corollaryroutine

Quantitative extension and Kakeya bounds

So Kakeya sets have dimension nn, with logarithmic maximal-function loss
payoff of the main theorem
Corollary·cor:quant-kakeya·§9 Spherical extension and Kakeya tubes

Statement (paper)

For n≥2n\ge2, R≥2R\ge2, z∈Rnz\in\RR^n, and 0<δ≤10<\delta\le1, ∥Eg∥Lpc(B(z,R))≤Cn[log⁡(2+R)]2/pc∥g∥Lpc(Sn−1),∥Kδf∥Ln(Sn−1)≤Cn[log⁡(2/δ)]4/pc∥f∥Ln(Rn).\begin{aligned} \|Eg\|_{L^{\pc}(B(z,R))} &\le C_n [\log(2+R)]^{2/\pc}\|g\|_{L^{\pc}(S^{n-1})},\\ \|K_\delta f\|_{L^n(S^{n-1})} &\le C_n [\log(2/\delta)]^{4/\pc}\|f\|_{L^n(\RR^n)}. \end{aligned} Every subset of Rn\RR^n containing a unit line segment in each direction has Hausdorff dimension nn.

In plain English

For n≥2n\ge2, the spherical extension operator satisfies ∥Eg∥Lpc(B(z,R))≤Cn[log⁡(2+R)]2/pc∥g∥Lpc(Sn−1)\|Eg\|_{L^{p_c}(B(z,R))}\le C_n[\log(2+R)]^{2/p_c}\|g\|_{L^{p_c}(S^{n-1})}, and the Kakeya maximal function satisfies ∥Kδf∥Ln(Sn−1)≤Cn[log⁡(2/δ)]4/pc∥f∥Ln(Rn)\|K_\delta f\|_{L^n(S^{n-1})}\le C_n[\log(2/\delta)]^{4/p_c}\|f\|_{L^n(\mathbb R^n)}. Consequently every subset of Rn\mathbb R^n containing a unit line segment in each direction has Hausdorff dimension nn.

Why it's here

The main local smoothing estimate has loss [log⁡]2/pc[\log]^{2/p_c}, and the local-extension lemma turns this into the logarithmic extension bound on balls. The overlap lemma, dualized, then gives the Kakeya maximal bound with the doubled exponent 4/pc4/p_c. The volume and gauge-measure claims convert that bound into the Kakeya set conjecture's dimension statement, all with only logarithmic losses.
Classified routine:
Direct consequence of the main log-loss local smoothing theorem combined with the local-extension and tube-overlap lemmas. It follows the well-known implication chain local smoothing ⇒ restriction/extension ⇒ Kakeya maximal ⇒ Hausdorff dimension (Bourgain, Wolff, Tao).

Uses

thmLocal smoothing on one frequency annuluslemLocal smoothing implies local spherical extensionlemKakeya overlap bound for tubes

Used by

Nothing in the paper

Inside the proof

claimKakeya maximal estimate by dualityclaimLogarithmic volume bound for neighborhoodsclaimKakeya sets have infinite gauge Hausdorff measure

Sources

PaperOpen
Section
§9 Spherical extension and Kakeya tubes
Page
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Label
cor:quant-kakeya
Line
sections/kakeya.tex:14
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