claimroutineinferred_dep

Mixed-norm Besov local smoothing for Schrödinger

Gives mixed bound

L^p_xL^r_t

from

B^s_{p,p}

with

s=d(1-2/p)-2/r

exact-order Schrödinger smoothing of the corollary

Claim·§12 The paraboloid and Schr\"odinger evolution·inside corParaboloid extension and Schrödinger local smoothing

Statement (paper)

Let

p_c<p<\infty

and

p\le r\le\infty

, and let

s=d(1-2/p)-2/r

. Then for every Schwartz

f

\|e^{it\Delta}f\|_{L^p(\RR^d;L^r(I))}\lesssim_{d,p,r,I}\|f\|_{B^s_{p,p}(\RR^d)}\lesssim_{d,p,s}\|(1-\Delta)^{s/2}f\|_p

, where the first inequality follows from with

b=a

for some

p_c<a<p

, and the second from spatial Littlewood--Paley theory and

\|\cdot\|_{\ell^p}\le\|\cdot\|_{\ell^2}

for

p>2

In plain English

For

p_c<p<\infty

p\le r\le\infty

and

s=d(1-2/p)-2/r

, every Schwartz

f

satisfies

\|e^{it\Delta}f\|_{L^p(\RR^d;L^r(I))}\lesssim\|f\|_{B^s_{p,p}}\lesssim\|(1-\Delta)^{s/2}f\|_p

, with

I

a compact interval.

Why it's here

This is the final, exact-order Schrödinger local smoothing statement of the corollary, covering the spacetime (

r=p

) and maximal (

r=\infty

) cases. The Besov bound comes from the global mixed extension estimate with

b=a

for some

p_c<a<p

, applied to the frequency-localized pieces. The Sobolev bound follows from spatial Littlewood–Paley theory together with

\ell^p\subset\ell^2

for

p>2

Classified routine:

Follows from the preceding global paraboloid extension estimate by rescaling frequency-localized pieces, plus Littlewood–Paley theory and the embedding

\ell^2\subset\ell^p

; this is the standard passage from restriction/extension bounds to Schrödinger local smoothing (as in Rogers' work relating restriction to Schrödinger smoothing).

Uses

claimGlobal mixed paraboloid extension estimate

Used by

Nothing in the paper

Sources

PaperOpen

Section: §12 The paraboloid and Schr\"odinger evolution
Page: —
Label: —
Line: sections/parabolic.tex:131

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Friction

info
inferred_dep
outline: "Mixed-norm Besov local smoothing" rests on claim-cor-parabolic-2