claimroutineinferred_dep

Limiting argument and extension to Lp

Passes to the limit and uses density to extend

\mathcal G_\alpha

L^p

closes the proof by a limiting argument

Claim·§11 Stein's square function·inside thmStein's square function bounded on Lp

Statement (paper)

For Schwartz

f

, the estimates established for finite sets of

k

and finite shell sums pass to the limit, giving

\|\mathcal G_\alpha f\|_p\le C_{n,p,\alpha}\|f\|_p

for all

f\in\mathcal S(\RR^n)

. Consequently the linear map

f\mapsto (r\partial_r B_r^\alpha f)_{r>0}

extends by density to a bounded map into

L^p_x(L^2(\dd r/r))

, which yields the extension of

\mathcal G_\alpha

L^p(\RR^n)

In plain English

The square-function bound holds for all Schwartz

f

with constant

C_{n,p,\alpha}

, by passing finite-index estimates to the limit. By density,

f\mapsto (r\partial_r B_r^\alpha f)_{r>0}

then extends to a bounded map into

L^p_x(L^2(dr/r))

, which extends

\mathcal G_\alpha

L^p(\mathbb R^n)

Why it's here

This closes the proof of the theorem. The smooth part and the summable dyadic shell pieces give uniform bounds on finite truncations, and letting the truncations grow recovers the full square function for Schwartz

f

. Density of Schwartz functions then yields the

L^p

extension.

Classified routine:

Standard limiting argument (monotone convergence over finite truncations) followed by density of Schwartz functions in L^p to extend a bounded linear operator; a direct consequence of the earlier shell and smooth-part bounds.

Uses

claimShell pieces sum to a convergent boundclaimSmooth part away from ball boundary is bounded

Used by

Nothing in the paper

Sources

PaperOpen

Section: §11 Stein's square function
Page: —
Label: —
Line: sections/stein_square.tex:233

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Friction

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outline: "Limiting argument and extension to Lp" rests on claim-thm-stein-square-3
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outline: "Limiting argument and extension to Lp" rests on claim-thm-stein-square-4