claimroutine

Kakeya maximal estimate by duality

Bounds Kakeya maximal function in LnL^n with loss L(1/δ)2A/nL(1/\delta)^{2A/n} by duality
maximal estimate behind volume and dimension claims
Claim·§9.3 The maximal estimate and Hausdorff dimension·inside corQuantitative extension and Kakeya bounds

Statement (paper)

For 0<δ≤10<\delta\le1, ∥Kδf∥Ln(Sn−1)≲nL(1/δ)2A/n∥f∥Ln(Rn)\|K_\delta f\|_{L^n(S^{n-1})}\lesssim_n L(1/\delta)^{2A/n}\|f\|_{L^n(\RR^n)} with A=2/pA=2/p. This is obtained by testing the discretized norm (δd∑jK2δf(ωj)n)1/n\big(\delta^d\sum_j K_{2\delta}f(\omega_j)^n\big)^{1/n} against weights aj≥0a_j\ge0 with δd∑jajq=1\delta^d\sum_j a_j^q=1, and bounding δd∑jaj∣Tj∣∫Tj∣f∣≲∫∣f∣∑jaj1Tj≲L(1/δ)2A∥f∥n\delta^d\sum_j \frac{a_j}{|T_j|}\int_{T_j}|f|\lesssim \int|f|\sum_j a_j\mathbf 1_{T_j}\lesssim L(1/\delta)^{2A}\|f\|_n for any choice of centers.

In plain English

For 0<δ≤10<\delta\le1, the Kakeya maximal function satisfies ∥Kδf∥Ln(Sn−1)≲nL(1/δ)2A/n∥f∥Ln(Rn)\|K_\delta f\|_{L^n(S^{n-1})}\lesssim_n L(1/\delta)^{2A/n}\|f\|_{L^n(\mathbb R^n)} with A=2/pA=2/p, so the logarithmic loss is L(1/δ)2A/nL(1/\delta)^{2A/n}.

Why it's here

The LnL^n norm of the discretized maximal function is dualized against weights aja_j in ℓq\ell^q, which converts the problem into bounding ∫∣f∣∑jaj1Tj\int|f|\sum_j a_j\mathbf 1_{T_j} for arbitrary tube centers. The overlap lemma controls this sum in LqL^q with loss L(1/δ)2AL(1/\delta)^{2A}, and Hölder's inequality then gives the claim. This maximal bound is the input for the volume and Hausdorff-dimension claims that follow.
Classified routine:
Standard duality argument (as in Bourgain 1991 and Wolff's treatment of Kakeya maximal functions) turning the tube overlap bound in L^q into a Kakeya maximal estimate on L^n via discretization and Hölder; it follows directly from the paper's overlap lemma.

Uses

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

Used by

claimLogarithmic volume bound for neighborhoodsclaimKakeya sets have infinite gauge Hausdorff measure

Sources

PaperOpen
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§9.3 The maximal estimate and Hausdorff dimension
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