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Multipliers from spaces of test functions to amalgams

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Maria Torres De Squire
Maria Torres De Squire
Affiliation:
Department of Mathematics and StatisticsUniversity of ReginaRegina SaskatchewanCanadaS4S 0A2
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Abstract

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In this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

MSC classification

Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 97 - 110
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Argabright, L., and de Lamadrid, J. Gil, ‘Fourier analysis of unbounded measures on locally compact Abelian groups’, Mem. Amer. Math. Soc. 145 (1974).Google Scholar
[2]Bertandias, J. P. and Dupuis, C., ‘Transformation de Fourier sur les espaces lp (Lp)’, Ann. Inst. Fourier (Grenoble) 29 (1979), 189–206.CrossRefGoogle Scholar
[3]Bourbaki, N., Eléments de Mathematique. Integration, Act. Sci. et md. 1175 (Hermann et Cie, Paris, 1973).Google Scholar
[4]Feichtinger, H. G., ‘Un espace de Banach de distributions tempérées sur les groupes localement compacts Abeliens’, C. R. Acad. Sci. Paris Sér. I Math. 290 (1980), 791–794.Google Scholar
[5]Feichtinger, H. G., ‘On a new Segal algebra’, Mh. Math. 92 (1981), 269–289.CrossRefGoogle Scholar
[6]Figá-Talamanca, A., ‘On the subspace of Lp invariant under multiplication of transform by bounded continuous functions’, Rend. Sem. Mat. Univ. Padova 35 (1965), 176–189.Google Scholar
[7]Figá-Talamanca, A., and Gaudry, G. I., ‘Multipliers and sets of uniqueness of Lp’, Michigan Math. J. 17 (1970), 179–191.CrossRefGoogle Scholar
[8]Fournier, J. J. F., ‘On the Hausdorff-Young theorem for amalgams’, Math. 95 (1983), 117–135.Google Scholar
[9]Fournier, J. J. F. and Stewart, J., ‘Amalgams of Lp and lq’, Bull. Amer. Math. Soc. 13 (1985), 1–21.CrossRefGoogle Scholar
[10]Holland, F., ‘On the representations of functions as Fourier transforms of unbounded measures’, Proc. London math. Soc. (3) 30 (1975), 347–365.CrossRefGoogle Scholar
[11]Larsen, R., An introduction to the theory of multipliers, Grundlehren Math. Wiss. 175 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[12]Stewart, J., ‘Fourier transforms of unbounded measures’, Canad. J. Math. 31 (1979), 1281–1292.CrossRefGoogle Scholar
[13]de Squire, M. L. Torres, ‘Amalgams of Lp and 1q’, (Ph. D. Thesis, McMaster University, Canada, 1985).Google Scholar
[14]de Squire, M. L. Torres, ‘Multipliers for amalgams and the algebra S0(G)’, Canad. J. Math. 39 (1987), 123–148.CrossRefGoogle Scholar
[15]de Squire, M. L. Torres, ‘Resonance classes of measures’, Internal. J. Math. Math. Sci. 10 (1987), 461–472.CrossRefGoogle Scholar