= Strichartz estimate for the free Schrödinger equation
{c}
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 are
$$
\|e^{it\Delta}f\|_{L^4(\mathbb R\times\mathbb R^2)}\le C\|f\|_2,\qquad
\left\|\int_{\mathbb R}e^{-is\Delta}F(s)\,ds\right\|_2\le C\|F\|_{L^{4/3}(\mathbb R\times\mathbb R^2)}.
$$
The 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.<Nonlinear dispersive equations: local and global analysis by Terence Tao>[Theorem 2.3, admissible pairs, the homogeneous estimate and its dual; rescale time from Laplacian coefficient one-half]{ref}
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