claimroutineinferred_dep

Shell pieces sum to a convergent bound

Bounds shell pieces on LpL^p; they sum over δ\delta since σ<α−1/2\sigma<\alpha-1/2
bounds the singular part of the square function
Claim·§11 Stein's square function·inside thmStein's square function bounded on Lp

Statement (paper)

For each dyadic \delta=2^{-j}, \ j\ge0, the boundary piece \delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta) of the differentiated multiplier, taken over all dyadic radius intervals together, satisfies the bound C\delta^{\alpha-1}\left(\delta^{1/2}+\delta+[1+\log(1/\delta)]^{1/2-1/p}\delta^{1/2-\sigma}\right)\|f\|_p with C independent of \delta and f. Since \alpha-1/2-\sigma>0, the sum of these bounds over j\ge0 converges.

In plain English

For each dyadic δ=2−j\delta=2^{-j}, j≥0j\ge0, the boundary piece δα−1A(s,ρ)φ((s−ρ)/δ)\delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta), taken over all dyadic radius intervals, is bounded on LpL^p by Cδα−1(δ1/2+δ+[1+log⁡(1/δ)]1/2−1/pδ1/2−σ)∥f∥pC\delta^{\alpha-1}\left(\delta^{1/2}+\delta+[1+\log(1/\delta)]^{1/2-1/p}\delta^{1/2-\sigma}\right)\|f\|_p, with CC independent of δ\delta and ff. Because α−1/2−σ>0\alpha-1/2-\sigma>0, these bounds are summable over j≥0j\ge0.

Why it's here

This applies the dilate square-function lemma to the shell multiplier, with the L2L^2 and LpL^p shell bounds supplying the constants B2∼δ1/2B_2\sim\delta^{1/2} and Bp∼δ1/2−σB_p\sim\delta^{1/2-\sigma}, so each dyadic shell is bounded with explicit dependence on δ\delta. The dominant term is δα−1/2−σ\delta^{\alpha-1/2-\sigma} up to a logarithm, which is summable because σ<α−1/2\sigma<\alpha-1/2, so the boundary part of the square function is controlled on LpL^p.
Classified routine:
A direct consequence of earlier nodes: the dilated square-function lemma (lem-square-dilates) is applied with the L^2 and L^p thin-shell constants from lem-square-shell, followed by summing a geometric series over dyadic δ. This is the standard argument for bounding Stein's square function from local smoothing.

Uses

lemThin-shell multiplier bounds on L^p and L^2lemSquare function bound for dilated vector kernelsclaimDyadic shell decomposition near the boundary

Used by

claimLimiting argument and extension to Lp

Sources

PaperOpen
Section
§11 Stein's square function
Page
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Label
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Line
sections/stein_square.tex:204
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Shell bounds summed over radius intervals" rests on claim-thm-stein-square-2