On Besov spaces and absolute convergence of the Fourier transform on Heisenberg groups

Leszek Skrzypczak

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 4, page 755-763
  • ISSN: 0010-2628

Abstract

top
In this paper the absolute convergence of the group Fourier transform for the Heisenberg group is investigated. It is proved that the Fourier transform of functions belonging to certain Besov spaces is absolutely convergent. The function spaces are defined in terms of the heat semigroup of the full Laplacian of the Heisenberg group.

How to cite

top

Skrzypczak, Leszek. "On Besov spaces and absolute convergence of the Fourier transform on Heisenberg groups." Commentationes Mathematicae Universitatis Carolinae 39.4 (1998): 755-763. <http://eudml.org/doc/248287>.

@article{Skrzypczak1998,
abstract = {In this paper the absolute convergence of the group Fourier transform for the Heisenberg group is investigated. It is proved that the Fourier transform of functions belonging to certain Besov spaces is absolutely convergent. The function spaces are defined in terms of the heat semigroup of the full Laplacian of the Heisenberg group.},
author = {Skrzypczak, Leszek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Besov spaces; Heisenberg groups; group Fourier transform; Besov space; Heisenberg group; Fourier transform; atomic decomposition},
language = {eng},
number = {4},
pages = {755-763},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On Besov spaces and absolute convergence of the Fourier transform on Heisenberg groups},
url = {http://eudml.org/doc/248287},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Skrzypczak, Leszek
TI - On Besov spaces and absolute convergence of the Fourier transform on Heisenberg groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 4
SP - 755
EP - 763
AB - In this paper the absolute convergence of the group Fourier transform for the Heisenberg group is investigated. It is proved that the Fourier transform of functions belonging to certain Besov spaces is absolutely convergent. The function spaces are defined in terms of the heat semigroup of the full Laplacian of the Heisenberg group.
LA - eng
KW - Besov spaces; Heisenberg groups; group Fourier transform; Besov space; Heisenberg group; Fourier transform; atomic decomposition
UR - http://eudml.org/doc/248287
ER -

References

top
  1. Bernstein S., Sur la convergence absolute des séries trigonométriques, C.R.A.S. Paris 158 (1994), 1661-1664. (1994) 
  2. Coulhon T., Saloff-Coste L., Semi-groupes d'opérateurs et espaces fonctionels sur les groupes de Lie, J. Approx. Theory 65 (1991), 176-198. (1991) MR1104158
  3. Folland G., Harmonic analysis in phase space, Ann. Math. Stud. 112, Princeton Univ. Press, 1989. Zbl0682.43001MR0983366
  4. Gaudry G.I., Meda S., Pini R., A heat semigroup version of Bernstein's theorem on Lie groups, Monatshefte Math. 110 (1990), 101-115. (1990) Zbl0719.43008MR1076325
  5. Geller D., Fourier analysis on the Heisenberg group, J. Funct. Anal. 36 (1980), 205-254. (1980) Zbl0433.43008MR0569254
  6. Geller D., Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group, Canad. J. Math. 36 (1984), 615-684. (1984) Zbl0596.46034MR0756538
  7. Herz C., Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transform, J. Math. Mechanics 18 (1968), 283-324. (1968) MR0438109
  8. Inglis I.R., Bernstein's theorem and the Fourier algebra of the Heisenberg group, Boll. Un. Mat. Ital. 6 (1983), 39-46. (1983) Zbl0528.43008MR0694742
  9. Saka K., Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J. 31 (1979), 383-437. (1979) Zbl0429.43004MR0558675
  10. Skrzypczak L., Atomic decompositions on Riemannian manifolds with bounded geometry, Forum Math. 10 (1998), 19-38. (1998) MR1490136
  11. Skrzypczak L., Heat and harmonic extensions for function spaces of Hardy-Sobolev-Besov type on symmetric spaces and Lie groups, J. Approx. Theory, to appear. Zbl0918.43007MR1659396
  12. Strichartz R.S., Analysis of the Laplacian on a complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48-79. (1983) Zbl0515.58037MR0705991
  13. Thangavelu S., On Paley-Wiener theorems for the Heisenberg group, J. Funct. Anal. 115 (1993), 24-44. (1993) Zbl0793.43006MR1228140
  14. Triebel H., Function spaces on Lie Groups, the Riemannian approach, J. London Math. Soc. 35 (1987), 327-338. (1987) Zbl0587.46036MR0881521
  15. Triebel H., Theory of Function Spaces II, Birkhäuser Verlag, 1992. Zbl0763.46025MR1163193
  16. Varopoulos N.Th., Saloff-Coste L., Coulhon T., Analysis and Geometry on Groups, Cambridge Univ. Press, 1992. MR1218884

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.