Abstract
This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood—Paley-type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.
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B. Barrios and J. J. Betancor, Characterizations of anisotropic Besov spaces, Math. Nachr. 284 (2011), 1796–1819.
D. Bernier and K. F. Taylor, Wavelets from square-integrable representations, SIAM J. Math. Anal. 27 (1996), 594–608.
M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003).
M. Bownik, Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z. 250 (2005), 539–571.
J. Bruna, J. Cufí, H. Führ and M. Miró, Characterizing abelian admissible groups, J. Geom. Anal. 25 (2015), 1045–1074.
J. Cheshmavar and H. Führ, A classification of anisotropic Besov spaces, Appl. Comput. Harmon. Anal. 49 (2020), 863–896.
J. G. Christensen, Sampling in reproducing kernel Banach spaces on Lie groups, J. Approx. Theory, 164 (2012), 179–203.
J. G. Christensen and G. Ólafsson, Examples of coorbit spaces for dual pairs, Acta Appl. Math. 107 (2009), 25–48.
J. G. Christensen and G. Ólafsson, Coorbit spaces for dual pairs, Appl. Comput. Harmon. Anal. 31 (2011), 303–324.
B. Currey, H. Führ and V. Oussa, A classification of continuous wavelet transforms in dimension three, Appl. Comput. Harmon. Anal. 46 (2019), 500–543.
B. Currey, H. Führ and K. Taylor, Integrable wavelet transforms with abelian dilation groups, J. Lie Theory 26 (2016), 567–596.
B. N. Currey, Admissibility for a class of quasiregular representations, Canad. J. Math. 59 (2007), 917–942.
M. Duflo and C. C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal. 21 (1976), 209–243.
H. Feichtinger, Banach convolution algebras of Wiener type, in Functions, Series, Operators, Vol. I, II (Budapest, 1980), North-Holland, Amsterdam, 1983, pp. 509–524.
H. G. Feichtinger, Banach spaces of distributions defined by decomposition methods. II, Math. Nachr. 132 (1987), 207–237.
H. G. Feichtinger and P. Gröbner, Banach spaces of distributions defined by decomposition methods. I, Math. Nachr. 123 (1985), 97–120.
H. G. Feichtinger and K. Gröchenig, A unified approach to atomic decompositions via integrable group representations, in Function Spaces and Applications (Lund, 1986), Springer, Berlin, 1988, pp. 52–73.
H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), 307–340.
H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), 129–148.
J. J. F. Fournier and J. Stewart, Amalgams of Lpand lq, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 1–21.
M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799.
M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34–170.
H. Führ, Continuous wavelet transforms with abelian dilation groups, J. Math. Phys. 39 (1998), 3974–3986.
H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer, Berlin, 2005.
H. Führ, Generalized Calderón conditions and regular orbit spaces, Colloq. Math. 120 (2010), 103–126.
H. Führ, Coorbit spaces and wavelet coefficient decay over general dilation groups, Trans. Amer. Math. Soc. 367 (2015), 7373–7401.
H. Führ, Vanishing moment conditions for wavelet atoms in higher dimensions, Adv. Comput. Math. 42 (2016), 127–153.
H. Führ and M. Mayer, Continuous wavelet transforms from semidirect products: cyclic representations and Plancherel measure, J. Fourier Anal. Appl. 8 (2002), 375–397.
H. Führ and R. R. Tousi, Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups, Appl. Comput. Harmon. Anal. 43 (2017), 449–481.
H. Führ and F. Voigtlaender, Wavelet coorbit spaces viewed as decomposition spaces, J. Funct. Anal. 269 (20150), 80–154.
K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1{42}
K. Gröchenig, E. Kaniuth and K. F. Taylor, Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups, Math. Proc. Cambridge Philos. Soc. 111 (1992), 545–556.
A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations. I, General results, J. Math. Phys. 26 (1985), 2473–2479.
F. Holland, Harmonic analysis on amalgams of Lpand ℓq, J. London Math. Soc. (2) 10 (1975), 295–305.
E. Kaniuth and K. F. Taylor, Minimal projections in L1-algebras and open points in the dual spaces of semi-direct product groups, J. London Math. Soc. (2) 53 (1996), 141–157, 1996.
E. Kaniuth and K. F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, Cambridge, 2013.
R. Koch, Analysis of Shearlet Coorbit Spaces, PhD thesis, RWTH Aachen University, 2015.
D. Larson, E. Schulz, D. Speegle and K. F. Taylor, Explicit cross-sections of singly generated group actions, in Harmonic Analysis and Applications, Birkhäuser, Boston, MA, 2006, pp. 209–203.
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 (2002), 89–102.
H. Liu and L. Peng, Admissible wavelets associated with the Heisenberg group, Pacific J. Math. 180 (1997), 101–123.
V. Oussa, Admissibility for quasiregular representations of exponential solvable Lie groups, Colloq. Math. 131 (2013), 241–264.
R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323.
J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, NC, 1976.
J. Roe, Lectures on Coarse Geometry, American Mathematical Society, Providence, RI, 2003.
J. L. Romero, J. T. Van Velthoven and F. Voigtlaender, Invertibility of frame operators on Besov-type decomposition spaces, arXiv:1905.04934 [math.FA].
W. Rudin, Functional Analysis, McGraw-Hill, New York—Düsseldorf—Johannesburg, 1973.
E. Schulz and K. F. Taylor, Projections in L1-algebras and tight frames, in Banach Algebras and their Applications, American Mathematical Society, Providence, RI, 2004. pp. 313–319.
B. Stöckert and H. Triebel, Decomposition methods for function spaces of B sp, q type and F sp, q type, Math. Nachr. 89 (1979), 247–267.
H. Triebel, Fourier Analysis and Function Spaces (selected topics), Teubner, Leipzig, 1977.
H. Triebel, General function spaces. I. Decomposition methods, Math. Nachr. 79 (1977), 167–179.
H. Triebel, Spaces of Besov—Hardy—Sobolev Type, Teubner, Leipzig, 1978.
F. Voigtlaender, Embeddings of decomposition spaces into Sobolev and BV spaces, ArXiv:1601.02201 [math.FA].
F. Voigtlaender, Embeddings of decomposition spaces, Mem. Amer. Math. Soc., to appear.
K. Yosida, Functional Analysis, Springer, Berlin, 1995.
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The authors thank the referee for his/her careful reading and the thoughtful comments. The second named author acknowledges support from the Austrian Science Fund (FWF): P 29462 — N35.
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Führ, H., van Velthoven, J.T. Coorbit spaces associated to integrably admissible dilation groups. JAMA 144, 351–395 (2021). https://doi.org/10.1007/s11854-021-0192-1
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DOI: https://doi.org/10.1007/s11854-021-0192-1