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A converse of Bernstein's inequality for locally compact groups

  • Walter R. Bloom (a1)

Abstract

Let G be a Hausdorff locally compact abelian group, Γ its character group. We shall prove that, if S is a translation-invariant subspace of Lp (G) (p ∈ [1, ∞]),

for each aG and , then is relatively compact (where Σ(f) denotes the spectrum of f). We also obtain a similar result when G is a Hausdorff compact (not necessarily abelian) group. These results can be considered as a converse of Bernstein's inequality for locally compact groups.

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References

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[1]Bloom, Walter R., “Bernstein's inequality for locally compact abelian groups”, J. Austral. Math. Soc. (to appear).
[2]Edwards, D.A., “On translates of L -functions”, J. London Math. Soc. 36 (1961), 431432.
[3]Edwards, R.E., “Translates of L functions and of bounded measures”, J. Austral. Math. Soc. 4 (1964), 403409.
[4]Edwards, R.E., Functional analysis: Theory and applications (Holt, Rinehart and Winston, New York, Chicago, San Francisco, Toronto, London, 1965).
[5]Edwards, R.E., Integration and harmonic analysis on compact groups (Notes on Pure Mathematics, 5. Australian National University, Canberra, 1970).
[6]Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis. Volume I (Die Grundlehren der mathematischen Wissenschaften, Band 115. Academic Press, New York; Springer-Verlag, Berlin, Göttingen, Heidelberg; 1963).
[7]Reiter, Hans, Classical harmonic analysis and locally compact groups (Clarendon Press, Oxford, 1968).
[8]Rudin, Walter, Fourier analysis on groups (interscience, New York, London, 1962; 2nd Printing, 1967).
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A converse of Bernstein's inequality for locally compact groups

  • Walter R. Bloom (a1)

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