claimroutine

Dyadic shell decomposition near the boundary

Splits multiplier into dyadic shells at distance δ\delta with gain δα−1\delta^{\alpha-1}
reduces to thin shells with a uniform profile
Claim·§11 Stein's square function·inside thmStein's square function bounded on Lp

Statement (paper)

Let p>2p>2 and choose ε>0\varepsilon>0 with σ=s+(p)+ε<α−1/2\sigma=s_+(p)+\varepsilon<\alpha-1/2. Writing r=2ksr=2^ks with 1≤s≤21\le s\le2 and ρ=2−k∣ξ∣\rho=2^{-k}|\xi|, the differentiated multiplier on this interval is m(s,ρ)=2αρ2s−2α(s+ρ)α−1(s−ρ)α−11{ρ<s}m(s,\rho)=2\alpha\rho^2s^{-2\alpha}(s+\rho)^{\alpha-1}(s-\rho)^{\alpha-1}\mathbf1_{\{\rho<s\}}, and splitting s−ρ>0s-\rho>0 into dyadic intervals of size δ=2−j\delta=2^{-j}, j≥0j\ge0, expresses each piece as δα−1A(s,ρ)φ((s−ρ)/δ)\delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta), where AA is a fixed amplitude supported away from ρ=0\rho=0 and φ∈Cc∞((1/2,2))\varphi\in C_c^\infty((1/2,2)).

In plain English

For p>2p>2 and σ=s+(p)+ε<α−1/2\sigma=s_+(p)+\varepsilon<\alpha-1/2, on the radius interval r=2ksr=2^ks with 1≤s≤21\le s\le2 and ρ=2−k∣ξ∣\rho=2^{-k}|\xi|, the differentiated multiplier is m(s,ρ)=2αρ2s−2α(s+ρ)α−1(s−ρ)α−11{ρ<s}m(s,\rho)=2\alpha\rho^2s^{-2\alpha}(s+\rho)^{\alpha-1}(s-\rho)^{\alpha-1}\mathbf1_{\{\rho<s\}}. Splitting s−ρ>0s-\rho>0 into dyadic pieces of size δ=2−j\delta=2^{-j} writes each piece as δα−1A(s,ρ)φ((s−ρ)/δ)\delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta), with AA a fixed amplitude supported away from ρ=0\rho=0 and φ∈Cc∞((1/2,2))\varphi\in C_c^\infty((1/2,2)).

Why it's here

This reduces the square function to a family of shells localized at distance δ\delta from the singular boundary ρ=s\rho=s. Each shell carries the explicit gain δα−1\delta^{\alpha-1} from the (s−ρ)α−1(s-\rho)^{\alpha-1} singularity and has a smooth, uniform profile. The following claim can then bound each shell and sum over δ\delta, where the condition σ<α−1/2\sigma<\alpha-1/2 gives the decay needed to make the sum converge.
Classified routine:
A direct calculation differentiating the Bochner–Riesz multiplier in the radius, followed by the standard dyadic decomposition of (s−ρ)α−1(s-\rho)^{\alpha-1} near the boundary sphere, as in the classical treatment of Stein's square function (Stein; Carbery).

Uses

Nothing else in the paper

Used by

claimShell pieces sum to a convergent bound

Sources

PaperOpen
Section
§11 Stein's square function
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sections/stein_square.tex:185
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