该【an optimal multiplier theorem for grushin operators in the plane, ii 2022 gian maria dall’ara 】是由【宝钗文档】上传分享，文档一共【21】页，该文档可以免费在线阅读，需要了解更多关于【an optimal multiplier theorem for grushin operators in the plane, ii 2022 gian maria dall’ara 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:.
ANOPTIMALMULTIPLIERTHEOREMFORGRUSHIN
OPERATORSINTHEPLANE,II
GIANMARIADALL’ARAANDALESSIOMARTINI
Mihlin–H¨ormandertypefortwo-dimensionalGrushinoperators−∂x2−V(x)∂y2,
whereVisadoublingsingle-wellpotential,yieldingthesurprisingresultthat
theoptimalsmoothnessrequirementonthemultiplierisindependentofV.
Herewereflnethisresult,byreplacingtheL∞Sobolevconditiononthemul-
,weobtainthesharp
rangeofL1boundednessfortheassociatedBochner–
newingredientoftheproofisaprecisepointwiseestimateinthetransition
regionforeigenfunctionsofone-dimensionalSchr¨odingeroperatorswithdou-
blingsingle-wellpotentials.
,begunin
[DM21],oftwo-dimensionalGrushinoperators
L=−∂2−V(x)∂2,()
xy
whereV:R→[0,∞)isa“single-wellpotential”satisfyingascale-invariantreg-
ularityconditionoforder1+,weassumethatViscontinuous,
notidenticallyzero,C1offtheorigin,andthat,forsomeθ∈(0,1),theestimates
V(−x)≃V(x)≃xV′(x),()
|V′(xeh)−V′(x)|.|V′(x)||h|θ()
holdforallx∈R\{0}andh∈[−1,1].
todenotetheestimateA≤CBforsomepositiveconstantC,andA≃Btodenote
;≃sBto
introductionof[DM21]foradiscussionofthescopeoftheassumptions();here
welimitourselvestopointingoutthattheyaresatisfledbypowerlawsV(x)=|x|d
ofanydegreed>0andappropriateperturbationsthereof.
In[DM21]weprovedaspectralmultipliertheoremofMihlin–H¨ormandertype
arXiv:[]11Oct2021forL,whosesmoothnessrequirementisindependentofVandformulatedinterms
ofanL∞Sobolevnormoforders>2/2,thatis,halfthetopologicaldimensionof
isparticularlystrikingwhencompared,.,totheclassicalresultsbasedonheat
kernelbounds[Heb95,DOS02,RS08],whichwouldgiveinsteadaconditionof
orders>(2+d/2)/2inthecaseV(x)≃|x|
[DM21]foranextensivediscussionoftherelevanceofsuchresult,inthecontextof
aprogramme(seealso[MMN21])aimedatunderstandingtheoptimalsmoothness
requirementinmultipliertheoremsforsub-ellipticoperators.
,35J70,35H20,42B15.
,spectralmultiplier,Schr¨odingeroperator.
TheauthorsaremembersoftheGruppoNazionaleperl’AnalisiMatematica,laProbabilit`ae
leloroApplicazioni(GNAMPA)oftheIstitutoNazionalediAltaMatematica(INdAM).
1:.
2GIANMARIADALL’ARAANDALESSIOMARTINI
Whiletheresultof[DM21]isoptimal,inthesensethatthesmoothnessthreshold
2/2cannotbelowered,itisstillpossibletoreflneit,byreplacingtheL∞Sobolev
writeLq(R)todenotetheLqSobolevspaceof(fractional)ordersonR.
s
()onR2,wherethecoefficientV
satisflestheestimates().Lets>2/2.
(i)Forallm:R→Csuchthatsuppm⊆[−1,1],
supkm(tL)kL1→.
t>0
(ii)Letη∈C∞((0,∞)):R→Candp∈(1,∞),
c
km(L)kL1→L1,∞.ssupkm(t·)ηkL2s2s,km(L)kLp→,psupkm(t·)ηkL2s2s.
t>0t>0
Toappreciatethenatureoftheimprovement,
givesthesharpL1boundednessrangeforBochner–Rieszmeansassociatedwiththe
GrushinoperatorL,aresultthatcannotbededucedfromthemultipliertheorem
of[DM21].
,theBochner–Riesz
means(1−rL)λoforderλassociatedwithLareboundedonL1(R2)uniformlyin
+
r≥0wheneverλ>1/2.
“trans-
plantation”technique(cf.[Mit74,KST82];seealso[Mar17,]).Indeed
Liselliptic(itsprincipalsymbolisapositivedeflnitequadraticform)wherex6=0,
andthereforetherangesofindicessandλforwhichtheboundednessresultsin
whenLisreplacedbytheEuclideanLaplacian−∂2−∂2onR2.
xy
,
thecaseV(x)=x2isin[MS12,MM14]andthecaseV(x)=|x|isin[CS13].
Moreover,inapreviousjointpaper[DM20],weestablishedthesameresultwhen
Visconvex,C3offtheorigin,and,forsomed∈(1,2],theestimates
|x2V′′(x)|+|x3V′′′(x)|.xV′(x)≃V(x)=V(−x)≃|x|d
holdforallx∈R\{0}.Thisappearstohavebeentheflrstoptimalmultiplier
theoremforanonelliptic(sub-elliptic)operatorenjoyingsomeformofstability
,therestrictionon
thepowerdcannotberemovedusingthemethodsof[DM20],andthedesireto
overcomethislimitationhasbeenthemainmotivationforthedevelopmentofanew
proofstrategyin[DM21]
works[RS08,MS12,MM14,CS13,DM20]treatalsohigher-dimensionalcases,and,
asamatteroffact,somehigher-dimensionalcasescouldbetreatedbyadaptingthe
,inthesamespiritasin[DM21],hereweconsider
onlytwo-dimensionalGrushinoperators.
,
itisconvenienttorecallthenotationfortheclassesofsingle-wellpotentialsde-
flnedin[DM21,],whichexpresstheassumptions()ina
quantitativeform.
≥1andθ∈(0,1).WedenotebyP1(κ)theclassofnon-
identicallyzerocontinuousfunctionsV:R→[0,∞)whichareC1offtheorigin
andsuchthat
κ−1V(x)≤xV′(x)≤κV(x),V(−x)≤κV(x):.
GRUSHINOPERATORSINTHEPLANE3
forallx6=+θ(κ)theclassoftheV∈P1(κ)thatsatisfythe
additionalinequality
|V′(ehx)−V′(x)|≤κ|h|θ
forallx6=0andh∈[−1,1].
Asinotherworksonthesubject,-
propriate“weightedPlancherelestimate”.Inthepresentcase,inlightof[DM21,
],itwillbeenoughtoprovethatforallV∈P1+θ(κ),γ∈[0,1/2),
r>0,andallcontinuousfunctionsm:R→Cwithsuppm⊆[1/4,1],
Z
2−2γ′1/2−γ′2γ ′ 2
esssuprmax{V(r),V(x)}|y−y|Km(r2L)(z,z)dz
z′∈R2R2
.kmk22.()
θ,κ,γLγ
Herez=(x,y)andz′=(x′,y′),whileKm(r2L)denotestheintegralkernelof
theoperatorm(r2L).Indeed,theestimate()provesassumption(A)of[DM21,
]forq=2,whileassumption(B)isalreadyprovedin[DM21,Theorem
].Wepointoutthat,inthespecialcaseV(x)=x2,theaboveestimateis
provedin[MM14],whilethetechniquesof[MS12,CS13,DM20]leadtoadifferent
Plancherelestimate,withaweightdependingonlyonx,x′inplaceof|y−y′|2γand
L2inplaceofL2intheright-handside.
γ
OurproofoftheweightedPlancherelestimate()largelyfollowsthelinesof
theanalogousestimateprovedin[DM21,],withtheadditionofakey
newingredient:universalpointwiseestimatesforeigenfunctionsofone-dimensional
Schr¨odingeroperatorswithpotentialsintheclassP1+θ(κ).Asin[DM21,Section
7],weconsidertheSchr¨odingeroperatorH[V]:=−∂2+VonRwithpotential
x
V∈P1+θ(κ),andwedenotebyEn(V)andψn(·;V)(n≥1)thecorresponding
needherehavetheform
|ψ(x;V)|.|{V≤E(V)}|−1/2min{nδ/2,E(V)β/2|V(x)−E(V)|−β/2}()
nnnn
forsomeδ,β∈(0,1),andtheyhavethecrucialfeaturethattheimplicitconstant
“universality”of
anestimatesuchas()liesinthefactthattheright-handsideissimplyexpressed
intermsofnaturalquantitiessuchasV,En(V),nanduniversalexponentsδ,β,and
doesnotdepend,.,onthedegreeofpolynomialgrowthofV.
IntheregionswhereV≪En(V)andV≫En(V),theestimate()isalready
containedinestimatesprovedin[DM21],whichactuallyholdforallV∈P1(κ).
Whatiscrucialforourpresentpurposesisthat()alsocoversthe“transition
region”{V≃En(V)},wheretheeigenfunctionψn(·;V)exhibitsachangeinbe-
themoregeneralproblemofapproximatingeigenfunctionsinthetransitionregion
(.,Olver’smethod[Olv74,Chapter11]andtheWKBmethod,bothyielding
approximationsintermsoftheAiryfunction),butitdoesnotseempossibletouse
anyofthemasablackboxtoprove()
usedhereisinfactsubstantiallydifferentandofamoredirectnature,establishing
theupperbound()viaamonotonicityargumentinspiredbywhatisdubbed
the“Sonin’sfunction”methodin[Kr08],whichinturnrefersitbacktothework
ofSzeg˝oonorthogonalpolynomials[Sz75,§§].
Theimportanceofthepointwiseestimate()isthatfromitonecandeduce
avariantofthe“spectralprojectorbound”provedin[DM21,],which
playsafundamentalroleintheproofoftheweightedPlancherelestimate().
Speciflcally,thedesiredspectralprojectorbound()isobtained:.
4GIANMARIADALL’ARAANDALESSIOMARTINI
bysumminginstancesoftheeigenfunctionestimate()correspondingtodifferent
valuesofnandsuitablyscaledversionsτVofthepotentialV,wherethescaling
,anotherimportant
ingredientisanapproximatedBohr–Sommerfeldidentitywithlogarithmicerror
term()validforSchr¨odingeroperatorswithpotentialsin
theclassP1(κ),whichprovidespreciseinformationonthe“gaps”betweenthe
quantitiesEn(τV)involvedintheestimate.
inaconditionalform,namely,byassumingthatsuitablepointwiseeigenfunction
estimatesoftheform()hold.
Section3isdevotedtotheproofoftherequiredpointwiseeigenfunctionesti-
,suitablepointwiseestimatescanbeproved
foralargerclassthanP1+θ(κ).Indeed,severalvariantsoftheaboveeigenfunction
estimates()arediscussed,whichmaybeofindependentinterest,withdifferent
valuesofδandβcorrespondingtodifferentassumptionsonthepotentialV.
Finally,inSection4,weprovetheweightedPlancherelestimate()withL2
Sobolevnorm,which,inlightof[DM21,],impliesourmainresult.
+=
(0,∞)andR+=[0,∞).Ndenotesthesetofnaturalnumbers(includingzero),
0
whileN+=N\{0}
V,wewriteV←todenoteitscompositionalinverse.#Idenotesthenumberof
⊆Rwedenoteby|A|its
.
,letEn(V)andψn(·;V)(n≥1)be
theeigenvaluesandnormalisedeigenfunctionsoftheSchr¨odingeroperatorH[V]=
−∂2+“virial
x
theorem”in[DM21,].UndermorerestrictiveassumptionsonV,
analogousestimatescanbefoundin[DM20,eq.()].
∈P1(κ)andn∈N+.Thenthefunction
R+∋τ7→E(τV)∈R+
n
isastrictlyincreasing,realanalyticbijection,and
En(τV).κτ∂τEn(τV)≤En(τV)
forallτ∈R+.Moreover,ifΞn(·;V):R+→R+denotesitsinverse,then
Ξn(λ;V)≃κλ∂λΞn(λ;V).
forallλ∈R+.
Theaimofthissectionistheproofofthefollowingbound,whichshouldbe
comparedtothe“spectralprojectorbound”of[DM21,].
,a>1andθ,δ∈(0,1).LetP˜beasubconeofP1(κ)such
thattheeigenfunctionestimate
|ψn(x;V)|≤a|{V≤En(V)}|−1/2min{nδ/2,En(V)θ/2|V(x)−En(V)|−θ/2}
holdsforallV∈P˜,n∈N+,andx∈,forallV∈P˜andλ,A∈R+,
X1/2
ψn(x;Ξn(λ;V)V),a,θ,δλ1/2(1V≤8A+e−cλ|x|1V>8A),
n∈N+
λ/Ξn(λ;V)∈[A,2A]
wherec=c(κ).:.
GRUSHINOPERATORSINTHEPLANE5
Themaindifferencebetweenthepreviousresultand[DM21,]isthat
theabovesuminvolveseigenfunctionscorrespondingtodifferentpotentials(that
is,potentialsτVwhereτdependsonthesummationindexn),socannotbeimme-
diatelyrelatedtopropertiesofthespectraldecompositionofasingleSchr¨odinger
[DM20,],undermore
restrictiveassumptionsonV.
.
akeytoolintheproofofthespectralprojectorbound.
∈R+,κ∈[1,∞),θ,β∈[0,1).LetI⊆N+and,foralln∈I,
lettn∈[κ−1,∞)besuchthat
|t−cn|≤κnβ.()
n
ThenX
supmin{aθ−1|t−b|−θ,a−β}.1.()
nκ,c,θ,β
a>0n∈I
0<b≤κatn≤κa
an optimal multiplier theorem for grushin operators in the plane, ii 2022 gian maria dall’ara 来自淘豆网www.taodocs.com转载请标明出处.