claimroutine

Angular band projections decay geometrically

Shows ∥Qmf∥p≲2−mB∥L(Ω)Bf∥p\|Q_mf\|_p\lesssim2^{-mB}\|L(\Omega)^Bf\|_p: angular bands decay geometrically
makes the sum over angular bands converge
Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

For every B>0B>0 and all m≥0m\ge0, ∥Qmf∥p≲B2−mB∥L(Λω)Bf∥p\|Q_m f\|_p\lesssim_B 2^{-mB}\|L(\Om)^B f\|_p.

In plain English

For every B>0B>0 and all m≥0m\ge0, the angular frequency band projection QmQ_m satisfies ∥Qmf∥p≲B2−mB∥L(Λω)Bf∥p\|Q_m f\|_p\lesssim_B 2^{-mB}\|L(\Om)^B f\|_p.

Why it's here

This turns angular smoothness, measured by L(Λω)BL(\Om)^B, into geometric decay in the band index mm. The later steps for unselected angular bands (and the reassembly) use it to sum over mm and to absorb the losses from handling each band separately.
Classified routine:
Standard Littlewood–Paley/spectral multiplier argument for the angular Laplacian: Q_m = \phi(2^{-m}L(\Omega)) L(\Omega)^{-B}\cdot L(\Omega)^B, with the multiplier bounded on L^p by spherical harmonic multiplier theorems (e.g. Strichartz/Seeger–Sogge-type results on compact manifolds).

Uses

Nothing else in the paper

Used by

claimStrong bound for unselected angular bandsclaimReassembly of pieces via synthesis operator

Sources

PaperOpen
Section
§B.3 Summation over the spatial and angular frequencies
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sections/angular.tex:286
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