12 problems

Constant-extremizer conjecture. Constant functions are the only extremizers, modulo symmetries, for the Tomas–Stein inequality for every $d\geq 2$. The conjecture concerns the clas…

Wave maximizer conjecture. The set of maximizers for which equality holds in the sharp wave Strichartz inequality coincides with the set of initial data of solutions in the orbit o…

For an integer $n\geq 1$, set $p=2+4/n$. Let $f_*$ be the Gaussian … and let $u_*(t,x)=\mathrm{e}^{-\mathrm{i}t\Delta}f_*(x)$ be the corresponding solution of the Schrödinger equat…

Let $(q,r)$ be an admissible Strichartz pair in dimension $d=1$, and consider the corresponding Schrödinger Strichartz ratio and its extremizers. Universal Gaussian extremizer conj…

Critical-extremizer characterization.

Sharp $L^2\to L^6$ extension conjecture. For all $f\in L^2(\mathbb{S}^1)\setminus\{0\}$,

Let $d\geq 4$ and set … Consider the corresponding Fourier extension inequality for the paraboloid in dimension $d$, with Gaussian functions as candidate extremizers. Gaussian maxi…

Tomas–Stein extremizer conjecture. Constant functions are extremizers for this inequality.

Let $\mathcal{E}$ be the extension operator for the paraboloid, and consider the adjoint Fourier restriction inequality at the Stein–Tomas exponent. An extremizer is a nonzero func…

Let $d\geq 1$ and consider the extension inequality for the paraboloid … Here a radial Gaussian means a function of the form $f(\xi)=c e^{-a|\xi|^2}$ with suitable constants, and s…

Constant-extremizer conjecture. Equality is attained in this inequality when $f$ is a constant function.

Let $d$ be the dimension, let $p_0(d)$ be the exponent used in the generalized Euler–Lagrange equation, and let $\upsilon$ be the weight … Let $\lambda\in\mathbb C$ and let…