On extremizing sequences for the adjoint restriction inequality on the cone

It is known that extremizers for the $L^2$ to $L^6$ adjoint Fourier restriction inequality on the cone in $\mathbb{R}^3$ exist. Here we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the cone. If we use the knowledge of the exact form of the extremizers, as found by Carneiro, then we can show that nonnegative extremizing sequences converge, after the application of symmetries.Comment: 27 pages. Includes referee comment

Smoothness of solutions of a convolution equation of restricted-type on the sphere

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 $$a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1},$$ 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 to 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 & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.Comment: 45 pages, v2: referee's suggestions incorporate

A sharp inequality in Fourier restriction theory

Recentemente, os autores provaram em [15, 16] que as funções constantes são os únicos maximizantes reais da desigualdade L2 -> L6 de extensão de Fourier na 2-esfera. Isto é um caso particular de [16, Teorema 1.1], cuja prova contém vários dos métodos e ideias-chave. Neste artigo, descrevemos a prova deste caso particular, e apresentamos algumas generalizações e problemas em aberto

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq 2$ and general even exponents.Comment: 64 pages, 4 figures, 3 tables; v2: referee's suggestions incorporate