corollaryroutinestructure_uncertain

Global spherical and cone extension estimates

So sphere and cone extension hold globally for b>pcb>p_c, b>n+1n−1a′b>\frac{n+1}{n-1}a'
global extension step used for the paraboloid corollary
Corollary·cor:spherical-restriction·§9.1 Global extension from the sphere and the truncated cone

Statement (paper)

Let 1<a≤∞1<a\le\infty and b<∞b<\infty. If b>pc,b>n+1n−1a′, b>\pc,\qquad b>\frac{n+1}{n-1}a', then ∥Eg∥Lb(Rn)≲n,a,b∥g∥La(Sn−1),∥EΣG∥Lb(Rn+1)≲n,a,b∥G∥La(Σ).\begin{aligned} \|Eg\|_{L^b(\RR^n)} &\lesssim_{n,a,b} \|g\|_{L^a(S^{n-1})},\\ \|E_\Sigma G\|_{L^b(\RR^{n+1})} &\lesssim_{n,a,b} \|G\|_{L^a(\Sigma)}. \end{aligned} These are strong estimates, including when a=∞a=\infty. For b=∞b=\infty both estimates hold for every 1≤a≤∞1\le a\le\infty.

In plain English

For 1<a≤∞1<a\le\infty and b<∞b<\infty with b>pcb>p_c and b>n+1n−1a′b>\frac{n+1}{n-1}a', the sphere extension operator satisfies ∥Eg∥Lb(Rn)≲∥g∥La(Sn−1)\|Eg\|_{L^b(\mathbb R^n)}\lesssim\|g\|_{L^a(S^{n-1})}, and the cone extension operator satisfies ∥EΣG∥Lb(Rn+1)≲∥G∥La(Σ)\|E_\Sigma G\|_{L^b(\mathbb R^{n+1})}\lesssim\|G\|_{L^a(\Sigma)}, with strong-type bounds even at a=∞a=\infty. If b=∞b=\infty, both hold for all 1≤a≤∞1\le a\le\infty.

Why it's here

The main theorem and the local extension lemma give a local sphere extension bound with only a logarithmic loss at pcp_c. Passing from local to global estimates on the ranges of (a,b)(a,b) stated here, with no loss, then yields the global sphere and truncated-cone extension bounds, which later results such as the paraboloid corollary build on.
Classified routine:
Follows from the paper's local extension lemma and main theorem by the standard epsilon-removal / local-to-global argument for extension operators (Tao's epsilon removal, Bourgain's interpolation trick). The resulting ranges match the known Stein–Tomas/restriction-conjecture endpoints, so this corollary is a consequence rather than a new idea.

Uses

lemLocal smoothing implies local spherical extensionthmLocal smoothing on one frequency annulus

Used by

corParaboloid extension and Schrödinger local smoothingclaimGlobal mixed paraboloid extension estimate

Sources

PaperOpen
Section
§9.1 Global extension from the sphere and the truncated cone
Page
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Label
cor:spherical-restriction
Line
sections/global-extension.tex:13
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Friction

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    Statement carries two labels (cor:spherical-restriction and cor:cone-extension); references to either point to the same node.