claimroutineinferred_dep

Littlewood-Paley setup and small-source removal

Sets up Littlewood–Paley pieces and removes sources near the origin
reduces to sources away from the origin for the envelope
Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

Let fjf_j be a smooth radial Littlewood--Paley decomposition of ff at frequencies Nj=2jN_j=2^j, let uj=U(t)fju_j=U(t)f_j, and let ΠNj\Pi_{N_j} be a fixed enlarged annular multiplier equal to one on the support of f^j\widehat f_j. Then the bounded frequency part satisfies a uniform fixed-time LpL^p estimate, and on the remaining bands the input may be restricted to s≥Njη−1s\ge N_j^{\eta-1}, with a total strong error O(∥f∥p)O(\|f\|_p) even when summed in jj.

In plain English

Decompose ff into radial Littlewood–Paley pieces fjf_j at frequencies Nj=2jN_j=2^j, with uj=U(t)fju_j=U(t)f_j and an enlarged annular multiplier ΠNj\Pi_{N_j} equal to one on the support of f^j\widehat f_j. The bounded-frequency part obeys a uniform fixed-time LpL^p estimate, and on the other bands the input can be restricted to s≥Njη−1s\ge N_j^{\eta-1} at a total strong-type cost of O(∥f∥p)O(\|f\|_p), even summed over jj.

Why it's here

This step fixes the Littlewood–Paley framework for the proof of the angular theorem and disposes of the low-frequency part. Using the small-source estimate of the angular envelope lemma, it shows that sources within Njη−1N_j^{\eta-1} of the origin contribute a geometrically decaying Nj−δN_j^{-\delta} amount that sums to O(∥f∥p)O(\|f\|_p), so later steps may assume the envelope lemma's support condition ∣y∣≥s0|y|\ge s_0 holds.
Classified routine:
Standard Littlewood–Paley bookkeeping: the low-frequency part is handled by a fixed-time multiplier bound, and small sources are removed via the small-source estimate of the paper's angular envelope lemma, which gives geometric decay Nj−δN_j^{-\delta} that sums in jj.

Uses

lemAngular envelope bound and small-source estimate

Used by

claimSelected low-pass dominated by envelope and maximal function

Sources

PaperOpen
Section
§B.3 Summation over the spatial and angular frequencies
Page
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Label
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Line
sections/angular.tex:278
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Littlewood-Paley setup and small-source removal" rests on lem:angular-envelope