Difference equations and pseudo-differential operators on ℤⁿ

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Difference equations and pseudo-differential operators on Z

LindaN.A. Botchway^a,P. Gaël Kibiti^a, Michael Ruzhansky^b,c,d,∗,1

aAfricanInstituteforMathematicalSciences,AIMS-GH,Biriwa,Ghana

bDepartmentofMathematics:Analysis,LogicandDiscreteMathematics,Ghent University,Belgium

cSchoolofMathematicalSciences,QueenMaryUniversityofLondon, United Kingdom

dDepartmentofMathematics,ImperialCollegeLondon,United Kingdom

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received22May2017 Accepted8January2020 Availableonline14January2020 CommunicatedbyL.Gross Dedicatedtothe85th birthdayof FrancisKofiAmpenyiAllotey MSC:

58J40 35S05 35S30 42B05 47G30

Keywords:

Pseudo-differentialoperators Calculus

Ellipticity

Inthispaper wedevelopthe calculus ofpseudo-differential operators on the lattice Zⁿ, which we can call pseudo- difference operators. Aninteresting feature of this calculus isthat the global frequency space (Tⁿ) is compactso the symbol classes are defined in terms of the behaviour with respect to the lattice variable. We establish formulae for composition, adjoint, transpose, and for parametrix for the ellipticoperators.Wealsogiveconditionsforthe²,weighted ²,and^pboundednessofoperatorsandfortheircompactness on^p.Wedescribealinktothetoroidalquantizationonthe torusTⁿ,andapplyittogiveconditionsforthemembership in Schatten classes on ²(Zⁿ). Furthermore, we discuss a versionof Fourierintegraloperators on thelatticeandgive conditionsfortheir ²-boundedness.Theresults areapplied togiveestimatesforsolutionstodifferenceequationsonthe latticeZⁿ.Moreover,weestablishGårdingandsharpGårding

* Correspondingauthor.

E-mailaddresses:[email protected](L.N.A. Botchway),[email protected](P. Gaël Kibiti), [email protected](M. Ruzhansky).

1 ThethirdauthorwassupportedinpartsbytheEPSRCgrantsEP/K039407/1,EP/R003025/1,bythe LeverhulmeGrantsRPG-2014-02,RPG-2017-151,andbytheFWOOdysseus1GrantG.0H94.18N:Analysis andPartialDifferentialEquations.

https://doi.org/10.1016/j.jfa.2020.108473

0022-1236/©2020TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

Differenceequations Fourierintegraloperators Gårdinginequality

inequalities, withanapplicationtotheunique solvabilityof parabolicequationsonthelatticeZⁿ.

©2020TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction . . . . 2

2. Symbols,kernels,andamplitudes . . . . 7

3. Symboliccalculus . . . 17

4. Relationbetweenlatticeandtoroidalquantizations . . . 23

5. Applications . . . 25

5.1. Boundednesson²(Zⁿ) . . . 25

5.2. Compactness,Gohberglemma,andSchatten-vonNeumannclasses . . . 27

5.3. Weighted²-boundedness . . . 29

5.4. GårdingandsharpGårdinginequalitiesonZⁿ. . . 30

5.5. Well-posednessoftheparabolicequations . . . 32

5.6. Boundednessandcompactnesson^p(Zⁿ) . . . 34

5.7. Fourierseriesoperators . . . 37

6. Examples . . . 38

References . . . 39

1. Introduction

Theaimofthispaperistodevelopacalculusofpseudo-differentialoperatorssuitable for the applications to solving difference equations on the lattice Zⁿ. Such equations naturallyappearinvariousproblemsofmodellingandinthediscretisationofcontinuous problems. Wecalltheappearingoperatorspseudo-difference operators.

As asimplemotivatingexample,consider theequation n

j=1

f(k+v_j)−f(k−v_j)

+af(k) =g(k), k∈Zⁿ, (1.1) with vj = (0,. . . ,0,1,0,. . . ,0) ∈ Zⁿ, where the jth element of vj is 1, and all other elements are0.Theideaofthispaper istousethesuitableFourier analysisforsolving differenceequationsofthistype.Thus,if,forexample,Rea= 0,thisequationissolvable forany g∈²(Zⁿ) andthesolutioncanbegivenbytheformula

f(k) =

Tⁿ

e^2πk·x 1 2in

j=1sin(2πxj) +ag(x)dx, (1.2) where

g(x) =

k∈Zⁿ

e^−2πik·xg(k), x∈Tⁿ, (1.3)

is the Fourier transform of g. Formula (1.2) alsoextends to give solutions to (1.1) for anytemperedgrowthfunctiong∈ S(Zⁿ).Inparticular,ifg∈²(Zⁿ) thenthesolution f tothedifferenceequation(1.1) givenby(1.2) satisfiesf ∈²(Zⁿ) and,moregenerally, ifgsatisfies

k∈Zⁿ

(1 +|k|)^s|g(k)|²<∞

forsomes∈R, thenthe solutionf to thedifference equation(1.1) given by(1.2) also satisfies

k∈Zⁿ

(1 +|k|)^s|f(k)|²<∞,

seeExample(3) inSection6.

From the point of view of the theory of pseudo-differential operators the operators ofthe form(1.2) extend theusual differenceoperators onthe lattice,thuswe feelthat thetermpseudo-differenceoperatorsmaybejustifiedtoemphasisethattheyextendthe classof difference operators into a∗-algebra.This agrees withthe terminologyalready existingintheliterature(seee.g.[29]).

Thetheoryofpseudo-differentialoperators isusuallyeffectiveinansweringanumber ofquestionssuchas:

• Whatkindofdifferenceequations, similarto(1.1),aresolvable inthisway?

• Giveng(k),whatarepropertiesoff(k) giventherepresentationformula(1.2)?

• Whataboutvariablecoefficientversionsofdifferenceequations,wherethecoefficients oftheequationsmayalsodependonk?

It is the purpose of this paper to answer these and other questions by developing a suitable theory of pseudo-differential operators on the lattice Zⁿ. There are several interestingfeaturesofthistheorymakingitessentiallydifferentfromtheclassicaltheory ofpseudo-differentialoperators onRⁿ,suchas

• ThephasespaceisZⁿ×Tⁿwiththefrequenciesbeingelementsofthecompactspace Tⁿ(thetorusTⁿ:=Rⁿ/Zⁿ).Theusualtheoryofpseudo-differentialoperatorsworks with symbol classes withincreasing decay ofsymbols after taking theirderivatives inthefrequencyvariable.Herewecannotexpectanyimprovingdecaypropertiesin frequencysincethefrequencyspaceiscompact.

• Wecannotworkwithderivativeswithrespect tothespacevariablek∈Zⁿ.There- fore,this needsto be replacedby workingwithappropriate differenceoperators on thelattice.

The developed theory is similar in spirit to the global theory of (toroidal)pseudo- differential operators on the torus Tⁿ consistently developed in [35], see also [1–3] as well as[32,33] forearlierworks.Inparticular,symbol classesinthispaper willcoincide with symbol classes developed in [35,34] but with a twist, swapping the order of the space and frequency variables. As aresult, wecan draw onpropertiesof these symbol classes developedinthe aboveworks.Severalattempts of developing asuitabletheory of pseudo-differential operators onthe latticeZⁿ have been donein theliterature, see e.g.[25,29],butwith nosymbolic calculus.Operatorson theone-dimensionallattice Z havebeenconsideredin[18,11,12],butagainwithnosymboliccalculus,and^pestimates wereconsideredin[30] and[5].Therearenumerousphysicalmodelsrealisedasdifference equations, seee.g.[28,29,26] fortheanalysis ofSchrödinger,Dirac,andotheroperators onlattices,andtheirspectral properties.

Our symbol classes exhibit improvement when differences are taken with respect to the space (lattice) variable, thus resembling in their behaviour the so-called SG pseudo-differential operators in Rⁿ, developed by Cordes [6], but again with a twist invariables.

In the recent work [17], a framework has been developed for the theory of pseudo- differential operators on general locally compact type I groups, with application to spectral properties of operators. The Kohn-Nirenberg type quantization formula that the analysisof thispaper relieson makesaspecialcaseoftheconstruction of[17], but there isonly limitedsymbolic calculus availablethere due to the generality of theset- ting.Thus,hereweareabletoprovidemuchdeeperanalysisintermsoftheasymptotic expansionsandformulaefortheappearingsymbolsandkernels.

Comparedtosituationswhenthestatespaceiscompact(forexample,[37] oncompact groupsor[31] oncompactmanifolds)thecalculushereisessentiallydifferentsincewecan notconstructitusingstandardmethodsrelyingonthedecaypropertiesinthefrequency componentofthephasespacesincethefrequencyspaceisourcaseisthetorusTⁿwhich is compact, so no improvement with respect to the decay of the frequency variable is possible.

To givesomefurtherdetails,theFouriertransformoff ∈¹(Zⁿ) isdefinedby

FZⁿf(x) :=f(x) :=

k∈Zⁿ

e^−2πik·xf(k), (1.4)

forx∈Tⁿ=Rⁿ/Zⁿ,where wewillbedenoting,throughout thepaper,

k·x= n j=1

kjxj,

where k= (k1,. . . ,kn) andx= (x1,. . . ,xn).The Fouriertransform extendsto ²(Zⁿ) andtheconstantsarenormalisedinsuchaway thatwehavethePlancherelformula

k∈Zⁿ

|f(k)|²=

Tⁿ

|f(x)|²dx. (1.5)

TheFourierinversionformulatakestheform f(k) =

Tⁿ

e^2πik·xf(x)dx, k∈Zⁿ. (1.6)

Forameasurable functionσ:Zⁿ×Tⁿ→C,we definethesequenceOp(σ)f by

Op(σ)f(k) :=

Tⁿ

e^2πik·xσ(k, x)f(x)dx, k∈Zⁿ. (1.7)

Theoperatordefinedbyequation(1.7) willbecalledthepseudo-differentialoperatoron Zⁿ corresponding to the symbol σ = σ(k,x), (k,x) ∈ Zⁿ×Tⁿ. We will also call it a pseudo-differenceoperator andthequantizationσ→Op(σ) thelatticequantization.

TheSchwartzspaceS(Zⁿ) onthelatticeZⁿisthespaceofrapidlydecreasingfunctions ϕ: Zⁿ →C, thatis, ϕ ∈ S(Zⁿ) if for any M <∞ there exists a constant C_ϕ,M such that

|ϕ(k)| ≤C_ϕ,M(1 +|k|)^−M, for allk∈Zⁿ.

The topology on S(Zⁿ) is given by the seminorms pj, where j ∈ N0 and pj(ϕ) :=

sup

k∈Zⁿ

(1+|k|)^j|ϕ(k)|.ThespaceoftempereddistributionsS(Zⁿ) isthetopologicaldual toS(Zⁿ),i.e.thespaceofalllinearcontinuousfunctionals onS(Zⁿ).

Asusual,thetheoryofpseudo-differentialoperatorsappliesnotonlyto specificclass of operators but to general linear continuous operators on the space. Indeed, let A : ^∞(Zⁿ)→ S(Zⁿ) beacontinuouslinearoperator.ThenitcanbeshownthatAcanbe writtenintheformA= Op(σ) withthesymbol σ=σ(k,x) definedby

σ(k, x) :=e₋x(k)Aex(k) =e^−2πik·xA

e^2πik·x ,

where e_x(k)= e^2πik·x for all k ∈ Zⁿ and x∈ Tⁿ. Indeed, using the Fourier inversion formula(1.6) intheusualway onecanjustifythesimplecalculation

Af(k) =A

Tⁿ

e^2πik·xf(x)dx

=

Tⁿ

A

e^2πik·x f(x)dx

=

Tⁿ

e^2πik·xσ(k, x)f(x)dx= Op(σ)f(k).

Wealsopresent thefollowing applicationsofthedevelopedcalculus:

• conditionsfor ²(Zⁿ)-boundedness, compactness, and membership inSchatten-von Neumannclassesfor operatorsintermsof theirsymbols; Gohberglemma andesti- matesfortheessentialspectrumof operators;

• conditionsforweighted²(Zⁿ)-boundednessandweighteda-prioriestimatesfordif- ferenceequations;

• Fourierseriesoperators andtheir²(Zⁿ)-boundedness;

• GårdingandsharpGårdinginequalities,withanapplicationtotheuniquesolvability ofparabolicequations onthelatticeZⁿ.

We alsopresentconditionsfor ^p(Zⁿ)-boundednessandcompactness, extendingresults of[18] and[30].Wecannotethatcomparedtotheexistingliteratureon²-boundedness, ourresultsdonotrequireanydecaypropertiesofthesymbol,thusalsoleadingtoa-priori estimatesforelliptic differenceequationswithoutanyloss ofdecay.

InSection2weintroducesymbolclassesanddiscussthekernelsofthecorresponding pseudo-difference operators. Aninteresting difference withthe usual theory of pseudo- differential operators isthatsincethe spaceZⁿ isdiscrete, theSchwartz kernelsof the corresponding pseudo-differenceoperators donothavesingularityatthediagonal.

The plan of the paper is as follows. We study the properties of pseudo-difference operator on Zⁿ by first discussing in Section 2 their symbols and kernels, as well as amplitudes. The symbolic calculusis developedinSection3. InSection 4we establish thelinkbetweenthequantizationsonthelatticeZⁿ andthetorus Tⁿ.InSection5we investigatetheboundednesson²(Zⁿ),weighted²(Zⁿ),^p(Zⁿ),compactnesson^p(Zⁿ), and give conditions for the membership inSchatten-von Neumann classes. Finally, in Section6we givesomeexamples.

Throughout thepaperwewillusethenotationN0=N∪ {0}. Acknowledgments

The authors wouldlike tothank AIMSGhana and itsacademic directorEmmanuel Essel forthe hospitalityduring thefirsttwo authors’study there and duringthethird author’s visitstoGhanaandtotheAfricanInstituteforMathematicalSciences(AIMS) when thisworkwascarriedout.Theauthors wouldalsolike tothankJulioDelgadofor discussions andvaluableremarks.

2. Symbols,kernels,andamplitudes

Forthedevelopingofthesymboliccalculusandforthedefinitionofthesymbolclasses we needto havesomeanalogues of derivatives inthe spacevariable. Forthis purpose, wewillbe usingthefollowingdifferenceoperators.

Definition2.1 (Difference operators).Wedefine Δ^α actingon functionsτ:Zⁿ →C by theformula

Δ^ατ(k) :=

Tⁿ

e^2πik·y(e^2πiy−1)^ατ(y)dy, (2.1)

whereα= (α₁,. . . ,α_n) and

(e^2πiy−1)^α= (e^2πiy1−1)^α1· · ·(e^2πiyn−1)^αn. Itiseasytosee thatwehavethedecomposition

Δ^α= Δ^α₁¹·. . .·Δ^α_nⁿ, (2.2) wheredenotingvj = (0,· · ·,0,1,0,· · ·,0) with 1 atthejth position,wehave

Δjτ(k) =

Tⁿ

e^{2πi(k+vj)·y}τ(y) dy−

Tⁿ

e^2πik·yτ(y)dy

=τ(k+vj)−τ(k) (2.3)

aretheusualdifferenceoperatorsonZⁿ.Therefore,formulae(2.2)-(2.3) givetheequiv- alentcharacterisationto(2.1),andcanbetakenasthedefinitionofdifferenceoperators Δ^α.

At the sametime, the representation (2.1) becomes useful for comparing operators (2.2)-(2.3) tomoregeneraldifferenceoperatorsthatwillbeintroducedinDefinition2.6.

Theformula(2.1) makessenseforτ∈ S(Zⁿ).Indeed,inthiscasewehaveτ∈ D(Tⁿ) andtheformula(2.1) canbe interpretedintermsofthedistributionaldualityonTⁿ,

Δ^ατ(k) =τ , e^2πik·y(e^2πiy−1)^α (2.4) actingonthey-variable.These operatorshavebeenintroduced,analysed andshownto satisfymanyusefulproperties,suchastheLeibnizformula,summationbypartsformula, Taylor expansion formula, and many others, in [35] and [34, Section 3.3] to which we referfordetaileddiscussions.

As usual,wewillbe usingthenotation

D^α_x =D_x^α₁¹· · ·D^α_xnⁿ, Dxj = 1 2πi

∂

∂x_j.

Inthesequelwewillbe alsousingthemulti-indexnotationα!=α₁!· · ·α_n! It willbealsoconvenient touseoperators

D^(α)_x =D^(α_x1¹⁾· · ·D^(α_xnⁿ⁾, D_x⁽⁾_j = m=0

1 2πi

∂

∂x_j −m

, ∈N. (2.5)

Asusual,D⁰_x=D⁽⁰⁾x =I.TheoperatorsD^(α)x becomeveryusefulintheanalysisrelated to thetorus as theyappear inthe periodicTaylor expansion,see (2.21). Theirprecise form in(2.5) isrelatedto propertiesofStirling numbers,see[34,Section3.4].

Definition 2.2 (Symbol classes S_ρ,δ^μ (Zⁿ×Tⁿ)).Let ρ,δ ∈ R. We say that a function σ:Zⁿ×Tⁿ →C belongsto S_ρ,δ^μ (Zⁿ×Tⁿ) ifσ(k,·)∈C^∞(Tⁿ) forallk∈Zⁿ,andfor allmulti-indicesα,β thereexists apositiveconstantC_α,β suchthatwehave

|D^(β)_x Δ^α_kσ(k, x)| ≤C_α,β(1 +|k|)^{μ−ρ|α|+δ|β|} (2.6) forallk∈Zⁿ andx∈Tⁿ.

Ifρ= 1 andδ= 0, wewilldenotesimplyS^μ(Zⁿ×Tⁿ):=S^μ_1,0(Zⁿ×Tⁿ).

Wedenote byOp(σ) theoperatorwithsymbolσgivenby Op(σ)f(k) :=

Tⁿ

e^2πik·xσ(k, x)f(x)dx, k∈Zⁿ, (2.7)

andbyOp(S^μ_ρ,δ(Zⁿ×Tⁿ)) thecollectionofoperatorsOp(σ) asσvariesoverthesymbol class S_ρ,δ^μ (Zⁿ×Tⁿ).

Here and everywherewe mayoftenwrite Δ^α= Δ^α_k to emphasisethatthedifference operatorsareactingonfunctionswithrespecttothevariablek.Wenotethatthesesym- bol classes, moduloswappingtheorderof thevariables xandk, havebeen extensively analysed andusedin[35] forthedevelopmentof theglobal toroidalcalculusof pseudo- differential operators on the torus Tⁿ. Wealso refer to [34, Chapter 4] for athorough presentationoftheirproperties.

The classes on the torus, similar to Definition 2.2, wereanalysed in[34], and their equivalence (also on generalcompact Lie groups) to the usual Hörmander classes was shownin[38].

Pseudo-differential operator can be represented in various forms. For example, for suitablefunctionsf,usingformula (1.4) wecanwrite

Op(σ)f(k) =

Tⁿ

e^2πik·xσ(k, x)f(x)dx

=

Tⁿ

m∈Zⁿ

e^{2πi(k−m)·x}σ(k, x)f(m)dx

=

m∈Zⁿ

Tⁿ

e^{2πi(k−m)·x}σ(k, x)f(m)dx

=

l∈Zⁿ

Tⁿ

e^2πil·xσ(k, x)f(k−l)dx

=

l∈Zⁿ

κ(k, l)f(k−l)

=

m∈Zⁿ

K(k, m)f(m),

withkernels

K(k, m) =κ(k, k−m) and κ(k, l) =

Tⁿ

e^2πil·xσ(k, x)dx. (2.8)

Wenowestablishsomepropertiesofthekernelsofpseudo-differenceoperatorsonZⁿ withsymbolsσ∈S_ρ,δ^μ (Zⁿ×Tⁿ).

Theorem2.3. Letσ∈S_ρ,δ^μ (Zⁿ×Tⁿ) andletδ≥0.ThenforeveryN ∈N₀ thereexists apositiveconstant CN>0suchthat wehave

|K(k, m)| ≤CN(1 +|k|)^{μ+2N δ}(1 +|k−m|)^−2N, (2.9) forallk,m∈Zⁿ.

Inparticularwenotethatincomparisontopseudo-differentialoperatorsonRⁿ oron Tⁿ,thekernelK(k,m) iswelldefinedfork=mandhasnosingularity atthediagonal since the space Zⁿ ×Zⁿ is discrete. We also note that we do not need any further restrictionsonρandδ inTheorem2.3.

Proof of Theorem2.3. Wenote thatfork=m wehave,using (2.8),that K(k, k) =κ(k,0) =

Tⁿ

σ(k, x)dx, (2.10)

satisfying(2.9) inthiscase.

Let usnow assumethatk=m, sothatalso l=k−m= 0. DenotingtheLaplacian onTⁿ byLx:=

n j=1

∂²

∂x²_j, wehave (1− Lx)e^2πil·x=

1 + 4π²|l|² e^2πil·x ; e^2πil·x= (1− Lx)

1 + 4π²|l|²e^2πil·x, (2.11) so thatforl= 0 we canwrite

κ(k, l) =

Tⁿ

e^2πil·xσ(k, x)dx

=

Tⁿ

(1− Lx)^N

1 + 4π²|l|^{2 N}e^2πil·x

σ(k, x)dx

= (1 + 4π²|l|²)^−N

Tⁿ

e^2πil·x 1− Lx

Nσ(k, x)dx.

Therefore, forallN ≥0 wehave

|κ(k, l)| ≤CN(1 + 4π²|l|²)^−N(1 +|k|)^{μ+2N δ}. It followsthenfrom (2.8) thatK(k,m) satisfies(2.9). 2

Similar to the classical cases, we have the formula extracting the symbol from an operator.

Proposition 2.4. Thesymbol of apseudo-differenceoperatorAisgiven by

σ(k, x) =e^−2πik·xAex(k), (2.12) where e_x(k)=e^2πik·x,forallk∈Zⁿ andx∈Tⁿ.

Proof. Forthefunctioney(l)=e^2πil·y,itsFouriertransformisgivenformally by

ey(x) =

l∈Zⁿ

e^−2πil·xe^2πil·y,

with the usual justification in terms of limits or distributions. Plugging this into the formula

Op(σ)f(k) =

Tⁿ

e^2πik·xσ(k, x)f(x)dx,

itfollows that

Op(σ)ey(k) =

Tⁿ

l∈Zⁿ

e^2πik·xσ(k, x)e^−2πil·xe^2πil·ydx

=

Tⁿ

l∈Zⁿ

e−2πi(l−k)·xσ(k, x)e^2πil·ydx

=

l∈Zⁿ

σ(k, l−k)e^2πil·y

=

m∈Zⁿ

σ(k, m)e^2πim·ye^2πik·y (where l−k=m)

=σ(k, y)e^2πik·y,

where σ stands forthe Fourier transformon Tⁿ inthe second variable,and where we usedthe toroidalFourierinversion formulabyastandarddistributionalinterpretation.

Thisgivesformula(2.12). 2

Fromthedefinition(1.7) ofpseudo-differentialoperators andwritingouttheFourier transformoffusingformula(1.4) givestheamplituderepresentationofpseudo-difference operatorsas

Op(σ)f(k) =

m∈Zⁿ

Tⁿ

e^{2πi(k−m)·x}σ(k, x)f(m)dx. (2.13)

Thismotivatesanalysingamplitudeoperators oftheform Af(k) =

m∈Zⁿ

Tⁿ

e^{2πi(k−m)·x}a(k, m, x)f(m)dx, (2.14)

withamplitudesa:Zⁿ×Zⁿ×Tⁿ →C.Wemaystilldenote suchoperatorsbyOp(a), whichisconsistentwith(2.13).

Definition 2.5(Amplitude classes A^μ_ρ,δ^1,μ2(Zⁿ×Tⁿ)).Let ρ,δ∈R.A functiona:Zⁿ× Zⁿ×Tⁿ→CissaidtobelongtotheamplitudeclassA^μρ,δ^1,μ2(Zⁿ×Zⁿ×Tⁿ) ifa(k,m,·)∈ C^∞(Tⁿ) for all k,m ∈ Zⁿ, and if for all multi-indices α,β,γ there exists a positive constantC_α,β,γ>0 suchthatforsomeJ ∈N0withJ ≤ |γ|wehave

|D_y^(γ)Δ^α_kΔ^β_ma(k, m, y)| ≤Cα,β,γ(1 +|k|)^{μ1−ρ|α|+δJ}(1 +|m|)^μ2−ρ|β|+δ(|γ|−J). (2.15) We note that clearly S_ρ,δ^μ (Zⁿ ×Tⁿ) ⊂ A^μ,0_ρ,δ(Zⁿ ×Zⁿ ×Tⁿ). The space of ampli- tude operators Op(a) with amplitudes a ∈ A^μρ,δ^1,μ2(Zⁿ×Zⁿ×Tⁿ) will be denoted by Op(A^μρ,δ^1,μ2(Zⁿ×Zⁿ×Tⁿ)).

Thedefinitionabove ismotivated by propertiesofsymbols inDefinition 2.2,by the propertythattheamplitudeoftheoperatoradjointtoOp(σ) willbegivenbya(k,m,x)=

σ(m, x), and in order to have Theorem 2.8. Here, for the inclusion S_ρ,δ^μ (Zⁿ ×Tⁿ) ⊂ A^μ,0ρ,δ(Zⁿ×Zⁿ×Tⁿ) wemaytakeJ =|γ|in(2.15),whilefortheamplitudea(k,m,x)= σ(m, x) wemaytakeJ = 0.

Wenow aimtoshowthat

Op(A^μ_ρ,δ^1,μ2(Zⁿ×Zⁿ×Tⁿ))⊂Op(S_ρ,δ^μ1+μ2(Zⁿ×Tⁿ)).

Forthis, weestablishausefulproperty ofmoregeneraldifferenceoperators.

Definition2.6(Generaliseddifferenceoperators).Letq∈C^∞(Tⁿ).Thenforτ :Zⁿ→C we definetheq-differenceoperatorby

Δ_qτ(k) :=

Tⁿ

e^2πik·xq(x)τ(x)dx. (2.16)

Whiletheintegralformulaabovemakessenseforsuitablefunctionsτ,similarto(2.4) it canbeextendedto allτ ∈ S(Zⁿ) bythedistributionalduality

Δqτ(k) =τ , e^2πik·xq(x) (2.17) acting onthe x-variable. Atthe sametime, formula(2.16) alsoextendsto non-smooth functionsq:forexample,(2.16) makessenseforτ ∈²(Zⁿ) andq∈L²(Tⁿ),orforother choicesofmatching conditionsonτ and q,for(2.16) to makesense.

Expandingτ(x) wecanalsonote theusefulformula Δ_qτ(k) =

l∈Zⁿ

Tⁿ

e^{2πi(k−l)·x}q(x)τ(l)dx

=

l∈Zⁿ

τ(l)FZ⁻ⁿ¹q(k−l) = (τ∗ FZ⁻ⁿ¹q)(k). (2.18) Werecordthefollowingpropertyofgeneraliseddifferenceoperatorsactingonsymbols.

Lemma2.7. Let0≤δ≤1andletσ∈S_ρ,δ^μ (Zⁿ×Tⁿ),μ∈R.Thenforanyq∈C^∞(Tⁿ) and anyβ ∈N₀ⁿ wehave

|Δ_qD^(β)_x σ(k, x)| ≤C_q,β(1 +|k|)^μ+δ|β|, (2.19) forallk∈Zⁿ and x∈Tⁿ.

Proof. Itisenoughtoprovethisforβ= 0.Using(2.18),wewrite Δ_qσ(k,x) as Δ_qσ(k, x) =

l∈Zⁿ

Tⁿ

e^{2πi(k−l)·y}q(y)σ(l, x)dy

= σ(k, x)

Tⁿ

q(y)dy+

l∈Zⁿ l=k

Tⁿ

e^{2πi(k−l)·y}q(y)σ(l, x)dy

=:I₁+I₂,

whereinthefirsttermweset l=k.Thenwehave

|I1| ≤(1 +|k|)^μ.

Ontheotherhand,forμ≥0,integrating byparts withtheoperator(2.11),wehave

|I₂|=

l∈Zⁿ l=k

Tⁿ

e^{2πi(k−l)·y} (2π)^2M|k−l|^2M

L^My q(y)

σ(l, x)dy

≤C

l∈Zⁿ l=k

1

|k−l|^2M(1 +|l|)^μ

≤C

m=0

1

|m|^2M(1 +|k−m|)^μ

≤C

m=0

1

|m|^2M

(1 +|k|)^μ+|m|^μ

≤C(1 +|k|)^μ,

(2.20)

where we used that μ ≥ 0 in the last lines and that if we take M > n+μ 2 , then 2M−μ> n,andtheseriesinthelastlinesof(2.20) converges.

Ifμ<0,wewillusethePeetreinequalitywhichsaysthatforalls∈Randξ,η∈Rⁿ wehave

(1 +|ξ+η|)^s≤2^|s|(1 +|ξ|)^s(1 +|η|)^|s|, see[34,Proposition3.3.31].Applyingthiswiths=μ,we have

(1 +|k−m|)^μ≤2^|μ|(1 +|k|)^μ(1 +|m|)^|μ|. Applyingthistothethirdlineof(2.20) wegetthat

|I2| ≤C(1 +|k|)^μ,

providedthatwetakeM suchthat2M− |μ|> n,sothattheseriesinmconverges. So weobtain(2.19) inallthecases. 2

Beforeprovingthatamplitudeoperatorsarepseudo-differenceoperatorsandaregiven bysymbols,letusrecalltheperiodicTaylorexpansionformulafrom[34,Theorem3.4.4].

It saysthatifh∈C^∞(Tⁿ) thenwehavetheperiodicTaylorexpansionforhgivenby h(x) =

|α|<N

1

α!(e^2πix−1)^αD_z^(α)h(z)|z=0+

|α|=N

hα(x)(e^2πix−1)^α, (2.21)

where hα∈C^∞(Tⁿ),withD^(α)z givenby(2.5),and

(e^2πix−1)^α= (e^2πix1−1)^α1· · ·(e^2πixn−1)^αn.

The functions h_α ∈ C^∞(Tⁿ) are products of one-dimensional functions h_j(y), y ∈ T, definedinductivelybyh₀=handthenby

hj+1(y) :=

⎧⎪

⎨

⎪⎩

h_j(y)−h_j(0)

e^2πiy−1 if y= 0, D_yh_j(y) ify= 0.

(2.22)

Inparticular, wehave

h₁(y) =

⎧⎪

⎨

⎪⎩

h(y)−h(0)

e^2πiy−1 if y= 0, D_yh(0) ify= 0,

h₂(y) =

⎧⎪

⎨

⎪⎩

h1(y)−h1(0) e^2πiy−1 =

h(y)−h(0)

e^2πiy−1 −D_yh(0)

e^2πiy−1 if y = 0, D_yh₁(0) ify = 0,

and so on. It can be shown thatthese functions are smooth everywhere, including at y = 0,andthathj dependsonthejth orderderivativeofh.Wereferto[34,Section3.4]

fortheproof aswellas fortheexpressionsandanalysisfortheremainderfunctionshα. Theorem2.8.Let0≤δ < ρ≤1.Leta∈ A^μ_ρ,δ^1,μ2(Zⁿ×Zⁿ×Tⁿ)andletthecorresponding amplitudeoperator Abe givenby

Af(k) =

l∈Zⁿ

Tⁿ

e^{2πi(k−l)·x}a(k, l, x)f(l)dx. (2.23)

Then we have A = Op(σA) for some σA ∈ S^μ_ρ,δ^1+μ2(Zⁿ ×Tⁿ). Moreover, σA has the following asymptoticexpansion

σA(k, x)∼

α

1

α!Δ^α_lD_x^(α)a(k, l, x)

l=k, (2.24)

whichmeansthat forallN∈N wehave

σA−

|α|<N

1

α!Δ^α_lD^(α)_x a(k, l, x)

l=k ∈S_ρ,δ^{μ1+μ2−N(ρ−δ)}(Zⁿ×Tⁿ). (2.25) Fromnow onwewillalways understandasymptoticsumsof type(2.24) inthesense of(2.25).

Proof of Theorem2.8. Using formula (2.12) from Proposition 2.4, the symbol of the operatorA in(2.23) isgiven by

σA(k, x) =e^−2πik·x

l∈Zⁿ

Tⁿ

e^{2πi(k−l)·y}a(k, l, y)e^2πil·xdy

=

l∈Zⁿ

Tⁿ

e2πi(k−l)·(y−x)a(k, l, y)dy

=

Tⁿ

e^{2πik·(y−x)}a(k, y−x, y)dy,

whereadenotestheFouriertransformofawithrespecttothesecondvariable.Replacing y byy+x, weobtain

σ_A(k, x) =

Tⁿ

e^2πik·ya(k, y, y+x)dy. (2.26)

TakingtheTaylor expansionofa(k,y,y+x) inthethirdvariable atxas in(2.21), we have

a(k, y, x+y) =

|α|≤N

1

α!(e^2πiy−1)^αD^(α)_x a(k, y, x) +R0, (2.27) whereR0isaremainderthatwewill studylater.Substituting(2.27) into(2.26) gives

σA(k, x) =

Tⁿ

e^2πik·y

|α|≤N

1

α!(e^2πiy−1)^αD^(α)_x a(k, y, x)dy+R, (2.28) withtheremainderRthatcanbe expressedintermsofR0. Sinceby(2.1) wehave

Δ^α_lτ(k) =

Tⁿ

e^2πik·y(e^2πiy−1)^ατ(y)dy,

theformula(2.28) becomes

σA(k, x) =

|α|≤N

1

α!Δ^α_lD_x^(α)a(k, l, x)

l=k+R, givingthetermsinthesumin(2.24).

Letus nowanalysetheremainderR.It isthesumoftermsoftheform Rj(k, x) =

Tⁿ

e^2πik·y(e^2πiy−1)^αbj(k, y, x)dy,

with |α|=N and somebj containingthetermswhicharecombination offunctions D^α_x⁰F2a(k, y, x) for |α0| ≤N,

where F2 means the Fourier transformwith respect to the second variable,multiplied by some smooth functions, using the expressions (2.22). It follows that for any β the functionsD^(β)x R_j(k,x) arethesumsoftermsoftheform

Tⁿ

e^2πik·yaj(y)(e^2πiy−1)^αD_x^(β)D^(α_x ⁰⁾F2a(k, y, x)dy= ΔajΔ^α_lD^(α_x ^0+β)a(k, l, x)

l=k,

for some smooth functions aj ∈ C^∞(Tⁿ). Since a∈ A^μρ,δ^1,μ2, by Lemma2.7 we obtain thatRj(k,x) satisfiestheestimate

|Rj(k, x)| ≤C(1 +|k|)^μ1(1 +|k|)^{μ2−ρ|α|+δ|α0|+δ|β|}, forany valueofJ in(2.15).Using that|α|=N and|α0|≤N wegetthat

|Rj(k, x)| ≤C(1 +|k|)^μ1+μ2−(ρ−δ)N+δ|β|.

Also,forthetermsΔ^β_kRj(k,x) wecanexpressthem assumsoftermsoftheform

Tⁿ

e^2πik·z(e^2πiz−1)^βaj(z)(e^2πiz−1)^αbj(k, z, x)dz,

withsimilarbjandajasabove.Anargumentsimilartotheoneaboveshowstheestimate

|Δ^β_kRj(k, x)| ≤C(1 +|k|)^μ1+μ2−ρ|β|−(ρ−δ)N.

BychoosingN largeenough,argumentslike intheclassicalpseudo-differentialcalculus imply thatwehave(2.24). 2

3. Symboliccalculus

In this section we develop elements of the symbolic calculus of pseudo-differential operatorsonZⁿbyderivingformulaeforthecomposition,adjoint,transpose,aswellas fortheparametrixforellipticoperators.

Theorem 3.1 (Composition formula). Let 0≤ δ < ρ≤ 1. Let σ ∈ S_ρ,δ^μ1(Zⁿ ×Tⁿ) and τ∈S^μ_ρ,δ²(Zⁿ×Tⁿ).ThenthecompositionOp(σ)◦Op(τ)isapseudo-differentialoperator withsymbol ς ∈S_ρ,δ^μ1+μ2(Zⁿ×Tⁿ),whichcan begivenasan asymptoticsum

ς(k, x)∼

α

1

α!D^(α)_x σ(k, x)Δ^α_kτ(k, x). (3.1) Wenotethattheorderoftakingdifferences andderivativesin(3.1) changesincom- parisonto theanalogouscompositionformulaeonRⁿ andTⁿ, see[35,34].

Proof of Theorem3.1. Letf,g∈ S(Zⁿ).Thepseudo-differentialoperatorswithsymbols σandτ aregiven by

Op(σ)f(k) =

m∈Zⁿ

Tⁿ

e^{2πi(k−m)·x}σ(k, x)f(m)dx, (3.2)

Op(τ)g(m) =

l∈Zⁿ

Tⁿ

e^{2πi(m−l)·y}τ(m, y)g(l)dy. (3.3)

Thecompositionof(3.2) and(3.3) gives

Op(σ)

Op(τ)g (k) =

l∈Zⁿ

m∈Zⁿ

Tⁿ

Tⁿ

e^{2πi(k−m)·x}σ(k, x)e^{2πi(m−l)·y}τ(m, y)g(l)dydx

=

l∈Zⁿ

Tⁿ

e^{2πi(k−l)·y}ς(k, y)g(l)dy,

where

ς(k, y) =

m∈Zⁿ

Tⁿ

e2πi(k−m)·(x−y)σ(k, x)τ(m, y)dx

=

m∈Zⁿ

Tⁿ

e^{2πik·(x−y)}e^{−2πim·(x−y)}σ(k, x)τ(m, y)dx