Restriction estimates and the missing Strichartz interface

Recorded orientation, not proved here. The Schrödinger initial-value problem ${\mathrm{\partial }}_{t}u+i\mathrm{\Delta }u=h$, $u(0)=f$ on ${\mathbb{R}}^{1+d}$ uses a paraboloid extension operator, as seen in Williams's formula (11.21); it is not the spherical extension theorem proved on this page. The two arguments share dispersive bounds, Plancherel, $T{T}^{\ast }$ and fractional integration. Williams, Definition 11.5 and Theorem 11.6, calls $(p,q)$ admissible when $2\mathrm{/}p+d\mathrm{/}q=d\mathrm{/}2$, $2\le p,q\le \mathrm{\infty }$, excluding $(2,\mathrm{\infty })$; for admissible pairs with $p>2$ and Schwartz data he proves $\mathrm{\parallel }u{\mathrm{\parallel }}_{L_t^p{L}_x^q}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }h{\mathrm{\parallel }}_{L_t^{p^{\mathrm{\prime }}}{L}_x^{q^{\mathrm{\prime }}}})$. This records that nonendpoint theorem with the same pair for solution and forcing. It does not assert endpoint Strichartz estimates or identify them with spherical restriction.

The comparison is anchored to Stein-Tomas spherical restriction theorem. No PDE page is commissioned in this run, and this sourced remark remains a non-load-bearing leaf; it supplies no proof to another item.