Decoupling inequality

A decoupling inequality controls an $L^p$ norm of a function with Fourier support near a curved set by an ${\ell }^2$ sum of norms of pieces supported near smaller caps.

For Fourier supports in $R^{-1/2}\times R^{-1}$ caps along the parabola,

Functions with pairwise disjoint, or uniformly finitely overlapping, Fourier supports are almost orthogonal in $L^2$ by the Plancherel theorem.