theoremroutine

Stein's square function bounded on Lp

Stein's square function is bounded on LpL^p for α>max⁡{1/2,n(1/2−1/p)}\alpha>\max\{1/2,n(1/2-1/p)\}
main square-function application of local smoothing
Theorem·thm:stein-square·§11 Stein's square function

Statement (paper)

Let n≥2n\ge2 and 2≤p<∞2\le p<\infty. If α>max⁡{12, n(12−1p)}, \alpha>\max\left\{\tfrac12,\,n\left(\tfrac12-\tfrac1p\right)\right\}, then ∥Gαf∥p≤Cn,p,α∥f∥p\|\mathcal G_\alpha f\|_p\le C_{n,p,\alpha}\|f\|_p for f∈S(Rn)f\in\mathcal S(\RR^n), and Gα\mathcal G_\alpha extends to Lp(Rn)L^p(\RR^n).

In plain English

For n≥2n\ge2, 2≤p<∞2\le p<\infty and α>max⁡{1/2, n(1/2−1/p)}\alpha>\max\{1/2,\,n(1/2-1/p)\}, Stein's square function satisfies ∥Gαf∥p≤Cn,p,α∥f∥p\|\mathcal G_\alpha f\|_p\le C_{n,p,\alpha}\|f\|_p for Schwartz ff. Hence Gα\mathcal G_\alpha extends to a bounded operator on Lp(Rn)L^p(\mathbb R^n).

Why it's here

The case p=2p=2 is an exact identity requiring α>1/2\alpha>1/2. For p>2p>2 the multiplier is split into a smooth part, handled at δ=1\delta=1, and dyadic shells at distance δ\delta from the singular boundary ρ=s\rho=s. The shells are controlled by combining the thin-shell L2L^2 and LpL^p bounds, which come from local smoothing, with the dilate square-function lemma. The shell bounds sum over δ\delta because σ<α−1/2\sigma<\alpha-1/2, and density then extends the operator to LpL^p.
Classified routine:
The bound on Stein's square function for α>max⁡{1/2,n(1/2−1/p)}\alpha>\max\{1/2,n(1/2-1/p)\} is a known consequence of local smoothing, going back to Stein's square function for Bochner–Riesz. Here it is derived by the standard dyadic-shell decomposition from the thin-shell local smoothing bounds and the dilate square-function lemma.

Uses

lemThin-shell multiplier bounds on L^p and L^2lemSquare function bound for dilated vector kernels

Used by

Nothing in the paper

Inside the proof

claimL2 identity for Stein's square functionclaimDyadic shell decomposition near the boundaryclaimShell pieces sum to a convergent boundclaimSmooth part away from ball boundary is boundedclaimLimiting argument and extension to Lp

Sources

PaperOpen
Section
§11 Stein's square function
Page
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Label
thm:stein-square
Line
sections/stein_square.tex:21
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