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Ciprian Demeter coauthored the decoupling inequality with Jean Bourgain. It bounds global Fourier contributions by a quadratic sum of contributions from small caps near a curved hypersurface. The theorem turns geometric curvature into an analytic inequality and yields consequences for Strichartz estimate for the free Schrödinger equation and discrete restriction problems.^[ref]

Jean Bourgain proved the decoupling inequality with Ciprian Demeter. Their theorem controls a function with Fourier support near a curved hypersurface through contributions from smaller frequency caps. The result connects curvature with improved summation estimates, with applications to restriction and Strichartz estimate for the free Schrödinger equation.^[ref]

For $T_2>T_1\ge -1$, the dual Strichartz estimate for the free Schrödinger equation givesThe assumed finite global $L_{t,x}^4$ norm makes the right side tend to zero as $T_1,T_2\to \infty$. The displayed family is therefore a Cauchy sequence in the complete space $L^2$ and has a strong limit $F_{\infty }$.

The Duhamel principle givesDefineSince the free Schrödinger group is unitary,This is scattering from a finite Strichartz norm.

The scaling in part 5 givesThus $u\in L_{t,x}^4([-1,\infty )\times {\mathbb{R}}^2)$. For $T<T^{\prime }$, the dual Strichartz estimate for the free Schrödinger equation giveswhich tends to zero as $T,T^{\prime }\to \infty$. The integrals in the question are therefore Cauchy in $L^2$.

The Duhamel principle in the interaction representation defines an $L^2$ limit $u_{\infty }$ and expresses $S(-t)u(t)-u_{\infty }$ as the tail integral from $t$ to infinity. The same estimate sends that tail to zero, proving the scattering from a finite Strichartz norm conclusion

In two dimensions, the exponents $(q,r)=(4,4)$ give the Strichartz estimate for the free Schrödinger equation below, since $2/q+2/r=1$. The homogeneous estimate and its dual areThe second operator is the adjoint of the first, so the bound follows by Lp duality. It also defines the time integral by continuous extension when an ordinary absolutely convergent integral is unavailable.

The pair $(4,4)$ is the critical cubic spacetime pair. The excluded two-dimensional endpoint is $(2,\infty )$, so the displayed bound should not be called that endpoint estimate.^[ref]