The bilinear and its four legs

For TT waves and free scalars with data in $H^2\times H^1$,

The exchange of integrations follows from the finite-slab estimates. Moreover,

Indeed, $H^2({\mathbb{R}}^3)\subset L^{\infty }$, so ${\Vert k_{ij}{\partial }_i{\partial }_j\varphi \Vert }_2\le C{\Vert k\Vert }_{H^2}{\Vert \varphi \Vert }_{H^2}$. Free evolution and Lemma lem:wave prove the estimate.

Use signs $u_p^{+}=u_p,u_p^{-}={\bar{u}}_p$, and similarly for ${v_q}^{\delta }$. The four kernels are

Here $a^{+}=a,a^{-}=a^{†}$, so the quadratic expression is

For real $f$,

On ${\mathcal{D}}_2$, these formulas initially define a symmetric sesquilinear form: every nonzero matrix element is a finite sum of kernel integrals over compact external supports away from gravitational zero momentum. The local bound

makes them finite.