theoremroutineinferred_dep

Sobolev local smoothing for the wave equation

Shows U(T)U(T) loses at most s+(p)+εs_+(p)+\varepsilon derivatives in LpL^p
sums annulus bound over frequencies for shell estimates
Theorem·thm:local-smoothing·§1 Introduction

Statement (paper)

For n≥2n\ge2, 2≤p<∞2\le p<\infty and ε>0\varepsilon>0, ∥U(T)f∥Lp(Rn×I)≤Cn,p,ε∥Js+(p)+εf∥Lp(Rn). \|U(T)f\|_{L^p(\RR^n\times I)} \le C_{n,p,\varepsilon}\|J^{s_+(p)+\varepsilon} f\|_{L^p(\RR^n)} . At p=2p=2 one can take ε=0\varepsilon=0.

In plain English

For n≥2n\ge2, 2≤p<∞2\le p<\infty and ε>0\varepsilon>0, the wave propagator satisfies ∥U(T)f∥Lp(Rn×I)≤Cn,p,ε∥Js+(p)+εf∥Lp(Rn)\|U(T)f\|_{L^p(\mathbb R^n\times I)}\le C_{n,p,\varepsilon}\|J^{s_+(p)+\varepsilon}f\|_{L^p(\mathbb R^n)}, i.e. it loses at most s+(p)+εs_+(p)+\varepsilon derivatives in LpL^p over space-time. At p=2p=2 the ε\varepsilon can be taken to be 00.

Why it's here

This is the Euclidean local smoothing theorem in Sobolev form, obtained from the single-annulus estimate of the main theorem, which loses only a logarithm in NN. The ε\varepsilon of room in the exponent absorbs that logarithmic loss when summing over dyadic frequencies. The result then feeds the later maximal and square-function estimates for thin shells and intervals of radii.
Classified routine:
Follows from the single-annulus estimate (thm-main) by Littlewood–Paley decomposition, summing dyadic pieces with the ε absorbing the log loss, and interpolation with the trivial p=2 and p=∞ bounds; this is the local smoothing conjecture of Sogge, known in this form from Guth–Wang–Zhang (2020) for n=2.

Uses

thmLocal smoothing on one frequency annulus

Used by

lemMaximal bound for thin shell multiplierslemThin-shell multiplier bounds on L^p and L^2

Sources

PaperOpen
Section
§1 Introduction
Page
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Label
thm:local-smoothing
Line
sections/introduction.tex:16
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Friction

  • info
    inferred_dep
    critique: The proof (transfer.tex:100-110) rests on , which transfer.tex:88-94 obtains by "Interpolating Theorem  with the two bounds above". Introduction.tex:106-107 says the same: "Interpolation and summation over annuli yield Theorem ".