “integrability exponents”(p1,q1)6= (p2,q2) (for nonequivalent coveringsQ 6≈ P) and
2. the case in which the two coverings P,Q cover different sets
O0 6= O. A special case of this are embeddings between homo-geneous and inhomohomo-geneous Besov spaces.
The first of these cases has of course been studied for concrete ex-amples in the case Q = P, but apparently not for Q 6= P. Even in the (relatively) well-understood case ofα-modulation spaces, only the
case (p1,q1) = (p2,q2)has been considered, at least in [64] and [49]. 1.5 a p p l i c at i o n s
Finally, we illustrate the power and generality of our approach to embeddings for decomposition spaces by considering a large number of concrete examples.
As a litmus test, we first consider the well-studied case of em-beddings between α-m o d u l at i o n s pa c e s for different values of
α∈[0, 1), i.e., we characterize the existence of embeddings
Mγ1,α1
p1,q1 ,→ M
γ2,α2
For the case (p1,q1) = (p2,q2) a complete solution to this problem
was found by Han and Wang[49]. We will see that their findings can be established conveniently using our general approach. Fur-thermore, we even achieve a complete characterization in case of
(p1,q1)6= (p2,q2), thereby extending the known results.
Even more importantly, our unified approach indicates why the re-sults in [49] actually hold. The main geometric fact that is used is that the α-covering O(α) of Rd, which is used to define the α-modulation
spaces, is almost subordinate to the β-covering O(β) for α ≤ β.
Fur-thermore, O(α) is relatively moderate with respect to O(β) and the same is true of the weights used to define theα-modulation spaces.
The next example that we consider is an even more classical class of spaces, the b e s ov s pa c e s. These come in two variants, the ho-mogeneous and the inhoho-mogeneous Besov spaces. But although the properties of each individual of the two types of Besov spaces are very well studied, it seems that not much is known about the relation between homogeneous and inhomogeneous Besov spaces.
The only result in this direction of which I am aware is the fact (cf. [67, Theorem in §2.3.3]) that forα>σp =d
1 p−1 +, the norm f 7→ kfkLp +kfkB˙p α,q
defines an equivalent quasi-norm on the inhomogeneous Besov space Bαp,q Rd
. In particular, this yields an embedding Bαp,q(R
d),
→B˙αp,q(Rd)
of the inhomogeneous Besov space Bαp,q Rd
into the homogeneous Besov space ˙Bαp,q Rd
. Note that this is only a sufficient criterion, not a necessary one. Furthermore, nothing seems to be known about the reverse embedding.
Using our general embedding results, we change this situation com-pletely. We will derive sufficient and necessary conditions for the existence of the embedding
Bp1
α1,q1(R
d),→B˙p2
α2,q2(R
d)
and the reverse embedding. Although in some cases there is still a gap between the necessary and the sufficient conditions, this gap is very small: The only thing that can happen is that the sufficient condition requires astrictinequality between certain of the quantities
α1,α2,p1,p2,q1,q2, whereas the necessary condition only yields a
non-strictinequality.
Finally, we also consider embeddings betweenα-modulation spaces
and inhomogeneous Besov spaces. Again, for (p1,q1) = (p2,q2),
these results coincide with the characterization given by Han and Wang[49], but we also handle the case(p1,q1)6= (p2,q2).
A completely new class of decomposition spaces is given by the next example. Here, we study decomposition spaces that are obtained from the covering S(c), which is an induced covering of the shearlet-type group Hc = ( ε a b 0 ac ! ε∈ {±1}, a ∈(0,∞), b∈R )
for arbitrary c ∈ R. For c = 12, this yields the usual shearlet group as considered in [13, 15]. By our results on the isomorphism of wavelet coorbit spaces and decomposition spaces, we know that the Fourier transform yields an isomorphism between the s h e a r l e t -t y p e c o o r b i t s pa c e Co Lmp,q R2oHc and the decomposition spaceD(S(c),Lp,`qu)for a suitable weight u. We will use this isomor-phism in conjunction with our embedding results for decomposition spaces to characterize (for a certain class of weights) the existence of embeddings of coorbit spaces
Co Lp1,q1 m1 R 2 oHc1 →Co Lp2,q2 m2 R 2 oHc2 .
As promised above, these results indicate that the decomposition space point of view can indeed be used to derive embeddings be-tween coorbit spaces “living” on different groups.
We remark that the coveringsS(c)are a challenging example, since
S(c1) is not almost subordinate to S(c2) for c
1 6= c2. Nevertheless,
for c1 < c2, one can show that S(c1) is almost subordinate to S(c2)
“away from the y-axis”, whereasS(c2) is almost subordinate to S(c1) “near the y-axis”. Thus, our results from Chapter 5 will suffice to
characterize the existence of the desired embeddings.
Finally, we use the isomorphism between wavelet coorbit spaces and decomposition spaces, as well as our embedding results for de-composition spaces, to derive necessary and sufficient criteria for the existence of embeddings between inhomogeneous Besov spaces and shearlet-type coorbit spaces. Since the geometries of the two cover-ings are very different (i.e. S(c) is almost subordinate to the inhomo-geneous Besov covering B, but notrelatively moderate), our criteria do not achieve a complete characterization in all cases. Neverthe-less, our results are an improvement of the known results, since to my knowledge no embeddings of this type have been known. The only result which is slightly comparable to the embeddings derived here is [15, Theorem4.7], where the authors consider embeddings of a strict subspace of Co Lmp,q R2oH1/2
into a sum of two inhomoge-neousBesov spaces.
As our final example of embeddings for decomposition spaces, we consider the newly introduced class of s h e a r l e t s m o o t h n e s s s pa c e s[53]. In [53, Proposition 4.3], the authors already give suffi-cientcriteria for the existence of embeddings between inhomogeneous
Besov spaces and shearlet smoothness spaces, at least for the special case (p1,q1) = (p2,q2). As a strong improvement, we will be able to
completely characterize the existence of these embeddings, even for
(p1,q1)6= (p2,q2). Furthermore, we are able to give a similar
charac-terization forα-modulation spaces instead of Besov spaces, at least in
the rangeα∈0,12. This seems to be a completely new result.
In the last section of Chapter 6, we show that embedding results for different decomposition spaces can also be used to study opera-tors which act on asinglefixed decomposition space or coorbit space. More precisely, we consider the question of dilation invariance of shearlet coorbit spaces andα-modulation spaces. As we saw in
equa-tion (1.2.1) above, invariance of a decomposition space under dilation with g ∈ GL Rd
can be reformulated as an embedding statement between the “original” decomposition space D Q,Lp,`qu
and the “dilated” decomposition spaceD gQ,Lp,`qu
. In view of the isomor-phism of wavelet coorbit spaces and decomposition spaces, a similar statement also holds for coorbit spaces.
These observations will allow us to asymptotically compute the norm of the isotropic dilation operator Dλ : f 7→ f ◦λid on the
shearlet-type coorbit spaces Co Lpv,αq R 2
oHc. More precisely, we
consider weights of the form
vα : Hc→(0,∞),ε
a b 0 ac
!
7→aα.
We will then show Here, we have
1 p±4 = minn1p, 1−1 p o . |||Dλ||| λ− α−2p ·λ (1+c)1 q−12 +(1−c) 1 q−p±41 +
for λ ∈ (1,∞) and c ∈ (−∞, 1]. A similar estimate also holds for λ∈ (0, 1).
For the shearlet-type coorbit spaces, our results are completely new. In contrast, the asymptotic norm of the isotropic dilation operator acting on theα-modulation spaceMsp,,αqhas been computed for almost
all cases in [49]. As a further application, we show that almost the same results can be derived using the criteria developed in Chapter5. For the range p ∈ [1,∞], our findings are identical to those in [49], but for p ∈ (0, 1), λ > 1 and certain values of s, the results in [49] turn out to be slightly incorrect. This indicates that checking known results using our new, systematic approach to embeddings can still be worthwhile.
Returning to the setting of shearlet coorbit spaces, we also char-acterize the class of transformations g ∈ GL R2 which leave all shearlet-type coorbit spaces Co Lpm,q R2oHcinvariant under
con-jugation. We will see that this holds precisely for upper triangular matrices with positive determinant. This is in sharp contrast to the
case of the similitude group described above. There, all coorbit spaces turned out to be invariant under arbitrarydilations. This is a further case in point where the decomposition space point of view, together with our criteria for embeddings between decomposition spaces, eas-ily yields results which are not obvious from the point of view of coorbit spaces.
2
C O O R B I T T H E O R Y F O R Q U A S I - B A N A C H S PA C E SIn this chapter, we discuss the general theory of coorbit spaces. This is a preparation for the first main result of the thesis, the isomorphism between wavelet coorbit spaces and certain decomposition spaces on Rd, which is presented in Chapter4.
We consider the coorbit spaces directly in the setting of Quasi-Banach spaces. We remark that this is strictly speaking not a new contribution of this thesis, but was established before by Rauhut[58]. The main problem with the generalization to Quasi-Banach spaces is that the whole theory of coorbit spaces is based on r e p r o d u c i n g k e r n e l b a na c h s pa c e s, which are Banach spacesY of functions for which f = f ∗g for all f ∈ Y and some specific function g is true. Hence, the theory is based largely on certain convolution re-lations. The problem is that many Quasi-Banach spaces like Lp(G)
(with 0< p<1) do not satisfy any meaningful convolution relations, although they are invariant under left- and right translations. The reason for this is that the embeddingLp(G),→ L1loc(G)doesnothold (for 0< p < 1 if Gis not discrete), so that a convolution f∗g is not defined in general, even for g ∈ Cc(G). To circumvent this problem,
Rauhut uses the w i e n e r a m a l g a m s pa c e W(L∞,Y)instead of us-ing the Quasi-Banach spaceYdirectly. For the spaceW(L∞,Y), some useful convolution relations are indeed true.
But one problem of [58] is that it is based (in part) on the convolu-tion relaconvolu-tion W L∞,Lwp ∗W L∞,Lpw∗ ∨ ,→W L∞,Lwp , (2.0.1) where w : G → (0,∞) is a submultiplicative weight and 0 < p ≤ 1, see [58, Theorem3.7]. As we will see below (cf. Example2.3.26), this convolution relation is false in general. To circumvent this problem, we develop a slight variant of this convolution relation and check that the rest of the paper remains valid with suitable changes.
The chapter is structured as follows: In Section2.1, we start with a brief introduction to the main properties of general Quasi-Banach spaces. In Section2.2, we specialize this to the case of s o l i d q ua s i -b a na c h s pa c e s o f f u n c t i o n s, of which the usual Lp-spaces are the most prominent examples. In particular, we study the connec-tion between pointwise convergence and convergence in the funcconnec-tion space under consideration.
Section 2.3 is devoted to studying w i e n e r a m a l g a m s pa c e s with respect to Quasi-Banach spaces, based on [59]. The general idea of these spaces is to “combine” two (Quasi-)Banach spaces, a l o c a l
c o m p o n e n t B and a g l o b a l c o m p o n e n t Y to obtain the space W(B,Y) of functions/distributions whose local properties are deter-mined by B, but whose global properties are determined by Y. In this thesis, we will only consider B = L∞(G)as the local component to avoid some technicalities. In particular, this section contains an adjusted version of the incorrect convolution relation (2.0.1) above.
Finally, Section 2.4 is concerned with coorbit theory for Quasi-Banach spaces, as originally developed by Rauhut[58]. Our contri-bution here is to check that most of the paper remains valid, even though it originally uses the incorrect convolution relation (2.0.1).
We also remark that Rauhut assumes the space W(L∞,Y) to be invariant under left-, as well as right translations, cf. [58, Definition 4.1]. In our modified approach, only right invariance ofW(L∞,Y)is needed.
2.1 a c r a s h c o u r s e o n q ua s i-b a na c h s pa c e s
In this section, we will cover the most important properties ofq ua s i -n o r m e d s pa c e s and q ua s i-b a na c h s pa c e s. In the first subsec-tion, we develop the most essential properties of these spaces, which will be used again and again in this chapter. The second subsection – which is not required for the rest of the thesis – illustrates some of the pathological properties of Quasi-Banach spaces, e.g. the (pos-sible) failure of the Hahn-Banach theorem as well as the (pos(pos-sible) discontinuity of the quasi-norm.
