Estimates for Cone Multipliers

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.