corollaryroutineinferred_dep

Logarithmic Sobolev local smoothing estimate

So U(T)U(T) loses s+(p)s_+(p) derivatives plus explicit logarithmic weights
payoff of the main theorem
Corollary·cor:log-sobolev·§7.2 Interpolation and summation

Statement (paper)

Let 2<p<∞2<p<\infty, δp=1/2−1/p\delta_p=1/2-1/p, and B∗(p)=β(p)+δpB_*(p)=\beta(p)+\delta_p. Write J=(1−Δ)1/2J=(1-\Delta)^{1/2}, ℓ1(r)=log⁡(2+r)\ell_1(r)=\log(2+r) and ℓ2(r)=log⁡(2+ℓ1(r))\ell_2(r)=\log(2+\ell_1(r)). Then ∥U(T)f∥Lp(I×Rn)≲p,B,C∥Js+(p)ℓ1(J)Bℓ2(J)Cf∥Lp(Rn) \|\U(T) f\|_{L^p(I\times\RR^n)} \lesssim_{p,B,C} \|J^{s_+(p)}\ell_1(J)^B\ell_2(J)^C f\|_{L^p(\RR^n)} whenever B>B∗(p)B>B_*(p) and C∈RC\in\RR, or B=B∗(p)B=B_*(p) and C>δpC>\delta_p. At p=pcp=\pc, B∗(pc)=1/2+1/pcB_*(\pc)=1/2+1/\pc and δpc=1/(2n)\delta_{\pc}=1/(2n). At p=2p=2 no logarithmic factor is needed.

In plain English

For 2<p<∞2<p<\infty, the wave propagator satisfies ∥U(T)f∥Lp(I×Rn)≲∥Js+(p)ℓ1(J)Bℓ2(J)Cf∥Lp\|U(T)f\|_{L^p(I\times\mathbb R^n)}\lesssim \|J^{s_+(p)}\ell_1(J)^B\ell_2(J)^C f\|_{L^p}, where J=(1−Δ)1/2J=(1-\Delta)^{1/2}, provided B>B∗(p)=β(p)+δpB>B_*(p)=\beta(p)+\delta_p with any real CC, or B=B∗(p)B=B_*(p) with C>δpC>\delta_p. At p=pcp=p_c the threshold is B∗=1/2+1/pcB_*=1/2+1/p_c with δpc=1/(2n)\delta_{p_c}=1/(2n), and at p=2p=2 no logarithm is needed.

Why it's here

This sums the single-annulus estimates (the logarithmic bound of the main theorem at pcp_c, interpolated with other exponents) over dyadic frequency scales, turning a per-annulus logarithmic loss into a Sobolev bound at regularity s+(p)s_+(p) with explicit log weights. The ℓ2\ell_2 factor with C>δpC>\delta_p is what makes the dyadic sum converge at the endpoint B=B∗(p)B=B_*(p).
Classified routine:
Follows from the single-annulus logarithmic bound (thm-main) by interpolation and Littlewood–Paley summation over dyadic frequencies, a standard way of turning per-annulus estimates into Sobolev-type bounds with log weights.

Uses

thmLocal smoothing on one frequency annulus

Used by

Nothing in the paper

Sources

PaperOpen
Section
§7.2 Interpolation and summation
Page
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Label
cor:log-sobolev
Line
sections/transfer.tex:115
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    critique: The proof (transfer.tex:157) says "By , the j-th norm on the right is at most...". That estimate is the interpolation of Theorem  (transfer.tex:88-94).