On wavelet coorbit spaces associated to different dilation groups

Felix Voigtlaender; Hartmut F\"uhr; Jordy Timo van Velthoven

REVIEW 3 major objections 5 minor 51 references

On wavelet coorbit spaces associated to different dilation groups

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two dilation groups induce identical wavelet coorbit spaces exactly when they share an essential frequency support and the transition map between their parameter spaces is a quasi-isometry.

desk verdict A solid, careful extension of the coarse-geometric coorbit classification to the reducible case, with a real open caveat about the connectivity-respecting hypothesis. read the letter →

arxiv 2411.08416 v1 pith:VEIPWATU submitted 2024-11-13 math.FA math.CA

Hartmut Führ , Jordy Timo van Velthoven , Felix Voigtlaender This is my paper · ORCID

classification math.FAmath.CA MSC 42B3542C1542C4043A6551F30

keywords coarsegeometrywaveletcoorbitspacesdilationgroupsintegrablyadmissibleessentialfrequencysupportanisotropicBesovquasi-isometrydecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks when two different matrix dilation groups produce the same wavelet coorbit spaces, the Banach function spaces defined by imposing norm conditions on wavelet coefficients. Its answer, for a broad class of admissible dilation groups, is geometric: two groups are coorbit equivalent exactly when they have the same essential frequency support and the map between their parameter spaces is a quasi-isometry. This turns a norm-by-norm comparison of function spaces into a checkable statement about large-scale geometry, and it also settles the open question of uniqueness of the essential frequency support. The same machinery proves that anisotropic Besov spaces for general expansive matrices require reducible dilation groups: they cannot be coorbit spaces of an irreducibly admissible dilation group unless the dilation is isotropic.

What carries the argument

The load-bearing object is the essential frequency support $O$, the open full-measure set with $O=H^{T}C$ for compact $C$; it is shown unique and serves as the common space on which coorbit norms live. The full orbit map $p:H\times C\to O$, $(h,\xi)\mapsto h^{-T}\xi$, is the carrier of the classification: under the connectivity-respecting conditions it is a surjective quasi-isometry from the product of a word metric on $H$ and a bounded metric on $C$ to the cover-induced chain metric on $O$. The connectivity-respecting conditions require $C$ to be compact connected and the stabilizer of the connected component containing $C$ to be compactly generated. The proof route goes through induced covers and their weak equivalence, the condition that each set of one cover meets only boundedly many sets of the other and conversely, because coorbit spaces are Besov-type decomposition spaces determined by such covers.

What would settle it

A decisive test would be to produce an integrably admissible dilation group that satisfies the first connectivity condition but whose component stabilizer is not generated by finitely many bounded pieces; the authors explicitly state they know no such group, and its existence would show the quasi-isometry classification is not universal. A second test would be to find two connectivity-respecting groups with the same essential frequency support whose transition map is not a quasi-isometry but whose coorbit norms are nevertheless equivalent for all $1\le p,q\le\infty$, which Theorem 4.4 says cannot happen.

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Extended reading notes

Core claim

The paper's central result (Theorem 4.4) is that for connectivity-respecting integrably admissible dilation groups $H_1,H_2$ with essential frequency supports $O_1=H_1^{T}C_1$, $O_2=H_2^{T}C_2$, coorbit equivalence is equivalent to $O_1=O_2$ and the transition map $p_2^{*}\circ p_1:(H_1\times C_1,d_{H_1\times C_1})\to(H_2\times C_2,d_{H_2\times C_2})$ being a quasi-isometry. To get there, the paper proves that every integrably admissible dilation group has a unique essential frequency support (Theorem 2.7) and that the full orbit map $p:H\times C\to O$, $(h,\xi)\mapsto h^{-T}\xi$, is a surjective quasi-isometry under the connectivity-respecting assumptions (Theorem 3.16). The applications include a subgroup criterion: for $H_1\subseteq H_2$, coorbit equivalence holds exactly when $H_2/H_1$ is compact, and an anisotropic Besov space for an expansive matrix admits an irreducible coorbit description only when the dilation is a scalar multiple of the identity.

Load-bearing premise

The classification rests on the assumption that each dilation group's frequency support is generated by one compact connected piece and that the group elements sending that piece's connected component to itself can be generated by finitely many bounded pieces; the authors know of no integrably admissible group that violates the second part.

Editorial extensions

If this is right

The essential frequency support of an integrably admissible dilation group is unique, so a single well-defined frequency set is attached to each such group.
Coorbit equivalence for connectivity-respecting groups can be decided by one quasi-isometry check between explicit metric spaces, eliminating norm-by-norm verification.
For nested dilation groups $H_1\subseteq H_2$, the coorbit spaces coincide exactly when the quotient $H_2/H_1$ is compact.
A one-parameter group $\exp(\mathbb{R}X)$ is coorbit equivalent to the isotropic $\mathbb{R}_+\cdot I$ exactly when $X=sI+Y$ with $s\neq 0$ and $\exp(\mathbb{R}Y)$ relatively compact.
Homogeneous anisotropic Besov spaces for general expansive matrices cannot be realized as coorbit spaces of irreducibly admissible dilation groups unless the dilation is isotropic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the uniqueness of the essential frequency support gives a fast necessary condition for coorbit equivalence, so new dilation groups can first be checked for support agreement before any quasi-isometry is computed.
Inference: the theorem's scope depends on the connectivity-respecting condition (c2), so a natural search is for an integrably admissible dilation group whose component stabilizer is not compactly generated; the authors state they do not know such an example.
Inference: the support-based picture suggests that coorbit spaces are ultimately determined by the shape of the frequency support together with the large-scale structure of the group action, so the same method could be adapted to dilation groups acting on cones or sectors rather than all of $\mathbb{R}^d\setminus\{0\}$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the paper is a genuinely useful advance in the coarse-geometric approach to wavelet coorbit spaces, and the proof quality is high. The main novelty is that it carries the quasi-isometric classification from irreducible to reducible integrably admissible dilation groups, and it settles an open uniqueness question for the essential frequency support. The negative result about anisotropic Besov spaces (Theorem 5.2) is a nice payoff: it shows you genuinely need reducible representations to cover the general anisotropic case.

The technical core is the quasi-isometry between the product H×C and the frequency support O for a 'connectivity-respecting' group. The orbit-map estimates in Theorems 3.14–3.16 are carefully argued, and the authors are explicit about standing assumptions. I also appreciate that they prove uniqueness of the frequency support with a clean integrability argument rather than hand-waving. The writing is heavy, but the burden is on the subject, not on them.

Where I'd push back is on the hypothesis. The classification in Theorem 4.4 and the subgroup criterion in Corollary 4.7 only apply to connectivity-respecting groups, and condition (c2)—that the stabilizer of the relevant connected component is compactly generated—is not known to hold for every integrably admissible dilation group. The authors say so themselves in Section 3.2. That is a real scope limitation, and it means the abstract's claim of a 'general subgroup characterization' is slightly stronger than what is established. If someone later finds an integrably admissible group failing (c2), the main equivalence theorem would not cover it. That doesn't make the paper wrong; it makes it conditional. I'd want the referee to ask the authors to either prove (c2) for all integrably admissible groups, or at least to state the classification with the caveat more prominently.

The other issue is minor: Corollary 4.5 is stated without proof. It looks like a routine adaptation of Theorem 4.4, but the authors should either prove it or say explicitly why it follows.

On citations: several load-bearing steps use the authors' own earlier work ([26], [30], [7]). That's not a problem—those are published results, and the dependence is transparent. The Besov classification from [7] is doing real work in Theorem 5.2, but that's a legitimate external machinery, not a loop.

Who is this for? Specialists in wavelet coorbit theory and decomposition spaces. It deserves a serious referee. I'd accept for peer review and let the referee decide whether the connectivity-respecting caveat is acceptable or needs more work.

Referee Report

3 major / 5 minor

Summary. The paper develops coarse-geometric methods for comparing wavelet coorbit spaces associated with different dilation groups H ≤ GL(d, R). It first proves that the essential frequency support of an integrably admissible dilation group is unique (Theorem 2.7), resolving an open point from [30]. It then shows that, under the 'connectivity-respecting' hypotheses of Definition 3.3, the full orbit map p : H × C → O is a quasi-isometry (Theorem 3.16), and uses this to characterize coorbit equivalence of two connectivity-respecting dilation groups in terms of a quasi-isometry between transition maps (Theorem 4.4). Applications include a subgroup criterion (Corollary 4.7), a classification of dilation groups coorbit equivalent to the isotropic group R_+ · Id (Theorem 5.1), and the result that anisotropic Besov spaces defined by general expansive matrices cannot arise from irreducibly admissible dilation groups unless the dilation is isotropic (Theorem 5.2). The proofs are detailed and the main hypotheses are stated explicitly, but the central classification is conditional on the connectivity-respecting condition, whose automatic validity is left open in Section 3.2.

Significance. If the main results hold, the paper provides a substantial extension of the coarse-geometric classification of wavelet coorbit spaces from irreducible dilation groups [26] to the full class of integrably admissible dilation groups, including reducible ones. The uniqueness of the essential frequency support is a clean and useful result in itself. The quasi-isometry framework of Section 3 is carefully developed, and Theorem 5.2 gives a sharp negative answer to a natural question about anisotropic Besov spaces. The paper is honest about the unresolved nature of condition (c2) in Definition 3.3, and it gives explicit and well-organized proofs for the main theorems. The significance is tempered by the fact that the abstract and introduction present the subgroup characterization and equivalence criterion without the connectivity-respecting qualification, although the theorems themselves carry that hypothesis.

major comments (3)

[Abstract; Definition 3.3; Section 3.2] The abstract and introduction present the subgroup characterization and the coorbit-equivalence criterion as general results, but Theorem 4.4 and Corollary 4.7 are proved only for connectivity-respecting dilation groups, i.e., groups satisfying conditions (c1) and (c2) of Definition 3.3. The paper explicitly states in Section 3.2 that no example is known of an integrably admissible dilation group failing (c2), the compact-generation of the stabilizer H0. If such a group exists, neither the transition-map quasi-isometry characterization nor the subgroup criterion is established for it. This is a genuine scope limitation, not an internal contradiction, but it is load-bearing for the paper's central claims. The authors should either prove that (c2) is automatic for integrably admissible groups, or explicitly qualify the statements in the abstract and introduction and list the validity of (c2) as an open problem.
[Corollary 4.5] Corollary 4.5 is stated without proof, with the comment that its proof is similar to the first part of the proof of Theorem 4.4. This corollary is then used in the proofs of Corollary 4.7 and Theorem 5.1(ii). Since it is a necessary-condition tool for the subgroup characterization and for the isotropic classification, the paper should include a complete proof or a precise derivation from Theorem 4.4, rather than leaving this step to the reader.
[Theorem 5.2] The proof of Theorem 5.2 relies on the classification results of [7] for anisotropic Besov spaces and on the identification of coorbit spaces with Besov-type spaces. The argument is coherent, but the step leading to H2 ⊆ SH1 = Z(B) depends on the nontrivial equality SH1 = {C : C^{-1}BC = B}. The authors should make explicit which statements in [7] justify this equality for all C ∈ GL(d, R), since this is the point where the expansive normal-form reduction and the Besov classification are combined.

minor comments (5)

[Theorem 3.15, Step 1] In Notation 3.12(A1), W is an open, precompact, symmetric generating set, but the proof of Theorem 3.15 states 'Since W ⊆ H is compact'. This is not literally true; the proof can be repaired by replacing W with its closure (which is compact and still generated by H0) or by using the standard fact that every compact subset of a compactly generated locally compact group lies in some power W^m. The manuscript should clarify this point.
[Lemma 3.8, proof] In the displayed formula for the cover induced by P, the symbol C^* appears in 'P = ⋃_{i∈I} h_i^{-T} V^{-T} C^*' but is never defined; it should be B_{\epsilon/2}(C) (or the corresponding set from the definition of P).
[Theorem 4.3, proof] The citation '[51, Theorem 6.9 1/2]' looks like an incorrect or incomplete reference label; it should be corrected to the precise theorem in [51] that is being invoked.
[Example 5.3] The reference to 'Example 3.5(a)' should be 'Example 3.5(1)'.
[Corollary 3.17, proof] The sentence 'By Lemma A.2, we can choose a compact, connected set C_ξ ⊆ O satisfying {ξ} ∪ C ⊆ C_ξ' is slightly elliptical, since Lemma A.2 applies to an open connected set; the argument should first restrict to the connected component O0 of O containing C and ξ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main classification is proved from internal quasi-isometry results and parameter-free prior theorems; the open (c2) condition affects scope, not circularity.

full rationale

No load-bearing step reduces to its own input. The uniqueness theorem for the essential frequency support (Theorem 2.7) and the orbit-map quasi-isometry theorems (Theorems 3.14-3.16) are proved directly in the paper. Theorem 4.3 uses the parameter-free identifications and classifications from [30, Theorem 5.5] and [51, Theorem 6.9/Lemma 6.11], which do not assume coorbit equivalence of dilation groups; these are legitimate external supports rather than circular imports. Theorem 4.4 combines those results with the internally proved Theorem 3.16, so the transition-map condition is not a restatement of coorbit equivalence. Theorem 5.2 uses [7]'s classification of anisotropic Besov spaces as an independent benchmark, not as a reconstituted version of the present claim. The passages that could look problematic are scope/completeness caveats, not circularity: Section 3.2 states 'we currently do not know of an example where such groups arise as stabilizer groups H0 obtained in the sense of (c2) from an integrably admissible matrix groups,' so the classification is conditional on connectivity-respecting; and Corollary 4.5 is stated without proof ('its proof is similar to the first part of the proof of Theorem 4.4 ..., and hence we skip it'). These limit the unconditional reach of the results but do not make any equation or derived quantity equivalent to its own input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper contains no fitted constants or free parameters. Its main additional assumption is the connectivity-respecting condition, while all other inputs are established theorems from coarse geometry, Lie theory, and the existing coorbit and decomposition-space literature. Several load-bearing cited results are authored by the same group, but they are published, parameter-free theorems and not restatements of the present claims. No new particles, forces, dimensions, or ad hoc entities are introduced.

assumptions (7)

standard math Coorbit spaces for integrably admissible dilation groups admit equivalent Besov-type decomposition space norms relative to induced covers.
Invoked in the proof of Theorem 4.3 via [30, Theorem 5.5]; this is the bridge from wavelet transforms to frequency covers.
standard math Weak equivalence of admissible covers is equivalent to equality of the associated decomposition spaces for (p,q) different from (2,2), and to quasi-isometry of cover-induced metrics.
Used in Theorems 4.3 and 4.4 through [51, Theorem 6.9] and [26, Theorem 3.22].
standard math Classification of anisotropic Besov spaces with respect to equivalence and conjugation of expansive matrices, including the centralizer identification S_H1 = Z(B).
Used in Theorems 5.1 and 5.2 via [7, Corollary 6.5, Theorem 7.9, Lemmas 7.7 and 7.8].
standard math One-parameter groups exp(RX) are integrably admissible iff the real parts of eigenvalues of X are all positive or all negative, with frequency support R^d minus {0}.
Used throughout Section 5 via [30, Proposition 6.3].
standard math A connected Lie group quasi-isometric to R has a closed cocompact noncompact one-parameter subgroup.
Proved as Proposition C.4 in Appendix C, relying on results of Losert [39,40] and Cornulier-de la Harpe [10]; used in Theorem 5.1(ii).
standard math Existence of admissible analyzing vectors with compactly supported Fourier transform for integrably admissible dilation groups.
Used to define the coorbit spaces in Section 4.2 via [11, Proposition 2.7].
domain assumption The connectivity-respecting conditions (c1) and (c2) hold: O is generated by a compact connected set, and the stabilizer of the relevant connected component is compactly generated.
Introduced in Definition 3.3 and assumed in Theorems 3.16, 4.4, 4.7, 5.1, and 5.2; the authors note in Section 3.2 that they do not know whether every integrably admissible group satisfies (c2).

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Pith. "Pith review of On wavelet coorbit spaces associated to different dilation groups." pith.science (2026). https://pith.science/paper/VEIPWATU

@misc{pith2026241108416,
  author       = {Pith},
  title        = {Pith review of: On wavelet coorbit spaces associated to different dilation groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEIPWATU}},
  note         = {Machine review of arXiv:2411.08416}
}

read the original abstract

This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular representations. We show that the use of reducible representations is essential to include a variety of examples, such as anisotropic Besov spaces defined by general expansive matrices, in a common framework. The obtained criteria yield, among others, a simple characterization of subgroups of a dilation group yielding the same coorbit spaces. They also allow to clarify which anisotropic Besov spaces have an alternative description as coorbit spaces associated to irreducible quasi-regular representations.

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Works this paper leans on

51 extracted references · 47 canonical work pages

[26]

F¨ uhr and R

H. F¨ uhr and R. Koch. Classifying decomposition and wav elet coorbit spaces using coarse geometry. J. Funct. Anal., 283(9):109637, 2022

work page 2022
[30]

F¨ uhr and J

H. F¨ uhr and J. T. van Velthoven. Coorbit spaces associa ted to integrably admissible dilation groups. J. Anal. Math. , 144(1):351–395, 2021

work page 2021
[7]

Cheshmavar and H

J. Cheshmavar and H. F¨ uhr. A classiﬁcation of anisotrop ic Besov spaces. Appl. Comput. Harmon. Anal. , 49(3):863–896, 2020

work page 2020
[1]

Abbaspour and M

H. Abbaspour and M. Moskowitz. Basic Lie theory . Hackensack, NJ: World Scientiﬁc, 2007

work page 2007
[2]

Berge and F

E. Berge and F. Luef. A large scale approach to decomposit ion spaces. Stud. Math. , 265(3):257–301, 2022

work page 2022
[3]

M. Bownik. Atomic and molecular decompositions of aniso tropic Besov spaces. Math. Z. , 250(3):539–571, 2005

work page 2005
[4]

Breuillard

E. Breuillard. Geometry of locally compact groups of pol ynomial growth and shape of large balls. Groups Geom. Dyn. , 8(3):669–732, 2014

work page 2014
[5]

Bruna, J

J. Bruna, J. Cuf ´ ı, H. F¨ uhr, and M. Mir´ o. Characterizin g abelian admissible groups. J. Geom. Anal. , 25(2):1045–1074, 2015

work page 2015

Show all 51 references

[6]

Burtscheidt

A. Burtscheidt. Vanishing moments conditions for atomic decompositions of coor- bit spaces on quasi-Banach spaces . PhD thesis, R WTH Aachen University, 2020. https://publications.rwth-aachen.de/record/808437

2020
[8]

J. G. Christensen, A. H. Darweesh, and G. ´Olafsson. Coorbits for projective representations with an application to Bergman spaces. Monatsh. Math. , 189(3):385–420, 2019

2019
[9]

J. G. Christensen and G. ´Olafsson. Coorbit spaces for dual pairs. Appl. Comput. Harmon. Anal. , 31(2):303– 324, 2011

2011
[10]

Cornulier and P

Y. Cornulier and P. de la Harpe. Metric geometry of locally compact groups , volume 25 of EMS Tracts Math. Z¨ urich: European Mathematical Society (EMS), 2016

2016
[11]

Currey, H

B. Currey, H. F¨ uhr, and K. Taylor. Integrable wavelet t ransforms with abelian dilation groups. J. Lie Theory, 26(2):567–595, 2016

2016
[12]

Dahlke, M

S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, and G. Tes chke. Generalized coorbit theory, Banach frames, and the relation to α -modulation spaces. Proc. Lond. Math. Soc. (3) , 96(2):464–506, 2008

2008
[13]

Dugundji

J. Dugundji. Topology. Allyn and Bacon Series in Advanced Mathematics. Allyn and B acon, Inc., Boston, Mass.-London-Sydney, 1978

1978
[14]

H. G. Feichtinger and P. Gr¨ obner. Banach spaces of dist ributions deﬁned by decomposition methods. I. Math. Nachr. , 123:97–120, 1985

1985
[15]

H. G. Feichtinger and K. Gr¨ ochenig. A uniﬁed approach t o atomic decompositions via integrable group representations. In Function spaces and applications (Lund, 1986) , volume 1302 of Lecture Notes in Math. , pages 52–73. Springer, Berlin, 1988

1986
[16]

H. G. Feichtinger and K. H. Gr¨ ochenig. Banach spaces re lated to integrable group representations and their atomic decompositions. I. J. Funct. Anal. , 86(2):307–340, 1989

1989
[17]

H. G. Feichtinger and K. H. Gr¨ ochenig. Banach spaces re lated to integrable group representations and their atomic decompositions. II. Monatsh. Math. , 108(2-3):129–148, 1989

1989
[18]

G. B. Folland. A course in abstract harmonic analysis . Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995

1995
[19]

Fornasier and H

M. Fornasier and H. Rauhut. Continuous frames, functio n spaces, and the discretization problem. J. Fourier Anal. Appl. , 11(3):245–287, 2005

2005
[20]

Frazier and B

M. Frazier and B. Jawerth. Decomposition of Besov space s. Indiana Univ. Math. J. , 34:777–799, 1985

1985
[21]

Frazier, B

M. Frazier, B. Jawerth, and G. Weiss. Littlewood-Paley theory and the study of function spaces , volume 79 of Reg. Conf. Ser. Math. Providence, RI: American Mathematical Society, 1991

1991
[22]

H. F¨ uhr. Continuous wavelet transforms with abelian d ilation groups. J. Math. Phys. , 39(8):3974–3986, 1998

1998
[23]

H. F¨ uhr. Generalized Calder´ on conditions and regular orbit spaces. Colloq. Math. , 120(1):103–126, 2010

2010
[24]

H. F¨ uhr. Coorbit spaces and wavelet coeﬃcient decay ov er general dilation groups. Trans. Amer. Math. Soc., 367(10):7373–7401, 2015

2015
[25]

F¨ uhr and K

H. F¨ uhr and K. Gr¨ ochenig. Sampling theorems on locally compact groups from oscillation estimates. Math. Z., 255(1):177–194, 2007

2007
[27]

F¨ uhr and M

H. F¨ uhr and M. Mayer. Continuous wavelet transforms fr om semidirect products: cyclic representations and Plancherel measure. J. Fourier Anal. Appl. , 8(4):375–397, 2002

2002
[28]

F¨ uhr and R

H. F¨ uhr and R. Raisi-Tousi. Dilational symmetries of d ecomposition and coorbit spaces. Appl. Comput. Harmon. Anal. , 69:22, 2024. Id/No 101610

2024
[29]

F¨ uhr and R

H. F¨ uhr and R. R. Tousi. Simpliﬁed vanishing moment cri teria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups. Appl. Comput. Harmon. Anal. , 43(3):449–481, 2017. W A VELET COORBIT SPACES ASSOCIATED TO DIFFERENT DILATIONS 3 3

2017
[31]

F¨ uhr and F

H. F¨ uhr and F. Voigtlaender. Wavelet coorbit spaces vi ewed as decomposition spaces. J. Funct. Anal. , 269(1):80–154, 2015

2015
[32]

Gr¨ ochenig

K. Gr¨ ochenig. Unconditional bases in translation and dilation invariant function spaces on Rn. In Con- structive theory of functions (Varna, 1987) , page 174–183. Publ. House Bulgar. Acad. Sci., Soﬁa, 1988

1987
[33]

Gr¨ ochenig

K. Gr¨ ochenig. Describing functions: Atomic decompos itions versus frames. Monatsh. Math. , 112(1):1–42, 1991

1991
[34]

Gr¨ ochenig, E

K. Gr¨ ochenig, E. Kaniuth, and K. F. Taylor. Compact ope n sets in duals and projections in L1-algebras of certain semi-direct product groups. Math. Proc. Cambridge Philos. Soc. , 111(3):545–556, 1992

1992
[35]

Hilgert and K.-H

J. Hilgert and K.-H. Neeb. Structure and geometry of Lie groups . Springer Monographs in Mathematics. Springer, New York, 2012

2012
[36]

Hochschild

G. Hochschild. The structure of Lie groups . Holden-Day, Inc., San Francisco-London-Amsterdam, 1965

1965
[37]

Kempka, M

H. Kempka, M. Sch¨ afer, and T. Ullrich. General coorbit space theory for quasi-Banach spaces and inho- mogeneous function spaces with variable smoothness and int egrability. J. Fourier Anal. Appl. , 23(6):1348– 1407, 2017

2017
[38]

R. S. Laugesen, N. Weaver, G. L. Weiss, and E. N. Wilson. A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. , 12(1):89–102, 2002

2002
[39]

V. Losert. On the structure of groups with polynomial gr owth. II. J. London Math. Soc. (2) , 63(3):640–654, 2001

2001
[40]

V. Losert. On the structure of groups with polynomial gr owth III. J. Algebra, 554:1–40, 2020

2020
[41]

P. W. Nowak and G. Yu. Large scale geometry. EMS Textb. Math. Z¨ urich: European Mathematical Society (EMS), 2012

2012
[42]

J. Peetre. New thoughts on Besov spaces , volume 1 of Duke Univ. Math. Ser. Mathematics Department, Duke University, Durham, NC, 1976

1976
[43]

H. Rauhut. Coorbit space theory for quasi-Banach space s. Stud. Math. , 180(3):237–253, 2007

2007
[44]

Rauhut and T

H. Rauhut and T. Ullrich. Generalized coorbit space the ory and inhomogeneous function spaces of Besov- Lizorkin-Triebel type. J. Funct. Anal. , 260(11):3299–3362, 2011

2011
[45]

J. Roe. Lectures on coarse geometry, volume 31 of Univ. Lect. Ser. Providence, RI: American Mathematical Society (AMS), 2003

2003
[46]

K. A. Ross. Closed subgroups of compactly generated LCA groups are compactly generated. Topology Appl., 259:378–383, 2019

2019
[47]

St¨ ockert and H

B. St¨ ockert and H. Triebel. Decomposition methods for function spaces of Bs p,q type and F s p,q type. Math. Nachr., 89:247–267, 1979

1979
[48]

H. Triebel. General function spaces. I: decomposition methods. Math. Nachr. , 79:167–179, 1977

1977
[49]

H. Triebel. Spaces of Besov-Hardy-Sobolev type . Teubner-Texte Math. Teubner, Leipzig, 1978

1978
[50]

J. T. Van Velthoven and F. Voigtlaender. Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition. M´ em. Soc. Math. Fr., Nouv. S´ er., To appear. Preprint: arXiv:2203.07959

arXiv
[51]

Voigtlaender

F. Voigtlaender. Embeddings of Decomposition Spaces. Mem. Amer. Math. Soc. , 287(1426):iii+255, 2023. Lehrstuhl f ¨ur Geometrie und Analysis, R WTH Aachen University, D-52056 Aachen, Germany Email address : fuehr@mathga.rwth-aachen.de F aculty of Mathematics, University of Vie...

2023

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Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [26]

F¨ uhr and R

H. F¨ uhr and R. Koch. Classifying decomposition and wav elet coorbit spaces using coarse geometry. J. Funct. Anal., 283(9):109637, 2022

work page 2022

[2] [30]

F¨ uhr and J

H. F¨ uhr and J. T. van Velthoven. Coorbit spaces associa ted to integrably admissible dilation groups. J. Anal. Math. , 144(1):351–395, 2021

work page 2021

[3] [7]

Cheshmavar and H

J. Cheshmavar and H. F¨ uhr. A classiﬁcation of anisotrop ic Besov spaces. Appl. Comput. Harmon. Anal. , 49(3):863–896, 2020

work page 2020

[4] [1]

Abbaspour and M

H. Abbaspour and M. Moskowitz. Basic Lie theory . Hackensack, NJ: World Scientiﬁc, 2007

work page 2007

[5] [2]

Berge and F

E. Berge and F. Luef. A large scale approach to decomposit ion spaces. Stud. Math. , 265(3):257–301, 2022

work page 2022

[6] [3]

M. Bownik. Atomic and molecular decompositions of aniso tropic Besov spaces. Math. Z. , 250(3):539–571, 2005

work page 2005

[7] [4]

Breuillard

E. Breuillard. Geometry of locally compact groups of pol ynomial growth and shape of large balls. Groups Geom. Dyn. , 8(3):669–732, 2014

work page 2014

[8] [5]

Bruna, J

J. Bruna, J. Cuf ´ ı, H. F¨ uhr, and M. Mir´ o. Characterizin g abelian admissible groups. J. Geom. Anal. , 25(2):1045–1074, 2015

work page 2015