论文标题
圆锥的端线几何条件上的尖锐无效估计
Sharp null form estimates on endline geometric conditions of the cone
论文作者
论文摘要
We prove $\mathcal{H}^{α_1}\times\mathcal{H}^{α_2}\to L^q_tL^r_x$ null form estimates for solutions to homogeneous wave equations with $(q,r)$ on the endline of the condition concerning geometry of the cone, except the critical index.这扩展了陶，数学的先前终点结果。 Z. 238，没有。 2，215-268，（2001）在对称规范中，以混合规范并改善了Tataru的局部时间结果，MR1979953在恒定变量系数方程的设置以及由Lee and Vargas建立的锐利离线估计的设置中全球化。 J. Math 130（2008），第1号。 5，1279-1326，相对于锥体条件而言。我们的证明是基于混合规范中的乘数理论，最终将问题降低到统一的双线性双线性限制估计值，包括圆锥类型表面家族的高低频率相互作用，具体取决于参数$σ$，该$σ$将其收敛于斜锥为$σ\ to $σ\至0 $。我们通过使用刻度方法的诱导版本的增强版本证明了这一统一估计。
We prove $\mathcal{H}^{α_1}\times\mathcal{H}^{α_2}\to L^q_tL^r_x$ null form estimates for solutions to homogeneous wave equations with $(q,r)$ on the endline of the condition concerning geometry of the cone, except the critical index. This extends the previous endpoint result of Tao, Math. Z. 238, no. 2, 215-268, (2001) in symmetric norms to mixed norms and improves the local in time result of Tataru, MR1979953, to be global in the setting of constant variable coefficient equations, as well as the sharp off-endline estimates established by Lee and Vargas, Amer. J. Math 130 (2008), no. 5, 1279-1326, to the borderline with respect to the cone condition. Our proof is based on the multiplier theory in mixed norms, which ultimately reduces the question to a uniform endline bilinear restriction estimates including high-low frequency interactions for a family of conic type surfaces depending on a parameter $σ$, which converges to the oblique cone as $σ\to 0$. We prove this uniform estimate by using the enhanced version of the induction on scale method.
