Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions

Colin Bennett; John E. Gilbert

Annales de l'institut Fourier (1972)

  • Volume: 22, Issue: 3, page 21-50
  • ISSN: 0373-0956

Abstract

top
This paper considers the Lipschitz subalgebras Λ ( α , p , 𝒜 ) of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in Λ ( α , p ; 𝒜 ) , α ≠ [ α ] . From these estimates the Ditkin and Analytic Ditkin conditions for Λ ( α , p ; 𝒜 ) follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to Λ ( α ; , p ; 𝒜 ) as does the theory developed in part I of this series which requires the Analytic Ditkin condition.Examples are discussed showing that many of the Banach algebras on the circle considered previously in isolation can be both generated and describes within this framework of interpolation space theory.

How to cite

top

Bennett, Colin, and Gilbert, John E.. "Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions." Annales de l'institut Fourier 22.3 (1972): 21-50. <http://eudml.org/doc/74089>.

@article{Bennett1972,
abstract = {This paper considers the Lipschitz subalgebras $\Lambda (\alpha ,p,\{\cal A\})$ of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in $\Lambda (\alpha ,p;\{\cal A\})$, $\alpha \ne [\alpha ]$. From these estimates the Ditkin and Analytic Ditkin conditions for $\Lambda (\alpha ,p;\{\cal A\})$ follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to $\Lambda (\alpha ;,p;\{\cal A\})$ as does the theory developed in part I of this series which requires the Analytic Ditkin condition.Examples are discussed showing that many of the Banach algebras on the circle considered previously in isolation can be both generated and describes within this framework of interpolation space theory.},
author = {Bennett, Colin, Gilbert, John E.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {21-50},
publisher = {Association des Annales de l'Institut Fourier},
title = {Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions},
url = {http://eudml.org/doc/74089},
volume = {22},
year = {1972},
}

TY - JOUR
AU - Bennett, Colin
AU - Gilbert, John E.
TI - Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions
JO - Annales de l'institut Fourier
PY - 1972
PB - Association des Annales de l'Institut Fourier
VL - 22
IS - 3
SP - 21
EP - 50
AB - This paper considers the Lipschitz subalgebras $\Lambda (\alpha ,p,{\cal A})$ of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in $\Lambda (\alpha ,p;{\cal A})$, $\alpha \ne [\alpha ]$. From these estimates the Ditkin and Analytic Ditkin conditions for $\Lambda (\alpha ,p;{\cal A})$ follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to $\Lambda (\alpha ;,p;{\cal A})$ as does the theory developed in part I of this series which requires the Analytic Ditkin condition.Examples are discussed showing that many of the Banach algebras on the circle considered previously in isolation can be both generated and describes within this framework of interpolation space theory.
LA - eng
UR - http://eudml.org/doc/74089
ER -

References

top
  1. [1] C. BENNETT, Harmonic Analysis of Rearrangement Invariant Banach Function spaces, Ph. D. thesis. Newcastle. (1971). 
  2. [2] O.V. BESOV, Investigation of a family of function spaces in connection with theorems of extension and embedding, Amer Math. Soc. Transl. (2) 40 (1964), 85-126. Zbl0158.13901
  3. [3] A. BEURLING, Construction and analysis of some convolution algebras, Ann. l'Inst. Fourier 14 (1964), 1-32. Zbl0133.07501MR32 #321
  4. [4] D.W. BOYD, Indices of function spaces and relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. Zbl0184.34802MR54 #909
  5. [5] P.L. BUTZER and H. BERENS, “Semi-groups of Operators and Approximation”, Springer-Verlag, New York 1967. Zbl0164.43702MR37 #5588
  6. [6] A.P. CALDERÓN, Intermediate spaces and interpolation, Studia Math. 24 (1964), 113-190. Zbl0204.13703MR29 #5097
  7. [7] J.E. GILBERT, Harmonic Analysis : Banach Algebras and Interpolation Theory, lecture notes (in preparation). 
  8. [8] P. GRISVARD, Commutativité de deux foncteurs d'interpolation et applications I, II, J. Math. Pure et Appl. 45 (1966), 143-290. Zbl0187.05803MR36 #4361
  9. [9] C. GOULAOUIC, Prolongements de foncteurs d'interpolation et applications, Ann. l'inst. Fourier, Grenoble 18 (1968), 1-98. Zbl0174.17301MR39 #4658
  10. [10] G.H. HARDY and J.E. LITTLEWOOD, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. für Math. 157 (1927), 141-158. Zbl53.0193.03JFM53.0193.03
  11. [11] C.S. HERZ, Lipschitz spaces and Bernstein's Theorem on Absolutely Convergent Fourier Transforms, J. Math. Mech. 18 (1968) 283-324. Zbl0177.15701MR55 #11028
  12. [12] E. HEWITT and K.A. ROSS, “Abstract Harmonic Analysis II”, Springer-Verlag, New York, 1970. Zbl0213.40103
  13. [13] J.L. LIONS and J. PEETRE, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Paris, 19, (1964), 5-68. Zbl0148.11403MR29 #2627
  14. [14] R.J. LOY, Maximal ideal spaces of Banach algebras of derivable elements, J. Aust. Math. Soc. 11 (1970), 310-312. Zbl0198.47401MR42 #3578
  15. [15] Y. MEYER, Prolongement des multiplicateurs d'ideaux fermes de L1 (Rn), C.R. Acad. Sci. Paris 262 (1966), 744-745. Zbl0166.11601MR34 #8099
  16. [16] H. MIRKIL, The work of Silov on commutative Banach algebras, Inst. Mat. Pura e Appl., Rio de Janeiro, (1959). Zbl0090.09301
  17. [17] J. PEETRE, On an interpolation theorem of Foias and Lions, Acta Sci. Math. 25 (1964), 255-261. Zbl0148.37301MR30 #1391
  18. [18] J. PEETRE, Interpolation of Lp-spaces with weight function, Acta Sci. Math. 28 (1967), 61-69. Zbl0166.10801MR35 #5919
  19. [19] J. PEETRE, Interpolation of Lipschitz operators and metric spaces, (preprint). Zbl0217.44504
  20. [20] H. REITER, “Classical Harmonic Analysis and Locally Compact Groups”, Oxford University Press, Oxford, 1968. Zbl0165.15601MR46 #5933
  21. [21] W. RUDIN, “Fourier Analysis on Groups”, J. Wiley, New York, 1962. Zbl0107.09603MR27 #2808
  22. [22] D.R. SHERBERT, Structure of ideals and point derivations in Banach Algebras of Lipschitz functions, Trans. Amer. Math. Soc III (1964), 240-272. Zbl0121.10204MR28 #4385
  23. [23] M. TAIBLESON, On the theory of Lipschitz spaces of distributions on Euclidean spaces, I. Principal Properties, J. Math. Mech. 13 (1964), 407-479. Zbl0132.09402MR29 #462

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.