claimroutineinferred_dep

Critical Schrödinger estimate with logarithmic loss

Gives critical LpcL^{p_c} Schrödinger bound with loss [log⁡(2+N)]A[\log(2+N)]^{A}
critical Schrödinger estimate of the corollary
Claim·§12 The paraboloid and Schr\"odinger evolution·inside corParaboloid extension and Schrödinger local smoothing

Statement (paper)

For h=f130Qh=f\mathbf 1_{30Q} with QQ a cube of side NN translated to be centered at zero, and f=PNff=P_Nf, the near contribution satisfies ∥S(t)PNh∥Lp(Q×I0)≤C[log⁡(2+N)]A∥f∥Lp(30Q)\|S(t)P_Nh\|_{L^p(Q\times I_0)}\le C[\log(2+N)]^A\|f\|_{L^p(30Q)} with p=pcp=p_c. Consequently, summing the ppth powers over QQ (bounded overlap of the cubes 30Q30Q) and adding the far contribution gives ∥eitΔf∥Lpc(Rd×I0)≲[log⁡(2+N)]A∥f∥pc\|e^{it\Delta}f\|_{L^{p_c}(\RR^d\times I_0)}\lesssim[\log(2+N)]^A\|f\|_{p_c} with I0=[1/2,1]I_0=[1/2,1], and by parabolic scaling on [a/2,a][a/2,a], dyadic summation and conjugation for negative times, this yields on II with power A=2/pcA=2/p_c.

In plain English

For data f=PNff=P_Nf localized to 30Q30Q on a cube QQ of side NN, the near part of S(t)fS(t)f satisfies ∥S(t)PNh∥Lpc(Q×I0)≤C[log⁡(2+N)]A∥f∥Lpc(30Q)\|S(t)P_Nh\|_{L^{p_c}(Q\times I_0)}\le C[\log(2+N)]^A\|f\|_{L^{p_c}(30Q)}. Summing over cubes and adding the far part gives ∥eitΔf∥Lpc(Rd×I0)≲[log⁡(2+N)]A∥f∥pc\|e^{it\Delta}f\|_{L^{p_c}(\RR^d\times I_0)}\lesssim[\log(2+N)]^A\|f\|_{p_c} for I0=[1/2,1]I_0=[1/2,1], and then the critical estimate on any compact II with A=2/pcA=2/p_c.

Why it's here

The Fresnel formula converts the near piece into a paraboloid extension operator on a ball of radius about NN, so the local logarithmic extension bound gives the [log⁡(2+N)]A[\log(2+N)]^A loss. Combined with the far-field kernel decay, bounded overlap of the cubes 30Q30Q assembles the local bounds into a global one on I0I_0. Parabolic scaling, dyadic summation in time and conjugation for negative times then extend it to the full interval II and give the critical Schrödinger estimate of the corollary.
Classified routine:
Standard transfer of extension bounds to Schrödinger local smoothing via the Fresnel formula, localization to cubes, far-field kernel decay, parabolic scaling and dyadic summation (as in Rogers' and Lee–Rogers–Vargas' extension-to-Schrödinger arguments). It follows directly from the earlier local paraboloid extension bound and kernel decay claims.

Uses

claimLocal paraboloid extension with logarithmic lossclaimFar-field Schrödinger kernel decay

Used by

Nothing in the paper

Sources

PaperOpen
Section
§12 The paraboloid and Schr\"odinger evolution
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sections/parabolic.tex:100
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Friction

  • info
    inferred_dep
    outline: "Near contribution via Fresnel formula; critical estimate" rests on claim-cor-parabolic-1
  • info
    inferred_dep
    outline: "Near contribution via Fresnel formula; critical estimate" rests on claim-cor-parabolic-3