Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-22T13:34:22.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorisation of Lipschitz functions on zero dimensional groups

Published online by Cambridge University Press:  17 April 2009

Walter R. Bloom
Walter R. Bloom
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6150.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G denote a locally compact metrisable zero dimensional group with left translation invariant metric d. The Lipschitz spaces are defined by

where af: x → f(ax) and α > 0; when r = ∞ the members of Lip(α; r) are taken to be continuous. For a suitable choice of metric it is shown that , where 1 ≤ p ≤ 2, α > q−1, p, q are conjugate indices and . It is also shown that for G infinite the range of values of α cannot be extended.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 2 , April 1981 , pp. 215 - 225
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Benke, George, “Smoothness and absolute convergence of Fourier series in compact totally disconnected groups”, J. Funct. Anal. 29 (1978), 319–327.CrossRefGoogle Scholar
[2]Bloom, Walter R., “Absolute convergence of Fourier series on totally disconnected groups”, submitted.Google Scholar
[3]Figà-Talamanca, Alessandro and Rider, Daniel, “A theorem of Littlewood and lacunary series for compact groups”, Pacific J. Math. 16 (1966), 505–514.Google Scholar
[4]Hahn, Liang-Shin, “On multipliers of p-integrable functions”, Trans. Amer. Math. Soc. 128 (1967), 321–335.Google Scholar
[5]Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis I. Structure of topological groups, integration theory, group representations (Die Grundlehren der mathematischen Wissenschaften, 115. Springer-Verlag, Berlin, Heidelberg, New York, 1963).Google Scholar
[6]Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis II. Structure and analysis for compact groups. Analysis on locally compact abelian groups (Die Grundlehren der mathematischen Wissenschaften, 152. Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar
[7]Lohoué, Noël “Une condition d'appartenance à A p (T)”, C.R. Acad. Sci. Paris Sér. A 270 (1970), 736–738.Google Scholar
[8]Uno, Yoshikazu, “Lipschitz functions and convolution”, Proc. Japan Acad. 50 (1974), 785–788.Google Scholar
[9]Uno, Yoshikazu, “Lipschitz functions and convolution on bounded Vilenkin groups”, Sci. Rep. Kanazawa Univ. 23 (1978), 1–6.Google Scholar
[10]Вилннин, ВЯ. [Vilenkin, N.Ja.], “Об одном нласс прлных ортонормальных систм” [On a class of complete orthonormal systems], Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 363–400; English Transl: Amer. Math. Soc. Transl. (2) 28 (1963), 1–35.Google Scholar
[11]Walker, P.L., “Lipschitz classes on 0-dimenslonal groups”, Proc. Cambridge Philos. Soc. 63 (1967), 923–928.Google Scholar