propositionroutineinferred_depstructure_uncertain

Weak-type endpoint smoothing fails logarithmically

Shows weak-type smoothing needs loss (log⁡N)d(1/2−1/p)(\log N)^{d(1/2-1/p)}
side result: weak-type obstruction at the endpoint
Proposition·prop:weak-obstruction·§8 Two obstructions to endpoint smoothing

Statement (paper)

For 2<p≤pc2<p\le\pc and every sufficiently large NN, there is a nonzero Schwartz function gNg_N with supp⁡g^N⊂{5N/4<∣ξ∣<7N/4}\supp\widehat g_N\subset\{5N/4<|\xi|<7N/4\} such that ∥U(t)gN∥Lp,∞(Rn×I)∥gN∥Lp(Rn)≳n,p(log⁡N)d(1/2−1/p). \frac{\|U(t)g_N\|_{L^{p,\infty}(\RR^n\times I)}}{\|g_N\|_{L^p(\RR^n)}} \gtrsim_{n,p}(\log N)^{d(1/2-1/p)}.

In plain English

For 2<p≤pc2<p\le\pc and all large NN, there is a nonzero Schwartz function gNg_N with Fourier support in {5N/4<∣ξ∣<7N/4}\{5N/4<|\xi|<7N/4\} for which the ratio ∥U(t)gN∥Lp,∞(Rn×I)/∥gN∥Lp(Rn)\|U(t)g_N\|_{L^{p,\infty}(\RR^n\times I)}/\|g_N\|_{L^p(\RR^n)} is at least a constant times (log⁡N)d(1/2−1/p)(\log N)^{d(1/2-1/p)}.

Why it's here

This is the first of the two obstructions to endpoint smoothing: even the weak-type Lp,∞L^{p,\infty} bound cannot hold without a logarithmic loss. The proof builds gNg_N from δ−d\delta^{-d} modulated packets (with δ=2−k\delta=2^{-k}, k≍log⁡Nk\asymp\log N) that start on disjoint tubes and whose outputs focus into a small set. Random signs keep ∥gN∥p≲1\|g_N\|_p\lesssim1 while giving an L2L^2 lower bound on that set, and Hölder converts this into the stated gain of kd(1/2−1/p)k^{d(1/2-1/p)}.
Classified routine:
A counterexample built with standard tools (Khintchine random signs, wave-packet focusing, Hölder), adapting known Kakeya/Besicovitch-type log-loss constructions for wave and Bochner–Riesz operators; it supports the paper's sharpness discussion rather than being a main new idea.

Uses

Nothing else in the paper

Used by

Nothing in the paper

Inside the proof

claimTube arrangement: disjoint start, compressed endclaimModulated packets transport along tubesclaimRandom signs give weak-norm lower bound

Sources

PaperOpen
Section
§8 Two obstructions to endpoint smoothing
Page
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Label
prop:weak-obstruction
Line
sections/sharpness.tex:83
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    "Both lower bounds" refers to this bound together with the earlier obstruction in the same section, Proposition 8.1 (the sharpness proposition).
  • info
    structure_uncertain
    Proved in Appendix C (sections/weak.tex 15-124) without a proof environment.