L p - L q multipliers on locally compact groups
aa r X i v : . [ m a t h . R T ] A p r L p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS
RAUAN AKYLZHANOV AND MICHAEL RUZHANSKY
Abstract.
In this paper we discuss the L p - L q boundedness of both spectral andFourier multipliers on general locally compact separable unimodular groups G forthe range 1 < p ≤ q < ∞ . We prove a Lizorkin type multiplier theorem for 1
The aim of this paper is to give sufficient conditions for the L p - L q boundedness ofFourier and spectral multipliers on locally compact separable unimodular groups. Itis known that in this case we must have p ≤ q and two classical results are availableon R n , namely, H¨ormander’s multiplier theorem [H¨or60] for 1 < p ≤ ≤ q < ∞ , andLizorkin’s multiplier theorem [Liz67] for 1 < p ≤ q < ∞ . There is a philosophicaldifference between these results: H¨ormander’s theorem does not require any regularityof the symbol and applies to p and q separated by 2, while Lizorkin theorem appliesalso for 1 < p ≤ q ≤ ≤ p ≤ q < ∞ but imposes certain regularity conditionson the symbol.In this paper we are able to prove versions of these theorems on general locallycompact separable unimodular groups based on developing a new approach relyingon the analysis in the noncommutative Lorentz spaces on the group von Neumannalgebra. This suggested approach seems very effective, implying as special casesknown results expressed in terms of symbols, in settings when the symbolic calculusis available. The obtained results are for general Fourier multipliers, in particularalso implying new results for spectral multipliers. Date : April 4, 2017.2010
Mathematics Subject Classification.
Primary 43A85; 43A15; Secondary 35S05;
Key words and phrases.
Hausdorff-Young-Paley inequalities, locally compact groups, Fourier mul-tipliers, spectral multipliers, H¨ormander theorem, Lizorkin theorem.The second author was supported by the EPSRC Grant EP/K039407/1 and by the LeverhulmeGrant RPG-2014-02. No new data was collected or generated during the course of research.
The class of groups covered by our analysis is very wide. In particular, it containsabelian, compact, nilpotent groups, exponential, real algebraic or semi-simple Liegroups, solvable groups (not all of which are type I, but we do not need to assumethe group to be of type I or II), and many others. As far as we are aware our resultsare new in all of these non-Euclidean settings.In this paper we focus on the L p - L q multipliers as opposed to the L p -multiplierswhen theorems of Mihlin-H¨ormander or Marcinkiewicz types provide results for bothFourier and spectral multipliers in some settings, based on the regularity of the mul-tiplier. L p -multipliers have been intensively studied on different kinds of groups,however, mostly L p spectral multipliers, for which a wealth of results is available:e.g. [MS94, MRS95] on Heisenberg type groups, [CM96] on solvable Lie groups,[MT07] on nilpotent and stratified groups, to mention only very very few. L p Fouriermultiplies have been also studied but to a lesser extent due to lack of symbolic calcu-lus that was not available until recently, e.g. Coifman and Weiss [CW71b, CW71a]on SU(2), [RW13, RW15] and then [Fis16] on compact Lie groups, or [FR14, CR16]on graded Lie groups. A characteristic feature of the L p - L q multipliers is that lessregularity of the symbol is required. Therefore, in this paper we concentrate on the L p - L q multiplier theorems, however aiming at obtaining unifying results for generallocally compact groups. We give several short applications of the obtained results toquestions such as embedding theorems and dispersive estimates for evolution PDEs.The approach to the L p -Fourier multipliers is different from the technique proposedin this paper allowing us to avoid making an assumption that the group is compactor nilpotent. In this paper we are interested in both Fourier multipliers and spectralmultipliers, for the latter some L p - L q results being available in some special settings,see e.g. [CGM93], and also [Cow74], as well as [ANR16a] for the case of SU(2),and for the discussion of some relations between those in the group setting we canrefer to [RW15] and references therein. Fourier multipliers in the context of group vonNeumann algebras have been studied in [GPJP15]. By the combinatorial method it ispossible to establish the L p - L q estimates for the Poisson-type semigroup P t on discretegroups G [JPPP13]. Finally we note that multiplier estimates on noncommutativegroups are in general considerably more delicate than those in the commutative case,recall e.g. the asymmetry problem and its resolution in [DGR00]. A link betweenFourier multipliers and Lorentz spaces on group von Neumann algebras has beenoutlined in [AR16].We now proceed to making a more specific description of the considered problems.1.1. H¨ormander’s theorem on locally compact groups.
To put this in context,we recall that in [H¨or60, Theorem 1.11], Lars H¨ormander has shown that for 1 < p ≤ ≤ q < ∞ , if the symbol σ A : R n → C of a Fourier multiplier A on R n satisfies thecondition sup s> s Z ξ ∈ R n : | σ A ( ξ ) |≥ s dξ p − q < + ∞ , (1.1) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 3 then A is a bounded operator from L p ( R n ) to L q ( R n ). Here, as usual, the Fouriermultiplier A on R n acts by multiplication on the Fourier transform side, i.e. c Af ( ξ ) = σ A ( ξ ) b f ( ξ ) , ξ ∈ R n . (1.2)Moreover, it then follows that k A k L p ( R n ) → L q ( R n ) . sup s> s Z ξ ∈ R n | σ A ( ξ ) |≥ s dξ p − q , < p ≤ ≤ q < + ∞ . (1.3)The L p - L q boundedness of Fourier multipliers has been also recently investigated inthe context of compact Lie groups, and we now briefly recall the result. Let G be acompact Lie group and b G its unitary dual. For π ∈ b G , we write d π for the dimensionof the (unitary irreducible) representation π . In [ANR16b] the authors have shownthat, for a Fourier multiplier A acting via c Af ( π ) = σ A ( π ) b f ( π )by its global symbol σ A ( π ) ∈ C d π × d π we have k A k L p ( G ) → L q ( G ) . sup s> s X π ∈ b G k σ A ( π ) k op ≥ s d π p − q , < p ≤ ≤ q ≤ ∞ . (1.4)Here for π ∈ b G , the Fourier coefficients are defined as b f ( π ) = Z G f ( x ) π ( x ) ∗ dx, and k σ A ( π ) k op is the operator norm of σ A ( π ) as the linear transformation of therepresentation space of π ∈ b G identified with C d π . For a general development of globalsymbols and the corresponding global quantization of pseudo-differential operatorson compact Lie groups we can refer to [RT13, RT10].One of the results of this paper generalises both multiplier theorems (1.3) and (1.4)to the setting of general locally compact separable unimodular groups G .By a left Fourier multiplier in the setting of general locally compact unimodulargroups we will mean left invariant operators that are measurable with respect to theright group von Neumann algebra VN R ( G ), see Section 2.1 for a discusson.Thus, in Theorem 7.1 we prove the following inequality k A k L p ( G ) → L q ( G ) . sup s> s Z t ∈ R + : µ t ( A ) ≥ s dt p − q , < p ≤ ≤ q < + ∞ , (1.5)where µ t ( A ) are the t -th generalised singular values of A , see [FK86] (and also Defi-nition 2.4) for definition and properties. The proof of inequality (1.5) is based on aversion of the Hausdorff-Young-Paley inequality on locally compact separable groupsthat we establish for this purpose. RAUAN AKYLZHANOV AND MICHAEL RUZHANSKY
The key idea behind this extension is that H¨ormander’s theorem (1.3) can bereformulated as k A k L p ( R n ) → L q ( R n ) . sup s> s Z ξ ∈ R n | σ A ( ξ ) |≥ s dξ p − q ≃ k σ A k L r, ∞ ( R n ) ≃ k A k L r, ∞ (VN( R n )) , (1.6)where r = p − q , k σ A k L r, ∞ ( R n ) is the Lorentz space norm of the symbol σ A , and k A k L r, ∞ (VN( R n )) is the norm of the operator A in the Lorentz space on the group vonNeumann algebra VN( R n ) of R n . In turn, our estimate (1.5) is equivalent to theestimate k A k L p ( G ) → L q ( G ) . k A k L r, ∞ (VN R ( G )) ≃ sup s> s Z t ∈ R + : µ t ( A ) ≥ s dt r , (1.7)where k A k L r, ∞ (VN R ( G )) is the norm of the operator A in the noncommutative Lorentzspace on the right group von Neumann algebra VN R ( G ) of G . Thus, the Lorentzspaces become a key point for the extension of H¨ormander’s theorem to the settingof locally compact (unimodular) groups.In Remark 7.3 and Proposition 7.5 we show that the multiplier theorem (1.5)implies both (1.3) and (1.4) in the respective settings of R n and compact Lie groups.We assume for simplicity that G is unimodular but we do not make assumption that G is either of type I or type II. The assumption for the locally compact group to beseparable and unimodular may be viewed as natural allowing one to use basic resultsof von Neumann-type Fourier analysis, such as, for example, Plancherel formula (seeSegal [Seg50]). However, the unimodularity assumption may be in principle avoided,see e.g. [DM76], but the exposition becomes much more technical. For a more detaileddiscussion of pseudo-differential operators in such settings we refer to [MR15], butwe note that compared to the analysis there in this paper we do not need to assumethat the group is of type I.1.2. Spectral multipliers on locally compact groups.
Let us illustrate the useof the Fourier multiplier theorem (1.5) in the important case of spectral multiplierson locally compact groups. Later, in Theorem 8.1 we will give a spectral multiplierresult on general semifinite von Neumann algebras, however, we now formulate itsspecial case for the case of group von Neumann algebras associated to locally compactgroups.Interestingly, this result asserts that the L p - L q norms of spectral multipliers ϕ ( |L| )depend essentially only on the rate of growth of traces of spectral projections of theoperator |L| : Theorem 1.1.
Let G be a locally compact separable unimodular group and let L be aleft Fourier multiplier on G . Assume that ϕ is a monotonically decreasing continuous p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 5 function on [0 , + ∞ ) such that ϕ (0) = 1 , lim u → + ∞ ϕ ( u ) = 0 . Then we have the inequality k ϕ ( |L| ) k L p ( G ) → L q ( G ) . sup u> ϕ ( u ) (cid:2) τ ( E (0 ,u ) ( |L| )) (cid:3) p − q , < p ≤ ≤ q < ∞ . (1.8)Here E (0 ,u ) ( |L| ) are the spectral projections associated to the operator |L| to theinterval (0 , u ), see Section 2 for precise definitions, and τ is the canonical trace onthe right group von Neumann algebra VN R ( G ), see Section 4 for a discussion.Also we note that more general statements, without the above assumptions on ϕ ,are possible, see Corollary 8.3.The estimate (1.8) says that if the supremum on the right hand side is finite thenthe operator ϕ ( |L| ) is bounded from L p ( G ) to L q ( G ). Moreover, the estimate for theoperator norm can be used for deriving asymptotics for propagators for equations on G . For example, we get the following consequences for the L p - L q norm for the heatkernel of L , applying Theorem 1.1 with ϕ ( u ) = e − tu , or embedding theorems for L with ϕ ( u ) = u ) γ .We note that estimates of the type (1.10) are exactly those leading to subsequentStrichartz estimates. Here, our method is very different from the usual ones as we donot get it by interpolation from the end-point case. Corollary 1.2.
Let G be a locally compact unimodular separable group and let L bea positive left Fourier multiplier such that for some α we have τ ( E (0 ,s ) ( L )) . s α , s → ∞ . (1.9) Then for any < p ≤ ≤ q < ∞ there is a constant C = C α,p,q > such that wehave k e − t L k L p ( G ) → L q ( G ) ≤ Ct − α ( p − q ) , t > . (1.10) We also have the embeddings k f k L q ( G ) ≤ C k (1 + L ) γ f k L p ( G ) , (1.11) provided that γ ≥ α (cid:18) p − q (cid:19) , < p ≤ ≤ q < ∞ . (1.12)The number α in (1.9) is determined based on the spectral properties of L . Forexample, we have(a) if L is the sub-Laplacian on a compact Lie group G then α = Q , where Q isthe Hausdorff dimension of G with respect to the control distance associatedto L ;(b) if L is the sub-Laplacian on the Heisenberg group G = H n then α = Q , where Q = 2 n + 2 is the homogeneous dimension of H n . RAUAN AKYLZHANOV AND MICHAEL RUZHANSKY
Consequently, in both of these sub-Laplacian cases, Corollary 1.2 implies that for any1 < p ≤ ≤ q < ∞ there is a constant C = C p,q > k e − t L k L p ( G ) → L q ( G ) ≤ Ct − Q ( p − q ) , t > . (1.13)The embeddings (1.11) under conditions (1.12) show that the statement of Theorem1.1 is in general sharp. Taking ϕ ( s ) = s ) a/ and applying (1.8) to the sub-Laplacian∆ sub in either of examples (a) or (b) above, we get that the operator ϕ ( − ∆ sub ) =( I − ∆ sub ) − a/ is L p ( G )- L q ( G ) bounded and the inequality k f k L q ( G ) ≤ C k (1 − ∆ sub ) a/ ) f k L p ( G ) (1.14)holds true provided that a ≥ Q (cid:18) p − q (cid:19) , < p ≤ ≤ q < ∞ . (1.15)However, this yields the Sobolev embedding theorem which is well-known to be sharpat least in the case (b) of the Heisenberg group ([Fol75]), showing the sharpness ofTheorem 1.1 and hence also of the Fourier multiplier theorem (1.7). More examplesare given in Section 9.1.3. Lizorkin theorem.
The classical Lizorkin theorem [Liz67] applies for the range1 < p ≤ q < ∞ . Let A be a Fourier multiplier on R with the symbol σ A as in (1.2).Assume that for some C < ∞ the symbol σ A ( ξ ) satisfies the following conditionssup ξ ∈ R | ξ | p − q | σ A ( ξ ) | ≤ C, (1.16)sup ξ ∈ R | ξ | p − q +1 (cid:12)(cid:12)(cid:12)(cid:12) ddξ σ A ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (1.17)Then A : L p ( R ) → L q ( R ) is a bounded linear operator and k A k L p ( R ) → L q ( R ) . C. (1.18)An extension to G = R n with sharper conditions on the symbol has been obtainedin [STT10]. A number of papers [YT16, PST08, PST12] deal with the same problemon G = T n and G = R n .In Section 6 we establish versions of this result in two settings: Lizorkin typeFourier multiplier theorem on general locally compact separable unimodular groups,and Fourier multiplier theorem, spectral multiplier theorem, and L p - L q boundednessfor general, not necessarily invariant operators, on compact Lie groups. Our proofsare based on several new ingredients: Nikolskii inequality, approximations by trigono-metric type functions, Abel transform, and a a new type of difference operators on theunitary duals of compact Lie groups for measuring the required regularity of symbols.1.4. Other results.
Our proof of Fourier (and then also spectral) multiplier the-orems are based on two major new ingredients which are of interest on its own:Paley/Hausdorff-Young-Paley and Nikolskii inequalities for the H¨ormander and Li-zorkin versions of multiplier statements, respectively. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 7
Recall briefly that in [H¨or60] H¨ormander has shown the following version of the
Paley inequality on R n : if a positive function ϕ ≥ |{ ξ ∈ R n : ϕ ( ξ ) ≥ t }| ≤ Ct for t > , (1.19)then (cid:18) Z R n | b u ( ξ ) | p ϕ ( ξ ) − p dξ (cid:19) p . k u k L p ( R n ) , < p ≤ . (1.20)For the special case of ϕ ( ξ ) = (1 + | ξ | ) − n , this inequality implies the classical Hardy-Littlewood inequality [HL27] giving necessary condition for u to be in L p in terms ofits Fourier coefficients for 1 < p ≤
2. For functions with monotone Fourier coefficientssuch results serve as an extension of the Plancherel identity to L p -spaces: for example,on the circle T , Hardy and Littlewood have shown that for 1 < p < ∞ , if the Fouriercoefficients b f ( m ) are monotone, then one has f ∈ L p ( T ) if and only if X m ∈ Z (1 + | m | ) p − | b f ( m ) | p < ∞ . (1.21)Hardy-Littlewood inequalities on locally compact groups have been studied e.g. byKosaki [Kos81], see Theorem 2.8. In Section 3 we establish a version of Paley in-equality, and consequently of the Hausdorff-Young-Paley inequality on locally com-pact groups, yielding extensions of the Euclidean version (1.20) as well as of Kosaki’sresults. The established Hausdorff-Young-Paley inequality (Theorem 3.3) is a crucialingredient in our proof of H¨ormander’s version of multiplier theorem in Section 7.The crucial idea for our proof of Lizorkin theorem is the Nikolskii inequality, some-times also called the reverse H¨older inequality in the literature. Originating in Nikol-skii’s work [Nik51] in 1951 for trigonometric inequalities on the circle, on R n , the Nikolskii inequality takes the form k f k L q ( R n ) ≤ C [vol[conv[supp( b f )]] p − q k f k L p ( R n ) , ≤ p ≤ q ≤ ∞ , (1.22)for every function f ∈ L p ( R n ) with Fourier transform b f of compact support, whereconv( E ) denotes the convex hull of the set E. The Nikolskii inequality plays an impor-tant role in many questions of function theory, harmonic analysis, and approximationtheory. Its versions on compact Lie groups (and compact homogeneous manifolds)and on graded Lie groups have been established in [NRT16] ([NRT15]) and in [CR16],with further applications to Besov spaces and to Fourier multipliers acting in Besovspaces in those settings.In Section 5 we prove a version of the Nikolskii inequality on general locally compactseparable unimodular groups. An interesting question in this setting already is howto understand trigonometric functions in such generality. In Theorem 5.1 we showthat k f k L q ( G ) . (cid:16) τ ( P supp R [ b f ] ) (cid:17) p − q k f k L p ( G ) , < p ≤ min(2 , q ) , ≤ q ≤ ∞ , (1.23)with trigonometric function interpreted as having τ ( P supp R [ b f ] ) < ∞ , where P supp R [ b f ] denotes the orthogonal projector onto the support supp R [ b f ] of the operator-valuedFourier transform of f . The estimate (1.23) will play a crucial role in proving Lizorkintype multiplier theorems in Section 6. RAUAN AKYLZHANOV AND MICHAEL RUZHANSKY Notation and preliminaries
In this section fix the notation and briefly recall some preliminaries on von Neu-mann algebras to be used for developing subsequent harmonic analysis on locallycompact groups. For exposition purposes it seems beneficial to recall several generalnotions in the context of general von Neumann algebras M . However, for our appli-cation to multipliers on locally compact groups G we will be later setting M to be theright group von Neumann algebra VN R ( G ). In particular, we will be able to readilyapply the notion of noncommutative Lorentz spaces on M as developed in [Kos81],one of the key ingredients for our analysis.Let M ⊂ L ( H ) be a semifinite von Neumann algebra acting in a Hilbert space H with a trace τ . The semifinite assumption simplifies the formulations and is satisfiedin our main example M = VN R ( G ). Definition 2.1 (Affiliated operators) . A linear closed operator A (possibly un-bounded in H ) is said to be affiliated with M , symbolically AνM , if it commuteswith the elements of the commutant M ! of M , i.e. AU = U A, for all U ∈ M ! . (2.1)This relation ν is a natural relaxation of the relation ∈ : if A is a bounded op-erator affiliated with M , then by the double commutant theorem A ∈ M . One ofthe original motivations [MVN36, MvN37] of John von Neumann was to build amathematical foundation for quantum mechanics. In this framework, the observableswith unbounded spectrum correspond to closed densely defined unbounded operators.Although the algebra M consists primarily of bounded operators, the technique ofprojections makes it possible to approximate unbounded operators. Definition 2.2 ( τ -measurable operators S ( M )) . A closeable operator A (possiblyunbounded) affiliated with M is said to be τ -measurable if for each ε > p in M such that p H ⊂ D ( A ) and τ ( I − p ) ≤ ε . Here D ( A ) is the domainof A in H . We denote by S ( M ) the set of all τ -measurable operators.We note that the notion of τ -measurability does not appear in the classical theoryof Schatten classes since for M = L ( H ) we have S ( L ( H )) = L ( H ). Example 2.3.
Let M = { M ϕ : L ( X, µ ) ∋ f M ϕ f = ϕf ∈ L ( X, µ ) } ϕ ∈ L ∞ ( X,µ ) and take τ ( M ϕ ) := R X ϕdµ , where ( X, µ ) is a measure space. Then an operator M ϕ is τ -measurable if and only if ϕ is a µ -almost everywhere finite function.The ∗ -algebra S ( M ) is a basic constructon for the noncommutative integration.Let A = U | A | be the polar decomposition. The spectral theorem yields that | A | = Z Sp( | A | ) λdE λ ( | A | ) , (2.2)where { E λ ( | A | ) } λ ∈ Sp( | A | ) are the spectral projections associated with the operator | A | . Here dE λ ( | A | ) should be understood as the relative dimension function first con-structed in [MVN36]. Since A is affiliated with M , the projections satisfy E λ ( | A | ) ∈ M . Now, we are ready to ‘measure the speed of decay’ of the operator A . p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 9
Definition 2.4 (Generalised t -th singular numbers) . For an operator A ∈ S ( M ),define the distribution function d λ ( A ) by d λ ( A ) := τ ( E ( λ, + ∞ ) ( | A | )) , λ ≥ , (2.3)where E ( λ, + ∞ ) ( | A | ) is the spectral projection of | A | corresponding to the interval( λ, + ∞ ). For any t >
0, we define the generalised t -th singular numbers by µ t ( A ) := inf { λ ≥ d λ ( A ) ≤ t } . (2.4) Example 2.5.
For the operator M ϕ in Example 2.3, from Defintion 2.4 we can showits generalised t -th singular numbers to be µ t ( M ϕ ) = ϕ ∗ ( t ) , where ϕ ∗ ( t ) is the classical function rearrangement (see e.g. [BS88]).As a noncommutative extension [Kos81] of the classical Lorentz spaces, we defineLorentz spaces L p,q ( M ) associated with a semifinite von Neumann algebra M asfollows: Definition 2.6 (Noncommutative Lorentz spaces) . For 1 ≤ p < ∞ , 1 ≤ q < ∞ ,denote by L p,q ( M ) the set of all operators A ∈ S ( M ) satisfying k A k L p,q ( M ) := + ∞ Z (cid:16) t p µ t ( A ) (cid:17) q dtt q < + ∞ . (2.5)For q = ∞ , we define L p, ∞ ( M ) as the space of all operators A ∈ S ( M ) satisfying k A k L p, ∞ ( M ) := sup t> t p µ t ( A ) . (2.6)With this, for 1 ≤ p < ∞ , we can also define L p -spaces on M by k A k L p ( M ) := k A k L p,p ( M ) = + ∞ Z µ t ( A ) p dt p . The classical Lorentz spaces L p,q ( X, µ ) correspond to the case of commutative vonNeumann algebra. Modulus technical details [Dix81, p. 132, Theorem 1], an arbitraryabelian von Neumann algebra in a Hilbert space H is isometrically isomorphic to thealgebra { M ϕ } ϕ ∈ L ∞ ( X,µ ) from Example 2.3. Then noncommutative Lorentz spacescoincide with the classical ones: Example 2.7 (Classical Lorentz spaces) . Let M be the algebra { M ϕ } ϕ ∈ L ∞ ( X,µ ) fromExample 2.3 consisting of all the multiplication operators M ϕ : L ( X, µ ) ∋ f M ϕ f = ϕf ∈ L ( X, µ ). By Example 2.5, we have µ t ( M ϕ ) = ϕ ∗ ( t ) . Thus, the Lorentz space L p,q ( M ) consists of all operators M ϕ such that + ∞ Z [ t p ϕ ∗ ( t )] q dtt < + ∞ , which gives the classical Lorentz space. Concerning the structure of semifinite von Neumann algebras, given an arbitrarysemifinite von Neumann algebra M with a trace τ , there is an isomorphism of M onto a certain Hilbert algebra U ([Dix81, p. 99, Theorem 2]). Thus, we constructthe trace on the Hilbert algebra yielding the trace on M due to ismomorphism. Werefer to [Dix81], [Naj72] as well as to Section 4 for more details on this.Let now G be a locally compact unimodular separable group. Denote by π L ( g ) and π R ( g ) the left and the right action of G on L ( G ), respectively: π L ( g ) f ( x ) := f ( g − x ) ,π R ( g ) f ( x ) := f ( xg ) , and by VN L ( G ) the group von Neumann algebra generated by all the π L ( g ) with g ∈ G , i.e. VN L ( G ) := { π L ( g ) } !! g ∈ G , and similary VN R ( G ) := { π R ( g ) } !! g ∈ G , where !! is the bicommutant of the self-adjoint subalgebras { π L ( g ) } g ∈ G , { π R ( g ) } g ∈ G ⊂L ( L ( G )). It has been shown in [Seg49] thatVN L ( G ) ! = VN R ( G ) , (2.7)VN R ( G ) ! = VN L ( G ) . (2.8)We do not make assumption that G is either of type I or type II. The decompositiontheory for unitary representations of locally compact separable unimodular groupshas been established in [Ern61, Ern62].From now on we take M = VN R ( G ).For f ∈ L ( G ) ∩ L ( G ), we say that f on G has a Fourier transform whenever theconvolution operator R f h ( x ) := ( h ∗ f )( x ) = Z G h ( g ) f ( g − x ) dg (2.9)is a τ -measurable operator with respect to VN R ( G ), i.e. R f ∈ S (VN R ( G )). ThePlancherel identity takes ([Seg50, Theorem 3 on page 282]) the form k R f k L (VN R ( G )) = k f k L ( G ) . (2.10)In this setting, the Hausdorff-Young inequality has been established in [Kun58] inthe form k R f k L p ′ (VN R ( G )) ≤ k f k L p ( G ) , < p ≤ . (2.11)In [Kos81], as an application of the technique of the t -th generalised singular values,the Hardy-Littlewood theorem ([HL27]) has been generalised to an arbitrary locallycompact separable unimodular group G : Theorem 2.8 ([Kos81]) . Let < p ≤ and f ∈ L p ( G ) . Then we have k R f k L p ′ ,p (VN R ( G )) ≤ k f k L p ( G ) . (2.12) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 11
Remark 2.9.
The Plancherel equality (2.10) by Segal [Seg50] and Kosaki’s version[Kos81] of Hardy-Littlewood inequality (2.12) have been originally established for theleft convolution L f h = f ∗ h . However, the same line of reasoning yields inequalities(2.12) and (2.10) with the right convolution R f . We work with the right convolutionoperators R f here since it naturally corresponds to left-invariant operators whenanalysing the Fourier multipliers on groups.Using the technique of the t -th generalised singular values developed in [FK86],we can formulate both the Hausdorff-Young (2.11) and Hardy-Littlewood (2.12) in-equalities in the forms (for 1 < p ≤ + ∞ Z µ t ( R f ) p ′ dt p ′ ≡ k R f k L p ′ (VN R ( G )) ≤ k f k L p ( G ) , (2.13) + ∞ Z t p − µ t ( R f ) p dt p ≡ k R f k L p ′ ,p (VN R ( G )) ≤ k f k L p ( G ) . (2.14)In the sequel, when we prove Paley inequality in Theorem 3.1, the Hardy-Littlewoodinequalities (2.12) and (2.14) (for the right convolution R f ) will also follow as its spe-cial cases.2.1. Fourier multipliers on locally compact groups.
Let G be a locally compactseparable unimodular group. The first question is how to understand the notion ofFourier multipliers. In the first instance we adopt the following definition: Definition 2.10.
A linear operator A is said to be a left Fourier multiplier on G if A ∈ S (VN R ( G )).If we now recall Definition 2.1 we can see that A is a left Fourier multiplier on G if and only if A is affiliated with the right group von Neumann algebra VN R ( G ) andis τ -measurable. We can then clarify Definition 2.10 further: Remark 2.11.
For M = VN R ( G ) the operators affiliated with M are precisely those A that are left-invariant on G , namely, A is affiliated with VN R ( G ) ⇐⇒ Aπ L ( g ) = π L ( g ) A, for all g ∈ G. (2.15)Summarising this observation with Definition 2.10, left Fourier multipliers on G areprecisely the left-invariant operators that are measurable (in the sense of Definition2.2). Proof of Remark 2.11. = ⇒ . By Definition 2.10, we have AU = U A, for all U ∈ VN R ( G ) ! . (2.16)Then by (2.8), and by taking U = π L ( g ), g ∈ G , we see that A must be left-invariant. ⇐ =. We have Aπ L ( g ) = π L ( g ) A, for all g ∈ G. By definition, the algebra VN L ( G ) is the closure of the involutive subalgebra { π L ( g ) } g ∈ G ⊂ L ( L ( G )) in the strong operator topology. Therefore, we obtain AU = U A, for all U ∈ VN R ( G ) ! , (2.17)where we used (2.8). This completes the proof of Remark 2.11. (cid:3) Paley and Hausdorff-Young-Paley inequalities
Our analysis of L p - L q multipliers will be based on a version of the Hausdorff-Young-Paley inequality that we establish in this section in the context of locally compactgroups. It will be obtained by interpolation between the Hausdorff-Young inequalityand Paley inequality that we discuss first.We start first with an inequality that can be regarded as a Paley type inequality. Theorem 3.1 (Paley inequality) . Let G be a locally compact unimodular separablegroup. Let < p ≤ . Suppose that a positive function ϕ ( t ) satisfies the condition M ϕ := sup s> s Z t ∈ R + ϕ ( t ) ≥ s dt < + ∞ . (3.1) Then for all f ∈ L p ( G ) we have + ∞ Z µ t ( R f ) p ϕ ( t ) − p dt p ≤ M − pp ϕ k f k L p ( G ) . (3.2)As usual, the integral over an empty set in (3.1) is assumed to be zero.We note that taking ϕ ( t ) = t we recover Kosaki’s Hardy-Littlewood inequality(2.14). In this sense, the Paley inequality can be viewed as an extension of (one of)the Hardy-Littlewood inequalities. As a small byproduct of our proof of Theorem 3.1we thus get a simple proof of Theorem 2.8. Proof of Theorem 3.1.
Let ν give measure ϕ ( t ) to the set consisting of the singlepoint { t } , t ∈ R + , i.e. ν ( t ) := ϕ ( t ) dt. (3.3)We define the corresponding space L p ( R + , ν ), 1 ≤ p < ∞ , as the space of complex(or real) valued functions f = f ( t ) such that k f k L p ( R + ,ν ) := Z R + | f ( t ) | p ϕ ( t ) dt p < ∞ . (3.4)We will show that the sub-linear operator T : L p ( G ) ∋ f T f := µ t ( R f ) /ϕ ( t ) ∈ L p ( R + , ν )is well-defined and bounded from L p ( G ) to L p ( R + , ν ) for 1 < p ≤
2. In other words,we claim that we have the estimate k T f k L p ( R + ,ν ) = (cid:18)Z R + (cid:18) µ t ( R f ) ϕ ( t ) (cid:19) p ϕ ( t ) dt (cid:19) p . M − pp ϕ k f k L p ( G ) , (3.5) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 13 which would give (3.2), and where we set M ϕ := sup t> t R t ∈ R + ϕ ( t ) ≥ s dt . We will show that T is of weak-type (2,2) and of weak-type (1,1). More precisely, with the distributionfunction ν , we show that ν ( y ; T f ) ≤ (cid:18) M k f k L ( G ) y (cid:19) with norm M = 1 , (3.6) ν ( y ; T f ) ≤ M k f k L ( G ) y with norm M = M ϕ , (3.7)where ν is defined in (3.3). Recall that the distribution function ν ( y ; T f ) with respectto the weight ϕ is defined as ν ( y ; T f ) = Z t ∈ R + µt ( Rf ) ϕ ( t ) ≥ y ϕ ( t ) dt. Then (3.5) would follow from (3.6) and (3.7) by the Marcinkiewicz interpolationtheorem. Now, to show (3.6), using Plancherel’s identity (2.10), we get y ν ( y ; T f ) ≤ k T f k L p ( R + ,ν ) = Z R + (cid:18) µ t ( R f ) ϕ ( t ) (cid:19) ϕ ( t ) dt = Z R + µ t ( R f ) dt = k R f k L ( V N R ( G )) = k f k L ( G ) . Thus, T is of weak-type (2,2) with norm M ≤
1. Further, we show that T is ofweak-type (1,1) with norm M = M ϕ ; more precisely, we show that ν { t ∈ R + : µ t ( R f ) ϕ ( t ) > y } . M ϕ k f k L ( G ) y . (3.8)The left-hand side here is the integral R ϕ ( t ) dt taken over those t ∈ R + for which µ t ( R f ) ϕ ( t ) > y . From the definition of the Fourier transform it follows that µ t ( R f ) ≤ k f k L ( G ) . (3.9)Indeed, from the Definition 2.4, we have µ t ( R f ) ≤ k R f k L ( G ) → L ( G ) . The Young inequality for convolution (e.g. [Fol16, p. 52, Proposition 2.39]) yields k R f g k L ( G ) ≤ k f k L ( G ) k g k L ( G ) . Thus k R f k L ( G ) → L ( G ) ≤ k f k L ( G ) . This proves (3.9). Therefore, we have y < µ t ( R f ) ϕ ( t ) ≤ k f k L ( G ) ϕ ( t ) . Using this, we get (cid:26) t ∈ R + : µ t ( R f ) ϕ ( t ) > y (cid:27) ⊂ (cid:26) t ∈ R + : k f k L ( G ) ϕ ( t ) > y (cid:27) for any y >
0. Consequently, ν (cid:26) t ∈ R + : µ t ( R f ) ϕ ( t ) > y (cid:27) ≤ ν (cid:26) t ∈ R + : k f k L ( G ) ϕ ( t ) > y (cid:27) . Setting v := k f k L G ) y , we get ν (cid:26) t ∈ R + : µ t ( R f ) ϕ ( t ) > y (cid:27) ≤ Z t ∈ R + ϕ ( t ) ≤ v ϕ ( t ) dt. (3.10)We now claim that Z t ∈ R + ϕ ( t ) ≤ v ϕ ( t ) dt . M ϕ v. (3.11)Indeed, first we notice that we have Z t ∈ R + ϕ ( t ) ≤ v ϕ ( t ) dt = Z t ∈ R + ϕ ( t ) ≤ v dt ϕ ( t ) Z dτ. We can interchange the order of integration to get Z t ∈ R + ϕ ( t ) ≤ v dt ϕ ( t ) Z dτ = v Z dτ Z t ∈ R + τ ≤ ϕ ( t ) ≤ v dt. Further, we make a substitution τ = s , yielding v Z dτ Z t ∈ R + τ ≤ ϕ ( t ) ≤ v dt = 2 v Z s ds Z s ∈ R + s ≤ ϕ ( t ) ≤ v dt ≤ v Z s ds Z t ∈ R + s ≤ ϕ ( t ) dt. Since s Z t ∈ R + s ≤ ϕ ( t ) dt ≤ sup s> s Z t ∈ R + s ≤ ϕ ( t ) dt = M ϕ is finite by the assumption that M ϕ < ∞ , we have2 v Z s ds Z t ∈ R + s ≤ ϕ ( t ) dt . M ϕ v. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 15
This proves (3.11) and hence also (3.8). Thus, we have proved inequalities (3.6) and(3.7). Then by using the Marcinkiewicz interpolation theorem with p = 1, p = 2and p = 1 − θ + θ we now obtain Z R + (cid:18) µ t ( R f ) ϕ ( t ) (cid:19) p ϕ ( t ) dt p = k Af k L p ( R + ,ν ) . M − pp ϕ k f k L p ( G ) . This completes the proof of Theorem 3.1. (cid:3)
Further, we recall a result on the interpolation of weighted spaces from [BL76]:
Theorem 3.2 (Interpolation of weighted spaces) . Let dµ ( x ) = ω ( x ) dµ ( x ) ,dµ ( x ) = ω ( x ) dµ ( x ) , and write L p ( ω ) = L p ( ωdµ ) for the weight ω . Suppose that < p , p < ∞ . Then ( L p ( ω ) , L p ( ω )) θ,p = L p ( ω ) , where < θ < , p = − θp + θp , and ω = ω p − θp ω p θp . From this, interpolating between the Paley-type inequality (3.2) in Theorem 3.1and Hausdorff-Young inequality (2.13), we readily obtain an inequality that will becrucial for our consequent analysis of L p - L q multipliers: Theorem 3.3 (Hausdorff-Young-Paley inequality) . Let G be a locally compact uni-modular separable group. Let < p ≤ b ≤ p ′ < ∞ . If a positive function ϕ ( t ) , t ∈ R + , satisfies condition M ϕ := sup s> s Z t ∈ R + ϕ ( t ) ≥ s dt < ∞ , (3.12) then for all f ∈ L p ( G ) we have Z R + (cid:16) µ t ( R f ) ϕ ( t ) b − p ′ (cid:17) b dt b . M b − p ′ ϕ k f k L p ( G ) . (3.13)Naturally, this reduces to the Hausdorff-Young inequality (2.13) when b = p ′ andto the Paley inequality in (3.2) when b = p .4. Traces and singular numbers on von Neumann algebras
Some properties of traces shall be used in the proofs of our theorems. Moreover, wewill need to use Haagerup’s version of traces in the case of non-unimodular groups.Therefore, we give a brief background on traces summarising the results that will beused in the sequel. For some description of measurable fields of operators and linksto the representation theory and general von Neumann and C ∗ -algebras we refer to[FR16, Appendices B and C]. The following definition is taken from [Dix81, DefinitionI.6.1, p.93]: Definition 4.1.
Let M be a von Neumann algebra. A trace on the positive part M + = { A ∈ M : A ∗ = A > } of M is a functional τ defined on M + , taking non-negative, possibly infinite, real values, possessing the following properties: • If A ∈ M + and B ∈ M + , we have τ ( A + B ) = τ ( A ) + τ ( B ); • If A ∈ M + and λ ∈ R + , we have τ ( λA ) = λτ ( A ) (with the convention that0 · + ∞ = 0); • If A ∈ M + and if U is a unitary operator of M , then τ ( U AU − ) = τ ( A ).We say that τ is faithful (or exact) if the condition A ∈ M + , τ ( A ) = 0, imply that A = 0. We say that τ is finite if τ ( A ) < + ∞ for all A ∈ M + . We say that τ issemifinite if, for each A ∈ M + , τ ( A ) is the supremum of the numbers τ ( B ) overthose B ∈ M + such that B ≤ A and τ ( B ) < + ∞ . We say that τ is normal if, foreach increasing filtering set S ⊂ M + with supremum S ∈ M + , τ ( S ) is the supremumof { τ ( B ) } B ∈S . A von Neumann algebra M is said to be semifinite if there exists asemifinite faithful normal trace τ on M + .Let X be a Borel space, ν a positive measure on X and let λ
7→ H λ be a measurablefield of Hilbert spaces H λ . For every λ ∈ X , let A λ be an element of B ( H λ ), i.e. alinear bounded operator on H λ . The mapping λ A λ is called a field of boundedlinear operators over X . Measurable fields of operators associated to group vonNeumann algebras have been discussed in detail in [FR16, Appendix B]. Definition 4.2.
A measurable field { A λ } λ ∈ X is said to be essentially bounded ifthe essential supremum of the function λ
7→ k A λ k B ( H λ ) is finite. A linear operator A : H → H is said to be decomposable if it is defined as an essentially boundedmeasurable field { A λ } λ ∈ X . We then write A = M Z A λ dν ( λ ) . For every λ ∈ X , let M λ be a von Neumann algebra in H λ . The mapping λ M λ is called a field of von Neumann algebras over X . Definition 4.3.
A von Neumann algebra M ⊂ B ( H ) is said to be decomposable ifit is defined by a measurable field λ M λ of von Neumann algebras. We then write M = M Z M λ dν ( λ ) , Every abelian algebra C is isometrically isomorphic [Dix81, Theorem 1, p.132] to L ∞ ( X, ν ), where X is a locally compact space with a positive measure ν on X . Theimportance of abelian subalgebras is motivated by the following Theorem 4.4 ([vN49, Theorem 7, p. 460]) . Let M be a von Neumann algebra andlet C = M ∩ M ! ∼ = L ∞ ( X, ν ) be its center. Then M = L X R M λ dν ( λ ) , where each M λ is a factor, i.e. M λ ∩ M ! λ = multiplies of the identity operator I . We recall the basic result on central decomposition of traces of von Neumannalgebras.
Theorem 4.5.
Let M = L M λ dν ( λ ) be a semifinite decomposable von Neumannalgebra. Suppose that ν is standard. Then we have p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 17 (1)
The M λ ’s are semifinite almost everywhere. (2) Let τ be a semifinite faithful normal trace on M + . Then there exists a mea-surable field λ τ λ of semifinite faithful normal traces on the ( M λ ) + ’s suchthat τ = Z τ λ dν ( λ ) . (4.1)It can be seen that A ∈ M + if and only if A = ( A / ) ∗ A / . Example 4.6.
Let G be a locally compact unimodular group with VN R ( G ) the groupvon Neumann algebra generated by the right regular representation π R of G . Let A be a linear bounded operator commuting with the left regular representation. Thenby the double commutant theorem A ∈ VN R ( G ) and its action is given L ( G ) ∋ h Ah = h ∗ K A ∈ L ( G ) , where K A is its convolution kernel. We can define a trace τ on VN + R ( G ) by τ ( A ) := ( k K A k L ( G ) , if K A / ∈ L ( G ) , ∞ , otherwise . (4.2)Let C = VN R ( G ) ∩ VN R ( G ) ! . The abelian algebra C can be identified with theunitary dual b G via a canonical map. For more details we refer to [Dix77] or [Fol16,Theorem 7.37, p. 227]. The reduction theory allows us to decompose τ with respectto the center VN R ( G ) ∩ VN R ( G ) ! of the group algebra VN R ( G ) τ ( A ) = Z b G τ π ( A π ) dπ, (4.3)where A = L R A π dπ .The trace τ on M can also be extended to the ∗− algebra S ( M )of all τ -measurableoperators. Proposition 4.7.
Let ( M, τ ) be a von Neumann algebra and let A be a τ -measurablelinear operator. Assume that ϕ is a Borel function on sp( |L| ) ⊂ [0 , + ∞ ) . Then wehave τ ( ϕ ( | A | )) = + ∞ Z ϕ ( t ) dµ ( t ) , (4.4) where µ t = τ ( E t ) and |L| = + ∞ Z tdE t ( | A | ) . Although the equality (4.4) has been used to define a trace on the algeba S ( M ).However, the authors prefer to prove (4.4) while fully acknowledging the influence of[Hay14]. Proof of Proposition 4.7.
For the spectral measure we can take the family { E [0 ,t ) } t ≥ of spectral projections E (0 ,t ) corresponding to the intervals [0 , t ). The reader cancheck that the spectral measure axioms hold true.The trace τ is continuous with respect to τ -measure. In the view of the monotoneconvergence theorem (see [FK86, Theorem 3.5]) we can assume, without loss of gen-erality, that A is a bounded τ -measurable operator. Indeed, for every τ -measurableoperator | A | there exists a sequence { A n } of of τ -measurable bounded operators A n = n Z tdE t ( | A | ) ≤ A converging to A in τ -measure topology. Then, taking the limitlim n →∞ τ ( A n ) = τ ( A ) , we justify the claim. We notice that every Borel function can be uniformly ap-proximated by bounded Borel functions. Thus, we concentrate to establish (4.4)for bounded measurable A and bounded Borel functions ϕ on [0 , k A k B ( H ) ]. By thespectral mapping theorem sp( ϕ ( A )) = ϕ ([0 , k A k B ( H ) ])Let 0 ≤ λ ≤ λ ≤ . . . λ N be a partition of the interval ϕ ([0 , k A k B ( H ) ]). Then theRiemannian-like sums R N = N X k =1 λ k E ϕ − ( λ k − ,λ k ) ( | A | )converge to ϕ ( A ) in τ -measure topology. The trace τ on R N is given by τ ( R N ) = N Z k =1 λ k τ ( E ϕ − ( λ k − ,λ k ) ( | A | )) . (4.5)One can notice that sum in (4.5) is a Lebesuge integral sum N X k =1 λ k µ ( λ k − ,λ k ) for the integral k A k Z ϕ ( t ) dµ ( t ) , where we set the measure µ (( a, b )) = τ ( E ( a,b ) ) , ( a, b ) ⊂ [0 , k A k B ( H ) ]. (cid:3) For the sake of the exposition clarity we now formulate some properties of thedistribution function d A which we will be using in the proofs. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 19
Proposition 4.8.
Let A ∈ S ( M ) . Then we have d A ( µ A ( t )) ≤ t ; (4.6) µ A ( t ) > s if and only if t < d A ( s ); (4.7)sup t> t α µ A ( t ) = sup s> s [ d A ( s )] α for < α < ∞ . (4.8)The proof of this proposition is almost verbatim to the proof of [Gra08, Proposition1.4.5 on page 46]. The word ‘almost’ stands for the right-continuity of the non-commutative distribution function d A ( s ) which is discussed after [FK86, Definition1.3 on page 272]. Therefore, in the following proof we shall use the right-continuityof d A ( s ) without any justification. Proof of Proposition 4.8.
Let s n ∈ { s > d A ( s ) ≤ t } be such that s n ց µ A ( t ).Then d A ( s n ) ≤ t and the right-continuity of d A implies that d A ( µ A ( t )) ≤ t . Thisproves (4.6). Now, we apply this property to derive (4.7). If s < µ A ( t ) = inf { s > d A ( s ) ≤ t } , then s does not belong to the set { s > d A ( s ) ≤ t } = ⇒ d A ( s ) > t .Conversely, if for some t and s , we had µ A ( t ) < s , then the application of d A andproperty (4.7) would yield the contradiciton d A ( s ) ≤ d A ( µ A ( t )) ≤ t . Property (4.7)is established. Finally, we show (4.8). Given s >
0, pick ε satisfying 0 < ε < s .Property (4.7) yields µ A ( d A ( s ) − ε ) > s which implies thatsup t> t α µ A ( t ) ≥ ( d A ( s ) − ε ) α µ A ( d A ( s ) − ε ) > ( d A ( s ) − ε ) α s. (4.9)We first let ε → s > t >
0, pick 0 < ε < µ A ( t ). Property (4.7) yields that d A ( µ A ( t ) − ε ) >t . This implies that sup s> s ( d A ( s )) α ≥ ( µ A ( t ) − ε )( d A ( µ A ( t ) − ε )) α > ( µ A ( t ) − ε ) t α .We first let ε → t > (cid:3) We recall the following result which will be partially used.
Theorem 4.9 ([Seg53, Theorem 4, p. 412]) . If operators A and B are τ -measurablewith respect to a von Neumann algebra M , then so are A ∗ , A + B and AB , i.e. themaps + : M × M ∋ ( A, B ) A + B ∈ M, (4.10) · : M × M ∋ ( A, B ) AB ∈ M, (4.11) ∗ : M ∋ A A ∗ ∈ M (4.12) are well-defined. Here we formulate some properties of µ t that we use in the proof of Theorem 7.1. Lemma 4.10 ([FK86, Lemma 2.5, p. 275]) . Let
A, B be τ -measurable operators.Then the following properties hold true. (1) The map (0 , + ∞ ) ∋ t µ t ( A ) is non-increasing and continuous from theright. Moreover, lim t → µ t ( A ) = k A k ∈ [0 , + ∞ ] . (4.13) (2) µ t ( A ) = µ t ( A ∗ ) . (4.14)(3) µ t + s ( AB ) ≤ µ t ( A ) µ s ( B ) . (4.15)(4) µ t ( ACB ) ≤ k A kk B k µ t ( C ) , for any τ -measurable operator C. (4.16)(5) For any continuous increasing function f on [0 , + ∞ ) we have µ t ( f ( | A | )) = f ( µ t ( | A | )) . (4.17)In Lemma 4.10, we formulate only the properties we use, whereas in [FK86, Lemma2.5, p. 275] the reader can find more details.5. Nikolskii inequality on locally compact groups
In this section we establish the Nikolskii inequality (sometimes called the reverseH¨older inequality) in the setting of locally compact groups. It will be instrumentalin proving the Lizorkin type multiplier theorem in Theorem 6.2.Let G be a locally compact unimodular separable group and VN R ( G ) its rightgroup von Neumann algebra with trace τ . We shall denote by F R [ f ] the right Fouriertransform of f ∈ L ( G ), i.e. F R [ f ] = R f : L ( G ) ∋ h
7→ F R [ f ]( h ) = R f [ h ] = h ∗ f ∈ L ( G ) . (5.1)The reason to introduce the new notation F R [ f ] is to emphasise the connection withthe classical Nikolskii inequality [Nik51].Let us denote by supp R ( b f ) the subspace of L ( G ) orthogonal to the kernel Ker( F R [ f ])of the Fourier transform F R [ f ], i.e.supp R [ b f ] := Ker (cid:0) F R [ f ] (cid:1) ⊥ , (5.2)where Ker( F [ f ]) ⊂ L ( G ) is the kernel of the operator F R [ f ] in (5.1).We note that the classical Nikolskii inequality is an L p - L q estimate for norms ofthe same functions for p < q so that the Fourier transforms of the functions underconsideration must have bounded support.In [NRT16, NRT15] Nikolskii inequality has been established on compact Lie groupsand on compact homogeneous manifolds, respectively, for functions with boundedsupport of the noncommutative Fourier coefficients. In [CR16] the Nikolskii inequal-ity was proved in the setting of graded groups: Let G be a graded Lie group ofhomogeneous dimension Q and let R be a positive Rockland operator of order ν . Forevery L > T L f := χ L ( R ) f, (5.3)where χ L is the characteristic function of the interval [0 , L ]. In these notations, itwas shown in [CR16, Theorem 3.1] that we have k T L k L q ( G ) ≤ CL Qν ( p − q ) k T L k L p ( G ) , ≤ p ≤ q ≤ ∞ , (5.4)with constant C explicitly depending on the spectral resolution of the Rocklandoperator R . p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 21
The main question in the setting of locally compact groups is to find an analogueof the condition for bounded support of Fourier transforms since we may not have acanonical operator to use its spectral decomposition for the definition of the boundedspectrum.The version that will work well for our purposes is the following.Let P supp R [ b f ] be the orthogonal projector onto the support supp R [ b f ]. We say that f ∈ L ( G ) has bounded spectrum if τ ( P supp R [ b f ] ) < + ∞ . Theorem 5.1.
Let G be a locally compact separable unimodular group. Let < q ≤∞ and < p ≤ min(2 , q ) . Assume that τ ( P supp R [ b f ] ) < ∞ , where P supp R [ b f ] denotes theorthogonal projector onto the support supp R [ b f ] . Then we have k f k L q ( G ) . (cid:16) τ ( P supp R [ b f ] ) (cid:17) p − q k f k L p ( G ) , (5.5) with the constant in (5.5) independent of f . We shall call (5.5) the
Nikolskii inequality on topological groups.In the case of graded Lie groups we have τ ( P supp R [ b f ] ) = L Qν , so that (5.5) in Theorem5.1 recovers (5.4) for the corresponding ranges of exponents p and q .In [NRT16, NRT15] Nikolskii inequality has been established on compact Lie groupsand on compact homogeneous manifolds, respectively. We prove Theorem 5.1 alongthe lines of the proof in [NRT16] adapting the latter to the setting of locally compactgroups. Proof of Theorem 5.1.
We will give the proof of (5.5) in three steps. We can abbre-viate F R [ f ] in the proof to simply writing F [ f ].Step 1. The case p = 2 and q = ∞ . We have (by e.g. [Hay14, Proposition A.1.2. p.216]) that (cid:12)(cid:12)(cid:12) Tr( b f ( π ) π ( x )) (cid:12)(cid:12)(cid:12) ≤ Tr (cid:12)(cid:12)(cid:12) b f ( π ) (cid:12)(cid:12)(cid:12) , x ∈ G. (5.6)We notice that F [ f ] P supp R [ b f ] = F [ f ] . Then by [FK86, Lemma 2.6, p. 277], we have µ s ( F [ f ]) = 0 , s ≥ τ ( P supp R [ b f ] ) . (5.7)From now on we shall denote t := τ ( P supp R [ b f ] ) throughout the proof. Further, theapplication of [FK86, Proposition 2.7, p.277] yields τ ( |F [ f ] | ) = ∞ Z µ s ( F [ f ]) ds = τ ( P supp R [ b f ] ) Z µ s ( F [ f ]) ds, (5.8) where we used (5.7) in the last equality. Combining (5.8) and (5.6), we obtain k f k L ∞ ( G ) ≤ Z b G Tr (cid:12)(cid:12)(cid:12) b f ( π ) (cid:12)(cid:12)(cid:12) dπ = τ ( |F [ f ] | ) = t Z µ s ( F [ f ]) ds ≤ t Z ds t Z µ s ( F [ f ]) ds = q τ ( P supp R [ b f ] ) k f k L ( G ) , (5.9)where in the last inequality we used the Plancherel identity.Step 2. The case p = 2 and 2 < q ≤ ∞ . We take 1 ≤ q ′ < q + q ′ = 1 . We set r := q ′ so that its dual index r ′ satisfies r ′ = 1 − q ′ . By the Hausdorff-Younginequality in (2.13), and by H¨older’s inequality, we obtain k f k L q ( G ) ≤ kF [ f ] k L q ′ (VN R ( G )) = t Z µ q ′ s ( F [ f ]) ds q ′ ≤ t Z ds q ′ r ′ t Z µ q ′ rs ( F [ f ]) ds q ′ r = t Z ds q ′ − t Z µ s ( F [ f ]) ds ≤ t Z ds q ′ − ∞ Z µ s ( F [ f ]) ds = τ ( P supp R [ b f ] ) − q k f k L ( G ) , where we have used that q ′ r = 1.Step 3. If p = min(2 , q ) and p = 2, then p = q and there is nothing to prove. For1 < p < min(2 , q ), we claim to have k f k L q ( G ) ≤ τ ( P supp R [ b f ] ) (1 /p − /q ) k f k L p ( G ) . Indeed, if q = ∞ , for f
0, we get k f k L = k| f | − p/ | f | p/ k L ≤ k| f | − p/ k L ∞ k| f | p/ k L = k f k − p/ L ∞ k| f | p/ k L = k f k L ∞ k f k − p/ L ∞ k| f | p/ k L = k f k L ∞ k f k − p/ L ∞ k f k p/ L p ≤ τ ( P supp R [ b f ] ) / k f k L k f k − p/ L ∞ k f k p/ L p , (5.10)where we have used (5.9) in the last line. Therefore, using that f
0, we have k f k L ∞ ≤ τ ( P supp R [ b f ] ) /p k f k L p . (5.11) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 23
For p < q < ∞ we obtain k f k L q = k| f | − p/q | f | p/q k L q ≤ k f k − p/qL ∞ k f k p/qL p ≤ τ ( P supp R [ b f ] ) /p (1 − p/q ) k f k − p/qL p k f k p/qL p = τ ( P supp R [ b f ] ) p − q k f k L p , (5.12)where we have used (5.11). (cid:3) Let us now discuss the closure of the space of functions having bounded spectrum.We show that the situation is quite subtle, already in the case of compact Lie groups.This space will naturally appear in the formulations of Lizorkin type theorems. Itwill be convenient to measure the rate of growth of the trace of projections to thesupport of the spectrum: given a function w = w ( t ) controlling such growth one canuse it when passing to the limit from functions with bounded spectrum. Definition 5.2 (The space L pw ( G )) . Let w = w ( t ) ≥ L pw ( G ) the space of functions f ∈ L p ( G ) for which there existsa sequence { f t } t> of functions f t ∈ L p ( G ) with bounded spectrum such that k f − f t k L p ( G ) → t → ∞ , (5.13) τ ( P supp R [ b f t ] ) ≤ w ( t ) . (5.14)It has been shown by Stanton [Sta76] that for class functions on semisimple compactLie groups the polyhedral Fourier partial sums S N f converge to f in L p provided that2 − s +1 < p < s . Here the number s depends on the root system R of the compactLie group G . We also note that the range of indices p as above is sharp, see Stantonand Tomas [ST76, ST78] as well as Colzani, Giulini and Travaglini [CGT89]. Werefer to Appendix A for more details. The mean convergence of polyhedral Fouriersums has been investigated in [ST76, Sta76, ST78]. These results allow us to describespaces L pw ( G ) on compact semisimple Lie groups for suitable choice of w in Example5.3 below. Example 5.3.
Let G be a compact semisimple Lie group of dimension n and rank l , and let T l be its maximal torus. We say that a function f is central (or class) if f ( xg ) = f ( gx ) , x ∈ T l , g ∈ G. We write L p inv ( G ) for the space of central functions f ∈ L p ( G ). Let λ , λ , . . . , λ , . . . denote the eigenvalues of the n -th order pseudo-differential operator ( I − L G ) n counted with multiplicities, where L G is the bi-invariant Laplacian (Casimir element)on G . We shall enumerate elements π of the unitary dual b G of G using the eigenvalues { λ k } ∞ k =1 , i.e. ( I − L G ) n π kmn = λ k π kmn . (5.15)Let Q N ⊂ b G be a finite polyhedron of N -th order and take w ( N ) to be the numberof the eigenvalues λ k enumerating the elements π k in Q N , i.e. w ( N ) = X k ∈ N λ k ∈ Q N . (5.16) Then L pw ( G ) = ( L p inv ( G ) , − s < p < s , ∅ , otherwise . (5.17)We refer to Appendix A for more details on the approximations by trigonometricfunctions on compact Lie groups.6. Lizorkin theorem
In this section we prove an analogue of the Lizorkin theorem for the L p - L q bound-edness of Fourier multipliers for the range of indices 1 < p ≤ q < ∞ . We recall theclassical Lizorkin theorem on the real line R : Theorem 6.1 ([Liz67]) . Let < p ≤ q < ∞ and let A be a Fourier multiplier on R with the symbol σ A , i.e. Af ( x ) = Z R e πixξ σ A ( ξ ) b f ( ξ ) dξ. Assume that the symbol σ A ( ξ ) satisfies the following conditions sup ξ ∈ R | ξ | p − q | σ A ( ξ ) | ≤ C < ∞ , (6.1)sup ξ ∈ R | ξ | p − q +1 (cid:12)(cid:12)(cid:12)(cid:12) ddξ σ A ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (6.2) Then A : L p ( R ) → L q ( R ) is a bounded linear operator and k A k L p ( R ) → L q ( R ) . C. (6.3)There have been recent works extending this statement to higher dimension, as wellas to the operators on the torus, see e.g. [STT10, YT16, PST08, PST12] deal with thesame problem on G = T n and G = R n . We start by proving an analogue of Theorem6.1 on general locally compact groups. Consequently, we also derive a version of suchtheorem on compact Lie groups when symbolic analysis is also possible.6.1. Locally compact groups.
The available information about operators (Fouriermultipliers) on locally compact groups is the spectral information measured in termsof the generalised t -singular numbers discussed in Section 2. The following statementgives a condition for the L p - L q boundedness in the spaces L pw which are the closureof the spaces of functions with bounded spectrum as discussed in Definition 5.2. Theorem 6.2.
Let G be a locally compact unimodular separable group and let A be aleft Fourier multiplier on G . Let < p ≤ min(2 , q ) and < q ≤ ∞ . Let w = w ( t ) ≥ be a locally integrable function. Then we have k A k L pw ( G ) → L qw ( G ) . sup t> w ( t ) p − q µ t ( A ) + + ∞ Z w ( t ) p − q (cid:18) − ddt (cid:19) µ t ( A ) dt. (6.4)The function µ t ( A ) is non-increasing and continuous from the right. Therefore, ithas derivative µ t ( A ) that exists and is finite almost everywhere with respect to theLebesgue measure on R + . p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 25
Proof of Theorem 6.2.
Let us decompose functions f ∈ L pw ( G ) and g ∈ L p ′ w ( G ) as f = f − f , g = g − g , into functions f i , g i with positive Fourier transform, i.e. with R f i ≥ , R g j ≥ , i, j = 1 , . By the linearity of the Fourier transform F : f R f , we have (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( − i + j ( Af i , g j ) L ( G ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X i,j =1 (cid:12)(cid:12) ( Af i , g j ) L ( G ) (cid:12)(cid:12) . Hence, without loss of generality, we may further assume that R f ≥ R g ≥ k Af t k L q ( G ) . w ( t ) p − q µ t ( A ) + t Z w ( s ) p − q (cid:20) − dds µ s ( A ) (cid:21) ds k f t k L p ( G ) (6.5)for arbitrary function f t with τ ( P supp R [ b f t ] ) ≤ w ( t ). By the Plancherel identity, wehave (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) = (cid:12)(cid:12) τ ( R Af R ∗ g ) (cid:12)(cid:12) . By using this and the hypothesis that A is a left Fourier multiplier, i.e. R Af = AR f and inequality | τ ( · ) | ≤ τ ( | · | ) from e.g. [Hay14, Proposition A.1.2. p. 216], we get (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) = (cid:12)(cid:12) τ ( R Af R ∗ g ) (cid:12)(cid:12) ≤ τ ( (cid:12)(cid:12) AR f R ∗ g (cid:12)(cid:12) )= + ∞ Z µ s ( AR f R ∗ g ) ds . + ∞ Z µ s ( A ) µ s ( R f R ∗ g ) ds, (6.6)where in the second line we used [FK86, Proposition 2.7, p. 227] to express τ ( (cid:12)(cid:12) AR f R ∗ g (cid:12)(cid:12) )via the s -th generalised singular values µ s ( AR f R ∗ g ) , s ∈ R + . In the third line in (6.6)we made the substitution s → s and used the sub-multiplicativity µ s ( · ) ≤ µ s ( · ) µ s ( · )of the generalised singular values µ t (see Lemma 4.10). By definition R f P supp R [ b f ] = R f . Hence, by [FK86, Lemma 2.6] we get µ s ( R f ) = 0 , s > w ( t ) ≥ τ ( P supp Rp [ b f ] ) . (6.7)Therefore, we get µ s ( R f R ∗ g ) = 0 , s > w ( t ) ≥ τ ( P supp R [ b f ] ) (6.8)in view of µ s ( R f R ∗ g ) ≤ µ s ( R f ) µ s ( R ∗ g ). Taking into account (6.8), we get from (6.6) (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) ≤ w ( t ) Z µ s ( A ) µ s ( R f R ∗ g ) ds. (6.9) Now, applying Abel transform to (6.9), we get w ( t ) Z µ s ( A ) µ s ( R f R ∗ g ) ds = µ s ( A ) s Z µ u ( R f R ∗ g ) du (cid:12)(cid:12)(cid:12) s =2 w ( t ) s =0 + w ( t ) Z (cid:20) − dds µ s ( A ) (cid:21) s Z µ u ( R f R ∗ g ) du ds. (6.10)By the Nikolskii inequality (5.5), we get w ( t ) Z µ s ( R f R ∗ g ) ds = ( f, g ) L ( G ) ≤ k f k L q ( G ) k g k L q ′ ( G ) . τ ( P supp R [ b f ] ) p − q k f k L p ( G ) k g k L q ′ ( G ) ≤ w ( t ) p − q k f k L p ( G ) k g k L q ′ ( G ) . (6.11)Combining (6.6), (6.11) and (6.10), we get (cid:12)(cid:12) ( Af t , g ) L ( G ) (cid:12)(cid:12) . w ( t ) p − q µ t ( A ) + w ( t ) Z w ( s ) p − q (cid:20) − dds µ s ( A ) (cid:21) ds k f t k L p ( G ) k g k L q ′ ( G ) . By the L p -space duality, we immediately get k Af t k L q ( G ) . w ( t ) p − q µ t ( A ) + w ( t ) Z w ( s ) p − q (cid:20) − dds µ s ( A ) (cid:21) ds k f t k L p ( G ) , (6.12)for every f with τ (supp R [ b f ]) ≤ w ( t ). Taking supremum over all such f , we finallyobtain k Af k L q ≤ sup t> w ( t ) p − q µ t ( A ) + + ∞ Z w ( s ) p − q (cid:20) − dds µ s ( A ) (cid:21) ds. (6.13)This completes the proof of Theorem 6.2. (cid:3) Remark 6.3.
It is not restrictive to take w = w ( t ) in Theorem 6.2 such that w ∈ C is increasing and w (0) = 0. In this case integrating by parts in (6.5) we get k Af t k L q ( G ) . t Z w ( s ) p − q − w ′ ( s ) µ s ( A ) ds k f t k L p ( G ) . (6.14)Passing to the limit, we get a sufficient condition for the L p - L q boundedness in termsof the derivative of w instead of the derivative of the µ s ( A ), namely k A k L pw ( G ) → L qw ( G ) . ∞ Z w ( s ) p − q − w ′ ( s ) µ s ( A ) ds. (6.15) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 27
We also have the following corollary of Theorem 6.2 and its proof, estimating the L p - L q norm by the Lorentz norm of the operator in the group von Neumann algebra,see Definition 2.6. Corollary 6.4.
Let G be a locally compact unimodular separable group and let A bea left Fourier multiplier on G . Let < p ≤ min(2 , q ) and take w ( t ) = t . Then wehave k A k L pw ( G ) → L qw ( G ) . k A k L r, (VN R ( G )) , (6.16) where r = p − q .Proof of Corollary 6.4. Let us recall inequality (6.5) from the proof of Theorem 6.2: k Af t k L q ( G ) k f t k L p ( G ) . t p − q µ t ( A ) − t Z s p − q (cid:20) dds µ s ( A ) (cid:21) ds. (6.17)Integrating by parts in the right-hand side of (6.17), we get k Af t k L q ( G ) k f t k L p ( G ) ≤ t p − q µ t ( A ) − t p − q µ t ( A ) + (cid:18) p − q (cid:19) t Z s p − q µ s ( A ) dss = (cid:18) p − q (cid:19) k A k L r, (VN R ( G )) , with r = p − q , in view of Definition 2.6. (cid:3) Compact Lie groups.
Since on compact Lie groups the symbolic calculus isavailable it is also natural to search for conditions expressed in terms of the matrix-valued symbol of the invariant operator under consideration.In order to measure the regularity of the symbol, we introduce a new family ofdifference operators b ∂ acting on Fourier coefficients and on symbols. These operatorsare used to formulate and prove a version of the Lizorkin theorem on compact groups.Let G be a compact Lie group of dimension n . Let λ , λ , . . . , λ N , . . . denote theeigenvalues of the n -th order elliptic pseudo-differential operator ( I − L G ) n countedwith multiplicities. We shall enumerate elements π of the unitary dual b G of G viathe eigenvalues { λ k } ∞ k =1 , i.e. ( I − L G ) n π kmn = λ k π kmn . (6.18)Let σ = { σ ( π ) } π ∈ b G be a field of operators and define b ∂ π σ ( π j ) := U π µ ( σ ( π j )) − µ ( σ A ( π j +1 )) 0 ... µ ( σ ( π j )) − µ ( σ A ( π j +1 )) ... ... ... ... ... µ k ( σ ( π j )) − µ k ( σ ( π j +1 )) ... ... µ dπj ( σ ( π j )) , (6.19)where U π is a partial isometry matrix in the polar decomposition σ ( π ) = U π | σ ( π ) | .Here µ k ( σ ( π )) are the singular numbers of σ ( π ) written in the descending order. Now, using the direct sum decomposition Op( σ ) = L π ∈ b G d π σ ( π ), we can also lift thedifference operators ∂ π to Op( σ ) by b ∂ Op( σ ) = M π ∈ b G d π b ∂σ A ( π ) . (6.20)Let G be a compact semisimple Lie group and let T be its maximal torus. Werecall that a function f is central if f ( xg ) = f ( gx ) , x ∈ T , g ∈ G. We write L p inv ( G ) for the space of central functions f ∈ L p ( G ). The number s in thetheorem below is defined by the root datum of G , see Appendix A for the precisedefinition. The appearance of the bounds for p and q involving s is caused by thedensity properties of the space of polyhedral trigonometric polynomials in L p for suchrange of indices and the failure of such density otherwise, see Stanton [Sta76], whichis explained in more detail in Appendix A. Theorem 6.5.
Let − s < p ≤ q < s and let A be a left Fourier multiplieron a compact semisimple Lie group G of dimension n . Let − p ≤ m < . Then wehave k A k L p inv ( G ) → L q inv ( G ) . sup π ∈ b G h π i n ( p − q ) k σ A ( π ) k op + sup π ∈ b G h π i n ( p − q + m ) k b ∂σ A ( π ) k op . (6.21) We also have k A k L p inv ( G ) → L q inv ( G ) . sup π ∈ b G h π i n ( p − q ) k σ A ( π ) k op + X π ∈ b G h π i n ( p − q ) k b ∂σ A ( π ) k op . (6.22)Here 2 − s < p < s is determined by the sharp conditions on mean summability([CGT89]), see Appendix A. Proof of Theorem 6.5.
We will sometimes denote r = p − q . It is sufficient to establishinequality (6.21) for polyhedral partial sums f N , i.e. for all N ∈ N , k Af N k L q inv ( G ) . sup π ∈ b G h π i n ( p − q ) k σ A ( π ) k op + X π ∈ b G h π i n ( p − q ) k b ∂σ A ( π ) k op k f N k L p inv ( G ) . (6.23)Let us decompose functions f ∈ L p ( G ) and g ∈ L p ′ ( G ) as f = f − f , g = g − g , into functions f i , g i with positive Fourier transform, i.e. R f i ≥ , R g j ≥ , i, j = 1 , , by taking f = X π ∈ Q d π Tr( b f ( π ) π ) ,f = X π ∈ Q d π Tr( b f ( π ) π ) , p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 29 where Q = { π ∈ b G : b f ( π ) ≥ } ,Q = { π ∈ b G : b f ( π ) < } . By the linearity of the Fourier transform F : f R f , we have (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( − i + j ( Af i , g j ) L ( G ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X i,j =1 (cid:12)(cid:12) ( Af i , g j ) L ( G ) (cid:12)(cid:12) . Hence, without loss of generality, we may assume that R f ≥ R g ≥
0. Then bythe Plancherel identity, we have (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) = (cid:12)(cid:12) τ ( R Af R ∗ g ) (cid:12)(cid:12) . By using this and the hypothesis that A is a left Fourier multiplier, i.e. R Af = AR f and inequality | τ ( · ) | ≤ τ ( | · | ) (see e.g. [Hay14, Proposition A.1.2. p. 216]), we get (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) τ ( R Af R ∗ g ) (cid:12)(cid:12) ≤ τ ( (cid:12)(cid:12) AR f R ∗ g (cid:12)(cid:12) ) . Now, we decompose the group von Neumann algebra VN R ( G ) with respect to itscenter C = VN R ( G ) ∩ VN R ( G ) ! . This yields the trace decomposition τ = ⊕ b G R τ π and we get τ ( (cid:12)(cid:12) AR f R ∗ g (cid:12)(cid:12) ) = X π ∈ b G d π τ π h σ A ( π ) b f ( π ) b g ( π ) ∗ i = X π ∈ b G d π X t =1 d π µ t h σ A ( π ) b f ( π ) b g ( π ) ∗ i ≤ X π ∈ b G [ dπ ] X t =1 d π µ t h σ A ( π ) b f ( π ) b g ( π ) ∗ i + X π ∈ b G [ dπ +12 ] X t =1 d π µ t − h σ A ( π ) b f ( π ) b g ( π ) ∗ i ≤ X π ∈ b G [ dπ ] X t =1 d π µ t h σ A ( π ) b f ( π ) b g ( π ) ∗ i + X π ∈ b G [ dπ +12 ] X t =1 d π µ t − h σ A ( π ) b f ( π ) b g ( π ) ∗ i ≤ X π ∈ b G [ dπ ] X t =1 d π µ t h σ A ( π )] µ t [ b f ( π ) b g ( π ) ∗ i + X π ∈ b G [ dπ +12 ] X u =1 d π µ u − h σ A ( π )] µ u − [ b f ( π ) b g ( π ) ∗ i ≤ X π ∈ b G d π X t =1 d π µ t [ σ A ( π )] µ t h b f ( π ) b g ( π ) ∗ i , where in the inequalities we made the substitution t → t and used the sub-multi-plicativity µ t ( · ) ≤ µ t ( · ) µ t ( · ) of the singular values µ t . We define all the singularnumbers µ t to be zero for t > d π , π ∈ b G , i.e. µ t ( b f ( π )) = 0 , µ t ( b g ( π )) = 0 , µ t ( σ A ( π )) = 0 , for t > d π , π ∈ b G. We have thus only to show that (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) . X π ∈ b G ∞ X t =1 d π µ t [ σ A ( π )] µ t h b f ( π ) b g ( π ) ∗ i , f ∈ L p ( G ) , g ∈ L q ′ ( G ) . (6.24) Further, we take f = f N = P π ∈ Q N d π Tr( b f ( π ) π ) in (6.24) and index the N -th orderpolyhedron Q N ⊂ b G by the eigenvalues of ( I − L G ) dim( G )2 , i.e. Q N = { π k } b N k = a N ⊂ b G and { a , . . . , a N } correspond to the indices k of such λ k ’s that π k ∈ Q N , i.e.( I − L G ) dim( G )2 π kmn = λ k π kmn , k = a , . . . , a N , π k ∈ Q N . Recall that the N -th order polyhedron Q N is defined via the highest weight theory Q N = { π ∈ b G : π i ≤ ρ i N, i = 1 , . . . , l } , where l is the rank of the group G and ( π , π , . . . , π l ) are the highest weightsof π and ρ is the half-sum of the positive roots of G . Hence, with every subset { λ k } a m k = a { a , . . . , a m } , m ≤ N we associate a polyhedron Q N m Q N m = { π ki ≤ ρ i N m } a m k = a , where N m is the minimum over all N ′ such that π ki ≤ ρ i N, k = a , . . . , a m . Weshall agree that the sum over Q N runs via the eigenvalues λ k , k = a , . . . , a N , withmultiplicities, i.e. X π ∈ Q N a N X k = a . (6.25)Changing the order of summation in (6.24) and using the convention above, we get (cid:12)(cid:12) ( Af N , g ) L ( G ) (cid:12)(cid:12) ≤ ∞ X t =1 k = a N X k = a d π k µ t (cid:2) σ A ( π k ) (cid:3) µ t h b f ( π k ) b g ( π k ) ∗ i . (6.26)Now, we shall write α t,k = µ t [ σ A ( π k )] , β t,k = µ t [ b f ( π k ) b g ( π k ) ∗ ] . (6.27)Let us apply the Abel transform with respect to k in the right hand side of (6.26): a N X k = a α t,k d π k β t,k = α t,a N a N X w = a d π w β t,w + a N X k = a (∆ k α t,k ) a N X w = a d π w β t,w , where ∆ k α t,k = α t,k − α t +1 ,k . Combining this with (6.24), we get (cid:12)(cid:12) ( Af, g ) L ( G ) (cid:12)(cid:12) ≤ ∞ X t =1 α t a N a N X w = a d π w β t,w + ∞ X t =1 a n − X k = a ∆ k α t,k k X w = a d π w β t,w ≤ α a N ∞ X t =1 a N X w = a d π w β t,w + a N − X k = a sup t ∈ N ∆ k α t,k ∞ X t =1 k X w = a d π w β t,w . p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 31
Interchanging the order of summation and applying the Plancherel formula, we get ∞ X t =1 k X w = a d π w µ t h b f ( π w ) b g ( π w ) ∗ i = k X w = a ∞ X t =1 d π w µ t h b f ( π w ) b g ( π w ) ∗ i = k X w = a d π w d πw X t =1 µ t h b f ( π w ) b g ( π w ) ∗ i = k X w = a d π w Tr[ b f ( π w ) b g ( π w ) ∗ ] = ( f ( a ,k ) , g ) L ( G ) ≤ k f ( a ,k ) k L q ( G ) k g k L q ′ ( G ) , where we write f k = k X w = a d π w Tr( b f ( π w ) π w ) . (6.28)Collecting these estimates, we obtain (cid:12)(cid:12) ( Af N , g ) L ( G ) (cid:12)(cid:12) ≤ α a N k f N k L q ( G ) + a N − X k = a sup t ∈ N ∆ k α t,k k f k k L q ( G ) ! k g k L q ′ ( G ) . By the duality of L p -spaces we immediately get k Af N k L q ( G ) . α a N k f N k L q ( G ) + a N − X k = a sup t ∈ N ∆ π α t,k k f k k L q ( G ) (6.29)We now claim that the following version of the Nikolskii-Bernshtein inequality holdstrue: k f N k L q ( G ) . λ p − q a N k f N k L p ( G ) , < p ≤ q ≤ ∞ , a, b ∈ N . (6.30)Moreover, the composition of the Banach-Steinhaus theorem and inequality (6.30)yields k f N k L q ( G ) . λ p − q a N k f k L p ( G ) , < p ≤ q ≤ ∞ , N ∈ N . (6.31)Assuming this to be true for a moment, and using inequality (6.30), we get k Af N k L q ( G ) . α a N λ p − q a N k f k L p ( G ) + a N − X k = a sup t ∈ N ∆ π α t,k k f k k L q ( G ) (6.32)At this point, we note that by Nikolsky inequality (6.30) and the Banach-Steinhaustheorem, we immediately get from (6.32) k Af N k L q ( G ) . α a N λ p − q a N k f k L p ( G ) + a N − X k = a sup t ∈ N ∆ π α t,k ! k f k L q ( G ) . We show how to pass to the limit N → ∞ in k Af N k L q ( G ) . The Fourier series X π ∈ Q N d π Tr σ A ( π ) b f ( π ) π ( x ) is absolutely convergent since k σ A ( π ) k op ≤ h π i n ( p − q ) and f ∈ C ∞ ( G ). Indeed, wehave X π ∈ Q N d π (cid:12)(cid:12)(cid:12) Tr σ A ( π ) b f ( π ) π ( x ) (cid:12)(cid:12)(cid:12) ≤ X π ∈ Q N d π k σ A ( π ) k op k b f ( π ) k HS k π k HS X π ∈ Q N d π d / π h π i nr k b f ( π ) k HS (6.33)Hence, for every x ∈ G lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X π ∈ Q N d π Tr σ A ( π ) b f ( π ) π ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X π ∈∈ b G d π Tr σ A ( π ) b f ( π ) π ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and by the Fatou theorem (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X π ∈ b G d π Tr σ A ( π ) b f ( π ) π ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( G ) ≤ lim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X π ∈ Q N d π Tr σ A ( π ) b f ( π ) π ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( G ) ≤ sup π ∈ b G h π i n ( p − q ) k σ A ( π ) k op + X π ∈ b G h π i nm k b ∂σ A ( π ) k op k f k L p ( G ) . (6.34)Now, we concentrate on establishing (6.21). Modulus technical details, we shallinterpolate between two Nikolsky inequalities in order to estimate the second sum in(6.32). Let p < p < p and p = − θp + θp , < θ <
1. We take γ = (1 − θ ) γ = (1 − θ ) (cid:18) p − p (cid:19) . (6.35)We divide and multiply the sum by λ γka N − X k = a sup t ∈ N ∆ π α t,k k f k k L q ( G ) = a N − X k = a λ p − q +1 − γk sup t ∈ N ∆ π α t,k λ γk k f k k L q ( G ) λ p − q λ k ≤ (cid:18) sup k ∈ N λ p − q +1 − γk sup t ∈ N ∆ π α t,k (cid:19) ∞ X k =1 λ γk k f k k L q ( G ) λ p − q k λ k . (6.36)Let us denote by f ( t ) the quantity given by f ( t ) = sup λ n ≥ t k f n k L q ( G ) λ δn , (6.37)where δ = 1 p − q . (6.38) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 33
Using the fact that the function t → f ( t ) is non-increasing function of t >
0, weget ∞ X k =1 λ γk k f k k L q ( G ) λ p − q k λ k ≤ ∞ X k =1 λ γk f ( λ k ) 1 λ k = X s ∈ Z X k ∈ N : 2 s ≤ λ k ≤ s +1 λ γk f ( λ k ) 1 λ k ≤ X s ∈ Z γ ( s +1) f (2 s ) X k ∈ N : 2 s ≤ λ k ≤ s +1 λ k = 2 γ X s ∈ Z γ ( s − f (2 s ) ≤ γ X s ∈ Z s Z s − t γ f ( t ) s Z s − dtt = 2 γ ∞ Z t γ f ( t ) dtt . = ∞ Z t γ f ( t ) dtt = ∞ Z t − θγ t γ f ( t ) dtt ≤ ∞ Z t − θγ (cid:18) sup u ≤ t u γ f ( u ) (cid:19) dtt = { v = t γ } ∞ Z v − θ sup u ≤ v γ u γ f ( u ) dtt . (6.39)Now, we shall interpolate between two Nikolsky inequalities. k f k k L q ( G ) ≤ λ p − q k k f k L p ( G ) , (6.40) k f k k L q ( G ) ≤ λ p − q k k f k L p ( G ) . (6.41)Rescalling in the second inequality in (6.40), we get k f k k L q ( G ) ≤ λ p − q k k f k L p ( G ) , (6.42) k f k k L q ( G ) ≤ λ p − p k λ p − q k k f k L p ( G ) . (6.43)Thus, using (6.37), we rewrite inequalities (6.42),(6.43) f ( λ k ) ≤ k f k L p ( G ) , (6.44) λ γ k f ( λ k ) ≤ k f k L p ( G ) . (6.45)Let f = f + f be an arbitrary decomposition. From (6.44) and (6.45) we obtainsup u ≤ v γ u γ f ( u ) ≤ sup u ≤ v γ u γ f ( u ) + sup u ≤ v γ u γ f ( u ) ≤ sup u ≤ v γ u γ f ( u ) + v sup u> f ( u ) ≤ k f k L p ( G ) + v k f k L p ( G ) . (6.46)Since the decomposition f = f + f is arbitrary, we take the infimum and getsup u ≤ v γ u γ f ( u ) ≤ K ( t, f ; L p ( G ) , L p ( G )) , (6.47) where the functional K ( t, f ) is given by K ( v, f ; L p ( G ) , L p ( G )) , = inf f = f + f (cid:8) k f k L p ( G ) + v k f k L p ( G ) (cid:9) . (6.48)Composing (6.39) and (6.47), we obtain ∞ X k =1 λ γk k f k k L q ( G ) λ p − q k λ k ≤ k f k L p ( G ) , (6.49)where in the last equality we used that L p q ( G ) are the interpolation spaces and theembedding of the Lorentz spaces. Composing (6.36) and (6.49), we obtain a N − X k = a sup t ∈ N ∆ π α t,k k f k k L q ( G ) ≤ (cid:18) sup k ∈ N λ p − q +1 − γk sup t ∈ N ∆ π α t,k (cid:19) k f k L p ( G ) . Using this and recalling (6.32), we get k Af N k L q ( G ) ≤ sup k ∈ N α k λ p − q k + a N − X k = a λ p − q +1 − γk sup t ∈ N ∆ π α t,k ! k f k L p ( G ) . (6.50)Let us denote m = 1 − γ. (6.51)Recalling (6.35), we obtain the range1 − p ≤ m < . (6.52)Recalling notation (6.27) we get k Af N k L q ( G ) . sup k ∈ N λ p − q k k σ A ( π k ) k op + ∞ X k =1 λ p − q + mk k b ∂σ A ( π k ) k op ! k f k L p ( G ) , where we used the fact that k b ∂σ A ( π k ) k op = sup t =1 ,...,d πk ∆ π α t,k . Returning to the‘unitary dual notation’, and using that λ k ∼ = (cid:10) π k (cid:11) n with n = dim G , we finally obtain k Af N k L q ( G ) . sup π ∈ b G h π i n ( p − q ) k σ A ( π ) k op + X π ∈ b G h π i nm k b ∂σ A ( π ) k op k f k L p ( G ) . (6.53)Passing to the limit as above, we obtain (6.21). Now, it remains to show that in-equality (6.30) holds true. It has been shown in [NRT16] that for every trigonometricpolynomial f L = X ξ ∈ b G h ξ i≤ L d π Tr( b f ( ξ ) ξ ) (6.54)a version of the Nikolskii-Bernshtein inequality can be written as k f L k L q ( G ) ≤ N ( ρL ) p − q k f k L p ( G ) , < p < q < ∞ , where N ( ρL ) := X ξ ∈ b G h ξ i≤ ρL d ξ , ρ = min(1 , [ p/ , p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 35 where [ p/
2] is the integer part of p/
2. The application of Weyl’s asymptotic law tothe counting function N ( ρL ) of the n -th order elliptic pseudo-differential operator( I − L G ) with L = h π i yields N ( ρ h π i ) ∼ = ρ n h π i n , n = dim( G ) . Hence, we immediately obtain k f h π i k L q ( G ) . h π i n ( p − q ) k f h π i k L p ( G ) . This completes the proof of (6.30). (cid:3)
Spectral multipliers and non-invariant operators.
Let A be a left Fouriermultiplier. We now give an illustration of Theorem 6.5 related to spectral multipliers ϕ ( A ) for a monotone continuous function ϕ on [0 , + ∞ ). In particular, the L p - L q boundedness is reduced to a condition involving the behaviour of the singular numbersof the symbol σ A of A compared to the eigenvalues of the Laplacian on G .For convenience of the following formulation we enumerate the representations of G according to the growth of the corresponding eigenvalues of the Laplacian. Moreprecisely, for the non-decreasing eigenvalues { λ j } j of ( I − L G ) dim( G )2 with multiplicitiestaken into account, we denote by π j the corresponding representations such that( I − L G ) dim( G )2 π jml = λ j π jml holds for all 1 ≤ m, l ≤ d π j . From the Weyl asymptotic formula for the eigenvaluecounting function we get that λ j ∼ = (cid:10) π j (cid:11) n ∼ = j, with n = dim G. (6.55)Therefore, we can formulate a spectral multipliers corollary of Theorem 6.5. Corollary 6.6.
Let A be a left Fourier multiplier on a compact semisimple Lie group G of dimension n . Let − s < p ≤ q < s . Assume that ϕ is a monotonefunction on [0 , + ∞ ) . Then ϕ ( A ) is a left-invariant operator and, moreover, we have k ϕ ( A ) k L p inv ( G ) → L q inv ( G ) . sup j ∈ N j p − q sup t =1 ,...,d πj | ϕ ( α t j ) | + X j ∈ N j p − q sup t =1 ,...,d πj | ϕ ( α t j ) − ϕ ( α t +1 j +1 ) | , (6.56) where α t j are the singular numbers of the symbol σ A ( π j ) , π j ∈ b G , t = 1 , . . . , d π j . It will be clear from the proof of Corollary 6.6 that the condition that ϕ is monotoneis not essential and is needed only for obtaining a simpler expression under the sumin (6.56). We leave it to the reader to formulate the analogous statement withoutassuming the monotonicity of ϕ . Example 6.7.
Let G be a compact semisimple Lie group of dimension n and let ϕ be as in Corollary 6.6. Let us also assume that ϕ is boundedly differentiable k ϕ ′ k L ∞ < + ∞ . As a very rough illustration of Corollary 6.6 assume that the symbol σ A ( π ) “decays” sufficiently fast with respect to h π i n , i.e. that k σ A ( π ) k op . h π i α , for some α > n (cid:18) p − q + 1 (cid:19) . (6.57) Then ϕ ( A ) is L p - L q bounded. Indeed, in this case the series in (6.56) is convergentin view of | ϕ ( α t j ) − ϕ ( α t +1 j +1 ) | . k ϕ ′ k L ∞ ( k σ A ( π j ) k op + k σ A ( π j +1 ) k op ) . h π j +1 i α ∼ = 1 j α/n . Proof of Corollary 6.6.
By the functional calculus for affiliated unbounded operators[DNSZ16, Proposition 4.2], we get ϕ ( A ) = ∞ M j =1 d π j d πj X k =1 ϕ ( σ A ( π j )) . (6.58)Let us denote by α t,π j the singular values of σ A ( π j ). From (6.58), we get that thesingular numbers β t,π j of ϕ ( σ A ( π j )) are given by { β t,π j } d πj t =1 = { ϕ ( α ,π j ) , ϕ ( α ,π j ) , . . . , ϕ ( α d πj ,π j ) } if ϕ is increasing, and by { β t,π j } d πj t =1 = { ϕ ( α d π ,π ) , ϕ ( α d π − ,π ) , . . . , ϕ ( α ,π ) } otherwise. By definition (6.19), we get b ∂ϕ ( σ A ( π j )) = diag( β ,j − β ,j +1 , · · · , β k,j − β k,j +1 , · · · , β d πj − ,j − β d πj ,j +1 , β d πj ) . (6.59)By Theorem 6.5 we have k ϕ ( A ) k L p inv ( G ) → L q inv ( G ) . sup π ∈ b G h π i nr k ϕ ( σ A ( π )) k op + X π ∈ b G h π i nr k b ∂ϕ ( σ A ( π )) k op . (6.60)Combining (6.60) and (6.59) we obtain k ϕ ( A ) k L p inv ( G ) → L q inv ( G ) . sup π ∈ b G h π i nr sup t =1 ,...,d π | ϕ ( α t π ) | + X π ∈ b G h π i nr sup t =1 ,...,d π | ϕ ( α t π ) − ϕ ( α t +1 π ) | . By using (6.55) and the numbering of the representations as explained before Corol-lary 6.6, it establishes (6.56) and completes the proof. (cid:3)
Finally we note that as a corollary of Theorem 6.5 on compact Lie groups we getthe boundedness result also for non-invariant operators. Indeed, a rather standardargument (see e.g. the proof of Theorem 7.9) immediately yields:
Theorem 6.8.
Let G be a compact semisimple connected Lie group of dimension n .Let − s < p ≤ q < s and suppose that l > qn is an integer. Let A be acontinuous linear operator on C ∞ ( G ) . Then we have k A k L pw ( G ) → L qw ( G ) . X | α |≤ l sup u ∈ G sup π ∈ b G h π i n ( p − q ) k ∂ αu σ A ( u, π ) k op + X | α |≤ l sup u ∈ G X π ∈ b G h π i n ( p − q ) k ∂ αu b ∂σ A ( u, π ) k op . (6.61) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 37 H¨ormander’s multiplier theorem on locally compact groups
It is possible to refine Theorem 6.2 for the range 1 < p ≤ ≤ q < + ∞ . Thestatement that we prove can be viewed as a locally compact groups analogue ofthe H¨ormander L p - L q multiplier theorem [H¨or60, p. 106, Theorem 1.11], however,because of the general setting of locally compact groups, the spectral rather thansymbolic information is used. However, our statement in Theorem 7.1 implies boththe H¨ormander theorem and the known results on compact Lie groups.In the following statements, to unite the formulations, we adopt the conventionthat the sum or the integral over an empty set is zero, and that 0 = 0. Theorem 7.1.
Let < p ≤ ≤ q < + ∞ and suppose that A is a Fourier multiplieron a locally compact separable unimodular group G . Then we have k A k L p ( G ) → L q ( G ) . sup s> s Z t ∈ R + : µ t ( A ) ≥ s dt p − q . (7.1) For p = q = 2 inequality (7.1) is sharp, i.e. k A k L ( G ) → L ( G ) = sup t ∈ R + µ t ( A ) . (7.2) Using the noncommutative Lorentz spaces L r, ∞ with r = p − q , p = q , we can alsowrite (7.1) as k A k L p ( G ) → L q ( G ) . k A k L r, ∞ ( V N R ( G )) . (7.3)We recall Definition 2.6 for the noncommutative Lorentz spaces. Remark 7.2.
We notice that inequality (7.1) holds true for L pθ − L qθ -Fourier mul-tipliers k A k L pθ ( G ) → L qθ ( G ) ≤ k A k L r ∞ (VN R ( G )) , ≤ θ < ∞ . (7.4) Proof of Remark 7.2.
Let us assume p < < q and fix p , p , q , q such that p < p < p , q < q < q , (7.5) p < < q , p < < q . (7.6)Applying inequality (7.1) for p = p , q = q and p = p , q = q , we get k Af k L qi ( G ) ≤ sup s> s Z t ∈ R + µ t ( A ) ≥ s dt pi − qi k f k L pi ( G ) , i = 0 , . (7.7)A standard interpolation argument yields k A k L pθ ( G ) → L qθ ( G ) ≤ k A k − θL r ∞ (VN R ( G )) k A k θL r ∞ (VN R ( G )) . (7.8)We show that k A k − θL r ∞ (VN R ( G )) k A k θL r ∞ (VN R ( G )) . ≤ k A k L r ∞ (VN R ( G )) , where r i = p i − q i and r = − θr + θr . Let us recall that k A k L r ∞ (VN R ( G )) = sup t> t r µ t ( A ) . Direct calculations yield that (cid:18) sup t> t r µ t ( A ) (cid:19) − θ (cid:18) sup t> t r µ t ( A ) (cid:19) θ ≤ sup t> t r µ t ( A ) . This completes the proof. This completes the proof. (cid:3)
Proof of Theorem 7.1.
Since the algebra S (VN R ( G )) of left Fourier multipliers A isclosed under taking the adjoint S (VN R ( G )) ∋ A A ∗ ∈ S (VN R ( G )) (see [Seg53,Theorem 4, p. 412] or [Ter81a, Theorem 28 on p. 4]), and k A k L p ( G ) → L q ( G ) = k A ∗ k L q ′ ( G ) → L p ′ ( G ) , (7.9)we may assume that p ≤ q ′ , for otherwise we have q ′ ≤ ( p ′ ) ′ = p and use (4.14)ensuring that µ t ( A ∗ ) = µ t ( A ). When f ∈ L p ( G ), dualising the Hausdorff-Younginequality (2.13) gives, since q ′ ≤ k Af k L q ( G ) ≤ + ∞ Z [ µ t ( R Af )] q ′ dt q ′ . (7.10)By the left-invariance of A (e.g. [Ter81b, Proposition 3.1 on page 31]) we have R Af = AR f , f ∈ L ( G ) . By our assumptions, A and R f are measurable with respect to VN R ( G ). Thismakes it possible to apply Lemma 4.10 to obtain the estimate µ t ( R Af ) = µ t ( AR f ) ≤ µ t ( A ) µ t ( R f ) . (7.11)Thus, we obtain k Af k L q ( G ) ≤ + ∞ Z [ µ t ( A ) µ t ( R f )] q ′ dt q ′ . (7.12)Now, we are in a position to apply the Hausdorff-Young-Paley inequality in Theo-rem 3.3. With ϕ ( t ) = µ t ( A ) r for r = p − q , the assumptions of Theorem 3.3 are thensatisfied, and since q ′ − p ′ = p − q = r , we obtain + ∞ Z [ µ t ( R f ) µ t ( A )] q ′ dt q ′ ≤ sup s> s Z t ∈ R + µ t ( A ) r ≥ s dt r k f k L p ( G ) . (7.13) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 39
Further, it can be easily checked that sup s> s Z t ∈ R + µ t ( A ) r ≥ s dt r = sup s> s r Z t ∈ R + µ t ( A ) ≥ s dt r = sup s> s Z t ∈ R + µ t ( A ) ≥ s dt r . (7.14)Thus, we have established inequality (7.1). This completes the proof. (cid:3) The case of R n . Here we relate the statement of Theorem 7.1 to the classicalH¨ormander theorem.
Remark 7.3.
As a special case with G = R n , Theorem 7.1 implies the H¨ormandermultiplier estimate (1.3) established in [H¨or60, p. 106, Theorem 1.11], and we have k A k L r, ∞ (VN R ( R n )) = k σ A k L r, ∞ ( R n ) . (7.15) Proof of Remark 7.3.
Indeed, we identify the algebra VN R ( R n ) via the Fourier trans-form F R n with the algebra Z = { M ϕ } ϕ ∈ L ∞ ( R n ) of the multiplication operators M ϕ : L ( b R n ) : h M ϕ h = ϕh ∈ L ( b R n ) , see Example 2.3. Given an element A of VN R ( R n ) which acts on L ( R n ) by theconvolution with its convolution kernel K A , A : L ( R n ) ∋ f Af = K A ∗ f, we associate with A the multiplication operator M σ A acting on L ( b R n ) via the mul-tiplication by the symbol σ A = c K A , M σ A : L ( b R n ) ∋ b f M σ A b f = σ A ( ξ ) b f ( ξ ) ∈ L ( b R n ) . Then by Example 2.5 with ϕ ( ξ ) = σ A ( ξ ), we get µ t ( M σ A ) = σ ∗ A ( t ) . (7.16)Thus, Definition 2.6 on the noncommutative Lorentz spaces yields k M σ A k L r, ∞ (VN R ( G )) = sup t> t r µ t ( A ) = sup t> t r σ ∗ A ( t )= sup t> s [ d M σA ( s ) ] p − q = sup t> s Z ξ ∈ R n | σ A ( ξ ) |≥ s dξ p − q , (7.17)where in the equality between the first and the second lines we used Proposition 4.8with M = VN R ( R n ). This completes the proof. (cid:3) The case of compact Lie groups.
In this section we compare Theorem 7.1with known results in the case of G being a compact Lie group. The global symboliccalculus for operators A acting on compact Lie groups has been introduced andconsistently developed in [RT13, RT10], to which we refer to further details on globalmatrix symbols on compact Lie groups. Here we also note that with this matrixglobal symbol, the Fourier multiplier A must act by multiplication on the Fouriertransform side c Af ( ξ ) = σ A ( ξ ) b f ( ξ ) , ξ ∈ b G, where b f ( ξ ) = R G f ( x ) ξ ( x ) ∗ dx is the Fourier coefficient of f at the representation ξ ∈ b G , where for simplicity we may identify ξ with its equivalence class. As we havementioned in (1.4), the L p - L q boundedness of Fourier multipliers on compact Liegroups can be controlled by its symbol σ A ( ξ ). However, Theorem 7.1 gives a betterresult than the known estimate (1.4); for completeness we recall the exact statement: Theorem 7.4 ([ANR16b]) . Let < p ≤ ≤ q < ∞ and suppose that A is a Fouriermultiplier on the compact Lie group G . Then we have k A k L p ( G ) → L q ( G ) . sup s ≥ s X ξ ∈ b G : k σ A ( ξ ) k op ≥ s d ξ p − q , (7.18) where σ A ( ξ ) = ξ ∗ ( g ) Aξ ( g ) (cid:12)(cid:12) g = e ∈ C d ξ × d ξ is the matrix symbol of A . The fact that Theorem 7.1 implies Theorem 7.4 follows from the following resultrelating the noncommutative Lorentz norm to the global symbol of invariant operatorsin the context of compact Lie groups:
Proposition 7.5.
Let < p ≤ ≤ q < ∞ and let p = q and r = p − q . Suppose G is a compact Lie group and A is a Fourier multiplier on G . Then we have k A k L r, ∞ ( V N R ( G )) ≤ sup s ≥ s X ξ ∈ b G k σ A ( ξ ) k op ≥ s d ξ p − q , (7.19) where σ A ( ξ ) = ξ ∗ ( g ) Aξ ( g ) (cid:12)(cid:12) g = e ∈ C d ξ × d ξ is the matrix symbol of A . Remark 7.6. If G is a compact Lie group, the sufficient condition (7.18) on theFourier multiplier A implies τ -measurability of A with respect to VN R ( G ), so we donot need to assume it explicitly in the setting of compact Lie groups. Indeed, thecondition of τ -measurability does not arise in the setting of compact Lie groups dueto the fact [Ter81a, Proposition 21, p. 16] that A is τ -measurable with respect to M if and only if lim λ → + ∞ d λ ( A ) = 0 . (7.20) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 41
Now, if the right hand side of (7.19) is finite, the latter condition holds. Indeed, byDefinition 2.6 we getsup s> s [ d s ( A )] r = sup t> t r µ t ( A )= k A k L r, ∞ ( V N L ( G )) ≤ sup s> s X ξ ∈ b G k σ A ( ξ ) k≥ s d ξ p − q < + ∞ , (7.21)where in the first equality we used (4.8) with α = r from Proposition 4.8. Thus, wehave d s ( A ) ≤ Cs r . (7.22)As a consequence, we obtain (7.20). This completes the proof. Proof of Proposition 7.5.
We first compute the norm k A k L r, ∞ ( V N R ( G )) with r = p − q , p = q . By definition, we have k A k L r, ∞ ( V N R ( G )) = sup t> t p − q µ A ( t ) . (7.23)The application of the property (4.8) from Proposition 4.8 yieldssup t> t r µ A ( t ) = sup s> s [ d A ( s )] p − q . Therefore, it is sufficient to show thatsup s> s [ d A ( s )] p − q ≤ sup s> s X ξ ∈ b G k σ A ( ξ ) k≥ s d ξ p − q . (7.24)The polar decomposition for arbitrary closed densely defined possibly unboundedoperators A acting on a Hilbert space H has been established in [vN32]. Thus, weapply [vN32, page 307, Theorem 7] to get A = W | A | , (7.25)where W is a partial isometry. This means that the operators W ∗ W and W W ∗ areprojections in H . If A is a left Fourier multiplier, then its modulus | A | is affiliatedwith VN R ( G ) as well: Lemma 7.7 ([MVN36, p. 33, Lemma 4.4.1]) . Let M be a von Neumann algebra.Suppose A is affiliated with M . Then | A | is affiliated with M as well and W ∈ M . To proceed, we will use the following property:
Claim 7.8.
Let A ∈ S (VN R ( G )) and let E [ s, + ∞ ) ( | A | ) be the spectral measure of | A | corresponding to the interval [ s, + ∞ ) . Then we have d A ( s ) = X ξ ∈ b G d ξ X n =1 ,...,d ξ s n,ξ ≥ s , (7.26) where for fixed n = 1 , . . . , d ξ , the number s n,ξ is the joint eigenvalue for the eigen-functions ξ kn , k = 1 , . . . , d ξ , of | A | . These functions ξ kn , k = 1 , . . . , d ξ , generate thesubspace H n,ξ = span { ξ kn } d ξ k =1 . We note that by Remark 2.11, in view of the left invariance of the operators A and | A | , by the Peter-Weyl theorem they leave the spaces H n,ξ invariant (for thediscussion of the spaces H n,ξ in the context of the Peter-Weyl theorem we refer to[RT10, Theorem 7.5.14 and Remark 7.5.16]).Assuming Claim 7.8 for the moment, the proof proceeds as follows. Without lossof generality, we can reorder, for each ξ ∈ b G , the numbers s n,ξ putting them in adecreasing order with respect to n = 1 , . . . , d ξ (thus, also reordering the correspondingeigenfunctions). Then we can estimate d A ( s ) = X ξ ∈ b G d ξ X n =1 ,...,d ξ s n,ξ ≥ s ≤ X ξ ∈ b G d ξ X n =1 ,...,d ξ s ,ξ ≥ s X ξ ∈ b Gs ,ξ ≥ s d ξ , where in the first inequality we used the inclusion { ξ ∈ b G, n = 1 , . . . , d ξ : s n,ξ ≥ s } ⊂ { ξ ∈ b G, n = 1 , . . . , d ξ : s ,ξ ≥ s } (7.27)since for fixed ξ ∈ b G the sequence { s n,ξ } d ξ n =1 monotonically decreases. We notice that s ,ξ = k σ A ( ξ ) k op . Thus, we obtain d A ( s ) ≤ X ξ ∈ b G k σ A ( ξ ) k op ≥ s d ξ . From this, we get s [ d A ( s )] p − q ≤ s X ξ ∈ b G k σ A ( ξ ) k op ≥ s d ξ p − q . (7.28)Taking supremum in the right-hand side of (7.28), we get s [ d A ( s )] p − q ≤ sup s> s X ξ ∈ b G k σ A ( ξ ) k op ≥ s d ξ p − q . (7.29)Then taking again the supremum in the left-hand side of (7.29), we finally obtainsup s> s [ d A ( s )] p − q ≤ sup s> s X ξ ∈ b G k σ A ( ξ ) k op ≥ s d ξ p − q . This proves (6.5). Now, it remains to justify (7.26) in Claim 7.8. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 43
Since G is compact, its von Neumann algebra M = V N R ( G ) is a type I factor.The trace τ for type I factors M (and we denote it by Tr in this case) can be given[Naj72, page 478] by Tr( A ) = + ∞ Z −∞ λ dD M ( E λ ) , A ∈ M, (7.30)where D M : M + → { , , , . . . } (7.31)is the dimension function introduced in [MVN36, MvN37] and M + is the set of allhermitian ( A ∗ = A ) operators A ∈ M . For each value of λ the projection E λ is thesum of minimal mutually orthogonal projection operators, hence the value D M ( E λ )can increase only in jumps and its points of growth s n are the characteristic valuesof the operator A . Thus, we get Tr( A ) = X n ∈ N m n s n , (7.32)where m n is the corresponding jump of the function D M ( E λ ). Further, we determinethe singular values of A , or equivalently we will look for the eigenvalues of | A | .Indeed, we recall that | A | (cid:12)(cid:12) L dξk =1 H k,ξ = σ | A | ( ξ ) and use the fact that s ,ξ = k σ | A | ( ξ ) k op .It is convenient to enumerate the singular values s k,ξ by two elements ( k, ξ ), k =1 , . . . , d ξ , in view of the decomposition into the closed subspaces invariant under thegroup action. We rewrite (7.32) once again as the usual traceTr( | A | ) = X π ∈ b G d ξ d ξ X n =1 s n,ξ , (7.33)where we write s n,ξ for the eigenvalue of the restriction | A | (cid:12)(cid:12) L dξn =1 H n,ξ of | A | to thesubspaces H k,ξ which are spanned by the eigenfunctions ξ kn , n = 1 , . . . , d ξ , corre-sponding to s k,ξ . In other words, the multiplicity of s k,ξ is d ξ . From this place, wewrite π rather than ξ to emphasize our choice of an element ξ from the equivalenceclass [ π ]. Each element π ∈ b G can be realised as a finite-dimensional matrix via somechoice of a basis in the representation space. Denote by π kn the matrix elements of π , i.e. π : G ∋ g π ( g ) = [ π kn ( g )] d π k,n =1 × C d π × d π . (7.34)By the Peter-Weyl theorem (see e.g. [RT10, Theorem 7.5.14]), we have the decom-position L ( G ) = M π ∈ b G d π M n =1 span { π kn } d π k =1 . (7.35)In other words, we can write L ( G ) ∋ f = X π ∈ b G d π d π X n =1 d π X k =1 ( f, π kn ) L ( G ) π kn ∈ M π ∈ b G d π M n =1 span { π kn } d π k =1 . (7.36) The action of A can be written in the form Af = X π ∈ b G d π d π X n =1 d π X k =1 d π X s =1 σ A ( π ) ns ( f, π ks ) L ( G ) π kn , (7.37)This implies A = M π ∈ b G d π M n =1 σ A ( π ) , (7.38)where σ A ( π ) is the global matrix symbol of A (cf. [RT13, RT10]). Then for themodulus | A | = √ AA ∗ we get | A | = M π ∈ b G d π M n =1 | σ A ( π ) | . (7.39)Choosing a representative ξ ∈ [ π ] from the equivalence class [ π ], we can diagonalisethe matrix | σ A ( π ) | as σ | A | ( ξ ) = s ,ξ . . . s ,ξ . . . . . . s d ξ ,ξ . (7.40)Thus, we obtain | A | f = X ξ ∈ b G d ξ d ξ X n =1 s k,ξ · d ξ X k =1 ( f, ξ kn ) L ( G ) ξ kn . (7.41)Each s n,ξ is a joint eigenvalue of | A | with the eigenfunctions ξ kn | A | ξ k,n = s k,ξ ξ k,n , n = 1 , . . . , d ξ . (7.42)Since each singular value s k,ξ , k = 1 , . . . , d ξ , has the multiplicity d ξ , we obtain E [ t, + ∞ ) ( | A | ) = M ξ ∈ b G M k =1 ,...,d ξ s k,ξ ≥ t E n,ξ , (7.43)where E n,ξ is the projection to the left-invariant subspace span { ξ kn } d ξ k =1 . Conse-quently, we have Tr( E [ t, + ∞ ) ( | A | )) = X ξ ∈ b G d ξ X k =1 ,...,d ξ s k,ξ ≥ t . (7.44)The proof is now complete. (cid:3) The case of non-invariant operators.
Theorem 7.1 can be extended to non-invariant operators, and also to the boundedness in Lorentz spaces.For the formulation it is convenient to use the Schwartz-Bruhat spaces S ( G ) thathave been developed by Bruhat [Bru61] as a way of doing distribution theory onlocally compact groups. We briefly mention its basic properties and refer to [Bru61]for further details. The space S ( G ) is a barrelled, bornological and complete locallyconvex topological vector space. It is continuously and densely contained in space p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 45 C c ( G ) of compactly supported continuous functions. The space S ( G ) is dense inevery L p ( G ), which follows from the fact that C c ( G ) is dense in L p ( G ). Theorem 7.9.
Let G be a locally compact unimodular separable group. Let D be aclosed densely defined operator affiliated with VN R ( G ) such that its inverse D − ismeasurable with respect to VN R ( G ) and such that for some < β ≤ we have kD − k L β (VN R ( G )) < + ∞ . (7.45) Let A be a linear continuous operator on the Schwartz-Bruhat space S ( G ) . Then forany < p ≤ ≤ q < ∞ and any < θ < we have k A k L p ( G ) → L q ( G ) . Z G (cid:0) kD ◦ A u k L r, ∞ (VN R ( G )) (cid:1) β du β , (7.46) where r = p − q . Here { A u } is the field of operators generated by varying the Schwartz kernel K A of A , for more details we refer to the proof of Theorem 7.9. But first we observethat choosing various D , we get different inequalities in (7.45). Thus, before provingTheorem 7.9, we illustrate it in a few examples. Example 7.10.
Let G be a compact Lie group of dimension n and let L G be theLaplace operator on G . Let us take D = ( I − L G ) n . By the Weyl’s asymptotic law,we get λ k ∼ = k, where λ k are the eigenvalues of D . Then, up to constant, we obtain kD − k βL β (VN R ( G )) ≃ ∞ X k =1 k β < + ∞ , for any β >
1. Thus, condition (7.45) is satisfied.
Example 7.11.
Let us take G to be the Heisenberg group H n with the homogeneousdimension Q = 2 n + 2, and let L sub H n be the canonical sub-Laplacian on H n . It can becomputed (see (9.32)) that τ ( E (0 ,s ) ( −L sub H n ) = C n s Q . Using this and Definition 2.4, it can be shown that µ t (( I − L sub H n ) − α ) = 1 (cid:16) t Q (cid:17) α . From this we obtain k ( I − L sub H n ) − α k βL β (VN R ( H n )) = + ∞ Z (cid:16) t Q (cid:17) αβ dt, (7.47) where we used the formula τ ( | A | p ) = + ∞ Z µ pt ( A ) dt established in [FK86, Corollary 2.8, p. 278]. The integral in (7.47) is convergent ifand only if αβ > Q . Proof of Theorem 7.9.
Let us define A u f ( g ) := L K A ( u ) f ( g ) = Z G K A ( u, gt − ) f ( t ) dt, so that A g f ( g ) = Af . For each fixed u ∈ G the operator A u is affiliated with VN R ( G ).Then k Af k L q ( G ) = Z G | Af ( g ) | q dg q ≤ Z G sup u ∈ G | A u f ( g ) | q dg q . (7.48)The subsequent proof will rely on the following theorem that we now assume tohold, and will prove it later: Theorem 7.12.
Let G be a locally compact unimodular separable group and let A bea left Fourier multipler on G . Let Let ≤ β ≤ . Then we have k A k L β ( G ) → L ∞ ( G ) ≤ k A k L β (VN R ( G )) . (7.49)By assuming Theorem 7.12 for a moment and applying it to A = D − we getsup u ∈ G | A u f ( g ) | = sup u ∈ G |D − D A u f ( g ) | ≤ kD − k L β (VN R ( G ) kD A u f k L βu ( G ) . (7.50)Therefore, using the Minkowski integral inequality to change the order of integra-tion, we obtain k Af k L q ( G ) . Z G Z G |D A u f ( g ) | β du qβ dg q = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G |D A u f ( g ) | β du (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qβ ( G ) β ≤ Z G (cid:13)(cid:13) |D A u f ( g ) | β (cid:13)(cid:13) L qβ ( G ) du β = Z G Z G |D A u f ( g ) | q dg βq du β ≤ Z G (cid:0) kD A u k L r, ∞ (VN R ( G )) (cid:1) β du β k f k L p ( G ) , where the last inequality holds due to Theorem 7.1. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 47
So, it now remains to prove Theorem 7.12:
Proof of Theorem 7.12. By b G we shall mean the quasi-dual in the sense of [Ern61,Ern62]. There is a canonical central decomposition Af ( g ) = Z b G τ π (cid:16) σ A ( π ) b f ( π ) π ( g ) (cid:17) dµ ( π ) . (7.51)The uniqueness in (7.51) is up to the quasi-equivalence [Ern61, Ern62]. For each( π, H π ) ∈ b G , the operator σ A ( π ) b f ( π ) π ( g ) acts in the Hilbert space H π . Since G isunimodular, every factor VN πR ( G ) = { π ( g ) } !! g ∈ G is either of type I or type II . Hence,there always exists a trace τ π on VN πR ( G ). By H¨older’s inequality, we have | τ π ( σ A ( π ) b f ( π ) π ( g )) | ≤ (cid:0) τ π | σ A ( π ) | β (cid:1) β (cid:16) τ π | b f ( π ) π ( g ) | β ′ (cid:17) β ′ . (7.52)The application of [FK86, Corollary 2.8 on p. 278] to τ π | b f ( π ) π ( g ) | β ′ yields τ π | b f ( π ) π ( g ) | β ′ = + ∞ Z µ t ( b f ( π ) π ( g )) β ′ dt. (7.53)Using property (4.15) of Lemma 4.10, we estimate µ t ( b f ( π ) π ( g )) ≤ µ t ( b f ( π )) , g ∈ G. (7.54)The absolute value trace τ π ( σ A ( π ) b f ( π ) π ( g )) of σ A ( π ) b f ( π ) π ( g ) can then be estimatedfrom above (cid:12)(cid:12)(cid:12) τ π ( σ A ( π ) b f ( π ) π ( g ) (cid:12)(cid:12)(cid:12) ) ≤ (cid:0) τ π | σ A ( π ) | β (cid:1) β (cid:16) τ π | b f ( π ) | β ′ (cid:17) β ′ . (7.55)Thus, we get | Af ( g ) | ≤ Z b G (cid:12)(cid:12)(cid:12) τ π ( σ A ( π ) b f ( π ) π ( g ) (cid:12)(cid:12)(cid:12) dµ ( π )) ≤ Z b G (cid:0) τ π | σ A ( π ) | β (cid:1) β (cid:16) τ π | b f ( π ) | β ′ (cid:17) β ′ dµ ( π ) ≤ Z b G τ π | σ A ( π ) | β dµ ( π ) β Z b G τ π | b f ( π ) | β ′ dµ ( π ) β ′ , (7.56)where the last inequality is H¨older inequality. Borel calculus and reduction theoryfor unbounded affiliated operators have been investigated in [DNSZ16, Section 4]. Itcan be shown [DNSZ16, Proposition 4.2, p.8] that | A | p = M b G Z | σ A ( π ) | p dπ. (7.57)Then, by [DNSZ16, Lemma 5.3, p.12], we get k A k L β (VN R ( G )) = (cid:16) τ ( | A | β ) (cid:17) β = Z b G τ π ( | σ A ( π ) | β ) dµ ( π ) β . (7.58) By the Hausdorff-Young inequality [Kun58] we have Z b G τ π | b f ( π ) | β ′ dµ ( π ) β ′ ≤ k f k L β ( G ) , < β ≤ . (7.59)Finally, collecting all the inequalities, we obtain k Af k L ∞ ( G ) ≤ k A k L β (VN R ( G )) k f k L β ( G ) , < β ≤ . (7.60)The argument above can be modified for the case β = 1 as well. This completes theproof of Theorem 7.12. (cid:3) And this also completes the proof of Theorem 7.9. (cid:3) Spectral multipliers on locally compact groups
In this and next section we will give an application of Theorem 7.1 to spectralmultipliers.The classical Laplace operator ∆ R n is affiliated with the von Neumann algebraVN( R n ) = VN L ( R n ) = VN R ( R n ) of all convolution operators, but is not measurableon VN( R n ). However, the Bessel potential ( I − ∆ R n ) − s is measurable with respectto VN( R n ). Therefore, one of the aims of spectral multiplier theorems is to “renor-malise” operators in Hilbert space H making them not only measurable but alsobounded. In the next theorem we first describe such a relation for general semifinitevon Neumann algebras, and then in Corollary 8.2 give its application to spectralmultipliers. Theorem 8.1.
Let L be a closed unbouned operator affiliated with a semifinite vonNeumann algebra M ⊂ B ( H ) . Assume that ϕ is a monotonically decreasing contin-uous function on [0 , + ∞ ) such that ϕ (0) = 1 , (8.1)lim u → + ∞ ϕ ( u ) = 0 . (8.2) Then for every ≤ r < ∞ we have the equality k ϕ ( |L| ) k L r, ∞ ( M ) = sup u> (cid:0) τ ( E (0 ,u ) ( |L| )) (cid:1) r ϕ ( u ) < + ∞ . (8.3)Let L be an arbitrary unbounded linear operator affiliated with ( M, τ ). ThenTheorem 8.1 says that the function ϕ ( |L| ) is necessarily affiliated with ( M, τ ) and ϕ ( |L| ) ∈ ( M, τ ) if and only if the r -th power ϕ r of ϕ grows at infitiy not faster than τ ( E (0 ,u ) ( |L| )) , i.e. if we have the estimate ϕ ( u ) r . τ ( E (0 ,u ) ( |L| )) . (8.4)We now give a corollary of Theorem 8.1 for M = VN R ( G ) being the right vonNeumann algebra of a locally compact unimodular group. This is formulated inTheorem 1.1 but we recall it here for readers’ convenience. p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 49
Corollary 8.2.
Let G be a locally compact unimodular separable group and let L be aleft Fourier multiplier on G . Let ϕ be as in Theorem 8.1 Then we have the inequality k ϕ ( |L| ) k L p ( G ) → L q ( G ) . sup u> ϕ ( u ) (cid:2) τ ( E (0 ,u ) ( |L| )) (cid:3) p − q , < p ≤ ≤ q < ∞ . (8.5)This corollary follows immediately from combining Theorem 7.1 and Theorem 8.1with M = VN R ( G ), also proving Theorem 1.1.For completeness, we give another corollary (of the proof of Theorem 8.1) withoutassuming that ϕ is monotone, continuous, and satisfies conditions (8.1)-(8.2). It isthese conditions that allow us to rewrite Corollary 8.3 in the more applcable form ofCorollary 8.2. Corollary 8.3.
Let G be a locally compact unimodular separable group and let L bea left Fourier multiplier on G . Let ϕ be a Borel measurable function on the spectrum Sp( |L| ) . Then we have the inequality k ϕ ( |L| ) k L p ( G ) → L q ( G ) . sup s> s [ τ ( E ( s, + ∞ ) )( ϕ ( |L| ))] p − q , < p ≤ ≤ q < ∞ . (8.6)We will prove this corollary together with the proof of Theorem 8.1. Proof of Theorem 8.1.
By defintion k ϕ ( |L| ) k L r, ∞ ( M ) = sup t> t p − q µ t ( ϕ ( |L| )) , r = 1 p − q . Using Property (4.8) from Proposition 4.8, we getsup t> t p − q µ t ( ϕ ( |L| )) = sup s> s [ τ ( E ( s, + ∞ ) )( ϕ ( |L| ))] p − q . Hence, we have k ϕ ( |L| ) k L r, ∞ ( M ) = sup s> s [ τ ( E ( s, + ∞ ) )( ϕ ( |L| ))] p − q . (8.7)Since L is affiliated with M the spectral projections E Ω ( |L| ) belong to M . Let hLi be an abelian subalgebra of M generated by the spectral projectors E ( λ, + ∞ ) ( |L| ). Let ϕ be a Borel measurable function on the spectrum Sp( |L| ). Then by Borel functionalcalculus [Arv06, Section 2.6] it is possible to construct the operator ϕ ( |L| ). Thisoperator is a strong limit of the spectral projections E Ω ( |L| ) ∈ M . Therefore ϕ ( |L| )is affiliated with M . The distribution function of the operator ϕ ( |L| ) is given by d s ( ϕ ( |L| )) = τ ( E ( s, + ∞ ) ( ϕ ( |L| )) . (8.8)This proves Corollary 8.3.Using [KR97, Corollary 5.6.29, p.363] and the spectral mapping theorem (see[KR97, Theorem 4.1.6]), we obtain τ ( E ( s, + ∞ ) ( ϕ ( |L| ))) = τ ( E ϕ − ( s, + ∞ ) ( ϕ − ◦ ϕ ( |L| ))) = τ ( E (0 ,ϕ − ( s )) ( |L| ) . (8.9)From the hypothesis (8.2) imposed on ϕ and using (8.9), we getlim s → + ∞ τ ( E ( s, + ∞ ) ( ϕ ( |L| ))) = lim s → + ∞ τ ( E (0 ,ϕ − ( s )) ( |L| ) = 0 . (8.10) Hence, the operator ϕ ( |L| ) is τ -measurable with respect to VN R ( G ). Combining (8.7)and (8.9), we finally obtain k ϕ ( |L| ) k L r, ∞ ( M ) = sup t> t p − q µ t ( ϕ ( |L| )) = sup s> s [ τ ( E ( s, + ∞ ) )( ϕ ( |L| ))] p − q = sup s> s [ τ ( E (0 ,ϕ − ( s )) ( |L| )] p − q = sup u> ϕ ( u )[ τ ( E (0 ,u ) ( |L| )] p − q , where in the last equality we used the monotonicity of ϕ . This completes the proofof Theorem 8.1. (cid:3) Heat kernels and embedding theorems
In this section we show that the spectral multipliers estimate (8.2) may be also usedto relate spectral properties of the operators with the time decay rates for propagatorsfor the corresponding evolution equations. We illustrate this in the case of the heatequation, when the the functional calculus and the application of Theorem 8.1 to afamily of functions { e − ts } t> yield the time decay rate for the solution u = u ( t, x ) tothe heat equation ∂ t u + L u = 0 , u (0) = u . For each t >
0, we apply Borel functional calculus [Arv06, Section 2.6] to get u ( t, x ) = e − t L u . (9.1)One can check that u ( t, x ) satisfies equation (9.1) and the initial condition. Then byTheorem 7.1, we get k u ( t, · ) k L q ( G ) ≤ k e − t L k L r, ∞ (VN R ( G ) k u k L p ( G ) , (9.2)reducing the L p - L q properties of the propagator to the time asymptotics of its non-commutative Lorentz space norm. Corollary 9.1 (The L -heat equation) . Let G be a locally compact unimodular sepa-rable group and let L be an unbounded positive operator affiliated with VN R ( G ) suchthat for some α we have τ ( E (0 ,s ) ( L )) . s α , s → ∞ . (9.3) Then for any < p ≤ ≤ q < ∞ we have k e − t L k L p ( G ) → L q ( G ) ≤ C α,p,q t − α ( p − q ) , t > . (9.4) Proof of Theorem 9.1.
The application of Theorem 8.1 yields k e − t L k L r, ∞ (VN R ( G )) = sup s> [ τ ( E (0 ,s ) ( |L| )] r e − ts . Now, using this and hypothesis (9.3), we get k e − t L k L r, ∞ (VN R ( G )) . sup s> s αr e − ts . The standart theorems of mathematical analysis yield thatsup s> s αr e − ts = (cid:16) αtr (cid:17) αr e − αr . (9.5) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 51
Indeed, let us consider a function ϕ ( s ) = s αr e − ts . We compute its derivative ϕ ′ ( s ) = s αr − e − ts (cid:16) αr − st (cid:17) . The only zero is s = αrt and the derivative ϕ ′ ( s ) changes its sign from positive tonegative at s . Thus, the point s is a point of maximum. This shows (9.5) andcompletes the proof. (cid:3) Let us now show an application of Theorem 8.1 in the case of ϕ ( s ) = s ) γ , s ≥ < p ≤ ≤ q < ∞ , the Sobolev type embeddingtheorems for an operator L depend only on the spectral behaviour of L . Corollary 9.2 (Embedding theorems) . Let G be a locally compact unimodular sepa-rable group and let L be an unbounded positive operator affiliated with VN R ( G ) suchthat for some α we have τ ( E (0 ,s ) ( L )) . s α , s → ∞ . (9.6) Then for any < p ≤ ≤ q < ∞ we have k f k L q ( G ) ≤ C k (1 + L ) γ f k L p ( G ) , (9.7) provided that γ ≥ α (cid:18) p − q (cid:19) , < p ≤ ≤ q < ∞ . (9.8) Proof.
By Theorem 8.1 with ϕ ( s ) = s ) γ and r = p − q we have k (1 + L ) − γ k L p ( G ) → L q ( G ) . k (1 + L ) − γ k L r, ∞ (VN R ( G )) . sup s> s αr (1 + s ) − γ . This supremum is finite for γ ≥ αr , giving the condition (9.8). (cid:3) Now, we illustrate Theorem 8.1 and Corollary 9.1 on a number of further examples,showing that the spectral estimate (9.3) required for the L p - L q estimate can be readilyobtained in different situations.In Example 9.3 below we illustrate condition (9.3) in Theorem 9.1 for the homo-geneous operator Op( a ) : L ( R n ) → L ( R n ) of order µ ∈ R . Example 9.3.
Let a ( ξ ) be a homogeneous function of degree µ and let Op( a ) be thelinear operator given by \ Op( a )( ξ ) = a ( ξ ) b f ( ξ ) , f ∈ S ( R n ) , ξ ∈ R n . According to the general theory (see [KR97, Theorem 5.6.26, p.360] and [KR97,Corollary 5.6.29, p.363]), the spectral projection E (0 ,s ) ( | Op( a ) | ) corresponds to themultiplication by χ (0 ,s ) ( | a ( ξ ) | ), where χ (0 ,s ) ( u ) is the characteristic function of theinteval (0 , s ). Then the trace τ ( E (0 ,s ) ( | Op( a ) | )) can be computed as follows τ ( E (0 ,s ) ( | Op( a ) | )) = Z R n | a ( ξ ) |≤ s dξ = Z u ∈ R n | a ( u ) |≤ s nµ d u = Cs nµ , (9.9) where we made the substitution ξ → s µ u . Hence, we get τ ( E (0 ,s ) ( | Op( a ) | ) = Cs nµ , (9.10)where C = R ξ ∈ R n | a ( ξ ) |≤ dξ . The application of Theorem 9.1 yields that if C = Z ξ ∈ R n | a ( ξ ) |≤ dξ < ∞ , then k e − t Op( a ) k L p ( R n ) → L q ( R n ) ≤ c α p q t − nµ ( p − q ) , < p ≤ ≤ q < ∞ . Sub-Riemannian structures on compact Lie groups.
First we considerthe example of sub-Laplacians on compact Lie groups in which case the number α in (9.3) can be related to the Hausdorff dimension generated by the control distanceof the sub-Laplacian. Moreover, we illustrate Theorem 8.1 with examples of otherfunctions ϕ than in Corollary 9.1, for example ϕ ( s ) = s ) α/ , leading to the Sobolevembedding theorems. Example 9.4.
Let L = − ∆ sub be the sub-Laplacian on a compact Lie group G , withdiscrete spectrum λ k . Then by [HK16] the trace of the spectral projections E (0 ,s ) ( L )has the following asymptotics τ ( E (0 ,s ) ( L )) . s Q , as s → + ∞ , (9.11)where Q is the Hausdorff dimension of G with respect to the control distance gener-ated by the sub-Laplacian. Let u ( t ) be the solution to ∆ sub -heat equation ∂∂t u ( t, x ) − ∆ sub u ( t, x ) = 0 , t > ,u (0 , x ) = u ( x ) , u ∈ L p ( G ) , < p ≤ . Then by Corollary 9.1, we obtain k u ( t, · ) k L q ( G ) ≤ C n,p,q t − Q ( p − q ) k u k L p ( G ) , < p ≤ ≤ q < + ∞ . (9.12)Let us now take ϕ ( s ) = s ) a/ , s ≥
0. Then by Theorem 8.1 the operator ϕ ( − ∆ sub ) = ( I − ∆ sub ) − a/ is L p ( G )- L q ( G ) bounded and the inequality k f k L q ( G ) ≤ C k (1 − ∆ sub ) a/ ) f k L p ( G ) (9.13)holds true provided that a ≥ Q (cid:18) p − q (cid:19) , < p ≤ ≤ q < ∞ . (9.14)Here the constant C in (9.13) is given by C := k ( I − ∆ sub ) − a/ k L r, ∞ (VN R ( G )) . One can always associate with ∆ sub a version of Sobolev spaces. Let us define k f k W a,p ∆ sub ( G ) := k ( I − ∆ sub ) a/ f k L p ( G ) . (9.15) p - L q MULTIPLIERS ON LOCALLY COMPACT GROUPS 53
Then the Borel functional calculus (see e.g. [Arv06]) together with (9.13)-(9.14)immediately yield k f k W b,q ∆ sub ( G ) ≤ C k f k W a,p ∆ sub ( G ) , a − b ≥ Q (cid:18) p − q (cid:19) . (9.16)Each sub-Riemannian structure yields a sub-Laplacian ∆ sub on G . If we fix a groupvon Neumann algebra VN R ( G ), then inequality (9.16) depends only on the values ofthe trace τ on the algebra VN R ( G ) and not on a particular choice of a sub-Laplacian∆ sub . Similarly, the Sobolev spaces W a,p ∆ sub ( G ) do not depend on a particular choiceof a sub-Laplacian.9.2. Rockland operators on the Heisenberg group.
Let G be the simply con-nected Heisenberg group and π be an irreducible unitary representation of G = H .Let X , X , . . . , X n , Y , Y , . . . , Y n , H be a basis in the Lie algebra h n of the Heisen-berg group H n such that [ X k , Y k ] = H . Let R = n P k =1 X jk + n P k =1 Y jk be the positiveRockland operator and its symbol σ R ( π ) is given [FR16, p.532] by σ R ( π ) = | λ | k n X k =1 ∂ ku − | u | k ! . (9.17)It can then be shown [ ? , Theorem 5.1] that the asymptotics of the eigenvalues s λm isas follows s λm ∼ = | λ | k n Y k =1 m k j . (9.18)Thus, we get τ ( E (0 ,s ) ( |R| ) = Z b H τ λ ( E (0 ,s ) ( (cid:12)(cid:12) σ R ( π λ ) (cid:12)(cid:12) ) = Z λ ∈ R λ =0 | λ | n dλ X m ∈ N n s λm ≤ s ∼ = s Q j (9.19)determining the value of α in (9.3).9.3. Sub-Laplacian on the Heisenberg group.
Here we look at the example ofthe Heisenberg group determining the value of α in (9.3) for the sub-Laplacian. Theinteresting point here is that while the spectrum of the sub-Laplacian is continuous,Theorem 8.1 can be effectively used in this situation as well. Example 9.5.
Let L be the positive sub-Laplacian on the Heisenberg group H n andlet Q = 2 n + 2 be the homogeneous dimension of H n . We claim that τ ( E (0 ,s ) ( L ) ≃ s Q/ . (9.20)Thus, under conditions of Theorem 8.1 on ϕ , the spectral multiplier ϕ ( L ) is τ -measu-rable with respect to VN R ( H n ) and k ϕ ( L ) k L r, ∞ (VN R ( H n )) ≃ sup u> u Q r ϕ ( u ) , r = 1 p − q . (9.21)For example, by choosing ϕ ( u ) = u ) a/ , α >
0, we recover the Sobolev embeddinginequalities k ( I + L ) b/ f k L q ( H n ) ≤ C k ( I + L ) a/ f k L p ( H n ) , (9.22) provided a − b ≥ Q (cid:18) p − q (cid:19) . (9.23)Inequality (9.22) has been established by Folland [Fol75], and it can be extendedfurther for Rockland operators on general graded Lie groups [FR16]. Proof of Example 9.5.
By Theorem 7.1, we get k ϕ ( L ) k L p ( H n ) → L q ( H n ) . k ϕ ( L ) k L r, ∞ (VN R ( H n )) . (9.24)Hence it is sufficient to find the conditions on ϕ so that the right-hand side in (9.24)is finite. By Theorem 8.1 we have k ϕ ( L ) k L r, ∞ (VN R ( G )) = sup u> [ τ ( E (0 ,u ) ( |L ))] r ϕ ( u ) . (9.25)We shall now show (9.20). Since L is affiliated with VN R ( H n ) it can be decomposed([Dix81, Theorem 1 on page 187]) L = M c H n Z L λ dν ( λ ) (9.26)with respect to the center C = VN R ( H n ) ∩ VN R ( H n ) ! of the group von Neumann algebra VN R ( H n ). Here the collection {L λ } λ ∈ c H n of the(densely defined) operators L λ : L ( R n ) → L ( R n ) can be interpreted as the globalsymbol of the operator L , as developed in [FR16].Hence, the spectral projections E (0 ,s ) ( L ) can be decomposed E (0 ,s ) ( L ) = M c H n Z E (0 ,s ) ( L λ ) | λ | n dλ. (9.27)As a consequence [Dix81, Theorem 1 on page 225], we get τ ( E (0 ,s ) ( L ) = Z H n τ ( E (0 ,s ) [ L λ ]) | λ | n dλ. (9.28)The global symbol L λ : S ( R n ) ⊂ L ( R n ) → L ( R n ) of the sub-Laplacian L can befound in [FR16, Lemma 6.2.1] L λ f ( u ) = −| λ | (∆ R n f ( u ) − | u | f ( u )) , f ∈ S ( R n ) , u ∈ R n , (9.29)and is a rescaled harmonic oscillator on R n , see also Folland [Fol89]. It is known thatfor each λ ∈ R \ { } the operator L λ has purely discrete spectrumSp( L λ ) = { s ,λ ≤ s ,λ ≤ . . . ≤ s m,λ ≤ . . . } . Since H n is of type I, we have τ ( E (0 ,s ) [ L λ ]) = X k ∈ N n s k,λ
The eigenvalues s k,λ are well-known and are given by s k,λ = λ n Y j =1 (2 k j + 1) , (9.31)see e.g. [NR10]. Thus, collecting (9.28), (9.30) and (9.31), we finally obtain τ ( E (0 ,s ) ( L )) = Z c H n X k ∈ N n s k,λ
It has been shown by Stanton [Sta76] that for class functions on semisimple compactLie groups the polyhedral Fourier partial sums S N f converge to f in L p provided that2 − s + 1 < p < s . (A.1)Here the number s depends on the root system R of the compact Lie group G , in theway we now describe. We also note that the range of indices p as above is sharp, seeStanton and Tomas [ST76, ST78] as well as Colzani, Giulini and Travaglini [CGT89].Let G be a compact semisimple Lie group and let T be a maximal torus of G ,with Lie algebras g and t , respectively. Let n = dim G and l = dim T = rank G .We define a positive definite inner product on t by putting ( · , · ) = − B ( · , · ), where B is the Killing form. Let R be the set of roots of g . Choose in R a system R + of positive roots (with cardinality r ) and let S = { α , . . . , α l } be the correspondingsimple system. We define ρ := P α ∈R + α .For every λ ∈ it ∗ there exists a unique H λ ∈ t such that λ ( H ) = i ( H λ , H ) forevery H ∈ t . The vectors H j = πiH αj α j ( H αj ) generate the lattice sometimes denoted byKer(exp). The elements of the setΛ = { λ ∈ it ∗ : λ ( H ) ∈ πi Z , for any H ∈ Ker(exp) } are called the weights of G and the fundamental weights are defined by the relations λ j = 2 πiδ jk , j, k = 1 , . . . , l . The subset D = { λ ∈ Λ : λ = l X j =1 m j ν j , m j ∈ N } of the set Λ with positive coordinates m j is called the set of dominant weights. Here,the word ‘dominant’ means that with respect to a certain partial order on the setΛ every weight λ = P lj =1 m j ν j with m j > b G and the semilattice D of the dominant weights of G , i.e. D ∋ λ = ( ν , . . . , ν l ) ←→ π ∈ b G. Therefore, we will not distinguish between π and the corresponding dominant weight λ and will write π = ( π , . . . , π l ) , where we agree to set π i = ν i . With ρ = P α ∈R + α , for a natural number N ∈ N , weset Q N := { ξ ∈ b G : ξ i ≤ N ρ i , i = 1 , . . . , l } . We call Q N a finite polyhedron of N th order and denote by M the set of all finitepolyhedrons in b G .Now, fix an arbitrary fundamental weight λ j , j = 1 , . . . , l , and set R ⊥ λ j := { α ∈R + : ( α, λ j ) = 0 } , and R + = R λ j ⊕ R ⊥ λ j . The number s appearing in (A.1) is definedby s := max j =1 ,...,l card R λ j . (A.2) References [ANR16a] R. Akylzhanov, E. Nursultanov, and M. Ruzhansky. Hardy-Littlewood-Paley inequalitiesand Fourier multipliers on SU (2). Studia Math. , 234(1):1–29, 2016.[ANR16b] R. Akylzhanov, E. Nursultanov, and M. Ruzhansky. Hardy-Littlewood-Paley type in-equalities on compact Lie groups.
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International Conference on Analysis and Applied Mathematics 2016 , volume1759, page 020125. AIP Publishing, 2016.
Rauan Akylzhanov:Department of MathematicsImperial College London180 Queen’s Gate, London SW7 2AZUnited Kingdom
E-mail address [email protected]
Michael Ruzhansky:Department of MathematicsImperial College London180 Queen’s Gate, London SW7 2AZUnited Kingdom