claimroutine

L2 identity for Stein's square function

Gives ∥Gαf∥22=α2α−1∥f∥22\|\mathcal G_\alpha f\|_2^2=\frac{\alpha}{2\alpha-1}\|f\|_2^2 for α>1/2\alpha>1/2
the L2L^2 case of Stein's theorem
Claim·§11 Stein's square function·inside thmStein's square function bounded on Lp

Statement (paper)

For p=2p=2, ∥Gαf∥22=2α2∫01u(1−u)2α−2  du ∥f∥22=α2α−1∥f∥22\|\mathcal G_\alpha f\|_2^2=2\alpha^2\int_0^1 u(1-u)^{2\alpha-2}\,\dd u\,\|f\|_2^2=\frac{\alpha}{2\alpha-1}\|f\|_2^2 for f∈S(Rn)f\in\mathcal S(\RR^n) and α>1/2\alpha>1/2.

In plain English

For p=2p=2 and α>1/2\alpha>1/2, the square function satisfies ∥Gαf∥22=α2α−1∥f∥22\|\mathcal G_\alpha f\|_2^2=\frac{\alpha}{2\alpha-1}\|f\|_2^2 for Schwartz ff, with the constant given by 2α2∫01u(1−u)2α−2 du2\alpha^2\int_0^1 u(1-u)^{2\alpha-2}\,du.

Why it's here

This gives the p=2p=2 endpoint of Stein's square function theorem as an exact identity, so Gα\mathcal G_\alpha is bounded on L2L^2 with constant α/(2α−1)\sqrt{\alpha/(2\alpha-1)}. The condition α>1/2\alpha>1/2 is exactly what makes the Beta-type integral converge, matching the threshold in the parent theorem. The parent's argument for p>2p>2 can start from this L2L^2 bound.
Classified routine:
Direct Plancherel computation for Stein's square function, as in Stein's classical treatment of Bochner–Riesz means (Stein–Weiss); it reduces to evaluating a Beta integral, which converges exactly when α>1/2\alpha>1/2.

Uses

Nothing else in the paper

Used by

Nothing in the paper

Sources

PaperOpen
Section
§11 Stein's square function
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sections/stein_square.tex:179
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