Pseudodifferential operators with rough symbols

0.28 MB·English

by  Atanas Stefanov

PSEUDODIFFERENTIAL OPERATORS WITH ROUGH SYMBOLS
ATANASSTEFANOV
7
0
0 ABSTRACT. In this work, we developLp boundednesstheoryfor pseudodifferentialop-
2 eratorswithrough(notevencontinuousingeneral)symbolsinthexvariable. Moreover,
n the B(Lp) operator norms are estimated explicitly in terms of scale invariant quantities
a involving the symbols. All the estimates are shown to be sharp with respect to the re-
J quiredsmoothnessin theξ variable. Asa corollary,we obtainLp boundsfor(smoothed

9 outversionsof)themaximaldirectionalHilberttransformandtheCarlesonoperator.
2

]
A
C 1. INTRODUCTION
.
h Inthispaper,weareconcernedwiththeLp mappingpropertiesofthepseudodifferential
t
a operators intheform
m
[ (1) T f(x) = σ(x,ξ)e2πiξxfˆ(ξ)dξ.
σ

1 RZn
v
6 The operators Tσ have been subject of continuous interest since the sixties. We should
5 mention that their usefullness in the study of partial differential equations have been re-
8
alized much earlier, but it seems that their systematic study began with the fundamental
1
0 works ofKohnandNirenberg, [10]and Ho¨rmander,[9].
7 To describetheresultsobtained intheseearly papers, definetheHo¨rmander’sclass Sm,
0
/ which consistsofall functionsσ(x,ξ), sothat
h
t (2) |DβDασ(x,ξ)| ≤ C (1+|ξ|)m−|α|.
a x ξ α,β
m
forallmultiindicesα,β. Aclassicaltheoremin[9]thenstatesthatOp(σ) : Hs+m,p → Hs,p
:
v for all s ≥ 0 and 1 < p < ∞. In particular, Op(σ) : Lp → Lp, 1 < p < ∞, wheneverthe
i
X symbolσ ∈ Sm. Subsequentimprovementsofthesemethodsestablishedtheboundedness
r of Op(σ) (basically under the assumption σ ∈ Sm for appropriate m) to various related
a
functionspaces,likeBesov,Triebel-Lizorkinspacestonameafew,butwewillnotreview
thosehere, sincetheyfalloutsideofthescopeofthispaper.
It isworthmentioninghowever,thatthesimpletoverifycondition(2)istheonearising
in manyapplications. TheL2 boundednessplaysspecial rolein thetheoryand thatis why
we discussit separately.
TheclassofsymbolsSm , defined via
ρ,δ
(3) |DβDασ(x,ξ)| ≤ C (1+|ξ|)m−ρ|α|+δ|β|.
x ξ α,β
Date:February2,2008.
2000MathematicsSubjectClassification. 35S05,47G30.
Keywordsandphrases. Pseudodifferentialoperators,Lpbounds,orthogonality.
Supportedinpartbynsf-dms0300511.
1
2 ATANASSTEFANOV
represents a larger set of symbolsthan Sm = Sm , which has subsequently found applica-
1,0
tionsin localsolvabilityforlinearPDE’s, [1].
Here, we have to mention the celbrated result of Caldero´n-Vaillancourt, [3], [4] which
states that L2 boundedness for T holds, whenever σ ∈ S0 , 0 ≤ ρ < 1, whereas S0
σ ρ,ρ 1,1
is a “forbidden” class, in the sense that there are symbols in that class, which give rise
to unbounded on L2 operators. We should mention here the work of Cordes [6], who
improvedtheresultforS0 by requiringthat(3) holdsonlyfor|α|,|β| ≤ [n/2]+1.
0,0
Regarding less regularin x symbols,for any modulusof continuityω : R → R (that
+ +
is,anincreasingandcontinuousfunction),definethespaceCω ofalluniformlycontinuous
and boundedfunctionsu : Rn → C, satisfying
|u(x+y)−u(x)| ≤ ω(|y|).
The followingclassof symbolswas introducedand studiedby Coifman-Meyer, [5]. More
precisely,let σ(x,ξ) ∈ CωS0 , whichmeansit satisfies
1,0
sup < |ξ| >|α| |Dα[σ(x+y,ξ)−σ(x,ξ)]| ≤ C ω(|y|)
ξ α
x,y,ξ
and assume that ω(2−j)2 < ∞. Then for all 1 < p < ∞, Op(σ) : Lp → Lp. The
j>0
condition ω(2−j)2 < ∞ is clearly very mild continuity assumption for the function
j P
x → σ(x,ξ). In particular, one sees that ∪ CγS0 ⊂ CωS0 . Related results can be
P γ>0 1,0
found in the work of M. Taylor, [23] (see Proposition 2.4, p. 23) and J. Marschall, [11]
where the spaces Cω are replaced by Hε,p spaces with p as large as onewish and 0 < ε =
ε(p) << 1 (seealso[23], p. 61)
One of the purposes of this work is to get away from the continuity requirements on
x → σ(x,ξ). Even more importantly, we would like to replace the pointwise conditions
on the derivatives of ξ by averaged ones. This particular point has not been thoroughly
exploredappropriately intheliteraturein theauthor’sopinion,see Theorem1 below.
Ontheotherhand,aparticularmotivationforsuchconsiderationsisprovidedbythere-
cent papers of Rodnianski-Tao [14] and the author [17], where concrete parametrices (i.e.
pseudodifferential operators, representing approximate solutions to certain PDE’s) were
constructedforthesolutionsofcertainfirstorderperturbationofthewaveandSchro¨dinger
equations. A very quick inspection of these examples shows that1 they do not obey point-
wiseconditionsonthederivativesonthesymbolsand thus,thesemethodsfailtoimplyL2
bounds for these (and related problems). Moreover, one often times has to deal with the
situation, where the maps ξ → σ(x,ξ) are not smooth in a pointwise sense. On the other
hand, onemaystillbeableto controlaveraged quantitieslike
(4) supkσ(x,ξ)k < ∞.
Hn/2
x ξ
ThiswillbeourtresholdconditionforL2 boundedness,whichwetry toachieve.
Heuristicallyatleast,(4)mustbe“enough”insomesense,sinceifwehadsimplesymbols
like σ(x,ξ) = σ (x)σ (ξ), then the L2 boundedness of Op(σ) is equivalent to kσ k <
1 2 1 L∞
x
1Mostreadersarelikelytohavetheirownfairlylonglistwithfavoriteexamples,forwhichtheHo¨rmander
conditionfails.
PSEUDODIFFERENTIALOPERATORSWITHROUGHSYMBOLS 3
∞,kσ k < ∞. Clearly, kσ k just fails to be controlled by (4), but on the other
2 L∞ 2 L∞(Rn)
ξ ξ
hand, thequanitityin(4)is controlledby theappropriateBesov spaceBn/2 norm.
2,1
A final motivation for the current study is to achieve a scale invariant condition, which
gives an estimateof the L2 → L2 (Lp → Lp) norm of Op(σ) in terms of a scale invariant
quantity,thatis,weaim at showingan estimate,
kOp(σ)k ≤ Ckσk kfk ,
Lp→Lp Y Lp
where foreveryλ 6= 0, onehas kσ(λ·,λ−1·)k = kσk .
Y Y
Inthatregard,notethatthecondition(whichisoneoftherequirementsoftheHo¨rmander
class S0)
(5) sup|Dασ(x,ξ)| ≤ C |ξ|−|α|
ξ α
x
is scale invariant in the sense described above. Moreover, by the standard Caldero´n-
Zygmund theory (see [21]), the pointwise condition (5) together with kT k < ∞
σ L2→L2
implies
T f(x) = K(x,x−y)f(y)dy,
σ
Z
whereK(x,·)satisfiestheHo¨rmander-Mihlinconditions,namely|K(x,z)| ≤ C|z|−n and
|∇ K(x,z)| ≤ C|z|−n−1, where the constant C depends on the constants C : |α| <
z α
[n/2]+1 in (5). This in turn is enough to conclude that T : Lp → Lp for all 1 < p ≤ 2
σ
and in fact thereistheendpointestimateT : L1 → L1,∞.
σ
1.1. Lp estimates for PDO with rough symbols - statement of results. We start now
with our main theorems, which concern the L2 and the Lp boundedness for pseudodiffer-
ential operators Op(σ) withrough symbols. Ourfirst result establishesthat aBesov space
versionof(4)isenough forL2 boundednessand theresult issharp.
Theorem 1. (L2 bounds) Let σ(x,ξ) : Rn ×Rn → C and T is the corresponding pseu-
σ
dodifferentialoperator. Then
(6) kT k ≤ C( 2ln/2supkPξσ(x,·)k ),
σ L2(Rn)→L2(Rn) l L2(Rn)
x
l
X
where Pξ is theLittlewood-Paleyoperatorin theξ variable.
l
Moreover,theresultissharpinthefollowingsense: foreveryp > 2,thereexistsσ(x,ξ)
so that sup |Dασ(x,ξ)| ≤ C |ξ|−|α| and sup kσ(x,·)k < ∞, but T fails to be
x ξ α x Wp,n/p σ
boundedon L2(Rn).
Remark:
(1) Notethattheestimateon T isscaleinvariant.
σ
(2) The sharpness claim of the theorem, roughly speaking, shows that in the scale of
spaces2 Wp,n/p, ∞ ≥ p ≥ 2, one may not require anything less than W2,n/2 =
Hn/2 ofthesymbolin ordertoensureL2 boundedness.
2NotethatthesespacesscalethesameandmoreoverbySobolevembeddingthesearestrictlydecreasing
sequence,atleastfor2≤p<∞.
4 ATANASSTEFANOV
(3) The counterexample to which we refer in Theorem 1 is a simple variation of the
well-known example of σ ∈ S0 , the “forbidden class”, which fails to be L2
1,1
bounded,see [21], p. 272and Section 6 below.
Ournextresult concernsLp boundednessforT .
σ
Theorem 2. (Lp bounds) For the pseudodifferential operator T there is the estimate for
σ
all 2 < p ≤ ∞,
(7) kT k ≤ C( 2ln/2supkPξσ(x,·)k ),
σ Lp(Rn)→Lp(Rn) l L2(Rn)
x
l
X
Fortherange1 < p < 2 andindeed fortheweak type(1,1),thereis
(8) kT k +kT k ≤ C 2lnsupkPξσ(x,·)k .
σ Lp→Lp σ L1→L1,∞ l L1(Rn)
x
l
X
Alternatively, if one assumes the L2 bound, together with (5), one still gets Lp → Lp,
1 < p ≤ 2, andin factweak type(1,1)bounds. Moreover,
kT k ≤ C( 2ln/2supkPξσ(x,·)k + sup sup|ξ||α||Dασ(x,ξ)|).
σ Lp→Lp l L2(Rn) ξ
x |α|<[n/2]+1 x,ξ
l
X
As we pointedout in Theorem 1, theestimates are essentiallysharp for Lp, 2 ≤ p < ∞
boundedness. The following corollary gives even more precise condition under which a
symbolσ willgiverisetoaboundedoperatoron Lq in thecase ofa given1 < q < 2
Corollary 1. Let 1 < q < 2. Then
kTσkLq→Lq ≤ C 2ln/qsupkPlξσ(x,·)kLq(Rn).
x
l
X
Clearly the proof follows by interpolation from the L2 estimates in Theorem 1 and the
weak type(1,1)estimatesofTheorem2.
1.2. PDO’s with homogeneous of degree zero symbols - statement of results. Re-
garding symbols that are homogeneous of degree zero, i.e. σ(x,ξ) = q(x,ξ/|ξ), where
q : Rn ×Sn−1 → C, weobtainmorepreciseresultsin termsofthesmoothnessofq.
NotethattheclassicalHo¨rmanderconditionrequirespointwisesmoothnessofthefunction
q in bothvariables. Ourresult ontheotherhand requires muchlessthan that.
Theorem 3. (Lp boundsforhomogeneousofdegree zerosymbols)
Let q : Rn ×Sn−1 → C. Let
T f(x) = q(x,ξ/|ξ|)e2πiξxfˆ(ξ)dξ.
q
RZn
Then Tq : L2 → L2, if l2l(n−1)/2kPlξ/|ξqkL2(Sn−1) < ∞ andinfact
(9) kTqkLP2→L2 ≤ C 2l(n−1)/2supkPlξ/|ξ|q(x,·)kL2(Sn−1).
x
l
X
PSEUDODIFFERENTIALOPERATORSWITHROUGHSYMBOLS 5
Concerning Lp bounds,we haveforevery2 ≤ p ≤ ∞.
(10) kTqkB0 →Lp ≤ Cn( 2l(n−1)/p′ supkPlξ/|ξ|qkLp′(Sn−1)).
p,1 x
l
X
Note thatin (10), theconstantC isindependentofp,r.
n
Remark:
(1) ItwouldbeinterestingtoseewhethertheusualLp → Lp boundednessholdstrue.
(2) Note that there is no weak type (1,1) statement in Theorem 3. This is a difficult
issueevenformultipliers.
Thesharpness statementassociated withTheorem3 is
Proposition 1. For every N > 1, there exists a homogeneous of degree zero symbol
σ(x,ξ) : R2 ×R2 → R1, so that sup |σ(x,ξ)| < ∞ and sup kσ(x,ξ)k < ∞,
x,ξ x W1,1(S1)
and sothatkT k > N.
σ L2→L2
The counterexample considered here is a smoothed out version of the maximal direc-
tional Hilbert transform in the plane H f(x) = sup |H f(x)|. We mention the spec-
∗ u∈S1 u
tacular recent result of Lacey and Li, [12] showing the boundedness of H on Lp(R2) :
∗
2 < p < ∞ with a H : L2(R2) → L2,∞(R2) as an endpoint estimate. Note that the
∗
L2 → L2 bound fails, as elementary examples show, see [12]. We verify later that the
condition (10) just fails for the (smoothed out) multiplier σ of H in two dimensions, but
∗
on theotherhand theconditionkσ(x,ξ)k < ∞ holds.
W1,1(S1)
ThisexamplewillshowthattheBesovspaces requirementsforσ in(9)and(10)cannot
bereplaced bySobolevspaces and/orspaces withlessderivatives.
1.3. PDO’swithradialsymbols. Finally,weconsiderthecaseofradialsymbols. Thatis
for ρ : Rn ×R1 → C and
+
T f(x) = ρ(x,|ξ|)e2πiξxfˆ(ξ)dξ.
ρ
Rn
Z
Theorem 4. TheoperatorT : L2 → L2, if 2l/2supkP|ξ|ρ(x,·)k < ∞. In fact,
ρ l l L2(R1)
x
P
kT k ≤ C 2l/2supkP|ξ|ρ(x,·)k .
ρ L2→L2 l L2(R1)
x
l
X
Clearly,establishingLp,p 6= 2boundsforsimpleradialsymbolsisalreadyanotoriously
difficult problem. One only needs to point out to the Bochner-Riez multiplier(1−|ξ|2)δ
+
(which satisfy Lp bounds only in certain range of p′s, depending on the dimension and δ)
or even the simpler “thin annulus” multiplier ϕ(2m(1−|ξ|2)) to understand the difficulty
oftheproblemingeneral.
6 ATANASSTEFANOV
2. APPLICATIONS
In this section, we demonstrate the effectiveness of the Lp boundedness theorems for
rough PDO’s. Wewill mostlyconcentrate on applicationtomaximalfunctionsand opera-
tors3Someofourexampleswillbewell-knownresultsformaximaloperators,whileothers
willbeahigherdimensionalextensionsofsuchresults.
2.1. AlmosteverywhereconvergenceforCesarosumsofLp functionsin1D. Westart
withCesaro’s sumfor Fourierseries inonespacedimension. Foranyδ > 0,define
C f(x) = sup (1−ξ2/u2)δ e2πiξxfˆ(ξ)dξ.
δ +
u>0 R1
Z
Clearly,asalimitasδ → 0,wegettheCarleson’soperator. Unfortunately,onecannotcon-
clude that sup kC k < ∞, for that would imply the famous Carleson-Hunt theorem.
δ>0 δ Lp
On theotherhand, definethemaximal”thin intervaloperator”
T f(x) = sup ϕ(2m(1−ξ2/u2))e2πiξxfˆ(ξ)dξ.
m
u>0 R1
Z
A simpleargumentbased on (theproofof)Theorem 2yields
Proposition2. Foranyε > 0,1 < p < ∞,thereexistsC , sothat
p,ε
(11) supkT k ≤ C .
m Lp(R1)→Lp(R1) p
m
In fact, there is the more general pointwise bound sup|T f(x)| ≤ CM(sup |P f|)(x),
m k k
m
which implies(11)as well as
(12) kC k +kC k ≤ C .
δ Lp→Lp δ F10,∞(R1)→L1,∞(R1) δ,p
Remark:
• Note that this result, while clearly inferior to the Carleson-Hunt theorem still im-
pliesa.e. convergenceforanyCesarosummabilitymethod,whenappliedtoLp(R1)
functions,and infact forthelarger classofF0 functions.
p,∞
• Usingthemethodofproofhere,onemayactuallyproveL2 estimates4forthemax-
imalBochner-Rieszoperator
BR f(x) = sup (1−|ξ|2/u2)δ e2πiξxfˆ(ξ)dξ.
δ +
u>0 Rn
Z
inanydimension.
Proof. It clearly suffices to show the pointiwise estimate |T f(x)| ≤ CM(sup P f)(x)
m k k
foranyk. ThestatementsaboutB0 → Lp boundsfollowbyelementaryLittlewood-Paley
p,1
theory and the lp bounds for the Hardy-Littlewood maximal function. The restricted-to-
weak estimateF0 → L1,∞ for C follows by summingan exponentiallydecaying series
1,∞ δ
in thequasi-Banach spaceL1,∞.
3Inaddition,theauthorhasalsoidentifiedseveralapplicationstobilinear/multilinearoperatorsofimpor-
tancetocertaindispersivePDE’s,whichwillbeaddressedinafuturepublication.
4AndinfactLp →Lpestimatesforp=p(δ)closeto2.
PSEUDODIFFERENTIALOPERATORSWITHROUGHSYMBOLS 7
By supportconsiderations,it isclearthat
T f(x) = sup ϕ(2m(1−ξ2/u2))e2πiξxϕ(2−kξ)fˆ(ξ)dξ =
m
u>0 R1
k Z
X
= sup ϕ(2m(1−ξ2/u2))e2πiξxϕ(2−kξ)fˆ(ξ)dξ =
k u∈(2k−2,2k+2)ZR1
X
= T f .
m,u(·)∈(2k−2,2k+2) k
k
X
Clearly, the requirement u ∈ (2k−2,2k+2) creates (almost) disjointness in the x support,
whence
(13) |T f(x)| ≤ Csup|T f (x)|.
m m,u(·)∈(2k−2,2k+2) k
k
Ourbasicclaimis that
(14) |T f (x)| ≤ CM(f ).
m,u(·)∈(2k−2,2k+2) k k
Clearly (13) and(14)implysup |T f(x)| ≤ CM(sup|f |), whencetheProposition2.
m m k
k
By scaleinvariance,(14)reduces tothecasek = 0,that isweneed toshow
|T P f(x)| ≤ CM(P f)(x).
m,u(x)∈(1/4,4) 0 0
foranySchwartzfunctionf andanym >> 1. By(31)(intheproofofTheorem2below),
it willsufficetoshow
(15) 2lsupkPξ[ϕ(2m(1−ξ2/u(x)2))ϕ(ξ)]k . 1.
l L1(R1)
x
l
X
for anymeasurablefunctionu, whichtakes itsvaluesin(1/4,4).
For(15), wehave
2lsupkPξ[ϕ(2m(1−ξ2/u(x)2))ϕ(ξ)]k
l L1(R1)
x ξ
l<m
X
≤ 2l sup |ϕ(2m(1−ξ2/u2))ϕ(ξ)|dξ . 2l−m . 1,
l<m u∈(1/4,4)Z l<m
X X
while
2lsupkPξ[ϕ(2m(1−ξ2/u(x)2))ϕ(ξ)]k
l L1(R1)
x ξ
l≥m
X
d2
≤ 2−l sup | ϕ(2m(1−ξ2/u2))ϕ(ξ)|dξ ≤ C 2−l+m . 1.
dξ2
l≥m u∈(1/4,4)Z l≥m
X X
(cid:3)
8 ATANASSTEFANOV
2.2. Maximaldirectional Hilberttransformsandthe Kakeyamaximalfunction. An-
otherinterestingapplicationisprovidedbythedirectionalHilberttransformindimensions
n ≥ 2. Namely,take
H∗ = sup (hu,ξ/|ξ|i)δe2πiξxfˆ(ξ)ϕ(ξ)dξ,
δ +
u∈Sn−1Z
where suppϕ ⊂ {1/2 < |ξ| < 2}.
As δ → 0, we obtain the operator f → sup e2πiξxfˆ(ξ)dξ,which is closely
u∈Sn−1 {hu,ξi>0}
related to themaximaldirectionalHilberttransform
R
H f(x) = sup|H f(x)| = sup| sgn(u,ξ)e2πiξxfˆ(ξ)ϕ(ξ)dξ|.
∗ u
u u
Z
H was of course shown to be Lp(R2),p > 2 bounded by Lacey and Li, [12] by very
∗
sophisticatedtime-frequencyanalysismethods.
Proposition3. Forthe“thinbig circle”multiplier
T f(x) = sup | ϕ(2mhu,ξ/|ξ|i)e2πiξxfˆ(ξ)ϕ(ξ)dξ|.
m
u∈Sn−1 ZRn
we have
(16) kT fk ≤ C 2m(n/2−1)
m L2→L2 ε
In particular
kH∗k ≤ C 2n/2−1.
δ L2(Rn)→L2(Rn) p,ε,δ
Remark:
• We believe that the operator T (m >> 1) has a particular connection to the
m
Kakeya maximal function and the corresponding Kakeya problem. Indeed, the
kernel of the corresponding singular integral behaves like a (L1 normalized) char-
acteristic function of a rectangle with long side along u of length 2m and (n−1)
shortsidesoflength1in thetransversedirections!
• Inrelation tothat,oneexpectstheconjectured Kakeyabounds
kT fk ≤ C 2m(n/p−1)
m Lp→Lp ε
forp ≤ n tohold,whileoneonlygets
kT fk ≤ C 2m(n/p−2/p)
m Lp→Lp ε
as a consequence of Theorem 3. Nevertheless, the two match when p = 2. So it
seems that (16), at least in principle, captures the Kakeya conjecture for p = 2 in
general andin particularthefullKakeyaconjectureintwo dimensions.
Sinceourestimatesdonotseemtocontributemuchtowardtheresolutionofany
new Kakeya estimates, we do not pursue here the exact relationship between T
m
and the Kakeya maximaloperator, although from our heuristicarguments above it
shouldbeclear thatitis acloseone.
PSEUDODIFFERENTIALOPERATORSWITHROUGHSYMBOLS 9
Proof. Weproceed as in theproofofProposition2. Weneed onlyshow
(17) 2l(n−1)/2supkPlξ/|ξ|ϕ(2mhu(x),ξ/|ξ|i)kL2(Sn−1) . 1.
x
l
X
We have
2l(n−1)/2supkPlξ/|ξ|ϕ(2mhu(x),ξ/|ξ|i)kL2(Sn−1)
x
l<m
X
≤ C 2l(n−1)/2supkϕ(2mhu(x),ξ/|ξ|i)kL2(Sn−1)
u
l<m
X
≤ C 2l(n−1)/22−m/2 . 2m(n/2−1).
l<m
X
2l(n−1)/2supkPlξ/|ξ|ϕ(2mhu(x),ξ/|ξ|i)kL2(Sn−1)
x
l≥m
X
C ≤ 2−l(n−1)/2supkΩ(n−1)ϕ(2mhu(x),ξ/|ξ|i)kL2(Sn−1)
u
l≥m
X
≤ C 2−l(n−1)/22m(n−1)2−m/2 . 2m(n/2−1).
l≥m
X
(cid:3)
2.3. Estimates onT , T etc. We now present a result, which allows us to treat pseu-
σ1σ2 eσ
dodifferential operators, whosesymbolsare products, exponentials(or moregenerally en-
tire functions) of symbols, which satisfy the requirements in Theorems 1, 2, 3. We would
liketopointoutthatsimilarinspirit(byessentiallyrequiringn/2+εderivativesinL2,but
in amoregeneral setting)functionalcalculus typeresult wasobtainedin [16].
Proposition 4. Let σ,σ ,σ : Rn × Rn → C, so that T ,j = 1,2 define Lp bounded
1 2 σj
operators,as in(6), (8). Then forevery1 < p < ∞,T isLp boundedand
σ1σ2
2
(18) kT k ≤ C ( 2ln/2supkPξσ (x,·)k ),
σ1σ2 L2(Rn)→L2(Rn) l j L2(Rn)
x
j=1 l
Y X
2
(19) kT k ≤ C ( 2lnsupkPξσ (x,·)k ).
σ1σ2 Lp(Rn)→Lp(Rn) l j L1(Rn)
x
j=1 l
Y X
In thesamespirit,T is alsoLp boundedandthereis
eσ
(20) kT k ≤ Cexp( 2lnsupkPξσ(x,·)k )
eσ Lp(Rn)→Lp(Rn) l L1(Rn)
x
l
X
10 ATANASSTEFANOV
Similarstatementscan bemadeforhomogeneousofdegreezerosymbols
µ (x,ξ/|ξ|),µ (x,ξ/|ξ|),
1 1
2
(21) kT k ≤ C ( 2l(n−1)/2supkPξ/|ξ|µ (x,·)k ),
µ1µ2 L2(Rn)→L2(Rn) l j L1(Rn)
x
j=1 l
Y X
(22) kT k ≤ Cexp( 2l(n−1)supkPξ/|ξ|µ(x,·)k ).
eµ L2(Rn)→L2(Rn) l L1(Rn)
x
l
X
TheproofofProposition4isbased onthecorrespondingTheoremforLp boundedness,
combined withthefact thatourrequirements form aBanach algebra underthemultiplica-
tion. Takeforexample(18). By Theorem 1,we have
kT k ≤ C( 2ln/2supkPξ(σ σ )(x,·)k )
σ1σ2 L2(Rn)→L2(Rn) l 1 2 L2(Rn)
x
l
X
We finishby invokingtheestimate
2
(23) 2ln/2supkPξ(σ σ )(x,·)k ≤ C ( 2ln/2supkPξσ (x,·)k ).
l 1 2 L2(Rn) l j L2(Rn)
x x
l j=1 l
X Y X
where thislastinequalityessentiallymeansthat Bn/2 is aBanach algebraoffunctions5.
2,1
The argument above can be performed for the proof of (19). For (20) (and more gener-
ally forany symbolsoftheform g(σ), whereg isentirefunction), oneiterates theproduct
estimate(23)to
(24) 2ln/2supkPξ(eσ)(x,·)k ≤ Cexp( 2ln/2supkPξσ(x,·)k ).
l L2(Rn) l L2(Rn)
x x
l l
X X
Fortheproofof(21), (22),onehas tousethefact thatB(n−1)/2(Sn−1) isaBanach algebra
2,1
as well, whenceonegetsan estimatesimilarto(23)and (24).
3. PRELIMINARIES
We startby introducingsomebasicconceptsin Fourieranalysis.
3.1. FourieranalysisonRn. First,define theFouriertransformand itsinverse
fˆ(ξ) = f(x)e−2πix·ξdx,
RZn
f(x) = fˆ(ξ)e2πix·ξdξ.
RZn
For a positive, smooth and even function χ : Rn → R1, supported in {ξ : |ξ| ≤ 2} and
+
so that χ(ξ) = 1 for all |ξ| ≤ 1. Define ϕ(ξ) = χ(ξ)− χ(2ξ), which is supported in the
annulus1/2 ≤ |ξ| ≤ 2. Clearly ϕ(2−kξ) = 1 forall6ξ 6= 0.
k∈Z
5This is well-known, but can be vePrified easily by means of the Kato-Ponce estimate k∂n/2(uv)kL2 ≤
C(k∂n/2ukL2kvkL∞+k∂n/2vkL2kukL∞),theembeddingB2n,/12 ֒→L∞andsomeLittlewood-Paleytheory.
6ThediscussionhenceforthwillbeforRn,unlessexplicitelyspecifiedotherwise.