lemmaroutineinferred_dep

Thin-shell multiplier bounds on L^p and L^2

Thin-shell multipliers: LpL^p loss δ1/2−σ\delta^{1/2-\sigma}, L2L^2 gain δ1/2\delta^{1/2}
shell constants for the square function sum
Lemma·lem:square-shell·§11 Stein's square function

Statement (paper)

Let 2<p<∞2<p<\infty, ε>0\varepsilon>0, σ=s+(p)+ε\sigma=s_+(p)+\varepsilon, and H=L2([1,2], ds)H=L^2([1,2],\dd s). Fix A∈Cc∞(R×(0,∞))A\in C_c^\infty(\RR\times(0,\infty)) and φ∈Cc∞(R)\varphi\in C_c^\infty(\RR). For 0<δ≤10<\delta\le1, let Tδf(x,s)T_\delta f(x,s) be the spatial Fourier multiplier A(s,∣ξ∣) φ((s−∣ξ∣)/δ) A(s,|\xi|)\,\varphi\bigl((s-|\xi|)/\delta\bigr) applied to ff. Then ∥Tδf∥Lxp(H)≲δ1/2−σ∥f∥p,∥Tδf∥Lx2(H)≲δ1/2∥f∥2, \|T_\delta f\|_{L^p_x(H)}\lesssim \delta^{1/2-\sigma}\|f\|_p, \qquad \|T_\delta f\|_{L^2_x(H)}\lesssim \delta^{1/2}\|f\|_2, with constants independent of δ\delta.

In plain English

For 2<p<∞2<p<\infty and σ=s+(p)+ε\sigma=s_+(p)+\varepsilon, the multiplier A(s,∣ξ∣)φ((s−∣ξ∣)/δ)A(s,|\xi|)\varphi((s-|\xi|)/\delta), which localizes to a shell of width δ\delta around ∣ξ∣=s|\xi|=s, maps LpL^p to Lxp(Ls2[1,2])L^p_x(L^2_s[1,2]) with norm ≲δ1/2−σ\lesssim\delta^{1/2-\sigma}. It maps L2L^2 to Lx2(Ls2)L^2_x(L^2_s) with norm ≲δ1/2\lesssim\delta^{1/2}, uniformly in 0<δ≤10<\delta\le1.

Why it's here

This gives the two endpoint bounds for a single thin shell, the LpL^p bound with loss δ1/2−σ\delta^{1/2-\sigma} and the L2L^2 bound with gain δ1/2\delta^{1/2}. They are the inputs for the shell estimates in Stein's square function argument, where the LpL^p and L2L^2 bounds are combined and summed over dyadic δ\delta. The LpL^p bound comes from local smoothing (Theorem 1.1), with the shell factor φ((s−∣ξ∣)/δ)\varphi((s-|\xi|)/\delta) providing the δ1/2\delta^{1/2} gain in L2L^2 over the ss-interval.
Classified routine:
Writes the thin-shell multiplier as an integral of wave propagators against the Fourier transform of φ\varphi and applies the local smoothing theorem, as in the known derivation of maximal Bochner–Riesz bounds from local smoothing (Mockenhaupt–Seeger–Sogge). The L2L^2 bound is a direct Plancherel computation.

Uses

thmSobolev local smoothing for the wave equation

Used by

thmStein's square function bounded on LpclaimShell pieces sum to a convergent bound

Sources

PaperOpen
Section
§11 Stein's square function
Page
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Label
lem:square-shell
Line
sections/stein_square.tex:39
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Friction

  • info
    inferred_dep
    critique: The proof (stein_square.tex:65-67) applies "the rescaled wave estimate ". That estimate is Theorem  rescaled (consequences.tex:34-47).