claimroutineinferred_dep

Strong and weak bounds for selected pieces

Bounds selected pieces strongly and weakly in LpL^p
selected-piece input to the reassembly
Claim·§B.3 Summation over the spatial and angular frequencies·inside thmSharp logarithm with angular regularity

Statement (paper)

For the selected pieces, the strong square function over jj of the selected parts of U(T)ΠNjfU(T)\Pi_{N_j}f is bounded in Lp(Rn×I)L^p(\RR^n\times I) by C∥L(J)1/pf∥pC\|L(\J)^{1/p}f\|_p, and the corresponding selected part of U(T)fU(T)f satisfies the weak Lp,∞(Rn×I)L^{p,\infty}(\RR^n\times I) bound C∥f∥pC\|f\|_p. In the region r>4r>4 the selected pieces satisfy a strong LpL^p bound of C∥f∥pC\|f\|_p.

In plain English

For the selected pieces, the ℓ2\ell^2 square function over jj of the selected parts of U(T)ΠNjfU(T)\Pi_{N_j}f is bounded in Lp(Rn×I)L^p(\RR^n\times I) by C∥L(J)1/pf∥pC\|L(\J)^{1/p}f\|_p. The corresponding selected part of U(T)fU(T)f obeys the weak bound Lp,∞(Rn×I)L^{p,\infty}(\RR^n\times I) with constant C∥f∥pC\|f\|_p. In the region r>4r>4 the selected pieces satisfy a strong LpL^p bound C∥f∥pC\|f\|_p.

Why it's here

This gives the estimates for the part of the decomposition where the angular cutoff is selected, using the angular polynomial bound (Proposition B.2) and the envelope domination by the maximal function (previous claim) to sum over jj without dependence on the cutoff m(j,r)m(j,r). These strong and weak bounds are the inputs that the later reassembly step combines, via the synthesis operator and interpolation, into the theorem's estimates.
Classified routine:
A bookkeeping step inside the proof of the angular-regularity theorem: it combines the angular polynomial local smoothing bound (Proposition B.2) with the envelope/maximal-function domination from the previous claim. The square-function and weak-type bounds then follow from standard maximal function arguments, such as the Fefferman–Stein vector-valued maximal inequality.

Uses

claimSelected low-pass dominated by envelope and maximal functionpropAngular polynomial local smoothing bound

Used by

claimReassembly of pieces via synthesis operator

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:337
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Strong and weak bounds for selected pieces" rests on claim-thm-angular-3
  • info
    inferred_dep
    outline: "Strong and weak bounds for selected pieces" rests on prop:angular-band