Kernel theorems in coorbit theory

Balazs, Peter; Gröchenig, Karlheinz; Speckbacher, Michael

doi:10.1090/btran/42

Kernel theorems in coorbit theory
HTML articles powered by AMS MathViewer

by Peter Balazs, Karlheinz Gröchenig and Michael Speckbacher HTML | PDF

Trans. Amer. Math. Soc. Ser. B 6 (2019), 346-364

Abstract:

We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable representation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger’s kernel theorem for modulation spaces and the recent generalizations by Cordero and Nicola. We also obtain a kernel theorem for operators between the Besov spaces $\dot {B}^0_{1,1}$ and $\dot {B}^{0}_{\infty , \infty }$.

References

S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Square integrability of group representations on homogeneous spaces. I. Reproducing triples and frames, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), no. 4, 829–855 (English, with French summary). MR 1144104
Peter Balazs, Matrix representation of operators using frames, Sampl. Theory Signal Image Process. 7 (2008), no. 1, 39–54. MR 2455829, DOI 10.1007/BF03549484
Isaac Pesenson, Frames: theory and practice, Frames and other bases in abstract and function spaces, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2017, pp. 3–12. MR 3700112
Shannon Bishop, Mixed modulation spaces and their application to pseudodifferential operators, J. Math. Anal. Appl. 363 (2010), no. 1, 255–264. MR 2559061, DOI 10.1016/j.jmaa.2009.08.032
Marcin Bownik, Lyapunov’s theorem for continuous frames, Proc. Amer. Math. Soc. 146 (2018), no. 9, 3825–3838. MR 3825837, DOI 10.1090/proc/14088
Jens Gerlach Christensen and Gestur Ólafsson, Coorbit spaces for dual pairs, Appl. Comput. Harmon. Anal. 31 (2011), no. 2, 303–324. MR 2806486, DOI 10.1016/j.acha.2011.01.004
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Elena Cordero and Fabio Nicola, Kernel theorems for modulation spaces, J. Fourier Anal. Appl. 25 (2019), no. 1, 131–144. MR 3901921, DOI 10.1007/s00041-017-9573-3
S. Dahlke, F. De Mari, E. De Vito, D. Labate, G. Steidl, G. Teschke, and S. Vigogna, Coorbit spaces with voice in a Fréchet space, J. Fourier Anal. Appl. 23 (2017), no. 1, 141–206. MR 3602813, DOI 10.1007/s00041-016-9466-x
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis 21 (1976), no. 2, 209–243. MR 0393335, DOI 10.1016/0022-1236(76)90079-3
Hans G. Feichtinger, Un espace de Banach de distributions tempérées sur les groupes localement compacts abéliens, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 17, A791–A794 (French, with English summary). MR 580567
Hans G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269–289. MR 643206, DOI 10.1007/BF01320058
H. G. Feichtinger. Modulation spaces on locally compact abelian groups. In Proceedings of “International Conference on Wavelets and Applications” 2002, pages 99–140, Chennai, India, 2003. Updated version of a technical report, University of Vienna, 1983.
Hans G. Feichtinger and Karlheinz Gröchenig, A unified approach to atomic decompositions via integrable group representations, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 52–73. MR 942257, DOI 10.1007/BFb0078863
Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), no. 2, 307–340. MR 1021139, DOI 10.1016/0022-1236(89)90055-4
Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), no. 2-3, 129–148. MR 1026614, DOI 10.1007/BF01308667
Hans G. Feichtinger and Karlheinz Gröchenig, Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 359–397. MR 1161258
H. G. Feichtinger and M. S. Jakobsen, The inner kernel theorem for a certain Segal algebra, arXiv:1806.06307, 2018.
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2, AMS Chelsea Publishing, Providence, RI, 2016. Spaces of fundamental and generalized functions; Translated from the 1958 Russian original [ MR0106409] by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer; Reprint of the 1968 English translation [ MR0230128]. MR 3469849, DOI 10.1090/chel/378
Karlheinz Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), no. 1, 1–42. MR 1122103, DOI 10.1007/BF01321715
Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Springer Study Edition, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065136, DOI 10.1007/978-3-642-61497-2
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR 1468229, DOI 10.1090/gsm/015
Viktor Losert, A characterization of the minimal strongly character invariant Segal algebra, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 129–139. MR 597020, DOI 10.5802/aif.795
Ivor J. Maddox, Infinite matrices of operators, Lecture Notes in Mathematics, vol. 786, Springer, Berlin, 1980. MR 568707, DOI 10.1007/BFb0088196
Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
Shahla Molahajloo, Kasso A. Okoudjou, and Götz E. Pfander, Boundedness of multilinear pseudo-differential operators on modulation spaces, J. Fourier Anal. Appl. 22 (2016), no. 6, 1381–1415. MR 3572906, DOI 10.1007/s00041-016-9461-2
A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. MR 858831, DOI 10.1007/978-3-642-61629-7
H.-J. Schmeisser and H. Triebel, Topics in Fourier analysis and function spaces, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. MR 900143
Winfried Sickel and Tino Ullrich, Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Theory 161 (2009), no. 2, 748–786. MR 2563079, DOI 10.1016/j.jat.2009.01.001
T. Tao, Lecture Notes 2 for 247 A: Fourier Analysis, www.math.ucla.edu/~tao/247a.1.06f/notes2.pdf.
Ernest B. Vinberg, Linear representations of groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010. Translated from the 1985 Russian original by A. Iacob; Reprint of the 1989 translation. MR 2761806