Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions
Annales de l'Institut Fourier, Volume 22 (1972) no. 3, pp. 21-50.

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.

Ce travail considère les sous-algèbres de Lipschitz Λ(α,p,𝒜) d’une algèbre homogène sur le cercle. La théorie des espaces d’interpolation est utilisée pour dériver des estimations pour les normes des multiplicateurs sur des idéaux primaires fermés de Λ(α,p;𝒜), α[α]. D’après ces estimations on déduit facilement la condition de Ditkin et la condition analytique de Ditkin pour Λ(α,p,𝒜). De cette façon la théorie familière des algèbres (régulières) de Banach qui satisfont à la condition de Ditkin et aussi la théorie développée dans la Note I de cette série qui exige que la condition analytique de Ditkin s’applique à Λ(α,p,𝒜).

On présente des exemples qui démontrent que beaucoup d’algèbres de Banach considérées précédemment séparément peuvent être introduites aussi bien qu’analysées dans ce cadre de la théorie des espaces d’interpolation.

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     author = {Bennett, Colin and Gilbert, John E.},
     title = {Homogeneous algebras on the circle. {II.} {Multipliers,} {Ditkin} conditions},
     journal = {Annales de l'Institut Fourier},
     pages = {21--50},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {22},
     number = {3},
     year = {1972},
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Bennett, Colin; Gilbert, John E. Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions. Annales de l'Institut Fourier, Volume 22 (1972) no. 3, pp. 21-50. doi : 10.5802/aif.423. http://www.numdam.org/articles/10.5802/aif.423/

[1] C. Bennett, Harmonic Analysis of Rearrangement Invariant Banach Function spaces, Ph. D. thesis. Newcastle. (1971).

[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. | Zbl

[3] A. Beurling, Construction and analysis of some convolution algebras, Ann. l'Inst. Fourier 14 (1964), 1-32. | Numdam | MR | Zbl

[4] D.W. Boyd, Indices of function spaces and relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. | MR | Zbl

[5] P.L. Butzer and H. Berens, “Semi-groups of Operators and Approximation”, Springer-Verlag, New York 1967. | MR | Zbl

[6] A.P. Calderón, Intermediate spaces and interpolation, Studia Math. 24 (1964), 113-190. | MR | Zbl

[7] J.E. Gilbert, Harmonic Analysis : Banach Algebras and Interpolation Theory, lecture notes (in preparation).

[8] P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications I, II, J. Math. Pure et Appl. 45 (1966), 143-290. | MR | Zbl

[9] C. Goulaouic, Prolongements de foncteurs d'interpolation et applications, Ann. l'inst. Fourier, Grenoble 18 (1968), 1-98. | Numdam | MR | Zbl

[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. | JFM

[11] C.S. Herz, Lipschitz spaces and Bernstein's Theorem on Absolutely Convergent Fourier Transforms, J. Math. Mech. 18 (1968) 283-324. | MR | Zbl

[12] E. Hewitt and K.A. Ross, “Abstract Harmonic Analysis II”, Springer-Verlag, New York, 1970. | Zbl

[13] J.L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Paris, 19, (1964), 5-68. | Numdam | MR | Zbl

[14] R.J. Loy, Maximal ideal spaces of Banach algebras of derivable elements, J. Aust. Math. Soc. 11 (1970), 310-312. | MR | Zbl

[15] Y. Meyer, Prolongement des multiplicateurs d'ideaux fermes de L1 (Rn), C.R. Acad. Sci. Paris 262 (1966), 744-745. | MR | Zbl

[16] H. Mirkil, The work of Silov on commutative Banach algebras, Inst. Mat. Pura e Appl., Rio de Janeiro, (1959). | Zbl

[17] J. Peetre, On an interpolation theorem of Foias and Lions, Acta Sci. Math. 25 (1964), 255-261. | MR | Zbl

[18] J. Peetre, Interpolation of Lp-spaces with weight function, Acta Sci. Math. 28 (1967), 61-69. | MR | Zbl

[19] J. Peetre, Interpolation of Lipschitz operators and metric spaces, (preprint). | Zbl

[20] H. Reiter, “Classical Harmonic Analysis and Locally Compact Groups”, Oxford University Press, Oxford, 1968. | MR | Zbl

[21] W. Rudin, “Fourier Analysis on Groups”, J. Wiley, New York, 1962. | MR | Zbl

[22] D.R. Sherbert, Structure of ideals and point derivations in Banach Algebras of Lipschitz functions, Trans. Amer. Math. Soc III (1964), 240-272. | MR | Zbl

[23] M. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean spaces, I. Principal Properties, J. Math. Mech. 13 (1964), 407-479. | MR | Zbl

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