Estimates for Cone Multipliers
Estimates for Cone Multipliers
复制标题
锥体乘数的估计
DOI：
10.1007/978-3-0348-9090-8_5
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
J. Bourgain
中科院分区：
文献类型：
--
作者：
J. Bourgain
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 o ∈ 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.
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 o ∈ 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.