claimroutineinferred_dep

Strong bound for unselected angular bands

Bounds unselected bands strongly; sum converges for B>A+1/2−1/pB>A+1/2-1/p
strong estimate for the unselected part
Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

The unselected angular frequencies, i.e. the pieces U(t)ΠNjQmfj1s≥Njη−1U(t)\Pi_{N_j}Q_m f_j\mathbf 1_{s\ge N_j^{\eta-1}} in the region where they are not selected, satisfy a strong LpL^p square-function bound: for each mm, the unselected mmth angular band has total strong square-function norm at most CL(Hm)A+1/2−1/p∥(bj)∥ℓp≲2m(A+1/2−1/p)∥Qmf∥pC L(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p}\lesssim 2^{m(A+1/2-1/p)}\|Q_m f\|_p, where bj=∥Qmfj∥pb_j=\|Q_m f_j\|_p. Consequently, by , summing over mm the unselected part is bounded in strong LpL^p by CB∥L(Λω)Bf∥pC_B\|L(\Om)^B f\|_p whenever B>A+1/2−1/pB>A+1/2-1/p.

In plain English

For each angular band index mm, the unselected pieces U(t)ΠNjQmfj1s≥Njη−1U(t)\Pi_{N_j}Q_m f_j\mathbf 1_{s\ge N_j^{\eta-1}} have strong LpL^p square-function norm at most CL(Hm)A+1/2−1/p∥(bj)∥ℓp≲2m(A+1/2−1/p)∥Qmf∥pC L(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p}\lesssim 2^{m(A+1/2-1/p)}\|Q_m f\|_p, where bj=∥Qmfj∥pb_j=\|Q_m f_j\|_p. Summing over mm, the unselected part is bounded in strong LpL^p by CB∥L(Λω)Bf∥pC_B\|L(\Om)^B f\|_p whenever B>A+1/2−1/pB>A+1/2-1/p.

Why it's here

This handles the unselected angular bands with a strong-type estimate, which is the part of the parent theorem's proof that needs no weak-type argument. Its proof uses the ball estimate (lem-angular-ball) and the single-annulus bound (thm-main), and the geometric decay of ∥Qmf∥p\|Q_m f\|_p (claim-thm-angular-2) makes the sum over mm converge once B>A+1/2−1/pB>A+1/2-1/p. The result feeds the reassembly step (claim-thm-angular-6).
Classified routine:
A bookkeeping step inside the proof of thm-angular: it combines the small-ball estimate (lem-angular-ball) and the single-annulus bound (thm-main) to get a strong square-function bound for each angular band. The geometric decay of the band projections (claim-thm-angular-2) then makes the sum over m converge.

Uses

thmLocal smoothing on one frequency annuluslemLocal smoothing bound on small ballsclaimAngular band projections decay geometrically

Used by

claimReassembly of pieces via synthesis operator

Sources

PaperOpen
Section
§B.3 Summation over the spatial and angular frequencies
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sections/angular.tex:358
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Strong bound for unselected angular bands" rests on thm:main
  • info
    inferred_dep
    outline: "Strong bound for unselected angular bands" rests on lem:angular-ball
  • info
    inferred_dep
    outline: "Strong bound for unselected angular bands" rests on claim-thm-angular-2