claimroutine

Smooth part away from ball boundary is bounded

Bounds the smooth part away from the ball boundary on Lxp(L2(dr/r))L^p_x(L^2(dr/r))
handles the nonsingular part of the multiplier
Claim·§11 Stein's square function·inside thmStein's square function bounded on Lp

Statement (paper)

The part of the multiplier away from the boundary of the ball is smooth, supported in ρ≤2\rho\le2, and vanishes to order two at zero. Decomposed into annuli ρ≍2ℓ\rho\asymp2^\ell, ℓ≤2\ell\le2, and rescaled to the unit annulus after extracting 22ℓ2^{2\ell}, its pieces have uniformly bounded derivatives and kernels satisfying Lemma  at δ=1\delta=1 with uniform constants. Hence, by the Littlewood--Paley estimate and ∑ℓ≤222ℓ<∞\sum_{\ell\le2}2^{2\ell}<\infty, this part is bounded on Lxp(L2( dr/r))L^p_x(L^2(\dd r/r)) by C∥f∥pC\|f\|_p.

In plain English

The portion of the multiplier supported away from the boundary of the unit ball (supported in ρ≤2\rho\le2, vanishing to order two at zero) gives an operator bounded on Lxp(L2(dr/r))L^p_x(L^2(dr/r)) with norm at most C∥f∥pC\|f\|_p.

Why it's here

Splitting into annuli ρ≍2ℓ\rho\asymp 2^\ell and rescaling to the unit annulus produces pieces whose kernels satisfy the dilate square function lemma at δ=1\delta=1, where no logarithmic loss occurs. The order-two vanishing at zero supplies the factor 22ℓ2^{2\ell}, which is summable over ℓ≤2\ell\le2, so the Littlewood–Paley estimate sums the pieces. This handles the non-singular part of the multiplier, leaving only the shells near the boundary for the δ\delta-dependent argument.
Classified routine:
This is the standard treatment of the smooth, non-singular part of Stein's square function multiplier. It decomposes the multiplier into dyadic annuli and rescales, then applies the earlier dilate square function lemma at δ=1 together with Littlewood–Paley theory, summing the pieces via the geometric factor 2^{2ℓ}.

Uses

lemSquare function bound for dilated vector kernels

Used by

claimLimiting argument and extension to Lp

Sources

PaperOpen
Section
§11 Stein's square function
Page
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sections/stein_square.tex:223
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