Sharp weighted absorption of modulated CGO remainders

Assume Countable Choice. Write $⟨\xi ⟩=(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ and use unitary Fourier transformation on ${\mathbb{R}}^3$. There is an absolute constant $C$ such that for every $b\ge 1$ and $Q\in H^{-1\mathrm{/}2}({\mathbb{R}}^3)$, ${\int }_{\mathrm{\mid }\xi \mathrm{\mid }\le b}⟨\xi {⟩}^{-1}\mathrm{\parallel }{e}^{-i\xi \cdot x}Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}^{2}d\xi \le C{b}^2\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}^2\mathrm{.}$ Consequently, if $t>0$, $L\ge 0$ and $a_{\xi }\in H^{1\mathrm{/}2}({\mathbb{R}}^3)$ is any family with $\mathrm{\parallel }{a}_{\xi }{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le L\mathrm{/}t$ for $\mathrm{\mid }\xi \mathrm{\mid }\le b$, its modulated bilinear distributional remainders $R(\xi )=⟨e^{-i\xi \cdot x}Q,a_{\xi }⟩$ satisfy ${({\int }_{\mathrm{\mid }\xi \mathrm{\mid }\le b}⟨\xi {⟩}^{-1}\mathrm{\mid }R(\xi ){\mathrm{\mid }}^{2}d\xi )}^{1\mathrm{/}2}\le C\frac{Lb}{t}\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}},$ whenever the scalar family $R$ is measurable. No continuity or derivative bound in $\xi$ is required.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Plancherel theorem

Weidling, Theorems 6.9–6.10

For $w\in L^2({\mathbb{R}}^3)$, set $[w{]}_{1\mathrm{/}2}^2:=\iint \mathrm{\mid }w(x)-w(y){\mathrm{\mid }}^2\mathrm{\mid }x-y{\mathrm{\mid }}^{-4}dxdy$. Translation, Plancherel and Tonelli [F1]–[F2] give

$[w{]}_{1\mathrm{/}2}^2=\int \mathrm{\mid }\widehat{w}(\xi ){\mathrm{\mid }}^2(\int \frac{\mathrm{\mid }{e}^{i\xi \cdot h}-1{\mathrm{\mid }}^2}{\mathrm{\mid }h{\mathrm{\mid }}^4}dh)d\xi =c_3\int \mathrm{\mid }\xi \mathrm{\mid }\mathrm{\mid }\widehat{w}(\xi ){\mathrm{\mid }}^{2}d\xi ,$

where rotation and dilation give the inner integral, and $0<c_3<\mathrm{\infty }$ follows from $\mathrm{\mid }{e}^{iu}-1\mathrm{\mid }\le \min \{2,\mathrm{\mid }u\mathrm{\mid }\}$. The identity holds with values in $[0,\mathrm{\infty }]$, since Plancherel applies to each $L^2$ translate difference and Tonelli to its nonnegative squared integrand. Thus $w\in H^{1\mathrm{/}2}$ exactly when $\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+[w{]}_{1\mathrm{/}2}^2<\mathrm{\infty }$, and this quantity is equivalent to $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2$. Cutoff remainders are whole-space $H^2$ functions by [F4], so this identity applies directly to them. The identity $f(x)g(x)-f(y)g(y)=f(x)(g(x)-g(y))+g(y)(f(x)-f(y))$ therefore yields

$\mathrm{\parallel }fg{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}+\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{1\mathrm{/}2}})$

for bounded cutoff remainders. For the exact CGO identity, take $a_{\xi }=\chi (r_1+r_2+r_1{r}_2)$, with a fixed smooth cutoff equal to one on the support of $Q$. For phases with $\mathrm{\mid }{\zeta }_j\mathrm{\mid }\asymp t\ge T$, [F3] gives $\mathrm{\parallel }\chi r_j{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathrm{/}t$ and uniformly bounded $L^{\mathrm{\infty }}$ norms. Insert another cutoff equal to one on the support of $\chi$ to apply the product estimate to $\chi r_1{r}_2$. Consequently $\mathrm{\parallel }{a}_{\xi }{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathrm{/}t$, uniformly in $\xi$. One may combine this bound with the separate exact boundary identity, high frequency cutoff and Liouville recovery suppliers. No measurable phase selection is necessary for that application: bound the remainder pointwise by $(C\mathrm{/}t)\mathrm{\parallel }{e}^{-i\xi \cdot x}Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}$, which is measurable by step 3.1, before integrating the resulting scalar Fourier inequality.

The source text at printed pages 172–179, including the full proofs of Theorems 6.8–6.11, and the norm convention (5.8b) and Theorem 5.8 were read. The source norm is an unmodulated localized Sobolev norm. The printed passage on page 176 bounds an absolute volume integral by negative-Sobolev duality and omits the actual Fourier modulation. The proof above uses neither that passage nor its pointwise conclusion: it keeps the modulation, moves it to $Q$, and uses the integrated kernel estimate in steps 1.1–3.1. No independent proof review or native workflow acceptance is asserted.