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Estimates for Cone Multipliers

  • Jean Bourgain
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Summary

In this paper we develop a technique to improve on [M]’s \(\tfrac{1}{8}\) result for the boundedness on L 4(ℝ3) of cone multipliers
$$ {m_\alpha }\left( {{x_1},{x_2},{x_3}} \right) = \phi \left( {{x_3}} \right)\left( {1 - \frac{{\sqrt {x_1^2 + x_2^2} }} {{{x_3}}}} \right)_ + ^\alpha $$
with ø ∈ C0 (l, 2). More precisely, we get this property for certain values of α < \(\tfrac{1}{8}\). There is a similarity in approach with estimates for the Bochner-Riesz problem in the case of the ball. Our argument shows also that if μ is a measure supported by \( {\Gamma_{(1)}} = \left\{ {x \in \left. {{\mathbb{R}^3}} \right|\left| {{x_3}} \right| = \sqrt {x_1^2 + x_2^2}, 1 < {x_3} < 2} \right\} \) and ρ = 0 on a neighborhood of the cone Γ, then if \( \frac{{d\mu }} {{d\sigma }} \in {L^2}\left( \sigma \right),\sigma = \) surface measure of T, one may bound ||(μ * μ) ρ|| p for certain p < 2. This fact and especially an understanding for what surfaces this phenomenon holds, seems of independent interest.

Keywords

Fourier Multiplier Bounded Number Angular Region Angular Restriction Cone Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [B1]
    J. Bourgain, On the restriction and multiplier problem in ℝ3, Springer LNM (1969), 179–191.Google Scholar
  2. [B2]
    J. Bourgain, Besicovitch type maximal functions and applications to Fourier Analysis, Geometric And Functional Analysis, Vol. 1, n0 2 (1991), 147–187.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [M]
    G. Mockenhaupt, A note on the cone multiplier, Proc. AMS, Vol. 117, 1 (1993), 145–152.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Jean Bourgain
    • 1
  1. 1.Institute of Advanced StudyPrincetonUSA

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