The case of non-invariant operators

G

f(u)RA(u⁻¹g)du. (5.26)

Then, byHölder’sinequality,wehave

|Af(g)| ≤ fL^β(G)RA(·⁻¹g)L^β(G)=fL^β(G)RAL^β(G), (5.27) where we usedthattheHaar measureis translation-invariant.Then, bythe Hausdorff-Younginequality,wehave

RAL^β(G)≤ F[RA]L^β(VNR(G))=AL^β(VNR(G)), 1≤β≤2, (5.28) whereweusedthattheFouriertransform ofthekernelR_AistheoperatorA,i.e.F[R_A]= A.Combininginequality(5.27) and(5.28),weget

|Af(g)| ≤ fL^β(G)AL^β(VNR(G)), g∈G,1≤β ≤2. (5.29) Taking supremum over g ∈ G in the left-hand side of (5.29), we obtain (5.25). This completes theproof. 2

5.4. Thecase ofnon-invariant operators

Theorem5.1canbeextendedtonon-invariantoperators,andalsototheboundedness fornon-invariant operatorsinLorentzspaces, inanalogy toRemark5.2.

FortheformulationitisconvenienttousetheSchwartz-BruhatspacesS(G) thathave been developedbyBruhat[10] asaway ofdoingdistributiontheoryonlocally compact groups. We briefly mention its basic properties and refer to [10] for further details.

The space S(G) is a barrelled, bornological and complete locally convex topological vector space.It iscontinuously anddenselycontainedinthespace C_c(G) ofcompactly supported continuous functions.The space S(G) is densein everyL^p(G),1≤p<∞, whichfollows fromthefactthatCc(G) isdenseinL^p(G).

Theorem5.10. LetGbealocallycompactunimodularseparablegroup. LetD:L²(G)→ L²(G)be aclosed denselydefined operatorsuchthat itsinverseD⁻¹ ismeasurablewith respecttoVN_R(G) andsuchthat forsome 1< β ≤2we have

D⁻¹L^β(VNR(G))<+∞. (5.30) LetAbealinearcontinuousoperatorontheSchwartz-BruhatspaceS(G).Thenforany 1< p≤2≤q <∞ wehave

AL^p(G)→L^q(G)

⎛

⎝

G

D ◦A_uL^r,∞(VNR(G))

β

du

⎞

⎠

1 β

, (5.31)

where ¹_r = ¹_p−¹_q.

Here{Au}isthefieldofoperatorsgeneratedbyvaryingtheSchwartzkernelKAofA, formoredetailswerefertotheproofofTheorem5.10.Butfirstweobservethatchoosing variousD,wegetdifferentinequalitiesin(5.30).Thus,beforeprovingTheorem5.10,we illustrateitinafewexamples.

Example5.11.LetGbeacompactLiegroupofdimensionnandletLG betheLaplace operatoronG.LetustakeD= (I− LG)ⁿ².BytheWeyl’sasymptoticlaw(seee.g.[63]), weget

λ_k ∼=k,

whereλk aretheeigenvaluesofD.Then,uptoconstant,weobtain D^−1β_L^β_(VNR(G))^∞

k=1

1

k^β <+∞, foranyβ >1.Thus,condition(5.30) issatisfied.

Example 5.12.Let us take G to be the Heisenberg group Hⁿ with the homogeneous dimension Q= 2n+ 2, and let L^subHⁿ be the canonical sub-Laplacian on Hⁿ. It canbe computed(see(7.29))that

τ(E_(0,s)(−L^subHⁿ)) =C_ns^Q2. UsingthisandDefinition2.8, itcanbeshownthat

μ_t((I− L^subHⁿ)^−α) = 1

1 +t^Q2 α.

Fromthis weobtain

(I− L^subHⁿ)^−αβ_Lβ(VNR(Hⁿ))=

+∞

0

1

1 +t^Q2αβdt, (5.32) whereweused theformula

τ(|A|^p) =

+∞

0

μ^p_t(A)dt

establishedin[67,Corollary 2.8,p. 278].Theintegralin(5.32) isconvergentifandonly ifαβ > ^Q₂.

Proof of Theorem5.10. Letusdefine

A_uf(g) :=L_KA(u)f(g) = isa functionon G.The operatorA =D⁻¹ is aleft Fourier multiplieron G.Then, by Theorem 5.9,weget

supg∈G|D⁻¹h(g)| ≤ D⁻¹L^β(VNR(G)h(g)L^β(G), 1< β≤2.

From this,forfunctionsoftheformh(g)=DA_uf(g),wefinallyobtain sup

u∈G|Auf(g)|= sup

u∈G|D⁻¹DAuf(g)| ≤ D⁻¹L^β(VNR(G)DAuf_L^β_u(G).

Therefore,usingtheMinkowskiintegralinequalitytochangetheorderofintegration, we obtain

≤

⎛

⎝

G

DA_uL^r,∞(VNR(G))

β

du

⎞

⎠

1 β

fL^p(G), wherethelastinequalityholdsdue toTheorem5.1.

ThiscompletestheproofofTheorem 5.10. 2 6. Spectralmultipliersonlocallycompactgroups

InthisandnextsectionwewillgiveanapplicationofTheorem5.1tospectral multi-pliers.

The classical Laplace operator ΔRⁿ is affiliated with the von Neumann algebra VN(Rⁿ) = VN_L(Rⁿ) = VN_R(Rⁿ) of all convolution operators, butis not measurable onVN(Rⁿ). However,the Besselpotential (I−ΔRⁿ)^−s2 ismeasurable withrespect to VN(Rⁿ).Therefore, oneof theaimsof spectral multipliertheoremsis to“renormalise”

operators in Hilbert space H making them not only measurable butalso bounded. In thenext theorem we first describe sucha relationfor general semifinitevon Neumann algebras,and theninCorollary 6.2giveitsapplicationtospectral multipliers.

Theorem6.1.LetLbe aclosed unbounded operatoraffiliatedwithasemifinitevon Neu-mann algebra M ⊂ B(H). Assume that ϕ is a monotonically decreasing continuous functionon[0,+∞) suchthat

ϕ(0) = 1, (6.1)

u→+∞lim ϕ(u) = 0. (6.2)

Thenforevery 1≤r <∞ wehavetheequality ϕ(|L|)L^r,∞(M)= sup

u>0

τ(E_(0,u)(|L|))_r¹

ϕ(u). (6.3)

LetL be anarbitraryunboundedlinearoperatoraffiliatedwith (M,τ).Then Theo-rem6.1saysthatthefunctionϕ(|L|) isnecessarilyaffiliatedwithMandϕ(|L|)∈(M,τ) ifandonlyifther-thpowerϕ^rofϕgrowsatinfinitynotfasterthan _τ(E ¹

(0,u)(|L|)),i.e.if wehavetheestimate

ϕ(u)^r 1

τ(E_(0,u)(|L|)). (6.4)

Wenow giveacorollaryof Theorem6.1 forM = VN_R(G) beingtheright von Neu-mannalgebraofalocallycompactunimodulargroup.ThisisformulatedinTheorem1.2 butwerecallithereforreaders’convenience.

Corollary6.2. LetGbe alocallycompactunimodularseparablegroupandletLbe aleft Fouriermultiplieron G.Letϕbe asinTheorem 6.1.Thenwehavetheinequality

ϕ(|L|)L^p(G)→L^q(G)sup

u>0

ϕ(u)

τ(E(0,u)(|L|))¹_p−¹q

, 1< p≤2≤q <∞. (6.5) This corollary follows immediately from combining Theorem 5.1 and Theorem 6.1 with M= VN_R(G),alsoprovingTheorem1.2.

For completeness, we give another corollary (of the proof of Theorem 6.1) without assumingthatϕismonotone,continuous,andsatisfiesconditions(6.1)-(6.2).Itisthese conditions thatallowus to rewriteCorollary 6.3inthe moreapplicableform of Corol-lary 6.2.

Corollary 6.3.LetGbealocallycompactunimodularseparablegroupandletLbealeft Fouriermultiplier onG.LetϕbeaBorelmeasurable functionon thespectrumSp(|L|).

Then wehavetheinequality ϕ(|L|)L^p(G)→L^q(G)sup

s>0

s[τ(E_(s,+∞)(ϕ(|L|)))]^1p−1q, 1< p≤2≤q <∞. (6.6) Wewill provethiscorollary togetherwiththeproofofTheorem6.1.

Proof of Theorem6.1. Bydefinition ϕ(|L|)L^r,∞(M)= sup

t>0

t^1p−1qμt(ϕ(|L|)), 1 r =1

p−1 q. Using property(2.13) fromProposition2.9,weget

sup

t>0

t^p1−1qμt(ϕ(|L|)) = sup

s>0

s[τ(E_(s,+∞)(ϕ(|L|)))]^p1−1q. Hence,wehave

ϕ(|L|)L^r,∞(M)= sup

s>0

s[τ(E_(s,+∞)(ϕ(|L|)))]^1p−1q. (6.7) SinceLisaffiliatedwithM thespectralprojectionsEΩ(|L|) belongtoM.LetLbean abelian subalgebraof M generated bythe spectralprojectors E_(λ,+∞)(|L|). Letϕbe a Borel measurable functionon thespectrumSp(|L|). Then byBorel functional calculus [5,Section2.6] itispossible toconstructtheoperatorϕ(|L|).This operatoris astrong limitofthespectralprojectionsEΩ(|L|)∈M.Thereforeϕ(|L|) isaffiliatedwithM.The distributionfunctionoftheoperatorϕ(|L|) isgivenby

ds(ϕ(|L|)) =τ(E_(s,+∞)(ϕ(|L|))). (6.8) This provesCorollary6.3.

Using [39,Corollary 5.6.29,p. 363] andthespectral mappingtheorem (see[39, The-orem 4.1.6]),weobtain

τ(E(s,+∞)(ϕ(|L|))) =τ(E_ϕ−1(s,+∞)(ϕ⁻¹◦ϕ(|L|))) =τ(E_(0,ϕ−1(s))(|L|)). (6.9) Fromthehypothesis(6.2) imposedonϕand using(6.9),weget

s→+∞lim τ(E_(s,+∞)(ϕ(|L|))) = lim

s→+∞τ(E_(0,ϕ−1(s))(|L|)) = 0. (6.10) Hence, theoperator ϕ(|L|) is τ-measurable with respect to VNR(G). Combining (6.7) and(6.9),wefinallyobtain

ϕ(|L|)L^r,∞(M)= sup

t>0

t^1p−1qμt(ϕ(|L|)) = sup

s>0

s[τ(E_(s,+∞)(ϕ(|L|)))]^1p−1q

= sup

s>0

s[τ(E_(0,ϕ−1(s))(|L|))]^p1−1q = sup

u>0

ϕ(u)[τ(E(0,u)(|L|))]^1p−1q,

where inthe last equality we used the monotonicityof ϕ. This completes the proof of Theorem6.1. 2