Estimates for Cone Multipliers

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


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.


Fourier Multiplier Bounded Number Angular Region Angular Restriction Cone Segment 
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  1. [B1]
    J. Bourgain, On the restriction and multiplier problem in ℝ3, Springer LNM (1969), 179–191.Google Scholar
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    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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