2 problems

Hyperbolic-paraboloid decoupling conjecture. For $p>3$, for every $\varepsilon>0$ there exists $C_\varepsilon>0$ such that for all $R\geq1$,

Let $d\geq 2$ and let $\vec{v}=(v_1,\ldots,v_{d-1})\in\{-1,1\}^{d-1}$. Consider the hyperbolic paraboloid … For $f$ on $[0,1]^{d-1}$, define its extension operator by … where…