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Parametrize the unit circle by $\omega (\theta )=(\cos \theta ,\sin \theta )$, with $-\pi \le \theta <\pi$ and $d\sigma =d\theta /(2\pi )$. Extract the constant phase at $\omega (0)$:On the support, $|\theta |\le 2\delta /C$. The frequency rectangle and the elementary bounds $|1-\cos \theta |\le {\theta }^2/2$, $|\sin \theta |\le |\theta |$ implyChoose the fixed constant $C$ large enough that this is less than $\pi /3$. Every phase then has real part at least $1/2$. Nonnegativity of $\psi$ prevents cancellation after this phase rotation, while its central plateau gives $\int \psi d\sigma \ge \delta /(\pi C)$. Hence the Fourier transform satisfiesThis is the circle cap Fourier lower bound. No upper bound on the values of $\psi$ is needed here; the support, plateau and nonnegativity suffice. The long radial scale ${\delta }^{-2}$ comes from the quadratic term $\cos \theta -1$, whereas the transverse scale ${\delta }^{-1}$ comes from the linear term $\sin \theta$.

Write $R_j$ for the rectangles, $x_j$ for their centers and $e_j$ for their long-axis directions. For the finite exponent $p$ in the displayed estimate, take smooth rotated cap functions ${\psi }_j$, with $0\le {\psi }_j\le 1$, equal to one on angular distance at most $\delta /C$ from $e_j$ and supported within $2\delta /C$. Choose $C$ sufficiently large once and for all. The direction separation makes these cap supports disjoint, and $\int |{\psi }_j{|}^{p}d\sigma \lesssim \delta$.

DefineThe Fourier modulation and translation identity gives the second equality. Rotating the circle cap Fourier lower bound then gives $|h_j(x)|\gtrsim \delta$ on $R_j$. The half-side lengths of $R_j$ are no larger than the two frequency bounds used in part (b).

Let ${\epsilon }_j$ be independent Rademacher random variables. Because the input cap supports are disjoint, for every choice of signswhere $M=\#\mathcal{R}$. Apply the assumed Fourier extension estimate to the sum. Average over signs and use the Khintchine inequality pointwise, followed by the Tonelli theorem:There is no requirement that the spatial rectangles be disjoint; disjointness is used only for the input caps on the unit circle. Their spatial overlaps are precisely what the square function measures. The cap lower bounds now implySince each rectangle has area ${\delta }^{-3}$, the restriction-to-rectangle overlap principle givesThe constants are independent of $\delta$, the centers and the collection. The finite-$p$ interpretation is the one for which the printed power integral is defined. A single cap also shows that the assumed diagonal Fourier extension estimate can hold only for $p\ge 4$: its output contributes at least ${\delta }^{p-3}$ to the $p$th-power norm, whereas its input contributes at most a constant times $\delta$.

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 $CM\delta$; the Khintchine inequality converts the averaged output norm into the square function. The circle cap Fourier lower bound gives ${\delta }^p\int ({\sum }_R{1}_R{)}^{p/2}{\lesssim }_{p}M\delta$. Each rectangle has area ${\delta }^{-3}$, giving the displayed form. Input cap disjointness is compatible with arbitrary spatial rectangle overlap.