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.

The Bourgain-Demeter decoupling theorem controls a function whose Fourier support lies near a positively curved hypersurface by the contributions from small frequency caps. For a compact $C^2$ hypersurface in ${\mathbb{R}}^n$, $n\ge 2$, with positive-definite second fundamental form, decompose its $\delta$-neighborhood into caps of tangential width ${\delta }^{1/2}$ and normal thickness $\delta$. If $f_{\theta }$ denotes the corresponding Fourier pieces, the critical estimate isThe constant is independent of $\delta \in (0,1)$ and $f$, for the fixed surface and chosen cap decomposition. The same arbitrarily small loss holds for $2\le p\le p_c$; supercritical exponents require an additional scale power.^[ref]

For a parabola in ${\mathbb{R}}^2$, $p_c=6$. A naive triangle inequality sums all cap norms, whereas decoupling sums their squares before taking a square root. Curvature makes this much stronger control possible. The theorem links Fourier restriction theory with estimates for oscillatory sums and dispersive equations.

Let $R\ge 1$ and decompose the Fourier support near a fixed compact arc of $(\xi ,{\xi }^2)$ into caps with tangential width $R^{-1/2}$ and normal width $R^{-1}$. For the associated pieces $f_{\theta }$, the decoupling inequality for $6\le p<\infty$ isFor $2\le p\le 6$, the corresponding loss is $R^{\epsilon }$. The constants are independent of $R$ and the input. This is the two-dimensional case of the Bourgain-Demeter decoupling theorem.^[ref]

At $p=2$, Fourier-support almost orthogonality already supplies a bounded constant. At the critical exponent $p=6$, the theorem keeps only the arbitrarily small power loss $R^{\epsilon }$, despite allowing many interacting caps. This is much stronger than applying the triangle inequality and then Cauchy-Schwarz to about $R^{1/2}$ pieces. The cap dimensions and the convention $\delta =R^{-1}$ determine the power of $R$; confusing $R$ with its square changes that exponent.

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