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A planar face on the unit sphere of the multiplier space $ M\sb{p}$, $ 1<p<\infty $


Authors: Charles Fefferman and Harold S. Shapiro
Journal: Proc. Amer. Math. Soc. 36 (1972), 435-439
MSC: Primary 42A18
DOI: https://doi.org/10.1090/S0002-9939-1972-0308669-X
MathSciNet review: 0308669
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Abstract | References | Similar Articles | Additional Information

Abstract: The unit sphere of the Banach space $ {M_p}$ of Fourier multipliers, $ 1 < p < \infty $, is shown to contain a flat portion, i.e. a portion of a plane having codimension one. The proof is based on an elementary inequality, a generalization of the classical Bernoulli inequality.


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  • [1] R. E. Edwards, Fourier series: a modern introduction. Vol. II, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1967. MR 0222538
  • [2] L. Leindler, On a generalization of Bernoulli's inequality, Acta Sci. Math. (Szeged) (to appear).
  • [3] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
  • [4] Harold S. Shapiro, Fourier multipliers whose multiplier norm is an attained value, Linear operators and approximation (Proc. Conf., Oberwolfach, 1971), Birkhäuser, Basel, 1972, pp. 338–347. Internat. Ser. Numer. Math., Vol. 20. MR 0390623

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308669-X
Keywords: Fourier multiplier, unit sphere, Bernoulli inequality
Article copyright: © Copyright 1972 American Mathematical Society