标题： 某些Radon变换不等式极值函数的光滑性 显示英文标题

标题： Smoothness of extremizers for certain inequalities of the Radon transform

摘要： The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in $\mathbb{R}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open. We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac1{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold. 证明是在$k$平面变换的框架下进行的，该变换通过在$k$维平面上对函数进行积分，将$\mathbb{R}^n$上的函数转换为所有仿射$k$平面在$\mathbb{R}^n$流形上的函数。 我们证明当 $q-1$ 和 $\frac1{p-1}$ 为整数时，泛函 \[ \|T_{n,k}f\|_{L^q(M)}/\|f\|_{L^p(\mathbb{R}^n)}\] 的所有非负临界点都是无限可微的，所有导数都属于 $L^p$ 并在加权 $L^p$-空间中表现出一些额外的衰减。 显示英文摘要

摘要： The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in $\mathbb{R}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open. We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac1{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold. The proof is carried out in the setting of the $k$-plane transform, which takes functions on $\mathbb{R}^n$ to functions on the manifold of all affine $k$-planes in $\mathbb{R}^n$ by integrating the function over the $k$-dimensional plane. We show that when $q-1$ and $\frac1{p-1}$ are intergers, all nonnegative critical points of the functional \[ \|T_{n,k}f\|_{L^q(M)}/\|f\|_{L^p(\mathbb{R}^n)}\] are infinitely differentiable, all derivatives are in $L^p$ and exhibit some additional decay measured in a weighted $L^p$-space.