claimroutineinferred_dep

Selected low-pass dominated by envelope and maximal function

Dominates selected low-pass pieces by a maximal function envelope

removes dependence on the cutoff for selected pieces

Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

For

r

with

m(j,r)\ge0

, the contribution

U(t)\Pi_{N_j}S_{m(j,r)}a_j

(t,r\theta)

is controlled by the profile

G_{\mathcal M_{\mathbb S^d}f_j}

of Lemma , with

\mathcal M_{\mathbb S^d}f_j

in place of

S_{m(j,r)}f_j

, so the profile does not depend on the cutoff

m(j,r)

. Moreover, for arbitrary nonnegative coefficients

c_j

\|(c_jG_{\mathcal M_{\mathbb S^d}f_j})_j\|_{L^p(\dd v\dd\theta;\ell^2)}\lesssim\|(c_jf_j)_j\|_{L^p(\RR^n;\ell^2)}

In plain English

Where the cutoff satisfies

m(j,r)\ge0

, the piece

U(t)\Pi_{N_j}S_{m(j,r)}a_j

(t,r\theta)

is bounded by the envelope profile

G_{\mathcal M_{\mathbb S^d}f_j}

, which does not depend on the cutoff

m(j,r)

. For any nonnegative coefficients

c_j

, the

\ell^2

-valued

L^p(\dd v\dd\theta)

norm of

(c_jG_{\mathcal M_{\mathbb S^d}f_j})_j

is at most a constant times

\|(c_jf_j)_j\|_{L^p(\RR^n;\ell^2)}

Why it's here

Applying the angular envelope lemma to the low-pass piece gives a pointwise bound by a profile of

S_{m(j,r)}f_j

. Replacing

S_{m(j,r)}f_j

by the spherical maximal function

\mathcal M_{\mathbb S^d}f_j

removes the dependence on

r

through the cutoff, and the vector-valued maximal inequality then converts the profile norm into the norm of

(c_jf_j)

. This lets the selected pieces be summed over

j

and bounded in the strong and weak estimates that follow.

Classified routine:

Follows from the angular envelope lemma by dominating the angular projection

S_{m}f_j

pointwise by the spherical maximal function, then applying the Fefferman–Stein vector-valued maximal inequality (Fefferman–Stein 1971) to bound the

f f f

-valued profile norm.

Uses

lemAngular envelope bound and small-source estimateclaimLittlewood-Paley setup and small-source removal

Used by

claimStrong and weak bounds for selected pieces

Sources

PaperOpen

Section: §B.3 Summation over the spatial and angular frequencies
Page: —
Label: —
Line: sections/angular.tex:308

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Selected low-pass via envelope and vector maximal" rests on lem:angular-envelope
info
inferred_dep
outline: "Selected low-pass via envelope and vector maximal" rests on claim-thm-angular-1