论文标题
限制了$ 2 $ - 平面变换与Codimension $ 2 $的傅立叶限制有关
A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$
论文作者
论文摘要
我们在P. Gressman构建的仿射不变的表面测量与与Codimension $ 2 $相关的某个几何平均操作员的界面之间进行了联系，并且与此类表面的傅立叶限制问题有关。对于由$（ξ，q_1（ξ），q_2（ξ））$提供的表面，$ q_1，q_2 $ quadratic forms上的$ \ mathbb {r}^d $上的特定操作员是$ 2 $ - 平面转换到表面正常的方向，这是\ [\ [\ m gathcal calcal cal calcal calcal calcal calcal calccal offormation \ iint_ {| s |，| t | \ leq 1} f（x -s \ nabla q_1（ξ） - t \ nabla q_2（ξ），s，t）\，ds \，dt，\]其中$ x，ξ\ in \ in \ in \ mathbb {r}^d $。我们表明，当表面以Gressman的意义（即，相关的仿射不变的表面度量不会消失）时，操作员满足了$ l^p \ to $ p，q $ p，q $ to the关键点的尖锐$ l^p \。我们还表明，曲折度的假设对于获得完整的估计范围是必要的。证明依赖于两种主要成分：根据多项式$ \ det的性质（s \ nabla^2 q_1 + t \ nabla^2 q_2）$的特性表征，并以几何不变理论技术和基督的精炼方法获得。使用后者，事项将减少为级别的集合估算，这是由线性编程论证证明的。
We draw a connection between the affine invariant surface measures constructed by P. Gressman and the boundedness of a certain geometric averaging operator associated to surfaces of codimension $2$ and related to the Fourier Restriction Problem for such surfaces. For a surface given by $(ξ, Q_1(ξ), Q_2(ξ))$, with $Q_1,Q_2$ quadratic forms on $\mathbb{R}^d$, the particular operator in question is the $2$-plane transform restricted to directions normal to the surface, that is \[ \mathcal{T}f(x,ξ) := \iint_{|s|,|t| \leq 1} f(x - s \nabla Q_1(ξ) - t \nabla Q_2(ξ), s, t)\,ds\,dt, \] where $x,ξ\in \mathbb{R}^d$. We show that when the surface is well-curved in the sense of Gressman (that is, the associated affine invariant surface measure does not vanish) the operator satisfies sharp $L^p \to L^q$ inequalities for $p,q$ up to the critical point. We also show that the well-curvedness assumption is necessary to obtain the full range of estimates. The proof relies on two main ingredients: a characterisation of well-curvedness in terms of properties of the polynomial $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$, obtained with Geometric Invariant Theory techniques, and Christ's Method of Refinements. With the latter, matters are reduced to a sublevel set estimate, which is proven by a linear programming argument.
