claimroutineinferred_dep

Reassembly of pieces via synthesis operator

Reassembles pieces via the synthesis operator for strong and weak bounds
closes the angular theorem
Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

The spatial synthesis operator (wj)↦∑jΠNjwj(w_j)\mapsto\sum_j\Pi_{N_j}w_j is bounded from Lq(I×Rn;ℓ2)L^q(I\times\RR^n;\ell^2) to Lq(I×Rn)L^q(I\times\RR^n) for 1<q<∞1<q<\infty, and by real interpolation the same holds for joint weak LpL^p. Consequently the square-function bounds for the unselected pieces (strong) and the selected pieces (strong and weak) transfer to the original uju_j, so that ∑juj=U(T)f\sum_j u_j=U(T)f satisfies the stated strong and weak LpL^p bounds, first for finite frequency sums uniformly and then in general by approximation.

In plain English

The map (wj)↦∑jΠNjwj(w_j)\mapsto\sum_j\Pi_{N_j}w_j is bounded from Lq(I×Rn;ℓ2)L^q(I\times\RR^n;\ell^2) to LqL^q for 1<q<∞1<q<\infty, and by real interpolation also in joint weak LpL^p. Hence the square-function bounds for the unselected and selected pieces give strong and weak LpL^p bounds for ∑juj=U(T)f\sum_j u_j=U(T)f, first for finite frequency sums uniformly and then in general by approximation.

Why it's here

This closes the proof of the angular theorem by passing from the square-function estimates on the individual pieces (the strong unselected bound and the strong and weak selected bounds) to the full solution U(T)fU(T)f. The spatial Littlewood–Paley synthesis operator converts ℓ2\ell^2 control over jj into control of the sum, and interpolation supplies the weak-type version, giving the two stated inequalities.
Classified routine:
The step combines standard Littlewood–Paley theory: the synthesis operator is the dual of the vector-valued Littlewood–Paley square function, bounded on L^q for 1<q<\infty. The weak-type version follows from real interpolation of L^q spaces. The rest is bookkeeping that combines the earlier piece estimates and finishes with a density/approximation argument.

Uses

claimStrong and weak bounds for selected piecesclaimStrong bound for unselected angular bandsclaimAngular band projections decay geometrically

Used by

Nothing in the paper

Sources

PaperOpen
Section
§B.3 Summation over the spatial and angular frequencies
Page
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sections/angular.tex:389
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Friction

  • info
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    outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-4
  • info
    inferred_dep
    outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-5
  • info
    inferred_dep
    outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-2