= Restriction-to-rectangle overlap principle
{title2=$\int(\sum_R\mathbf1_R)^{p/2}\lesssim_p\delta^{4-p}\sum_R|R|$}
Assume a finite diagonal <Fourier extension estimate> on the <unit circle>. For direction-separated rectangles of dimensions $\delta^{-2}\times\delta^{-1}$, choose disjoint frequency caps and translate their transforms to the rectangles by the <Fourier modulation and translation identity>. Randomize their signs. Disjointness gives an input $p$th-power <norm> at most $C M\delta$; the <Khintchine inequality> converts the averaged output <norm> into the square function. The <circle cap Fourier lower bound> gives $\delta^p\int(\sum_R\mathbf1_R)^{p/2}\lesssim_p M\delta$. Each rectangle has area $\delta^{-3}$, giving the displayed form. Input cap disjointness is compatible with arbitrary spatial rectangle overlap.
Back to article page
