Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$ , $d\geq 2$ , equipped with surface measure $\sigma _{d-1}$ . An instance of our main result concerns the regularity of solutions of the convolution equation $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$ where $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$ . We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$ . In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $ -smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].

中文翻译：

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球面上限制型卷积方程解的光滑度

让 $\mathbb {S}^{d-1}$ 表示欧几里得空间中的单位球体 $\mathbb {R}^d$ , $d\geq 2$ , 配备表面测量 $\sigma _{d-1}$ . 我们主要结果的一个例子涉及卷积方程解的规律性 $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}} =f,\text{ ae on }\mathbb{S}^{d-1}, \end{align*}$$ 在哪里 $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ 是一个整数，唯一的先验假设是 $f\in L^2(\mathbb {S}^{d-1})$ . 我们证明任何这样的解决方案都属于该类 $C^\infty (\mathbb {S}^{d-1})$ . 特别是，我们证明了与相应的伴随傅里叶限制不等式的尖锐形式相关的所有临界点 $\mathbb {S}^{d-1}$ 是 $C^\infty $ -光滑的。这将 Christ 和 Shao [4] 之前的工作扩展到任意维度和一般甚至指数，并在配套论文 [24] 中发挥关键作用。