The Strichartz conjecture on the real line times the three-sphere

Let $N$ denote the spatial frequency scale, let $P_N$ be the corresponding frequency projection on $\mathbb{R}\times\mathbb{S}^3$, and let $f\in L^2(\mathbb{R}\times\mathbb{S}^3)$. Define

$$\sigma(p)= \begin{cases} 2-\frac{6}{p},&\text{if }p\geq\frac{10}{3},\\\\ \frac{1}{2}-\frac{1}{p},&\text{if }2\leq p\leq\frac{10}{3}. \end{cases}$$

The Strichartz conjecture on $\mathbb{R}\times\mathbb{S}^3$. One has

$$\\|e^{it\Delta}P_Nf\\|_{L^p([0,1]\times\mathbb{R}\times\mathbb{S}^3)}\lesssim N^{\sigma(p)}\\|f\\|_{L^2(\mathbb{R}\times\mathbb{S}^3)}.$$

This estimate is identified as a natural open problem and would have consequences for energy-critical nonlinear Schrödinger equations on product spaces, including the setting involving $\mathbb{T}\times\mathbb{S}^3$.