claimroutineinferred_dep

Conserved current with Gaussian multiplier symbols

Builds a conserved current with ATT0≥∣S0gT∣2A^0_{TT}\ge|S_0g_T|^2 via Gaussian multipliers
concrete positive current for pencil waves
Claim·§E.2 The current and its Gaussian multiplier·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

For a finite periodic Fourier sum gT=e−iTϕ(D)fg_T=e^{-\mathrm iT\phi(D)}f supported in the annulus ∣ξ∣≍h−1|\xi|\asymp h^{-1} in the chosen chart, with ∣v(ξ)∣≤γ|v(\xi)|\le\gamma, the vector wave ∑ξ∣ξ∣d/2E(ξ)uξ\sum_\xi|\xi|^{d/2}E(\xi)u_\xi solves the symmetric system with matrices C0=IC^0=I, Cj=CjC^j=C_j. The holomorphic flux map Θ(w,Φ)=(12wtCμw)μ=0n+(0,ω)Φ\Theta(w,\Phi)=\big(\tfrac12 w^{\mathsf t}C^\mu w\big)_{\mu=0}^n+(0,\omega)\Phi has output (τ,P)(\tau,P) and real Jacobian J=∣R∣2nj(W)J=|R|^{2n}j(W) with j(W)≍1j(W)\asymp1. The associated column current χμ=J−1/2∑aMμai(Gsys)a\chi_\mu=J^{-1/2}\sum_a M_{\mu a}\mathbf i(G_{\rm sys})_a, A0=χχ∗A^0=\chi\chi^*, is column divergence free, satisfies dΦ χ=0d\Phi\,\chi=0, and is single valued. Its rows are, up to conjugation and multiplication by i\mathbf i, pairs of scalar multipliers on gTg_T whose real symbols are the two Ck\CC_{\mathbf k}-components of j(W)−1/2e(W)tCμE(η)∣η∣d/2(hy)−d/2e−Ψj(W)^{-1/2}e(W)^{\mathsf t}C^\mu E(\eta)|\eta|^{d/2}(hy)^{-d/2}e^{-\Psi}, with Ψ=y2h(W−v(η))∗S⊥(η)(W−v(η))\Psi=\frac{y}{2h}(W-v(\eta))^*S_\perp(\eta)(W-v(\eta)) and η=hξ\eta=h\xi. In particular ATT0≥∣S0gT∣2A^0_{TT}\ge|\mathsf S_0g_T|^2, where S0\mathsf S_0 is the multiplier whose symbol is the real part of this expression for μ=0\mu=0, and it can be inverted on the cap.

In plain English

For a frequency-localized wave gT=e−iTϕ(D)fg_T=e^{-\mathrm iT\phi(D)}f, the holomorphic flux map built from the system with matrices C0=IC^0=I, Cj=CjC^j=C_j has Jacobian ∣R∣2nj(W)|R|^{2n}j(W) with j(W)≍1j(W)\asymp1, and yields a single-valued, column-divergence-free current χ\chi with dΦ χ=0d\Phi\,\chi=0 and A0=χχ∗A^0=\chi\chi^*. Its rows are scalar Gaussian-weighted Fourier multipliers on gTg_T, so ATT0≥∣S0gT∣2A^0_{TT}\ge|\mathsf S_0g_T|^2 for an explicit multiplier S0\mathsf S_0 that is invertible on the cap.

Why it's here

This is where the abstract positive divergence-free tensor of the current-flow lemma becomes concrete for the pencil wave. It uses the chart and Jacobian bounds of the previous claim to show the current has an explicit multiplier form, so its time component dominates a squared multiplier of gTg_T. The following claims use this conserved positive tensor, and its lower bound by ∣S0gT∣2|\mathsf S_0g_T|^2, for the determinant-energy balance and the inversion estimates.
Classified routine:
Routine relative to the paper's central ideas (downgraded from novel at the paper level). This is the concrete construction of the paper's holomorphic-flux positive divergence-free current for pencil waves, with Gaussian multiplier symbols giving the lower bound ATT0≥∣S0gT∣2A^0_{TT}\ge|\mathsf S_0g_T|^2. It specializes the paper's own current-flow lemma and has no prior counterpart in the local smoothing literature (e.g. Guth–Wang–Zhang's decoupling and incidence methods).

Uses

lemDivergence-free positive source tensor from holomorphic flowclaimFiber chart and geometric bounds

Used by

claimDeterminant energy and balance on pencil fibersclaimWall decay and interior lower bound

Sources

PaperOpen
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§E.2 The current and its Gaussian multiplier
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Friction

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    outline: "Conserved current and Gaussian multiplier symbols" rests on lem:current-flow
  • info
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    outline: "Conserved current and Gaussian multiplier symbols" rests on claim-thm-pencil-2