Algebraic Calderón-Zygmund theory
aa r X i v : . [ m a t h . F A ] J u l ALGEBRAIC CALDER ´ON-ZYGMUND THEORY
MARIUS JUNGE, TAO MEI,JAVIER PARCET AND RUNLIAN XIA
Abstract.
Calder´on-Zygmund theory has been traditionally developed onmetric measure spaces satisfying additional regularity properties. In the lackof good metrics, we introduce a new approach for general measure spaces whichadmit a Markov semigroup satisfying purely algebraic assumptions. We shallconstruct an abstract form of ‘Markov metric’ governing the Markov processand the naturally associated BMO spaces, which interpolate with the L p -scaleand admit endpoint inequalities for Calder´on-Zygmund operators. Motivatedby noncommutative harmonic analysis, this approach gives the first form ofCalder´on-Zygmund theory for arbitrary von Neumann algebras, but is alsovalid in classical settings like Riemannian manifolds with nonnegative Riccicurvature or doubling/nondoubling spaces. Other less standard commutativescenarios like fractals or abstract probability spaces are also included. Amongour applications in the noncommutative setting, we improve recent resultsfor quantum Euclidean spaces and group von Neumann algebras, respectivelylinked to noncommutative geometry and geometric group theory. Introduction
The analysis of linear operators associated to singular kernels is a central topicin harmonic analysis and partial differential equations. A large subfamily of thesemaps is under the scope of Calder´on-Zygmund theory, which exploits the relationbetween metric and measure in the underlying space to provide sufficient conditionsfor L p boundedness. The Calder´on-Zygmund decomposition [6] or the H¨ormandersmoothness condition for the kernel [29] combine the notions of proximity in termsof the metric with that of smallness in terms of the measure. The doubling and/orpolynomial growth conditions between metric and measure yield more general formsof the theory [14, 46, 47, 62, 63]. To the best of our knowledge, the existence of ametric in the underlying space is always assumed in the literature.In this paper, we introduce a new form of Calder´on-Zygmund theory for generalmeasure spaces admitting a Markov semigroup which only satisfies purely algebraicassumptions. This is especially interesting for measure spaces where the geometricinformation is poor. It includes abstract probability spaces or fractals like theSierpinski gasket, where a Dirichlet form is defined. It is also worth mentioning thatour approach recovers Calder´on-Zygmund theory for classical spaces and providesalternative forms over them. In spite of these promising directions —very littleexplored here— our main motivation has been to develop a noncommutative formof Calder´on-Zygmund theory for noncommutative measure spaces (von Neumannalgebras) where the notions of quantum metric [37, 55, 56] seem inefficient. A great effort has been done over the last years to produce partial results towardsa noncommutative Calder´on-Zygmund theory [24, 28, 33, 45, 49]. The model casesconsidered so far are all limited to (different) noncommutative forms of Euclideanspaces, described as follows:
A) Tensor products.
Let f = ( f jk ) : R n → B ( ℓ ) be a matrix-valued function andconsider the tensor product extension of a standard Calder´on-Zygmund operatoracting on f , formally given by Tf ( x ) = Z R n k ( x, y ) f ( y ) dy = (cid:16) Tf jk ( x ) (cid:17) jk for x / ∈ supp f. The L p -boundedness of this map in the associated (tensor product) von Neumannalgebra M = L ∞ ( R n ) ¯ ⊗B ( ℓ ) trivially follows for p > p -classes. On the contrary, no endpointestimate for p = 1 is possible using vector-valued methods. The original argumentin [49] —also in a recent simpler form [4]— combines noncommutative martingaleswith a pseudolocalization principle for classical Calder´on-Zygmund operators. Moreprecisely, a quantification of how much L -mass of a singular integral is concentratedaround the support of the function on which it acts. This inequality has been thekey tool in the recent solution of the Nazarov-Peller conjecture [8], a strengtheningof the celebrated Krein conjecture [59] on operator Lipschitz functions. B) Crossed products.
New L p estimates for Fourier multipliers in group vonNeumann algebras have recently gained considerable momentum for its connectionsto geometric group theory. The first H¨ormander-Mikhlin type theorem in thisdirection [33] exploited finite-dimensional cocycles of the given group G to transferthe problem to the cocycle Hilbert space H = R n . To find sufficient regularityfor L p -boundedness amounts to study Calder´on-Zygmund operators in the crossedproducts L ∞ ( R n ) ⋊ G induced by the cocycle action. Nonequivariant extensions ofCZOs on these von Neumann algebras were investigated in [33], after identifyingthe right BMO space for the length function determined by the cocycle. Theseoperators have the form Z G f g ⋊ λ ( g ) dµ ( g ) Z G T g ( f g ) ⋊ λ ( g ) dµ ( g ) . Here µ, λ respectively denote the Haar measure and left regular representation onthe (unimodular) group G, whereas T g = α g T α g − is a twisted form of a classicalCZO T on R n by the cocycle action α . We refer to [34, 50] for further results. C) Quantum deformations.
PDEs in matrix algebras and ‘noncommutativemanifolds’ appear naturally in theoretical physics. Pseudodifferential operatorswere introduced by Connes in 1980 to study a quantum form of the Atiyah-Singerindex theorem over these algebras. These techniques have been underexploited overthe last 30 years, due to fundamental obstructions to understand singular integraltheory in this context. The core of singular integrals and pseudodifferential operator L p -theory was developed in [24] over the archetypal algebras of noncommutativegeometry. It includes quantum tori, Heisenberg-Weyl algebras and other quantumdeformations of R n of great interest in quantum field theory, string theory andquantum probability. This was the first approach to a ‘fully noncommutative’Calder´on-Zygmund theory for CZOs not acting on copies of R n as tensor or crossedproduct factors, but still related to Euclidean methods. LGEBRAIC CALDER ´ON-ZYGMUND THEORY 3
We introduce in this paper the first form of Calder´on-Zygmund theory validfor general (semifinite) von Neumann algebras. As we explained above, the maindifficulty arises from the lack of very standard geometric tools, like the existenceof a nice underlying metric or the construction of suitable covering lemmas. Weshall circumvent it using a very different approach based on algebraic properties ofa given Markov process. Our applications cover a wide variety of scenarios whichwill be discussed, giving especial emphasis to noncommutative forms of Euclideanspaces and locally compact abelian groups, which are our main classical models. Inthe first case, we shall weaken/optimize the CZ conditions on quantum Euclideanspaces [24]. In the second case, LCA groups correspond to quantum groups whichare both commutative and cocommutative [60]. We shall give CZ conditions forconvolution maps over quantum groups. In the cocommutative (non necessarilycommutative) context, this includes group von Neumann algebras.
Calder´on-Zygmund extrapolation
Based on the behavior of the Hilbert transform in the real line, the main goalof Calder´on-Zygmund theory is to establish regularity properties on the kernel ofa singular integral operator, so that L -boundedness automatically extrapolates to L p boundedness for 1 < p < ∞ . A singular integral operator in a Riemannianmanifold (X , d , µ ) admits the kernel representation T k f ( x ) = Z X k ( x, y ) f ( y ) dµ ( y ) for x / ∈ supp f. Namely, T k is only assumed a priori to send test functions into distributions, so thatit admits a distributional kernel in X × X which coincides in turn with a locallyintegrable function k away from the diagonal x = y , where the kernel presentscertain singularity. This already justifies the assumption x / ∈ supp f in the kernelrepresentation. The work in [6, 29] culminated in the following sufficient conditionson a singular integral operator in R n for its L p -boundedness:i) L -boundedness (cid:13)(cid:13) T k : L ( R n ) → L ( R n ) (cid:13)(cid:13) < ∞ . ii) H¨ormander kernel smoothnessess sup x,y ∈ R n Z | x − z | > | x − y | (cid:12)(cid:12) k ( x, z ) − k ( y, z ) (cid:12)(cid:12) + (cid:12)(cid:12) k ( z, x ) − k ( z, y ) (cid:12)(cid:12) dz < ∞ . Historically, this was used to prove a weak endpoint inequality in L . The sameholds for Riemannian manifolds with nonnegative Ricci curvature [1]. Alternativelyit is simpler to use L -boundedness and the kernel smoothness condition to prove L ∞ → BMO boundedness. The result then follows by well-known duality andinterpolation arguments. Our strategy resembles this approach:P1. Identify the appropriate BMO spaces.P2. Prove the expected interpolation results with L p spaces.P3. Provide conditions on CZO’s which yield L ∞ → BMO boundedness.In the classical setting, we typically find H /BMO spaces associated to a metric or amartingale filtration. Duong and Yan [19, 20] extended this theory replacing someaverages over balls in the metric space by semigroups of positive operators, although JUNGE, MEI, PARCET AND XIA the existence of a metric was still assumed. This assumption was removed in [32, 44]providing a theory of semigroup type BMO spaces with no further assumptions onthe given space. In particular, we could say that Problems 1 and 2 were solved in[32], but it has been unclear since then how to provide natural CZ conditions whichimply L ∞ → BMO estimates. In this paper we solve P3 by splitting it into:P3a. Construct a ‘metric’ governing the Markov process.P3b. Define ‘metric BMO’ spaces which still interpolate with the L p scale.P3 c. Provide CZ conditions giving L ∞ → BMO boundedness for metric BMO’s.
P3a. Markov metrics.
Given a Markov semigroup S = ( S t ) t ≥ on the semifinitevon Neumann algebra ( M , τ ) —in other words, formed by normal self-adjoint cpumaps S t — we introduce a Markov metric for it as any family Q = n(cid:0) R j,t , σ j,t , γ j,t (cid:1) : ( j, t ) ∈ Z + × R + o composed of completely positive unital (cpu) maps R j,t : M → M and elements σ j,t , γ j,t of M with γ j,t ≥ M , such that the following estimates (which show howthe Markov metric governs the Markov semigroup in a controlled way) hold:i) Hilbert module majorization : (cid:10) ξ, ξ (cid:11) S t ≤ X j ≥ σ ∗ j,t (cid:10) ξ, ξ (cid:11) R j,t σ j,t .ii) Metric integrability condition : k Q = sup t> (cid:13)(cid:13)(cid:13) X j ≥ σ ∗ j,t γ j,t σ j,t (cid:13)(cid:13)(cid:13) M < ∞ .Here h , i Φ is the M -valued inner product on M ¯ ⊗M for any cpu map Φ, given by h a ⊗ b, a ′ ⊗ b ′ i Φ = b ∗ Φ( a ∗ a ′ ) b ′ . Markov metrics are a priori unrelated to Rieffel’squantum metric spaces [55, 56]. They present on the contrary vague similaritieswith abstract formulations of classical CZ theory in the absence of CZ kernelsand/or doubling measures [2, 62]. We shall explain what motivates our definitionbelow and we shall also illustrate how Euclidean and other classical metrics fit in. P3b. Metric type BMO spaces.
Let k f k BMO c S = sup t ≥ (cid:13)(cid:13)(cid:13)(cid:16) S t ( f ∗ f ) − ( S t f ) ∗ ( S t f ) (cid:17) (cid:13)(cid:13)(cid:13) M and k f k BMO S = max {k f k BMO c S , k f ∗ k BMO c S } . We shall define the semigroup typeBMO space BMO S ( M ) as the weak- ∗ closure of M in certain direct sum of Hilbertmodules determined by S = ( S t ) t ≥ . These spaces interpolate with the L p scale[32]. Given a Markov metric Q associated to this semigroup, let us define in addition k f k BMO Q = max n k f k BMO c Q , k f ∗ k BMO c Q o , k f k BMO c Q = sup t> inf M t cpu M t : M→M sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) γ − j,t (cid:2) R j,t | f | − | R j,t f | + | R j,t f − M t f | (cid:3) γ − j,t (cid:17) (cid:13)(cid:13)(cid:13) M . Theorem A1.
Let ( M , τ ) be a semifinite von Neumann algebra equipped with aMarkov semigroup S = ( S t ) t ≥ . Let us consider a Markov metric Q associated to S = ( S t ) t ≥ . Then, we find k f k BMO S . k Q k f k BMO Q . Thus, defining
BMO Q ( M ) as a subspace of BMO S ( M ) , it interpolates with L p ( M ) . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 5
Theorem A1 solves P3b. Its proof is not hard after having defined the rightnotion of Markov metric and the right BMO norm. Let us note in passing that theterm R j,t f − M t f is there to accommodate nondoubling spaces to our definition inthe spirit of Tolsa’s RBMO space [62]. As a consequence of Theorem A1, proving L ∞ → BMO boundedness for metric BMO’s (Problem 3c) implies the same resultfor semigroup BMO spaces (Problem 3). Of course, one could try to prove such astatement directly, but it seems that the metric/measure relation found with thesenew notions is crucial for a noncommutative CZ theory.
P3c. Calder´on-Zygmund operators.
The commutative idea behind the notionof Markov metric (explained in more detail in the body of the paper) is to findpointwise majorants of the integral kernels of our semigroup S = ( S t ) t ≥ , so thatwe can dominate S t by certain sum of averaging operators over a distinguishedfamily of measurable sets Σ j,t ( x ). These sets may be considered as the ‘balls’ inthe Markov metric. In the noncommutative setting, this pointwise estimates mustbe written in terms of the given Hilbert module majorization and the cpu maps R j,t must be averages over certain projections q j,t . Making this precise in full generalityis one of the challenges of our algebraic approach and too technical to be explainedat this point of the paper. A simple model case is given by(Avg) R j,t f = ( id ⊗ τ )( q j,t ) − ( id ⊗ τ ) (cid:0) q j,t ( ⊗ f ) q j,t (cid:1) ( id ⊗ τ )( q j,t ) − for certain family of projections q j,t ∈ M ¯ ⊗M . The linear map b R j,t ( ⊗ f ) = R j,t f trivially amplifies to M ¯ ⊗M . We may also consider similar formulas for the cpumaps M t in the metric BMO norm. (Avg) allows to identify the Markov metric interms of the ‘balls’ q j,t instead of the corresponding averaging maps R j,t .As it happens in classical Calder´on-Zygmund theory, we need to impose someadditional properties in the Markov metric to establish a good relation with theunderlying (noncommutative) measure. We have split these into algebraic and analytic conditions, further details will be given in the text. Let us just mentionthat the algebraic ones are inherent to noncommutativity and hold trivially incommutative cases. The analytic ones provide forms of Jensen’s inequality anda measure/metric growth condition. Once we know the Markov metric satisfiesthese conditions, we may introduce Calder´on-Zygmund operators. Assume that T ( A M ) ⊂ M for a map T acting on a weak- ∗ dense subalgebra A M ⊂ M . Thegoal is to establish sufficient Calder´on-Zygmund conditions on T for L ∞ → BMO c boundedness. These are noncommutative forms of standard properties. Again, itis unnecessary to introduce them here in full generality, we do it in Section 2. Inthe model case above, our CZ conditions are:i) L ∞ ( L c ) -boundedness (cid:13)(cid:13)(cid:13) ( id ⊗ τ ) (cid:16) ( id ⊗ T )( x ) ∗ ( id ⊗ T )( x ) (cid:17) (cid:13)(cid:13)(cid:13) M . (cid:13)(cid:13) ( id ⊗ τ )( x ∗ x ) (cid:13)(cid:13) M . ii) Size ‘kernel’ conditions • c M t (cid:16)(cid:12)(cid:12) ( id ⊗ T ) (cid:0) ( ⊗ f )( A j,t − a t ) (cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k M , • b R j,t (cid:16)(cid:12)(cid:12) ( id ⊗ T ) (cid:0) ( ⊗ f )( A j,t − a j,t ) (cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k M ,for certain family of operators A j,t , a j,t ∈ M ¯ ⊗M with A j,t ≥ a j,t . JUNGE, MEI, PARCET AND XIA iii)
H¨ormander ‘kernel’ conditions • Φ j,t (cid:16)(cid:12)(cid:12) δ (cid:0) ( id ⊗ T ) (cid:0) ( ⊗ f )( − a j,t ) (cid:1)(cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k M , • Ψ j,t (cid:16)(cid:12)(cid:12) δ (cid:0) ( id ⊗ T ) (cid:0) ( ⊗ f )( − A j,t ) (cid:1)(cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k M , for certain family of cpu linear maps Φ j,t , Ψ j,t : M ¯ ⊗M → M .In condition ii), A j,t and a j,t play the role of ‘dilated balls’ from q j,t . In the lastcondition, δ is the derivation x x ⊗ − ⊗ x acting on the second leg of thetensor product. In the Euclidean case, these conditions reduce to L -boundednessand the classical size/smoothness conditions for the kernel. Our general conditionsinclude many more amplification algebras and derivations, other than M ¯ ⊗M and δ . Any map T : A M → M satisfying the above CZ-conditions will be called a column CZ- operator . Theorem A2 .
Let ( M , τ ) be a semifinite von Neumann algebra equipped witha Markov semigroup S = ( S t ) t ≥ with associated Markov metric Q fulfilling ouralgebraic and analytic assumptions. Then, any column CZ -operator T defines abounded operator T : A M → BMO c Q ( M ) . Interpolation and duality give similar ( symmetrized ) conditions for L p -boundedness. A generalized form of Theorem A2 is the main result of this paper. It is easyto recover Euclidean CZ-extrapolation from it. In the Euclidean and many otherdoubling scenarios, the size kernel condition ii) does not play any role. Our nextgoal is to explore how the general form of Theorem A2 applies in concrete vonNeumann algebras with specific Markov metrics.
Applications
Algebraic Calder´on-Zygmund theory applies in classical and noncommutativemeasure spaces. In the commutative context, we shall limit ourselves to provethat algebraic and classical theories match in three important cases: Euclideanspaces with both Lebesgue or Gaussian measures and Riemannian manifolds withnon-negative Ricci curvature. We shall not explore further implications in newcommutative scenarios, like abstract probability spaces or fractals equipped withspecific Dirichlet forms. In the noncommutative context, we start by analyzingthe model case of matrix-valued functions from a very general viewpoint. We alsoconsider Calder´on-Zygmund operators over matrix algebras, generalizing triangulartruncations as the archetype of singular integral operator. Most importantly, ourabstract theory applies to quantum Euclidean spaces and quantum groups, whichconstitute our main motivations in this paper.It will be useful to specify the form that our Calder´on-Zygmund operators takewhen come associated to a concrete kernel. Our applications below include CZconditions on the kernel. In the basic model case above, we set(Ker 1) T k f = ( id ⊗ τ ) (cid:0) k ( ⊗ f ) (cid:1) for some kernel k affiliated to M ¯ ⊗M op . Recall that the opposite structure ( M op is the same algebra M endowed with the reversed product a · b = ba ) in the second LGEBRAIC CALDER ´ON-ZYGMUND THEORY 7 tensor leg of the kernel for this (standard) model was already justified in [24]. It isa feature of CZ theory which can only be witnessed in noncommutative algebras. Itwill also be useful to generalize a bit our model case before analyzing any concreteapplication. Consider an auxiliary von Neumann algebra A equipped with a n.s.f.trace ϕ , a ∗ -homomorphism σ : M → A ¯ ⊗M and the representation(Ker 2) S ˜ k f = ( id ⊗ ϕ ) (cid:0) ˜ k flip ◦ σ ( f ) (cid:1) for some kernel ˜ k affiliated to M ¯ ⊗A op . Of course, when A = M and σ ( f ) = ⊗ f we recover our model case above, with kernel representation (Ker 1). This moregeneral framework requires to redefine R j,t in (Avg) and the CZ conditions, as weshall do in the body of the paper. The advantage is to take A as an elementary(commutative) algebra, from which we can transfer metric information. One maythink of σ as a corepresentation in the terminology of quantum groups. TheoremA2 still holds in this case. We shall refer to intrinsic or transferred theories whenusing the model case A = M or its generalization respectively. Quantum Euclidean spaces.
As geometrical spaces with noncommuting spatialcoordinates, quantum Euclidean spaces have appeared frequently in the literatureof mathematical physics, in the contexts of string theory and noncommutativefield theory. These algebras play the role of a central and testing example innoncommutative geometry as well. The singular integral operators on quantumEuclidean spaces naturally appear in the recent study of Connes’ quantized calculus[40, 42, 58] and noncommutative harmonic analysis [11, 24, 25, 65]. LetΘ ∈ M n ( R )be anti-symmetric. Briefly, the quantum Euclidean space R Θ is the von Neumannalgebra generated by certain family of unitaries { u j ( s ) : 1 ≤ j ≤ n, s ∈ R } satisfying u j ( s ) u j ( t ) = u j ( s + t ) ,u j ( s ) u k ( t ) = e πi Θ jk st u k ( t ) u j ( s ) . Define λ Θ ( ξ ) = u ( ξ ) u ( ξ ) · · · u n ( ξ n ) and set f = Z R n ˇ f Θ ( ξ ) λ Θ ( ξ ) dξ = λ Θ ( ˇ f Θ ) . for ˇ f Θ ∈ C c ( R n ). The trace on R Θ is determined by τ Θ ( f ) = τ Θ (cid:18)Z R n ˇ f Θ ( ξ ) λ Θ ( ξ ) dξ (cid:19) = ˇ f Θ (0) . When Θ = 0, L p ( R Θ , τ Θ ) reduces to L p ( R n ) with the Lebesgue measure. Precisedefinitions and a theory of singular integrals for R Θ appears in [24]. The main resultrelies on gradient kernel conditions for the intrinsic model (Ker 1). Remarkably, weshow in this paper that the transference model (Ker 2) σ Θ : R Θ ∋ λ Θ ( ξ ) exp ξ ⊗ λ Θ ( ξ ) ∈ L ∞ ( R n ) ¯ ⊗R Θ goes further, since it just requires H¨ormander type smoothness for the kernel. Hereexp ξ stands for the ξ -th character exp(2 πi h ξ, ·i ) in R n . There is a close relationbetween both models in this case T k ( f ) = S ˜ k ( f ) for k = ˜ π Θ (˜ k ) and ˜ π Θ ( m ⊗ exp ξ ) = mλ Θ ( ξ ) ∗ ⊗ λ Θ ( ξ ) . JUNGE, MEI, PARCET AND XIA
Another crucial map is the ∗ -homomorphism π Θ : L ∞ ( R n ) ∋ exp ξ λ Θ ( ξ ) ⊗ λ Θ ( ξ ) ∗ ∈ R Θ ¯ ⊗R opΘ . If B R denotes the Euclidean R -ball centered at the origin, define the projections a R = π Θ (1 R ) and a ⊥ R = − a R . Set k σ = ( σ Θ ⊗ id R opΘ )( k ) ∈ L ∞ ( R n ) ¯ ⊗R Θ ¯ ⊗R opΘ and define the derivation δϕ ( x, y ) = ϕ ( x ) − ϕ ( y ) to set the kernel condition in L ∞ ( R n ) ¯ ⊗R Θ ¯ ⊗R opΘ (H¨or) sup | x |≤ R, | y |≤ R (cid:12)(cid:12)(cid:12) δ (cid:16) ( id ⊗ id ⊗ τ Θ ) (cid:2) k σ ( ⊗ ⊗ f )( ⊗ a ⊥ R ) (cid:3)(cid:17) ( x, y ) (cid:12)(cid:12)(cid:12) . k f k R Θ . As we shall justify in the paper, (H¨or) is the right form of H¨ormander kernelcondition in this framework. The column BMO-norm admits in R Θ an equivalentform k f k BMO c ( R Θ ) ≈ k σ Θ ( f ) k BMO c ( R n ; R Θ ) for the operator-valued BMO space BMO c ( R n ; R Θ ) from [43]. These are all theingredients to obtain Calder´on-Zygmund extrapolation over quantum Euclideanspaces. Namely, the general form of Theorem A2 then yields the following theorem. Theorem B1. T k is bounded from R Θ to BMO c ( R Θ ) provided :i) T k is bounded on L ( R Θ ) . ii) The kernel condition (H¨or) holds.Interpolation and duality give similar ( symmetrized ) conditions for L p -boundedness. Theorem B1 improves the main CZ extrapolation theorem in [24] by reducing thegradient kernel condition there to the (more flexible) H¨ormander integral conditionabove, as we shall prove along the paper. In fact, the result which we shall finallyprove is slightly more general than the statement above.
Quantum groups.
Let G be a locally compact group with a left invariant Haarmeasure µ . When G is abelian, the Fourier transform carries the convolution algebra L (G , µ ) into the multiplication algebra L ∞ ( b G , b µ ) associated to the dual groupwith its (normalized) Haar measure. However, when G is not abelian, we can notconstruct the dual group and the multiplication algebra above becomes the groupvon Neumann algebra which is generated by the left regular representation of G.These algebras are basic models of (noncommutative, but still cocommutative)quantum groups, over which we shall study singular integrals.Let G be a locally compact quantum group —precise definitions in the bodyof the paper— with comultiplication ∆ and left-invariant and right-invariant Haarweights ψ , ϕ . Given a weak- ∗ dense subspace A of L ∞ ( G ) and a linear map T satisfying T ( A ) ⊂ L ∞ ( G ), it is is a convolution map when( T ⊗ id G ) ◦ ∆ = ∆ ◦ T = ( id G ⊗ T ) ◦ ∆ . To simplify the problem, we shall consider the case where G admits an α -doublingintrinsic Markov metric. That is, the projections which generate the cpu maps R j,t ’s satisfy ψ ( q α ( j ) ,t ) ψ ( q j,t ) ≤ c α for a strictly increasing function α : N → N with α ( j ) > j and a constant c α . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 9
Theorem B2.
Let G be a locally compact quantum group and assume it comesequipped with a convolution semigroup S = ( S t ) t ≥ which admits an α -doublingintrinsic Markov metric. Let T : A → L ∞ ( G ) be a convolution map defined on aweakly dense ∗ -subalgebra A of L ∞ ( G ) such that i) T is bounded on L ( G ) . ii) 1 | ψ ( q j,t ) | ( ψ ⊗ ψ ) (cid:16) ( q j,t ⊗ q j,t ) (cid:12)(cid:12) δ (cid:0) T ( f q ⊥ α ( j ) ,t (cid:1)(cid:12)(cid:12) (cid:17) . k f k L ∞ ( G ) .Then, the linear map T extends to a bounded map T : L ∞ ( G ) → BMO c S ( L ∞ ( G )) . As usual, L p estimates follow from symmetrized conditions by interpolation andduality. In fact, we shall prove a more general statement which incorporates tensorproducts with an additional algebra ( M , τ ). Theorem B2 is proved one more timefrom Theorem A2. In fact, it is conceivable to remove the α -doubling restriction andstill make the convolution map bounded under an additional size kernel conditionas Theorem A2 indicates. Noncommutative transference.
In a different direction, we shall finish thispaper with a section devoted to noncommutative forms of Calder´on-Cotlar methodof transference [5, 13, 15]. The basic idea is to transfer L p estimates of convolutionmaps on quantum groups to a much wider class of maps which arise by transference.We refer to [7, 9, 11, 48, 50, 54] for other forms of transference in the context ofgroup von Neumann algebras and quantum tori.1. Markov metrics
An abstract form of Calder´on-Zygmund theory incorporating noncommutativealgebras lacks standard geometrical tools. Given a Markov semigroup on a vonNeumann algebra —a semigroup of normal cpu self-adjoint maps on the givenalgebra— we shall construct a ‘metric’ governing the Markov process. Our modelcase in a commutative measure space (Ω , µ ) is a Markov semigroup of linear mapsof the form S t f ( x ) = Z Ω s t ( x, y ) f ( y ) dµ ( y ) . The idea is to find pointwise majorants of the form(1.1) s t ( x, y ) ≤ ∞ X j =1 | σ j,t ( x ) | µ (Σ j,t ( x )) χ Σ j,t ( x ) ( y ) , so that S t f ( x ) is dominated by a given combination of averaging operators overcertain measurable sets Σ j,t ( x ). These sets will determine some sort of metric on(Ω , µ ) under additional integrability properties. Naively, we may think of them asballs or coronas around x in the hidden metric with radii depending on ( j, t ). Inthis section we formalize this idea and construct BMO spaces with respect to theassociated ‘Markov metric’ which satisfy the expected interpolation results.1.1. Hilbert modules.
A noncommutative measure space is a pair ( M , τ ) formedby a semifinite von Neumann algebra M and a n.s.f. trace τ . We assume in whatfollows that the reader is familiar with basic terminology from noncommutativeintegration theory [36, 61]. Nonexpert readers may proceed by fixing a measure space (Ω , µ ) with M = L ∞ (Ω) and τ the integral operator associated to µ . Givena cpu map Φ : M → M we may construct the Hilbert module M ¯ ⊗ Φ M . Namelyconsider the seminorm on M ⊗ Mk ξ k M ¯ ⊗ Φ M = (cid:13)(cid:13)p h ξ, ξ i Φ (cid:13)(cid:13) M determined by the M -valued inner product D X j a j ⊗ b j , X k a ′ k ⊗ b ′ k E Φ = X j,k b ∗ j Φ( a ∗ j a ′ k ) b ′ k . Then M ¯ ⊗ Φ M will stand for the completion in the topology determined by ξ α → ξ when τ ( h ξ − ξ α , ξ − ξ α i Φ g ) → g ∈ L ( M ). When Φ is normal, the abstractcharacterization of Hilbert modules [51] yields a weak- ∗ continuous right M -modulemap ρ : M ¯ ⊗ Φ M → H c ¯ ⊗M satisfying h ξ, η i Φ = ρ ( ξ ) ∗ ρ ( η ). Let us collect a fewproperties which will be instrumental along this paper. Lemma 1.1.
Given a cpu map
Φ :
M → M :i) (cid:10) ξ + ξ , ξ + ξ (cid:11) Φ ≤ (cid:10) ξ , ξ (cid:11) Φ + 2 (cid:10) ξ , ξ (cid:11) Φ , ii) (cid:13)(cid:13) f ⊗ M − M ⊗ Φ f (cid:13)(cid:13) M ¯ ⊗ Φ M = (cid:13)(cid:13) Φ | f | − | Φ f | (cid:13)(cid:13) M , iii) (cid:12)(cid:12) Φ f − g (cid:12)(cid:12) ≤ (cid:10) f ⊗ M − M ⊗ g, f ⊗ M − M ⊗ g (cid:11) Φ , iv) (cid:13)(cid:13) f ⊗ M − M ⊗ Φ f (cid:13)(cid:13) M ¯ ⊗ Φ M ∼ inf g ∈M (cid:13)(cid:13) f ⊗ M − M ⊗ g (cid:13)(cid:13) M ¯ ⊗ Φ M , v) If Φ ≤ cp P k β k Ψ k , then k ξ k M ¯ ⊗ Φ M ≤ (cid:16) X k β k k ξ k M ¯ ⊗ Ψ k M (cid:17) . Proof.
The first inequality follows from hermitianity of the inner product and theidentity h ξ, η i Φ = ρ ( ξ ) ∗ ρ ( η ) explained above. The second one is straightforwardfrom the definition of M ¯ ⊗ Φ M . The third inequality follows from Kadison-Schwarzinequality after expanding both sides. The lower estimate in iv) holds trivially withconstant 1, while the upper estimate holds with constant 2 since f ⊗ M − M ⊗ Φ f = ( f ⊗ M − M ⊗ g ) − ( M ⊗ (Φ f − g ))and the second term on the right hand side is estimated using iii). Finally, for thelast inequality let ξ = P k a k ⊗ B j and define the column matrices A ∗ = P k a ∗ k ⊗ e k and B = P k B j ⊗ e k . Then we find (cid:10) ξ, ξ (cid:11) Φ = X j,k b ∗ j Φ( a ∗ j a k ) B j = B ∗ Φ( A ∗ A ) B ≤ X k β k B ∗ Ψ k ( A ∗ A ) B = X k β k (cid:10) ξ, ξ (cid:11) Ψ k . (cid:3) Let ( M , τ ) denote a noncommutative measure space equipped with a Markovsemigroup S = ( S t ) t ≥ acting on it. A Markov metric associated to ( M , τ ) and S is determined by a family Q = n(cid:0) R j,t , σ j,t , γ j,t (cid:1) : ( j, t ) ∈ Z + × R + o where R j,t : M → M are completely positive unital maps and σ j,t , γ j,t are elementsof the von Neumann algebra M with γ j,t ≥ M , so that the estimates below hold:i) Hilbert module majorization : (cid:10) ξ, ξ (cid:11) S t ≤ X j ≥ σ ∗ j,t (cid:10) ξ, ξ (cid:11) R j,t σ j,t , LGEBRAIC CALDER ´ON-ZYGMUND THEORY 11 ii)
Metric integrability condition : k Q = sup t> (cid:13)(cid:13)(cid:13) X j ≥ σ ∗ j,t γ j,t σ j,t (cid:13)(cid:13)(cid:13) M < ∞ .Our notion of Markov metric is easily understood for our (commutative) modelcase above. Let S = ( S t ) t ≥ be a Markov semigroup on (Ω , µ ) with associatedkernels s t ( x, y ) satisfying the pointwise estimate (1.1). Given ξ : Ω × Ω → C essentially bounded, we have(1.2) (cid:10) ξ, ξ (cid:11) S t = Z Ω s t ( x, y ) | ξ ( x, y ) | dµ ( y ) ≤ ∞ X j =1 | σ j,t ( x ) | µ (Σ j,t ( x )) Z Σ j,t ( x ) | ξ ( x, y ) | dµ ( y ) . This means that R j,t f ( x ) is the average of f over the set Σ j,t ( x ). Reciprocally, if wetake ξ k ( x, y ) = φ k ( y − y ) to be an approximation of identity around y , we recoverthe pointwise estimates for the kernel s t ( x, y ). In other more general contexts, theupper bounds for the kernel or even the kernel description of the semigroup mightnot have the same form. As we shall see, many of these cases can still be handledvia Hilbert module majorization. We shall provide along the paper a wide varietyof examples which fall into these possible classes.1.2. Semigroup
BMOs . Given a noncommutative measure space ( M , τ ) and aMarkov semigroup S = ( S t ) t ≥ acting on ( M , τ ), we may define the semigroup BMO S - norm as k f k BMO S = max n k f k BMO r S , k f k BMO c S o , where the row and column BMO norms are given by k f k BMO r S = sup t ≥ (cid:13)(cid:13)(cid:13)(cid:16) S t ( f f ∗ ) − ( S t f )( S t f ) ∗ (cid:17) (cid:13)(cid:13)(cid:13) M , k f k BMO c S = sup t ≥ (cid:13)(cid:13)(cid:13)(cid:16) S t ( f ∗ f ) − ( S t f ) ∗ ( S t f ) (cid:17) (cid:13)(cid:13)(cid:13) M . This definition makes sense since we know from the Kadison-Schwarz inequalitythat | S t f | ≤ S t | f | . The null space of this seminorm is ker A ∞ , the fixed-pointsubspace of our semigroup. Indeed, if k f k BMO S = 0 we know from [12] that f belongs to the multiplicative domain of S t , so that τ ( gf ) = τ ( S t/ ( gf )) = τ ( S t/ ( g ) S t/ ( f )) = τ ( gS t ( f )) . This proves that f is fixed by the semigroup. Reciprocally, ker A ∞ is a ∗ -subalgebraof M by [35]. Thus, the seminorm vanishes on ker A ∞ . In particular, we obtain anorm after quotienting out ker A ∞ . Letting w t ( f ) = f ⊗ − ⊗ S t f , this providesus with a map f ∈ M w (cid:0) w t ( f ) (cid:1) t ≥ ∈ M t ≥ M ¯ ⊗ S t M which becomes isometric when we equip M with the norm in BMO c S . Define BMO c S as the weak- ∗ closure of w ( M ) in the latter space. Similarly, we may define BMO S as the intersection BMO r S ∩ BMO c S , where the row BMO follows by taking adjointsabove. The natural operator space structure is given by M m (BMO S ( M )) = BMO b S ( M m ( M )) with b S t = id M m ⊗ S t . Remark 1.2.
Incidentally, we note that BMO S is written as bmo ( S ) in [32]. It will be essential for us to provide interpolation results between semigroup typeBMO spaces and the corresponding noncommutative L p spaces. It is a hard problemto identify the minimal regularity on the semigroup S = ( S t ) t ≥ which suffices forthis purpose. The first substantial progress was announced in a preliminary versionof [31], where the gradient form 2Γ( f , f ) = A ( f ∗ ) f + f ∗ A ( f ) − A ( f ∗ f ) with A the infinitesimal generator of the semigroup, was a key tool in finding sufficientregularity conditions in terms of nice enough Markov dilations. However we knowafter [32] an even sharper condition. Consider the sets A S f = n B t f = 1 t (cid:0) S t ( f ) + f − S t ( f ) f − f S t ( f ) (cid:1) : t > o , Γ M = n f ∈ M s . a . : A S f is relatively compact in L ( M ) o , where M s . a . denotes the self-adjoint part of M . The family A S f is called uniformlyintegrable in L ( M ) if for all ε > δ > k ( B t f ) q k < ε for every projection q satisfying τ ( q ) < δ . It is well-known that A S f is relativelycompact in L ( M ) if and only if it is bounded and uniformly integrable. Let usalso recall that B t f → f, f ) as t →
0. Define L ◦ p ( M ) = n f ∈ L p ( M ) : lim t →∞ S t f = 0 o . As it was explained in [32], the space L ◦ p ( M ) is complemented in L p ( M ) and[ L ◦ ( M ) , BMO S ] form an interpolation couple. A Markov semigroup S = ( S t ) t ≥ satisfying that Γ M is weak- ∗ dense in M s . a . is called regular . All the semigroupsthat we handle in this paper are regular. The following result will be crucial inwhat follows, we refer the reader to [32] for a detailed proof. Theorem 1.3. If S = ( S t ) t ≥ is regular on ( M , τ ) (cid:2) BMO S , L ◦ p ( M ) (cid:3) p/q ≃ cb L ◦ q ( M ) for all ≤ p < q < ∞ . Note that interpolation against the full space L p ( M ) is meaningless since BMO S does not distinguish the fixed-point space of the semigroup. Very roughly, weshall typically apply the above result to a CZO which is bounded on L ( M ) andsends a weak- ∗ dense subalgebra A of M to BMO S . Recalling the projection map J p : L p ( M ) → L ◦ p ( M ) and letting T denote the CZO, we find by interpolation andthe weak- ∗ density of A that J p T : L p ( M ) = (cid:2) A , L ( M ) (cid:3) /p → (cid:2) BMO S , L ◦ ( M ) (cid:3) /p = L ◦ p ( M ) ⊂ L p ( M ) . To obtain L p boundedness of T , it suffices to assume that T leaves the fixed-pointspace invariant and is bounded on it. It should be noticed though, that in manycases the L p boundedness of the CZO follows automatically. For instance, in R n with the Lebesgue measure and the heat semigroup, it turns out that L p = L ◦ p . Onthe other hand, the fixed-point space for the Poisson semigroup on the n -torus is justcomposed of constant functions and the corresponding projection can be estimatedapart regarded as a conditional expectation. Moreover, the same applies for Fouriermultipliers on arbitrary discrete groups. The L p boundedness for 1 < p < LGEBRAIC CALDER ´ON-ZYGMUND THEORY 13
Markov metric
BMOs . Let us now introduce a Markov metric type BMOspace for von Neumann algebras and relate it with the semigroup type BMO spacesdefined above. Given a Markov semigroup S = ( S t ) t ≥ acting on ( M , τ ), considera Markov metric Q = { ( R j,t , σ j,t , γ j,t ) : ( j, t ) ∈ Z + × R + } as defined above anddefine k f k BMO Q = max {k f k BMO c Q , k f ∗ k BMO c Q } , where the column BMO-norm isgiven bysup t> inf M t cpu sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) γ − j,t (cid:2) R j,t | f | − | R j,t f | + | R j,t f − M t f | (cid:3) γ − j,t (cid:17) (cid:13)(cid:13)(cid:13) M and the infimum runs over cpu maps M t : M → M . Since γ j,t ≥ M , the inversesexist and L ∞ ( M ) embeds in BMO Q . Indeed, using that R j,t and M t are cpu, thesquare braket above is bounded by 5 k f k ∞ M and γ − j,t ≤ M . The row norm isestimated in the same way. Now, recalling the value of the constant k Q in ourdefinition of Markov metric, we prove that BMO Q embeds in BMO S . Theorem 1.4.
Let ( M , τ ) be a noncommutative measure space equipped with aMarkov semigroup S = ( S t ) t ≥ . Let us consider a Markov metric Q associated to S = ( S t ) t ≥ . Then, we find k f k BMO S . k Q k f k BMO Q . In particular, we see that L ∞ ( M ) ⊂ BMO Q ⊂ BMO S and (cid:2) BMO Q , L ◦ p ( M ) (cid:3) p/q ≃ L ◦ q ( M ) for all ≤ p < q < ∞ for any Markov metric Q associated to a regular semigroup S = ( S t ) t ≥ on ( M , τ ) . Proof.
Let us set ξ t = f ⊗ M − M ⊗ M t f = ( f ⊗ M − M ⊗ R j,t f ) + ( M ⊗ ( R j,t f − M t f )) = ξ j,t + ξ j,t . The assertion follows from Lemma 1.1 and our definition of Markov metric k f k BMO c S = sup t> (cid:13)(cid:13)(cid:13)(cid:16) S t | f | − | S t f | (cid:17) (cid:13)(cid:13)(cid:13) M = sup t> (cid:13)(cid:13) f ⊗ M − M ⊗ S t f (cid:13)(cid:13) M ¯ ⊗ St M . sup t> (cid:13)(cid:13) f ⊗ M − M ⊗ M t f (cid:13)(cid:13) M ¯ ⊗ St M = sup t> kh ξ t , ξ t i S t k M . sup t> (cid:13)(cid:13)(cid:13) X j ≥ σ ∗ j,t (cid:2) h ξ j,t , ξ j,t i R j,t + h ξ j,t , ξ j,t i R j,t (cid:3) σ j,t (cid:13)(cid:13)(cid:13) M ≤ k Q k f k BMO c Q . The identities are clear. The first inequality follow from Lemma 1.1 iv), the secondone from the Hilbert module majorization associated to the Markov metric andLemma 1.1 i). To justify the last inequality, note that the square bracket inside theterm on the left equals R j,t | f | −| R j,t f | + | R j,t f − M t f | . Hence, left multiplicationby γ j,t γ − j,t and right multiplication by γ − j,t γ j,t yields the given inequality with k Q the metric integrability constant. The interpolation result follows from Theorem1.3 and the embeddings L ∞ ( M ) ⊂ BMO Q ⊂ BMO S . The proof is complete. (cid:3) Remark 1.5.
Let ξ = P j A j ⊗ B j , with A j = (cid:16) a jαβ (cid:17) and B j = (cid:16) b jαβ (cid:17) elements of M m ( M ). If b S t = id M m ⊗ S t , it turns out that (cid:10) ξ, ξ (cid:11) b S t = m X α =1 D X j,β a jαβ ⊗ b jβγ | {z } η α,γ , X j,β a jαβ ⊗ b jβγ | {z } η α,γ E S t ! γ ,γ ∈ M m ( M ) . This can be used to provide an operator space structure on BMO Q . Namely, thecanonical choice for the matrix norms is M m (BMO Q ( M )) = BMO b Q ( M m ( M ))where the Markov metric on M m ( M ) b Q = n(cid:0) id M m ⊗ R j,t , M m ⊗ σ j,t , M m ⊗ γ j,t (cid:1)o is associated to the extended semigroup ( b S t ) t ≥ . Then, we trivially obtain thatk b Q = k Q < ∞ . However, according to the identity above for h ξ, ξ i b S t , the Hilbertmodule majorization takes the form m X α =1 (cid:10) η α,γ , η α,γ (cid:11) S t ! γ ,γ ≤ X j ≥ σ ∗ j,t m X α =1 (cid:10) η α,γ , η α,γ (cid:11) R j,t σ j,t ! γ ,γ . This gives a matrix-valued generalization of our Hilbert module majorization for S = ( S t ) t ≥ on M , to be checked when we use this o.s.s. Theorem 1.4 yieldsa cb-embedding of BMO Q into BMO S under this assumption. According to thecharacterization (1.2), it holds for Markov metrics on commutative spaces (Ω , µ ).1.4. The Euclidean metric.
Before using Markov metrics in our approach toCalder´on-Zygmund theory, it is illustrative to recover the Euclidean metric from asuitable Markov semigroup. Let S = ( H t ) t ≥ denote the classical heat semigroupon R n , with kernels h t ( x, y ) = 1(4 πt ) n exp (cid:16) −| x − y | t (cid:17) . Take Q = (cid:8) ( R j,t , σ j,t , γ j,t ) : ( j, t ) ∈ Z + × R + (cid:9) determined by • σ j,t ≡ e √ π j n e − j and γ j,t ≡ j n ≥ • R j,t f ( x ) = 1 | B √ jt ( x ) | Z B √ jt ( x ) f ( y ) dy .Note that σ j,t and γ j,t are allowed to be essentially bounded functions in R n , butin this case it suffices to take constant functions. In the definition of R j,t , we writeB r ( x ) to denote the Euclidean ball in R n centered at x with radius r . It is clear that R j,t defines a cpu map on L ∞ ( R n ). To show that Q defines a Markov metric, weneed to check that it provides a Hilbert module majorization of the heat semigroupand the metric integrability condition holds. The latter is straightforward, whilethe Hilbert module majorization reduces to check that h t ( x, y ) ≤ e √ π X j ≥ j n e − j | B √ jt ( x ) | χ B √ jt ( x ) ( y ) . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 15
This can be justified by determining the unique corona centered at x with radii p j − t and √ jt where y lives, details are left to the reader. Note that wecould have taken γ j,t ≡ γ j,t ≡ j n tocompare BMO Q with other BMO spaces which interpolate. Before that, our onlyevidences that this is the right Markov metric in the Euclidean case are the factthat the R j,t ’s are averages over Euclidean balls and the isomorphismBMO Q = BMO R n , where the latter space is the usual BMO space in R n k f k BMO R n = sup B ⊂ R n (cid:16) | B | Z B (cid:12)(cid:12) f ( x ) − f B (cid:12)(cid:12) dx (cid:17) . Here, the supremum is taken over all Euclidean balls B in R n and f B stands for theaverage of f over B. Let us justify this isomorphism. If we pick M t f ( x ) = R ,t f ( x )it follows from a standard calculation that (cid:12)(cid:12) R j,t f ( x ) − M t f ( x ) (cid:12)(cid:12) (1.3)= (cid:12)(cid:12)(cid:12) | B √ t ( x ) | Z B √ t ( x ) (cid:0) f ( y ) − f B √ jt ( x ) (cid:1) dy (cid:12)(cid:12)(cid:12) ≤ j n | B √ jt ( x ) | Z B √ jt ( x ) (cid:12)(cid:12) f ( y ) − f B √ jt ( x ) (cid:12)(cid:12) dy = j n (cid:0) R j,t | f | − | R j,t f | (cid:1) ( x ) . This automatically yields the following inequality k f k Q . sup j,t ess sup x ∈ R n | B √ jt ( x ) | Z B √ jt ( x ) (cid:12)(cid:12) f ( y ) − f B √ jt ( x ) (cid:12)(cid:12) dy ≤ k f k R n . The converse is even simpler, since taking j = 1 we obtain k f k R n = sup t> ess sup x ∈ R n | B √ t ( x ) | Z B √ t ( x ) (cid:12)(cid:12) f ( y ) − f B √ t ( x ) (cid:12)(cid:12) dy = sup t> (cid:13)(cid:13)(cid:13) γ − ,t (cid:2) R ,t | f | − | R ,t f | (cid:3) γ − ,t (cid:13)(cid:13)(cid:13) L ∞ ( R n ) ≤ k f k Q . Remark 1.6.
The term | R j,t f − M t f | did not play a significant role at this point.More generally, the above argument also works for any doubling metric space Ωequipped with a Borel measure µ : µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) for every x ∈ Ω and r >
0, with B ( x, r ) = { y ∈ Ω : dist( x, y ) ≤ r } . As we shall see later, the additionalterm | R j,t f − M t f | in the BMO Q -norm appears to include Tolsa’s RBMO spaces[62] in those measure spaces (Ω , µ ) for which we can find an appropriate Dirichletform which provides us with a Markov semigroup acting on (Ω , µ ). Remark 1.7.
A related semigroup BMO norm is k f k BMO c S = sup t ≥ (cid:13)(cid:13)(cid:13)(cid:16) H t (cid:2) | f − H t f | (cid:3)(cid:17) (cid:13)(cid:13)(cid:13) ∞ . All the norms consider so far are equivalent for the heat semigroup S = ( H t ) t ≥ on R n , generated by the Laplacian ∆ = P nj =1 ∂ x j . In fact, we may also consider bysubordination the Poisson semigroup P = ( P t ) t ≥ on R n generated by the square root √− ∆, or even other subordinations [23] . Then, elementary calculations givethe following norm equivalences up to dimensional constants k f k BMO R n ∼ k f k BMO P ∼ k f k BMO P ∼ k f k BMO S ∼ k f k BMO S ∼ k f k BMO Q . Moreover, let R = L ∞ ( R n ) ¯ ⊗M denote the von Neumann algebra tensor product of L ∞ ( R n ) with a noncommutative measure space ( M , τ ). Define the norm in BMO R as k f k BMO R = max {k f k BMO c R , k f ∗ k BMO c R } , where k f k BMO c R = sup B balls (cid:13)(cid:13)(cid:13)(cid:16) | B | Z B (cid:12)(cid:12) f ( x ) − f B (cid:12)(cid:12) dx (cid:17) (cid:13)(cid:13)(cid:13) M . Then, the same norm equivalences hold in the semicommutative case k f k BMO R ∼ k f k BMO P⊗ ∼ k f k BMO P⊗ ∼ k f k BMO S⊗ ∼ k f k BMO S⊗ , where S ⊗ ,t = S t ⊗ id M and P ⊗ ,t = P t ⊗ id M . Moreover, by Remark 1.5, all thesenorms are in turn equivalent to the norm in BMO Q R , with the Markov metricwhich arises tensorizing the canonical one with the identity/unit of M .2. Algebraic CZ theory
In classical Calder´on-Zygmund theory, L p boundedness of CZOs follows from L boundedness under a smoothness condition on the kernel. Our next goal is toidentify which are the analogues of these conditions for semifinite von Neumannalgebras equipped with a Markov metric, and to show L p boundedness of CZOsfulfilling them. Our new conditions are certainly surprising. The boundednessfor p = 2 must be replaced by a certain mixed-norm estimate (which reduces inthe classical theory to L boundedness), while H¨ormander kernel smoothness willbe formulated intrinsically without any reference to the kernel. These abstractassumptions will adopt a more familiar form in the specific cases that we shallconsider in the forthcoming sections.In order to give a Calder´on-Zygmund framework for von Neumann algebras westart with some initial considerations, which determine the general form of Markovmetrics that we shall work with. Consider a Markov metric Q associated to aMarkov semigroup S = ( S t ) t ≥ acting on ( M , τ ). Then, we shall assume that thecpu maps R j,t from Q are of the following form(2.1) M ρ j −→ N ρ E ρ −→ ρ ( M ) ≃ M ,R j,t f = E ρ ( q j,t ) − E ρ (cid:0) q j,t ρ ( f ) q j,t (cid:1) E ρ ( q j,t ) − , where ρ , ρ : M → N ρ are ∗ -homomorphisms into certain von Neumann algebra N ρ , the map E ρ : N ρ → ρ ( M ) is an operator-valued weight and the q j,t ’s areprojections in N ρ . In particular, we shall assume that our formula for R j,t f makessense so that q j,t and q j,t ρ ( f ) q j,t belong to the domain of E ρ , see Section 2.1 forfurther details. Our model provides a quite general form of Markov metric whichincludes the Markov metric for the heat semigroup considered before. Indeed, take N ρ = L ∞ ( R n × R n ) with ρ f ( x, y ) = f ( x ) and ρ f ( x, y ) = f ( y ). Let E ρ be theintegral in R n with respect to the variable y and set q j,t ( x, y ) = χ B √ jt ( x ) ( y ) = χ B √ jt ( y ) ( x ) = χ | x − y | < √ jt . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 17
Then, it is straightforward to check that we recover from (2.1) the R j,t ’s for theheat semigroup. Note that the q j,t ( x, · )’s reproduce in this case all the Euclideanballs in R n . Morally, this is why we call Q a Markov metric, since it codifies somesort of underlying metric in ( M , τ ). According to our definition of BMO Q , we shallalso consider projections q t in N ρ and cpu maps(2.2) M t f = E ρ ( q t ) − E ρ (cid:0) q t ρ ( f ) q t (cid:1) E ρ ( q t ) − . Operator-valued weights.
In this subsection we briefly review the definitionand basic properties of operator-valued weights from [26, 27]. A unital, weaklyclosed ∗ -subalgebra is called a von Neumann subalgebra. A conditional expectation E M : N → M onto a von Neumann subalgebra M is a positive unital projectionsatisfying the bimodular property E M ( a f a ) = a E M ( f ) a for all a , a ∈ M . Itis called normal if sup α E M ( f α ) = E M (sup α f α ) for bounded increasing nets ( f α )in N + . A normal conditional expectation exists if and only if the restriction of τ to the von Neumann subalgebra M remains semifinite [61]. Any such conditionalexpectation is trace preserving τ ◦ E M = τ .The extended positive part c M + of the von Neumann algebra M is the set oflower semicontinuous maps m : M ∗ , + → [0 , ∞ ] which are linear on the positivecone, m ( λ φ + λ φ ) = λ m ( φ ) + λ m ( φ ) for λ j ≥ φ j ∈ M ∗ , + . Theextended positive part is closed under addition, increasing limits and is fixed bythe map x a ∗ xa for any a ∈ M . It is clear that M + sits in the extended positivepart. When M is abelian, we find M ≃ L ∞ (Ω , µ ) for some measure space (Ω , µ )and the extended positive part corresponds in this case to the set of µ -measurablefunctions on Ω (module sets of zero measure) with values in [0 , ∞ ]. A hardercharacterization of the extended positive part for arbitrary von Neumann algebraswas found by Haagerup in [26]. Assume that M acts on H and consider a positiveoperator A affiliated with M with spectral resolution A = R R + λde λ . Then, we mayconstruct an associated element in c M + m A ( φ ) = Z R + λd ( φ ( e λ )) . In general, any m ∈ c M + has a unique spectral resolution m ( φ ) = Z R + λd ( φ ( e λ )) + ∞ φ ( p )where the e λ ’s form an increasing family of projections in M and p is the projection M − lim λ e λ . Moreover, the map λ e λ is strongly continuous from the right andwe find that e = 0 iff m does not vanish on M + ∗ \ { } , while p = 0 iff the familyof φ ∈ M + ∗ with m ( φ ) < ∞ is dense in M + ∗ .Operator-valued weights appear as “unbounded conditional expectations” andthe simplest nontrivial model is perhaps a partial trace E M = tr A ⊗ id M with N = A ¯ ⊗M and A a semifinite non-finite von Neumann algebra. In general, an operator-valued weight from N to M is just a linear map E M : N + → c M + satisfying E M ( a ∗ f a ) = a ∗ E M ( f ) a for all a ∈ M . As usual, E M is called normal when sup α E M ( f α ) = E M (sup α f α ) forbounded increasing nets ( f α ) in N + . Since a ∗ f b = P k =0 i − k ( a + i k b ) ∗ f ( a + i k b ) by polarization, we see that bimodularity of conditional expectations is equivalentto E M ( a ∗ f a ) = a ∗ E M ( f ) a for a ∈ M . In particular, the fundamental propertieswhich operator-valued weights loose with respect to conditional expectations areunitality and the fact that unboundedness is allowed for the image. Additionally,when M = C the map E M becomes an ordinary weight on N . In analogy withordinary weights, we take L c ∞ ( N ; E M ) = n f ∈ N : (cid:13)(cid:13) E M ( f ∗ f ) (cid:13)(cid:13) M < ∞ o . Note that when E M = tr A ⊗ id M with N = A ¯ ⊗M , L c ∞ ( N ; E M ) are the Hilbertspace valued noncommutative L ∞ spaces defined in [30], which we denote by L c ( A ) ¯ ⊗M . Let N E M be the linear span of f ∗ f with f , f ∈ L c ∞ ( N ; E M ). Thenwe findi) N E M = span { f ∈ N + : k E M f k < ∞} ,ii) L c ∞ ( N ; E M ) and N E M are two-sided modules over M ,iii) E M has a unique linear extension E M : N E M → M satisfying E M ( a f a ) = a E M ( f ) a with f ∈ N E M and a , a ∈ M . In particular, if E M ( ) = we recover a conditional expectation onto M . Anoperator-valued weight E M is called faithful if E M ( f ∗ f ) = 0 implies f = 0 andsemifinite when L c ∞ ( N ; E M ) is σ -weakly dense in N . It is of interest to determinefor which pairs ( N , M ) we may construct n.s.f. operator-valued weights. Amongother results, Haagerup proved in [27] that this is the case when both von Neumannalgebras are semifinite and there exists a unique trace preserving one. Note thatconditional expectations do not always exist in this case. He also proved that given E M j n.s.f. operator-valued weights in ( N j , M j ) for j = 1 ,
2, there exists a uniquen.s.f. operator-valued weight E M ⊗M associated to ( N ¯ ⊗N , M ¯ ⊗M ) such that( φ ⊗ φ ) ◦ E M ⊗M = ( φ ◦ E M ) ⊗ ( φ ◦ E M ) for any pair ( φ , φ ) of normalsemifinite faithful weights on ( M , M ).2.2. Algebraic/analytic conditions.
The identity
T f ( x ) = Z Ω k ( x, y ) f ( y ) dµ ( y )is just a vague expression to consider classical Calder´on-Zygmund operators. It iswell-known that particular realizations as above are only meaningful outside thesupport of f and understanding k as a distribution which coincides with a locallyintegrable function on R n × R n \ ∆. Instead of that, we shall not specify any kernelrepresentation of our CZOs since our conditions below will be formulated in a moreintrinsic way. These kernel representations will appear later on in this paper withthe concrete examples that we shall consider.Let T be a densely defined operator on M , which means that T f ∈ M for all f in a weak- ∗ dense subalgebra A M of M . Our assumption does not necessarilyhold for classical Calder´on-Zygmund operators defined in abelian von Neumannalgebras ( M , τ ) = L ∞ (Ω , µ ), but it is true for the truncated singular integraloperators satisfying the standard size condition for the kernel, take for instance A M = M ∩ L ( M ). In particular, this is not a crucial restriction since we shall beable to take L p -limits as far as our estimates below are independent of T . Our aim is LGEBRAIC CALDER ´ON-ZYGMUND THEORY 19 to settle conditions on T of CZ type assuring that T : L ∞ ( M ) → BMO c Q , provided( M , τ ) comes equipped with a Markov metric Q . In this paragraph, we establishsome preliminary algebraic and analytic conditions on the Markov metric and theCZO. Consider ∗ -homomorphisms π , π : M → N π and an operator-valued weight E π : N π → π ( M ) which may or may not coincide with ρ , ρ and E ρ from (2.1).Assume there exists a (densely defined) map(2.3) b T : A N π ⊂ N π → N ρ satisfying b T ◦ π = ρ ◦ T on A M . Algebraic conditions: i) Q -monotonicity of E ρ E ρ ( q j,t | ξ | q j,t ) ≤ E ρ ( | ξ | )for all ξ ∈ N ρ and every projection q j,t determined by Q via the identity in(2.1). Similarly, we assume the same inequality holds when we replace the q j,t ’s by the q t ’s appearing in (2.2).ii) Right B -modularity of b T b T (cid:0) η π ρ − ( b ) (cid:1) = b T ( η ) b for all η ∈ A N π and all b lying in some von Neumann subalgebra B of ρ ( M ) which includes E ρ ( q t ), E ρ ( q j,t ) and ρ ( γ j,t ) for every projection q t and q j,t determined by Q via the identities in (2.1) and (2.2).As we shall see both conditions trivially hold in the classical theory, where thefirst condition essentially says that integrating a positive function over a “Markovmetric ball” is always smaller than integrating it over the whole space, while thesecond condition allows to take out x -dependent functions from the y -dependentintegral defining T . Our conditions remain true in many other situations, which willbe explored below in this paper. Nevertheless, condition i) suggests that certainamount of commutativity might be necessary to work with Markov metrics.To state our analytic conditions we introduce an additional von Neumann algebra N σ containing N ρ as a von Neumann subalgebra. Then, we consider derivations δ : N ρ → N σ given by the difference δ = σ − σ of two ∗ -homomorphisms, sothat δ ( ab ) = σ ( a ) σ ( b ) − σ ( a ) σ ( b ) = δ ( a ) σ ( b ) + σ ( a ) δ ( b ) as expected. We alsoconsider the natural amplification maps b R j,t : N ρ ∋ ξ E ρ ( q j,t ) − E ρ ( q j,t ξq j,t ) E ρ ( q j,t ) − ∈ ρ ( M ) , c M t : N ρ ∋ ξ E ρ ( q t ) − E ρ ( q t ξq t ) E ρ ( q t ) − ∈ ρ ( M ) . Analytic conditions: i) Mean differences conditions • b R j,t ( ξ ∗ ξ ) − b R j,t ( ξ ) ∗ b R j,t ( ξ ) ≤ Φ j,t (cid:0) δ ( ξ ) ∗ δ ( ξ ) (cid:1) , • (cid:2) b R j,t ( ξ ) − c M t ( ξ ) (cid:3) ∗ (cid:2) b R j,t ( ξ ) − c M t ( ξ ) (cid:3) ≤ Ψ j,t (cid:0) δ ( ξ ) ∗ δ ( ξ ) (cid:1) ,for some derivation δ : N ρ → N σ and cpu maps Φ j,t , Ψ j,t : N σ → ρ ( M ).ii) Metric/measure growth conditions • ≤ π ρ − E ρ ( q t ) − E π ( a ∗ t a t ) π ρ − E ρ ( q t ) − . π ρ − ( γ j,t ), • ≤ π ρ − E ρ ( q j,t ) − E π ( a ∗ j,t a j,t ) π ρ − E ρ ( q j,t ) − . π ρ − ( γ j,t ),for some family of operators a t , a j,t ∈ N π to be determined later on.A complete determination of the operators a t and a j,t is only possible after imposingadditional size and smoothness conditions in our definition of Calder´on-Zygmundoperator below. Nevertheless, we shall see that these operators will play the roleof “dilated Markov balls” as it is the case in classical CZ theory. In fact, in theclassical case our last condition trivially holds for doubling measures, and also formeasures of polynomial or even exponential growth provided we find a Markovmetric with large enough γ j,t ’s. Our assertions will be illustrated below. The firstcondition takes the form in the classical case of a couple of easy consequences ofJensen’s inequality, namely(2.4) − Z B | f | dµ − (cid:12)(cid:12)(cid:12) − Z B f dµ (cid:12)(cid:12)(cid:12) ≤ − Z B × B (cid:12)(cid:12) f ( y ) − f ( z ) (cid:12)(cid:12) dµ ( y ) dµ ( z ) , (cid:12)(cid:12)(cid:12) − Z B f dµ − − Z B f dµ (cid:12)(cid:12)(cid:12) ≤ − Z B × B (cid:12)(cid:12) f ( y ) − f ( z ) (cid:12)(cid:12) dµ ( y ) dµ ( z ) . CZ extrapolation.
Now we introduce CZOs in this context. As we alreadymentioned, we consider a priori densely defined (unbounded) maps T : A M → M whose amplified maps are right B -modules according to our algebraic assumptionsabove. In addition, we impose three conditions generalizing L boundedness, thesize and the smoothness conditions for the kernel. Calder´on-Zygmund type conditions: i) Boundedness condition b T : L c ∞ ( N π ; E π ) → L c ∞ ( N ρ ; E ρ ) . ii) Size “kernel” condition • c M t (cid:16)(cid:12)(cid:12) b T ( π ( f )( A j,t − a t )) (cid:12)(cid:12) (cid:17) . γ j,t k f k ∞ , • b R j,t (cid:16)(cid:12)(cid:12) b T ( π ( f )( A j,t − a j,t )) (cid:12)(cid:12) (cid:17) . γ j,t k f k ∞ ,for a family of operators A j,t ∈ N π with A j,t ≥ a j,t , a t to be determined.iii) Smoothness “kernel” condition • Φ j,t (cid:16)(cid:12)(cid:12) δ (cid:0) b T ( π ( f )( − a j,t )) (cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k ∞ , • Ψ j,t (cid:16)(cid:12)(cid:12) δ (cid:0) b T ( π ( f )( − A j,t )) (cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k ∞ .Let T : A M → M be a densely defined map which admits an amplification b T satisfying (2.3). Any such T will be called an algebraic column CZO whenever theamplification map is right B -modular and satisfies the CZ conditions we have givenabove. At first sight, our boundedness assumption might appear to be unrelatedto the classical condition. The reader could have expected the L boundednessof T , but our assumption is formally equivalent to it in the classical case andgives the right condition for more general algebras. On the other hand, our size LGEBRAIC CALDER ´ON-ZYGMUND THEORY 21 and smoothness conditions are intrinsic in the sense that the kernel is not specifiedunder this degree of generality. We shall recover classical kernel type estimates fromour conditions in our examples below. As explained above, the operators a t , a j,t and A j,t play the role of dilated Markov balls and our conditions were somehowmodeled by Tolsa’s arguments in [62]. Perhaps a significant difference —in contrastto Tolsa’s approach— is that our smoothness condition is analog to a H¨ormandertype condition, more than the (stronger) Lipschitz regularity assumption. Theorem 2.1.
Let ( M , τ ) be a noncommutative measure space equipped with aMarkov semigroup S = ( S t ) t ≥ with associated Markov metric Q which fulfills ouralgebraic and analytic assumptions. Then, any algebraic column CZO T defines abounded operator T : A M → BMO c Q . Proof.
The first goal is to estimate the norm ofA = γ − j,t (cid:16) R j,t | T f | − | R j,t T f | (cid:17) γ − j,t . The map Π j,t : a ⊗ b ∈ M ¯ ⊗ R j,t M 7→ ⊗ R j,t ( a ) b ∈ ⊗ M extends to a right( ⊗ M )-module projection, which is well-defined in the sense that h ξ, ξ i R j,t = 0implies Π j,t ( ξ ) = 0. Now, sinceA = γ − j,t D T f ⊗ − ⊗ R j,t T f , T f ⊗ − ⊗ R j,t T f E R j,t γ − j,t , we may use Π j,t to deduce the following identityA = D ( id − Π j,t )( T f ⊗ γ − j,t ) , ( id − Π j,t )( T f ⊗ γ − j,t ) E R j,t . Consider the amplification maps b R j,t and b Π j,t determined by R j,t = b R j,t ◦ ρ and Π j,t = b Π j,t ◦ ( ρ ⊗ id ) . By (2.3), it turns out that A = h a , a i b R j,t where a = ( id − b Π j,t )( ρ T f ⊗ γ − j,t )= ( id − b Π j,t )( b T π f ⊗ γ − j,t )= ( id − b Π j,t ) (cid:0) b T ( π ( f ) a j,t ) ⊗ γ − j,t (cid:1) + ( id − b Π j,t )( b T (cid:0) π ( f )( − a j,t )) ⊗ γ − j,t (cid:1) = a + a According to Lemma 1.1 i), we may estimate A as followsA . (cid:10) a , a (cid:11) b R j,t + (cid:10) a , a (cid:11) b R j,t = A + A . Since b Π j,t ( n ⊗ b ) = ⊗ b R j,t ( n ) b , the Kadison-Schwarz inequality yields Db Π j,t ( n ⊗ b ) , b Π j,t ( n ⊗ b ) E b R j,t . (cid:10) n ⊗ b, n ⊗ b (cid:11) b R j,t . In conjunction with Lemma 1.1 i) again, we deduce the following estimate for A A . D b T ( π ( f ) a j,t ) ⊗ γ − j,t , b T ( π ( f ) a j,t ) ⊗ γ − j,t E b R j,t = γ − j,t b R j,t (cid:16)(cid:12)(cid:12) b T ( π ( f ) a j,t ) (cid:12)(cid:12) (cid:17) γ − j,t . In order to bound the term in the right hand side, we apply (2.1) and the propertiesof the operator-valued weight E ρ together with our algebraic conditions. Indeed, wefirst use the Q -monotonicity of E ρ ; then the fact that it commutes with the left/right multiplication by elements affiliated to M (like γ − j,t or E ρ ( q j,t ) − / ); finally we usethe right B -modularity of the amplification of T : γ − j,t b R j,t (cid:16)(cid:12)(cid:12) b T ( π ( f ) a j,t ) (cid:12)(cid:12) (cid:17) γ − j,t ≤ γ − j,t E ρ ( q j,t ) − E ρ (cid:16)(cid:12)(cid:12) b T ( π ( f ) a j,t ) (cid:12)(cid:12) (cid:17) E ρ ( q j,t ) − γ − j,t = E ρ (cid:16) γ − j,t E ρ ( q j,t ) − (cid:12)(cid:12) b T (cid:0) π ( f ) a j,t (cid:1)(cid:12)(cid:12) E ρ ( q j,t ) − γ − j,t (cid:17) = E ρ (cid:12)(cid:12)(cid:12) b T (cid:16) π ( f ) a j,t π ρ − (cid:0) E ρ ( q j,t ) − γ − j,t (cid:1)| {z } ξ (cid:17)(cid:12)(cid:12)(cid:12) = E ρ | b T ( ξ ) | . Now, our first CZ condition i) gives the boundedness we need since k A k M ≤ (cid:13)(cid:13) b T ( ξ ) (cid:13)(cid:13) L c ∞ ( N ρ ; E ρ ) . (cid:13)(cid:13) ξ (cid:13)(cid:13) L c ∞ ( N π ; E π ) = (cid:13)(cid:13)(cid:13) E π (cid:16)(cid:12)(cid:12) π ( f ) a j,t π ρ − (cid:0) E ρ ( q j,t ) − γ − j,t (cid:1)(cid:12)(cid:12) (cid:17)(cid:13)(cid:13)(cid:13) M ≤ (cid:13)(cid:13)(cid:13) π ρ − (cid:0) γ − j,t E ρ ( q j,t ) − (cid:1) ∗ E π ( a ∗ j,t a j,t ) π ρ − (cid:0) E ρ ( q j,t ) − γ − j,t (cid:1)(cid:13)(cid:13)(cid:13) M k f k ∞ . The last term on the right is dominated by k f k ∞ according to our second analyticcondition on metric/measure growth. The estimate for A is simpler. Indeed, if weset ξ = b T ( π ( f )( − a j,t )) thenA = D ( id − b Π j,t )( ξ ⊗ γ − j,t ) , ( id − b Π j,t )( ξ ⊗ γ − j,t ) E b R j,t = γ − j,t (cid:16) b R j,t | ξ | − (cid:12)(cid:12) b R j,t ( ξ ) (cid:12)(cid:12) (cid:17) γ − j,t ≤ γ − j,t Φ j,t (cid:0) | δξ | (cid:1) γ − j,t . k f k ∞ , where the first inequality holds for some derivation δ : N ρ → N σ and some cpu mapΦ j,t : N σ → ρ ( M ) by our first analytic condition on mean differences. Then ourCZ condition iii) on kernel smoothness justifies our last estimate. Our estimates sofar prove the desired estimatesup t> sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) γ − j,t (cid:2) R j,t | T f | − | R j,t T f | (cid:3) γ − j,t (cid:17) (cid:13)(cid:13)(cid:13) M . k f k ∞ . Therefore, it remains to estimate the norm ofB = γ − j,t (cid:16)(cid:12)(cid:12) R j,t T f − M t T f (cid:12)(cid:12) (cid:17) γ − j,t . To do so, we decompose the middle term using (2.3) as follows R j,t T f − M t T f = b R j,t ( ρ T f ) − c M t ( ρ T f )= b R j,t (cid:16) b T (cid:0) π ( f ) a j,t (cid:1)(cid:17) − c M t (cid:16) b T (cid:0) π ( f ) a t (cid:1)(cid:17) + h b R j,t (cid:16) b T (cid:0) π ( f )( − a j,t ) (cid:1)(cid:17) − c M t (cid:16) b T (cid:0) π ( f )( − a t ) (cid:1)(cid:17)i = b − b + b . Letting B j = γ − j,t | b j | γ − j,t we get B . B + B + B . By Kadison-Schwarz we getB ≤ γ − j,t b R j,t (cid:16)(cid:12)(cid:12) b T ( π ( f ) a j,t ) (cid:12)(cid:12) (cid:17) γ − j,t . k f k ∞ , LGEBRAIC CALDER ´ON-ZYGMUND THEORY 23 where the last inequality was justified in our estimate of A above. Replacing q j,t by q t , the same argument serves to control the term B . To estimate B we decompose b as follows b = h b R j,t (cid:16) b T (cid:0) π ( f )( − A j,t ) (cid:1)(cid:17) − c M t (cid:16) b T (cid:0) π ( f )( − A j,t ) (cid:1)(cid:17)i + b R j,t (cid:16) b T (cid:0) π ( f )( A j,t − a j,t ) (cid:1)(cid:17) − c M t (cid:16) b T (cid:0) π ( f )( A j,t − a t ) (cid:1)(cid:17) = b + b − b . Taking ξ = b T (cid:0) π ( f )( − A j,t ) (cid:1) and applying our analytic condition i) on meandifferences together with our CZ condition iii) on kernel smoothness, we obtain that γ − j,t | b | γ − j,t . γ − j,t Ψ j,t (cid:0) | δξ | (cid:1) γ − j,t . k f k ∞ . It remains to estimate the terms B and B . Applying the Kadison-Schwarzinequality, it is easily checked that these terms are also dominated by k f k ∞ bymeans of our CZ size kernel condition ii). Altogether, we have justified thatsup t> inf M t cpu sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) γ − j,t (cid:2) | R j,t f − M t f | (cid:3) γ − j,t (cid:17) (cid:13)(cid:13)(cid:13) M . k f k ∞ . Combining our estimates for A and B, we deduce that T : A M → BMO c Q . (cid:3) The A M → BMO r Q boundedness of the map T is equivalent to the A M → BMO c Q boundedness of the map T † ( f ) = T ( f ∗ ) ∗ . According to this, an algebraic CZO isany column CZO T for which T † remains a column CZO. By Theorem 2.1, weknow that any algebraic CZO T associated to ( M , τ, Q ) as above is automatically A M → BMO Q bounded. Assuming L boundedness and regularity of the Markovsemigroup, we may interpolate via Theorem 1.4. Under the same assumptions for T ∗ , we may also dualize and obtain the following extrapolation result. Corollary 2.2.
Let ( M , τ ) be a noncommutative measure space equipped with aMarkov regular semigroup S = ( S t ) t ≥ and a Markov metric Q = ( R j,t , σ j,t , γ j,t ) fulfilling our algebraic and analytic assumptions. Then, every L -bounded algebraic CZO T satisfies that J p T : L p ( M ) → L ◦ p ( M ) for p > . Applying duality, similarconditions for T ∗ yield L p -boundedness of T J p for every < p < . Remark 2.3.
Theorem 2.1 admits a completely bounded version in the category ofoperator spaces. Since the operator space structure [22, 52] of BMO is determinedby M m (BMO S ( M )) = BMO b S ( M m ( M ))for b S = ( id M m ⊗ S t ) t ≥ , we just need to replace M by M m ( M ) everywhere, amplifyall the involved maps by tensorizing with id M m and require that the hypotheseshold with constants independent of m . Then, we obtain the cb-boundedness of T . Remark 2.4.
As noticed in the Introduction, a common scenario is given by thechoice N ρ = M ¯ ⊗M with ρ ( f ) = f ⊗ and ρ ( f ) = ⊗ f , together with E ρ = id ⊗ τ and π j = ρ j for j = 1 ,
2. In this case, it is clear that the amplification map is givenby b T = id M ⊗ T so that b T π = ρ T. In particular, it turns out that the L boundedness of T in Corollary 2.2 followsautomatically from our CZ boundedness condition i). This is the case in classicalCalder´on-Zygmund theory. It is also true when N ρ = M ¯ ⊗A for an auxiliaryalgebra A and ρ = flip ◦ σ , where σ : M → A ¯ ⊗M is a ∗ -homomorphism satisfying E ρ ◦ ρ ( f ) = τ ( f ) M . This leads to another significant family of examples. It ishowever surprising that in general, the L boundedness and the CZ boundednessassumptions are a priori unrelated. Thus, CZ extrapolation requires in this contextto verify two boundedness conditions. It would be quite interesting to explore thecorresponding “ T (1) problems” that arise naturally.2.4. The classical theory revisited.
We now illustrate our algebraic approachin the classical context of Euclidean spaces with the Lebesgue measure. This willhelp us to understand some of our conditions and will show how some others areautomatic in a commutative framework. Take M = L ∞ ( R n ) with the Lebesguemeasure and S = ( H t ) t ≥ the heat semigroup H t = exp( t ∆). In Paragraph 1.4 weintroduced the Markov metric Q given by • σ j,t ≡ e √ π j n e − k and γ j,t ≡ j n ≥ • R j,t f ( x ) = 1 | B √ jt ( x ) | Z B √ jt ( x ) f ( y ) dy .Moreover, as we explained at the beginning of this section R j,t f = E ρ ( q j,t ) − E ρ (cid:0) q j,t ρ ( f ) q j,t (cid:1) E ρ ( q j,t ) − satisfies our basic assumption (2.1). Here the amplification von Neumann algebrais N ρ = L ∞ ( R n × R n ), the ∗ -homomorphisms ρ j f ( x , x ) = f ( x j ), the projections q j,t ( x, y ) = χ B √ jt ( x ) ( y ), and the operator-valued weight E ρ is the integration mapwith respect to the second variable. The cpu map M t which appears in the definitionof BMO Q is still taken by M t = R ,t as in Paragraph 1.4.Taking N π = N ρ and T a standard CZO in R n , the algebraic conditions triviallyhold in this case. Let E xj,k,t = B √ jt ( x ) × B √ kt ( x ). Taking Φ j,t and Ψ j,t to be theaveraging maps over E xj,j,t and E xj, ,t respectively and the family of dilated balls( A j,t ( x, y ) , a j,t ( x, y )) = ( χ α B √ jt ( x ) ( y ) , χ √ jt ( x ) ( y )) with a ≥
5, we may recoverthe conditions as we explained right after stating them. Let us now show how ouralgebraic CZ conditions hold from the classical ones. The boundedness conditionreduces to the classical one, see Remark 4.2 A). Our size conditions can be rewrittenas follows: • ess sup x ∈ R n − Z B √ t ( x ) (cid:12)(cid:12)(cid:12) Z a B √ jt ( x ) \ √ t ( x ) k ( y, z ) f ( z ) dz (cid:12)(cid:12)(cid:12) dy . j n k f k ∞ , • ess sup x ∈ R n − Z B √ jt ( x ) (cid:12)(cid:12)(cid:12) Z a B √ jt ( x ) \ √ jt ( x ) k ( y, z ) f ( z ) dz (cid:12)(cid:12)(cid:12) dy . j n k f k ∞ .The above conditions follow from the usual size condition | k ( y, z ) | . | z − y | n . Next, taking E xj,k,t as above, our smoothness conditions are: • ess sup x ∈ R n − Z E xj, ,t (cid:16) Z (5B √ jt ( x )) c (cid:0) k ( y , z ) − k ( y , z ) (cid:1) f ( z ) dz (cid:17) dy dy . j n k f k ∞ , • ess sup x ∈ R n − Z E xj,j,t (cid:16) Z ( a B √ jt ( x )) c (cid:0) k ( y , z ) − k ( y , z ) (cid:1) f ( z ) dz (cid:17) dy dy . j n k f k ∞ . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 25
The above conditions easily follow from the usual H¨ormander conditioness sup y ,y ∈ R n Z | y − z |≥ | y − y | | k ( y , z ) − k ( y , z ) | dz < ∞ . Note that our algebraic CZ conditions are slightly weaker than the classical CZconditions, but still sufficient to provide L ∞ → BMO boundedness of the CZ map.
Remark 2.5.
Our size condition is only used to estimate B in the proof of Theorem2.1. We saw in Paragraph 1.4 that B . A for the Euclidean metric. Thus, our sizecondition is not necessary here, as it also happens in the classical formulation.3.
Applications I — Commutative spaces
In this section we give specific constructions of Markov metrics on two basiccommutative spaces: Riemannian manifolds with nonnegative Ricci curvature andGaussian measure spaces. Beyond the Euclidean-Lebesguean setting consideredabove, these are the most relevant settings over which Calder´on-Zygmund theoryhas been studied. As a good illustration of our algebraic method, we shall recoverthe extrapolation results. Noncommutative spaces will be explored later on.3.1.
Riemannian manifolds.
Let (Ω , µ ) be a measure space equipped with aMarkov semigroup, so that we may construct the corresponding semigroup typeBMO space. In order to study the L ∞ → BMO boundedness of CZOs in (Ω , µ ) itis essential to identify a Markov metric to work with. Now we provide sufficientconditions for a semigroup on a Riemannian manifold to yield a Markov metricsatisfying our algebraic/analytic conditions, so that Theorem 2.1 is applicable. Letus consider an n -dimensional complete Riemannian manifold ( M, g ) equipped withthe geodesic distance d determined by the Riemannian metric g . Denote the volumeof a geodesic ball centered at x with radius r by vol g (B r ( x )). Let S M be a Markovsemigroup on M given by S M,t f ( x ) = Z M s t ( x, y ) f ( y ) dy. Proposition 3.1.
Assume that i) M has Ricci curvature ≥ . ii) The kernel admits an upper bound s t ( x, y ) . φ ( t ) n + ε vol g (B φ ( t ) ( x ))( d ( x, y ) + φ ( t )) n + ε , for some strictly positive function φ and some parameter ε > .Then S M admits a Markov metric satisfying the algebraic/analytic conditions. Proof.
If Σ j,t ( x ) = B j φ ( t ) ( x ), our assumption. gives(3.1) s t ( x, y ) . ∞ X j =1 − j ( n + ε ) vol g (Σ ,t ( x )) χ Σ j,t ( x ) ( y ) . According to Davies [17, Theorem 5.5.1], non-negative Ricci curvature impliesvol g (B r ( x )) ≤ c n r n , vol g (B γr ( x )) ≤ γ n vol g (B r ( x ))for all x ∈ M , r > γ >
1. In particular, vol g (Σ j,t ( x )) ≤ jn vol g (Σ ,t ( x )). By(3.1), this implies that ( σ j,t , γ j,t ) = (2 − jε/ ,
1) forms a Markov metric for S M inconjunction with the averaging maps R j,t f ( x ) = − Z Σ j,t ( x ) f ( y ) dy for ( j, t ) ∈ Z + × R + . Our construction for M = L ∞ ( M ) follows the basic model in the Introductionand the one used above in the Euclidean setting: N ρ = N π = M ¯ ⊗M with ρ j the canonical inclusion maps and q j,t ( x, y ) = χ Σ j,t ( x ) ( y ) = χ Σ j,t ( y ) ( x ). Then, thealgebraic conditions for the Markov metric are obviously satisfied. Let us now checkthe analytic conditions. Taking N σ = M ¯ ⊗M ¯ ⊗M , the derivation δ : N ρ → N σ given by δ ( a ⊗ b ) = a ⊗ ( ⊗ b − b ⊗ ) and the maps M t = R ,t , it turns outthat the mean difference conditions follow from Jensen’s inequality on normalizedballs of ( M, g ) as it follows from our comments after the definition of the analyticconditions. It remains to consider the metric/measure growth conditions. By taking a j,t ( x, y ) = χ Σ j +1 ,t, ( x ) ( y ) and ( q t , a t ) = ( q ,t , a ,t ), these conditions reduce to showthat vol g (B j +1 φ ( t ) ( x )) ≈ vol g (B j φ ( t ) ( x )) . This follows in turn from the fact that M has a non-negative Ricci curvature. (cid:3) Let (
M, g ) be a complete Riemannian manifold with non-negative Ricci curvatureand let ∆ be the Laplace-Beltrami operator. The heat semigroup S ∆ generated by∆ admits a kernel on ( M, g ) satisfying the upper bound estimate mentioned in theabove proposition. We know from Davies [17, Theorem 5.5.11] that the heat kernelsatisfies(3.2) h t ( x, y ) ≤ a δ vol g (B √ t ( x )) exp (cid:16) − d ( x, y ) δ ) t (cid:17) for any δ > a δ . This implies that h t ( x, y ) . a δ vol g (B √ t ( x )) (4(1 + δ ) t ) n + ε ( d ( x, y ) + 4(1 + δ ) t ) n + ε . ( p δ ) t ) n + ε vol g (B √ δ ) t ( x ))( d ( x, y ) + p δ ) t ) n + ε , which gives the expected upper bound with φ ( t ) = p δ ) t . Remark 3.2.
Once we have confirmed that algebraic and analytic conditions holdfor the Markov process generated by the Laplace-Beltrami operator ∆, it should benoticed that our CZ conditions are again implied by the classical ones. Arguing as inRemarks 1.6 and 2.5, we see that the Ricci curvature assumption allows us to ignoreour size kernel conditions. Next, it is straightforward to check that the boundednesscondition reduces in this case to standard L -boundedness. Finally, our discussionin section 2.4 shows that our smoothness kernel condition is guaranteed under theclassical H¨ormander kernel condition. Note in addition that our conditions alsohold in the row case. In particular, classical CZOs in ( M, g ) become algebraicCZOs. Moreover, the gaussian upper estimate (3.2) indicates that in (
M, g ) withthe heat semigroup S ∆ we have L ◦ p ( M ) = L p ( M, g ) for 1 < p < ∞ . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 27
By the discussion above, we have all the ingredients to apply Theorem 2.1 andCorollary 2.2. Let us illustrate it for the Riesz transforms on (
M, g ). Considerthe Riemannian gradient ∇ = ( ∂ , ∂ , . . . , ∂ n ) on ( M, g ). The Riesz transform on(
M, g ) is formally defined by R = ( R j ) = ∇ ( − ∆) − with R j = ∂ j ( − ∆) − . Then we may recover Bakry’s theorem [1] using our algebraic approach. Indeedintegration by parts gives k|∇ f |k = k ∆ f k which implies L -boundedness ofRiesz transforms. Moreover, the H¨ormander condition follows from [10, 41]. Corollary 3.3.
Let ( M, g ) be a complete n -dimensional Riemannian manifold withnon-negative Ricci curvature. Then for all < p < ∞ , there exists a constant C p > such that k R j f k L p ( M,g ) ≤ C p k f k L p ( M,g ) for all ≤ j ≤ n. The Gaussian measure.
Now we study the Ornstein-Uhlenbeck semigroupon the Euclidean space equipped with its Gaussian measure. We shall first constructa Markov metric for it. Then, we shall prove that our algebraic/analytic andCalder´on-Zygmund conditions hold for the standard CZOs in this setting. Theinfinitesimal generator of the Ornstein-Uhlenbeck semigroup O = ( O t ) t ≥ is theoperator L = ∆2 − x · ∇ on ( R n , µ ) with dµ ( y ) = exp( −| y | ) dy . We have O t f ( x ) = 1( π − πe − t ) n Z R n exp (cid:16) − | e − t x − y | − e − t (cid:17) f ( y ) dy = 1( π − πe − t ) n Z R n exp (cid:16) | x | − | e t x − y | e t − (cid:17) f ( y ) dµ ( y ) . First, we establish a useful lemma showing that the local behavior —i.e. forsmall values of t — of the semigroup type BMO norm for the Ornstein-Uhlenbecksemigroup determines it completely. Lemma 3.4.
Given δ > , there exists C δ > such that sup t ≥ (cid:13)(cid:13) O t | f | − | O t f | (cid:13)(cid:13) ∞ ≤ C δ sup t<δ (cid:13)(cid:13) O t | f | − | O t f | (cid:13)(cid:13) ∞ . Proof.
It is easy to check that(3.3) O t f ( x ) = H v ( t ) f ( e − t x ) , for v ( t ) = (1 − e − t ) and the heat semigroup H t = exp( t ∆). Given t > f ∈ L ∞ ( R n ), let F ( s ) = H s | H t − s f | for 0 ≤ s ≤ t . According to the definition of H t , we obtain the following identity ∂ s F = ( ∂ s H s ) | H t − s f | + H s [( ∂ s H t − s f ) ∗ ( H t − s f )] + H s [( H t − s f ) ∗ ( ∂ s H t − s f )]= ∆ H s | H t − s f | − H s [(∆ H t − s f ) ∗ ( H t − s f )] − H s [( H t − s f ) ∗ (∆ H t − s f )]= H s [∆ | H t − s f | − (∆ H t − s f ) ∗ ( H t − s f ) − ( H t − s f ) ∗ (∆ H t − s f )]= 2 H s |∇ H t − s f | . Kadison-Schwarz inequality gives for 0 < u < sH u |∇ H t − u f | = H u | H s − u ∇ x H t − s f | ≤ H s |∇ x H t − s f | which implies that ∂ s F is increasing and F is convex. Rearranging the inequality F ( s ) ≤ ( F (0) + F (2 s )), we get H t | f | − | H t f | ≤ H t ( H t | f | − | H t f | ) for any t ≥
0. Then, the L ∞ contractivity of H t gives(3.4) (cid:13)(cid:13) H k t | f | − | H k t f | (cid:13)(cid:13) ∞ ≤ k (cid:13)(cid:13) H t | f | − | H t f | (cid:13)(cid:13) ∞ . On the other hand, choosing k δ such that 2 k δ v ( δ ) ≥ and applying (3.3) and (3.4)sup t ≥ (cid:13)(cid:13) O t | f | − | O t f | (cid:13)(cid:13) ∞ = sup t< (cid:13)(cid:13) H t | f | − | H t f | (cid:13)(cid:13) ∞ ≤ sup t< kδ v ( δ ) (cid:13)(cid:13) H t | f | − | H t f | (cid:13)(cid:13) ∞ ≤ k δ sup t<δ (cid:13)(cid:13) O t | f | − | O t f | (cid:13)(cid:13) ∞ . (cid:3) By the lemma above, it suffices to construct a Markov metric for ( O t ) t ≥ with0 < t < . Let v = √ e t − j, t ) ∈ Z + × R + Σ j,t ( x ) = B( e t x, p jv ) and Ω j,t ( x ) = Σ j,t ( x ) \ Σ j − ,t ( x ) . Let j = j ( x, t ) be the smallest possible integer j satisfying that 0 ∈ Σ j,t ( x ). The case n = 1 . If 1 ≤ j < j , let D − j,t ( x ) = n y ∈ Ω j,t ( x ) : e t | x | − p jv ≤ | y | ≤ e t | x | − p j − v o ,D + j,t ( x ) = n y ∈ Ω j,t ( x ) : e t | x | + p j − v ≤ | y | ≤ e t | x | + p jv o . Then, D − j,t ( x ) ∪ D + j,t ( x ) = Ω j,t ( x ) and we get(3.5) O t f ( x ) . v (cid:16) X ε = ± ≤ j 4. Therefore σ j,t ≤ p j exp( | x | − j ) < p j exp (cid:16) − j (cid:17) . Now we are ready to choose the optimal γ ’s for the metric integrability condition inthe definition of Markov metric. We respectively define for 1 ≤ j < j and j ≥ j γ j,t,ε ( x ) = exp (cid:16) | v | x | + e t √ j − | (cid:17) and γ j,t ( x ) = 1 √ j exp (cid:16) j (cid:17) . Then it turns out thatsup x ∈ R Corollary 3.5. Let O = ( O t ) t ≥ be the Ornstein-Uhlenbeck semigroup and T be asingular integral operator defined on L ∞ ( R n , dµ ) with kernel k . More precisely, wehave the kernel representation T f ( x ) = Z R n k ( x, y ) f ( y ) dµ ( y ) for x / ∈ supp f. Suppose T is bounded on L ( R n , µ ) and it satisfies (3.8) sup B ball sup z ∈ B j ≥ Z j +1 B \ j B | k ( z, y ) | dµ ( y ) < ∞ , (3.9) sup B ball sup z ,z ∈ B Z (5B) c | k ( z , y ) − k ( z , y ) | dµ ( y ) < ∞ . Then T is a bounded map from L ∞ ( R n , µ ) to the semigroup BMO O space. Proof. It suffices to prove that our CZ conditions hold. The row and columnboundedness conditions reduce to L -boundedness. Let M t be the averaging mapin L ∞ ( R n , µ ) over the ball Σ ,t ( x ). Given 1 ≤ j < j , define A j,t,s ( x, · ) as thecharacteristic function over the ball Σ j +1 ,t ( x ). Then A j,t,s ≤ χ r D sj,t ( x ) ∩ Σ ,t ( x ) with r = [2 log ( j + 1)] + 3, where [ ] stands for the integer part. Therefore, applying(3.8), we havesup z ∈ Σ ,t ( x ) Z Σ j +1 ,t ( x ) \ ,t ( x ) | k ( z, y ) | dµ ( y ) . r . γ j,t,s ( x ) , sup z ∈ D sj,t ( x ) Z Σ j +1 ,t ( x ) \ D sj,t ( x ) | k ( z, y ) | dµ ( y ) . r . γ j,t,s ( x ) . For j ≥ j , let A j,t ( x, · ) = χ Σ j,t ( x ) ≤ χ u Σ ,t ( x ) with u = [log (2 √ j )] + 1. Applying(3.8) as above, we see that T satisfies our size conditions. Moreover, (3.9) impliesour smoothness conditions as in the Euclidean-Lebesguean setting, Section 2.4. (cid:3) Remark 3.6. Since the Gaussian measure is non-doubling, the term R j,t f − M t f in the Markov metric BMO space BMO Q is essential to characterise the changesof the mean values of the function f . This explains the relevance of the size kernelcondition in the Calder´on-Zygmund theory for the gaussian measure.4. Applications II — Noncommutative spaces In this section we apply our algebraic approach to study Calder´on-Zygmundoperators in flag von Neumann algebras which originally motivated us and includematrix algebras, quantum Euclidean spaces and quantum groups. We start byreconstructing and refining the semicommutative theory, which deals with tensorand crossed products with metric measure spaces.4.1. Operator-valued theory. Let (Ω , µ ) be a doubling metric space —as inRemark 1.6— and consider a Markov semigroup S t : L ∞ (Ω) → L ∞ (Ω). Let M bea semifinite von Neumann algebra with a n.s.f. trace τ . Then we call the semigroup S = ( S t ⊗ id M ) t ≥ a semicommutative Markov semigroup . Consider the algebra ofessentially bounded functions f : Ω → M equipped with the trace ϕ ( f ) = Z Ω τ ( f ( y )) dµ ( y ) . Its weak- ∗ closure R = L ∞ (Ω) ¯ ⊗M is a von Neumann algebra. Assume that thereexists a Markov metric Q = { ( R j,t , σ j,t , γ j,t ) : ( j, t ) ∈ Z + × R + } associated tothe original Markov semigroup on L ∞ (Ω). Let q j,t ( x, y ) = χ Ω xj,t ( y ) stand for theprojections determined by Q via (2.1). We assume in addition that Q satisfies themetric/measure growth condition(4.1) µ (Σ xj,t ) µ (Ω xj,t ) ≤ γ j,t ( x )by choosing a j,t ( x, y ) = χ Σ xj,t ( y ). The remaining algebraic and analytic conditionstrivially hold in this case. Indeed, the algebraic conditions follow by commutativityand analytic conditions just require to pick the right averaging maps according toJensen’s inequality, as explained in (2.4). Note that Q satisfies an operator-valuedgeneralization of the Hilbert module majorization in the line of Remark 1.5. Thus Q extends to a Markov metric in R by tensorizing with id M and M respectively.Our goal is to study CZO’s formally given by T f ( x ) = Z Ω k ( x, y ) ( f ( y )) dµ ( y ) with ( f : Ω → M and x / ∈ supp Ω f,k ( x, y ) ∈ L ( L ( M ) , L ( M )) . That is, k ( x, y ) is linear from τ -measurable to τ -measurable operators. If weset R j = L ∞ (Ω) ¯ ⊗M j , we should emphasize that L p ( R j ) = L p (Ω; L p ( M j )). Inparticular, this framework does not fall in the vector-valued theory because wetake values in different Banach spaces for different values of p , see [49] for furtherexplanations. This class of operators is inspired by two distinguished examples with M = M = M : • Operator-valued case T f ( x ) = Z Ω k ov ( x, y ) · f ( y ) dµ ( y ) . • Noncommutative model T f ( x ) = Z Ω ( id M ⊗ τ ) (cid:2) k nc ( x, y ) · ( M ⊗ f ( y )) (cid:3) dµ ( y ) . In the first case, the kernel takes values in M or even in the complex field and actson f ( y ) by left multiplication k ( x, y )( f ( y )) = k ov ( x, y ) · f ( y ). It is the canonicalmap when L p ( R ) is regarded as the Bochner space L p (Ω; L p ( M )). On the contraryif we simply think of L p ( R ) as a noncommutative L p space, a natural CZO shouldbe an integral map with respect to the full trace ϕ = R Ω ⊗ τ and the kernel shouldbe a ϕ ⊗ ϕ -measurable operator k : Ω × Ω → M ¯ ⊗M . The noncommutative modelprovides the resulting integral formula. Note that this model also falls in our generalframework by taking k ( x, y )( f ( y )) = ( id M ⊗ τ )[ k nc ( x, y ) · ( M ⊗ f ( y ))]. Theorem 4.1. Let S = ( S t ) t ≥ be a Markov semigroup on (Ω , µ ) which admitsa Markov metric Q = { ( R j,t , σ j,t , γ j,t ) : ( j, t ) ∈ Z + × R + } satisfying the above LGEBRAIC CALDER ´ON-ZYGMUND THEORY 33 assumptions. Let a j,t ( x, y ) = χ Σ xj,t ( y ) be the projections determined by Q via (4.1) .Consider the CZO formally given by T f ( x ) = Z Ω k ( x, y )( f ( y )) dµ ( y ) . Then, T maps L ∞ ( R ) to BMO c S ( R ) provided the conditions below hold i) L c -boundedness condition, (cid:13)(cid:13)(cid:13)(cid:16) Z Ω | T f | dµ (cid:17) (cid:13)(cid:13)(cid:13) M . (cid:13)(cid:13)(cid:13)(cid:16) Z Ω | f | dµ (cid:17) (cid:13)(cid:13)(cid:13) M . ii) Smoothness condition for the kernel, Z Ω \ Σ xj,t (cid:13)(cid:13)(cid:0) k ( y , z ) − k ( y , z ) (cid:1)(cid:0) f ( z ) (cid:1)(cid:13)(cid:13) M dµ ( z ) . k f k R uniformly in j ≥ , t > , x ∈ Ω and y , y ∈ Ω xj,t . Proof. The proof follows from Theorem 2.1. Since the underlying space (Ω , µ ) is adoubling metric space, the size kernel condition is unnecessary. Thus, it remains tocheck the L c -boundedness condition and the kernel smoothness condition. Consider N π = L ∞ (Ω × Ω) ¯ ⊗M , N ρ = L ∞ (Ω × Ω) ¯ ⊗M , ω ( ϕ )( x, y ) = ϕ ( y ) for ϕ ∈ L ∞ (Ω)and ( π , ρ ) = ( ω ⊗ id M , ω ⊗ id M ). Let b T = id Ω ⊗ T , Φ j,t be the averaging mapover Ω xj,t × Ω xj,t and ∆ = δ ⊗ id M with δϕ ( x, y ) = ϕ ( x ) − ϕ ( y ). Then condition i)yields the L c -boundedness condition. It is also easy to see that condition ii) impliesour kernel smoothness condition. Thus, the result follows from Theorem 2.1. (cid:3) Remark 4.2. We continue with a few comments: A) When M = M = M and the kernel k ( x, y )( f ( y )) = k ( x, y ) · f ( y ) acts byleft multiplication, the boundedness condition i) becomes equivalent to the usual L boundedness. Indeed, using that M ⊂ B ( L ( M )) we obtain (cid:13)(cid:13)(cid:13)(cid:16) Z R n | T f ( y ) | dy (cid:17) (cid:13)(cid:13)(cid:13) M = sup k h k ≤ (cid:16) Z R n (cid:10) h, | T f ( y ) | h (cid:11) dy (cid:17) = sup k h k≤ (cid:13)(cid:13) ( T f ) ( R n ⊗ h ) (cid:13)(cid:13) L ( R ) = sup k h k≤ (cid:13)(cid:13) T ( f ( R n ⊗ h )) (cid:13)(cid:13) L ( R ) ≤ k T k B ( L ( R )) (cid:13)(cid:13)(cid:13)(cid:16) Z R n | f ( y ) | dy (cid:17) (cid:13)(cid:13)(cid:13) M . B) We have used so far semigroup type BMO’s. When (Ω , µ ) comes equippedwith a doubling metric, we may replace it by other standard (equivalent) formsof BMO, as pointed in Remark 1.7. By well-known arguments [49], our kernelsmoothness condition reduces to(Sm λ ) sup R> ess sup y ,y ∈ B R (cid:13)(cid:13)(cid:13) Z (B λR ) c (cid:0) k ( y , z ) − k ( y , z ) (cid:1) ( f ( z )) dz (cid:13)(cid:13)(cid:13) M . k f k R . for λ > 1. The classical H¨ormander condition(Hr λ ) ess sup y ,y ∈ R n Z d ( y ,z ) >λd ( y ,y ) (cid:13)(cid:13) k ( y , z ) − k ( y , z ) (cid:13)(cid:13) M dz < ∞ . satisfies (Hr λ ) ⇒ (Sm λ +1 ). In fact, an even weaker condition sufficessup R> (cid:13)(cid:13)(cid:13) − Z B R × B R (cid:12)(cid:12)(cid:12) Z (B λR ) c (cid:0) k ( y , z ) − k ( y , z ) (cid:1) f ( z ) dz (cid:12)(cid:12)(cid:12) dy dy (cid:13)(cid:13)(cid:13) M . k f k R . C) We recall that L ∞ ( R ) → BMO S boundedness requires that T † f = T ( f ∗ ) ∗ satisfies the same assumptions as T . If k ( x, y ) ∈ M is given by left multiplicationthe only effect in T † is that k ( x, y ) is replaced by k ( x, y ) ∗ and now operates byright multiplication. This left/right condition was formulated in [49] in terms of M -bimodular maps. Moreover, a counterexample was constructed to show thatthe bimodularity is indeed essential. It is also quite interesting to note that in the‘noncommutative model’ we have Z R n ( id M ⊗ τ ) (cid:2) k ( x, y ) · ( M ⊗ f ( y )) (cid:3) dy = Z R n ( id M ⊗ τ ) (cid:2) ( M ⊗ f ( y )) · k ( x, y ) (cid:3) dy by traciality and this pathology does not occur. Finally, the L p boundedness isguaranteed for 2 < p < ∞ since the classical heat semigroup has a regular Markovmetric and J p = id L p ( R n ) in this case. As for 1 < p < 2, it suffices to take adjointswhich leads to H¨ormander smoothness in the second variableess sup z ,z ∈ R n Z | y − z | >λ | z − z | (cid:13)(cid:13) k ( y, z ) − k ( y, z ) (cid:13)(cid:13) M dy < ∞ . Of course, this is still consistent with the classical CZ theory M = C . D) Our analysis of the semicommutative case from our basic Theorem 4.1 doesnot recover the weak type (1 , 1) inequality from [49]. It requires quasi-orthogonalitymethods which are still missing for general von Neumann algebras.We now study the L ∞ → BMO boundedness of twisted CZO’s on homogeneousspaces. Given a discrete group G with left regular representation λ : G 7→ B ( ℓ (G))let L (G) denote its group von Neumann algebra. Let ( M , τ ) with M ⊂ B ( H ) bea noncommutative probability space and α : G Aut( M ) be a trace preservingaction. Consider two ∗ -representations ρ : M ∋ f X h ∈ G α h − ( f ) ⊗ e h,h ∈ M ¯ ⊗B ( ℓ (G)) , Λ : G ∋ g X h ∈ G M ⊗ e gh,h ∈ M ¯ ⊗B ( ℓ (G)) , where e g,h is the matrix unit for B ( ℓ (G)). Now we define the crossed productalgebra M ⋊ α G as the weak operator closure in M ⊗ B ( ℓ (G)) of the ∗ -algebragenerated by ρ ( M ) and Λ(G). A generic element of M ⋊ α G can be formallywritten as P g ∈ G f g ⋊ α λ ( g ) with f g ∈ M . With this convention, we may embedthe crossed product algebra M ⋊ α G into M ¯ ⊗B ( ℓ (G)) via the map j = ρ ⋊ Λ.Indeed, we have j (cid:16) X g ∈ G f g ⋊ α λ ( g ) (cid:17) = X g ∈ G ρ ( f g )Λ( g )= X g ∈ G (cid:16) X h,h ′ ∈ G ( α h − ( f g ) ⊗ e h,h )( M ⊗ e gh ′ ,h ′ ) (cid:17) = X g ∈ G (cid:16) X h ∈ G α h − ( f g ) ⊗ e h,g − h (cid:17) LGEBRAIC CALDER ´ON-ZYGMUND THEORY 35 = X g ∈ G (cid:16) X h ∈ G α ( gh ) − ( f g ) ⊗ e gh,h (cid:17) . Since the action α will be fixed, we relax the terminology and write P g ∈ G f g λ ( g )instead of P g ∈ G f g ⋊ α λ ( g ). We say that a Markov semigroup S = ( S t ) t ≥ in M is G -equivariant if α g S t = S t α g for ( t, g ) ∈ R + × G . If S is a G-equivariant Markov semigroup on M , let S ⋊ = ( S t ⋊ id G ) t ≥ and S ⊗ = ( S t ⊗ id B ( ℓ (G)) ) t ≥ denote the crossed/tensor product amplification of oursemigroup on M ⋊ G and M ¯ ⊗B ( ℓ (G)) respectively. Note that S ⋊ is Markoviandue to the G-equivariance of S . In the following result, our CZO’s are of the form T f ( x ) = Z Ω k ( x, y )( f ( y )) dµ ( y )for all f ∈ ( R , ϕ ), where ( R j , ϕ j ) = L ∞ (Ω , µ ) ¯ ⊗ ( M j , τ j ) and k ( x, y ) : M → M .In other words, we keep the same terminology as for Theorem 4.1. We shall alsouse the notation c M j = M j ¯ ⊗B ( ℓ (G)) and b R j = R j ¯ ⊗B ( ℓ (G)) . Corollary 4.3. Let G y L ∞ (Ω , µ ) be an action α which is implemented by ameasure preserving transformation β , so that α g f ( x ) = f ( β g − x ) . Let S = ( S t ) t ≥ be a G -equivariant Markov semigroup on (Ω , µ ) which admits a Markov metric Q = { ( R j,t , σ j,t , γ j,t ) : ( j, t ) ∈ Z + × R + } satisfying the assumptions above. Let usconsider a family of CZO’s formally given by T g f ( x ) = Z Ω k g ( x, y )( f ( y )) dµ ( y ) for g ∈ G . Then, P g f g λ ( g ) P g T g ( f g ) λ ( g ) is bounded R ⋊ G → BMO c S ⋊ ( R ⋊ G) if i) L c -boundedness condition, (cid:13)(cid:13)(cid:13)(cid:16) Z Ω (cid:12)(cid:12) ( T gh − ) • ξ (cid:12)(cid:12) dµ (cid:17) (cid:13)(cid:13)(cid:13) c M . (cid:13)(cid:13)(cid:13)(cid:16) Z Ω | ξ | dµ (cid:17) (cid:13)(cid:13)(cid:13) c M , where • stands for the generalized Schur product of matrices. In otherwords, the CZO T gh − only acts on the ( g, h ) -th entry of ξ for each g, h ∈ G . ii) Smoothness condition for the kernel, Z Ω (cid:13)(cid:13)(cid:0) K ( y , z ) − K ( y , z ) (cid:1) • (cid:0) ξ ( z )( − a j,t ( x, z )) (cid:1)(cid:13)(cid:13) c M dµ ( z ) . k ξ k b R , uniformly on j ≥ , t > , x ∈ Ω and y , y ∈ Ω xj,t . Here, the CZ kernel K ( y, z ) = P g,h k gh − ( β g y, β g z ) ⊗ e g,h acts once more as a Schur multiplier. Proof. Letting ξ = P g,h a g,h ⊗ e g,h ∈ b R , we define the mapΦ : b R → BMO c S ⊗ ( b R ) , Φ( ξ )( x ) = X g,h α g − Z Ω k gh − ( x, y )( a g,h ( β g − ( y ))) dµ ( y ) ⊗ e g,h . By the definition of j , it is easy to check that j (cid:16) X g T g ( f g ) λ ( g ) (cid:17) = Φ (cid:16) j (cid:0) X g f g λ ( g ) (cid:1)(cid:17) . Since S is G-equivariant, according to [33, Lemma 2.1], we have k g k BMO c S ⋊ ( R ⋊ G) = sup t ≥ (cid:13)(cid:13)(cid:13)(cid:16) S ⊗ ,t | j ( g ) | − | S ⊗ ,t j ( g ) | (cid:17) (cid:13)(cid:13)(cid:13) b R . Therefore, it suffices to show that Φ is b R → BMO c S ⊗ ( b R ) bounded. We findΦ( ξ )( x ) = Z Ω K ( x, y )( ξ ( y )) dµ ( y ) . Thus, we may regard Φ as a semicommutative CZO and apply Theorem 4.1 where M j is replaced by c M j . Since Φ( ξ ) = P g,h ( α g − ) • ( T gh − ) • ( α g ) • ξ and β is measurepreserving, we immediately find that the L c -boundedness assumption implies thatthe map Φ : L c (Ω) ¯ ⊗ c M → L c (Ω) ¯ ⊗ c M is bounded. Moreover, the smoothness condition matches that of Theorem 4.1. (cid:3) Remark 4.4. Our work so far yields sufficient conditions for the L ∞ → BMOboundedness of T ⋊ id G in more general settings. In particular, if T g = T and α g T = T α g for all g ∈ G, then we find for any T fulfilling the assumption ofTheorem 4.1, T ⋊ id G : R ⋊ G BMO c S ⋊ ( R ⋊ G) is bounded.4.2. Matrix algebras. In this paragraph, we introduce a Markov metric for thematrix algebra B ( ℓ ). The triangular truncation plays the noncommutative formof the Hilbert transform on B ( ℓ ). We shall reprove the L p -boundedness of thetriangular truncation for 1 < p < ∞ and a new BMO → BMO estimate by meansof this Markov metric and our algebraic approach. Consider the ∗ -homomorphism u : B ( ℓ ) → L ∞ ( R ) ¯ ⊗B ( ℓ ) determined by u ( e mk ) = e πi ( m − k ) · e mk . Given A = P m,k a mk e mk , define the semigroup S t ( A ) = X m,k e − t | m − k | a mk e mk . It is not difficult to see that it defines a Markov semigroup of convolution type. Infact, u is a corepresentation of L ∞ ( R ) (equipped with its natural comultiplicationmap ∆ f ( x, y ) = f ( x + y )) in B ( ℓ ) and it turns out that S = ( S t ) t ≥ is thetransferred semigroup associated to the heat semigroup on R u ◦ S t = ( H t ⊗ id B ( ℓ ) ) ◦ u. Define the cpu map R j,t on B ( ℓ ) by u ◦ R j,t = ( e R j,t ⊗ id B ( ℓ ) ) ◦ u , where e R j,t f ( x )denotes the average of f ∈ L ∞ ( R ) over the interval B √ jt ( x ). Now, given a matrix A = P m,k a mk e mk we find u ◦ R j,t ( A )( x ) = − Z B √ jt ( x ) u ( A )( y ) dy = − Z B √ jt ( x ) X m,k e πi ( m − k ) y a mk e mk dy LGEBRAIC CALDER ´ON-ZYGMUND THEORY 37 = X m,k sin(4 √ jtπ ( m − k ))4 √ jtπ ( m − k ) e πi ( m − k ) x a mk e mk . Thus, we find the following identity R j,t ( A ) = X m,k sin(4 √ jtπ ( m − k ))4 √ jtπ ( m − k ) a mk e mk . Taking σ j,t = 2 e p j/πe − j B ( ℓ ) and γ j,t = √ j B ( ℓ ) , we obtain a Markov metric in B ( ℓ ). Indeed, the metric integrability condition holds trivially, as for the Hilbertmodule majorization it reduces to prove that B ≤ B with B = u (cid:0) h ξ, ξ i S t (cid:1) = (cid:10) u ⊗ u ( ξ ) , u ⊗ u ( ξ ) (cid:11) H t ⊗ id B ( ℓ ,B = X j σ j,t u (cid:0) h ξ, ξ i R j,t (cid:1) = X j σ j,t (cid:10) u ⊗ u ( ξ ) , u ⊗ u ( ξ ) (cid:11) e R j,t ⊗ id B ( ℓ . In other words, it suffices to note that the canonical Markov metric in R —whichrecovers the Euclidean metric, as proved in Paragraph 1.4— admits a matrix-valuedextension, as it was justified in Remark 1.5. Let us now consider the triangulartruncation △ ( A ) = X m>k a mk e mk . Corollary 4.5. We have k△ ( A ) k BMO S . k A k BMO S . In particular, given < p < ∞ we obtain k△ ( A ) k S p . p p − k A k S p . Proof. Recall that u ◦ △ = ( L ⊗ id B ( ℓ ) ) ◦ u. for L = ( id + iH ) and d Hf ( ξ ) = − i sgn( ξ ) b f ( ξ ), the Hilbert transform in the realline. We may also regard u : B ( ℓ ) → L ∞ ( T ) ¯ ⊗B ( ℓ ) as a corepresentation of T instead of R and the above identity holds replacing H by the Hilbert transformin the torus. In this case, u becomes a trace preserving ∗ -homomorphism andthe well-known S p inequalities for △ reduce to the boundedness of the Hilberttransform in L p ( T ; S p ( ℓ )), which is also well-known and follows in passing fromthe semicommutative theory in the previous paragraph. Alternatively, the secondassertion follows from the first one by interpolation and duality. According toRemark 1.7, to prove the first assertion it suffices to show that the map T = i ( id B ( ℓ ) − △ )is BMO → BMO bounded for the semigroup BMO space which is associated to thetransferred Poisson semigroup P t on B ( ℓ ) given by P t : ( a ij ) ( e − t | i − j | a ij ) . Given A = ( a jk ) j,k in B ( ℓ ) then | A | = A ∗ A = (cid:16) X k a ki a kj (cid:17) i,j T ( A ) = i (cid:16) sgn( k − j ) a jk (cid:17) j,k ( T A ) ∗ = i (cid:16) sgn( k − j ) a kj (cid:17) j,k Then (cid:0) P t | A | − | P t A | (cid:1) ij = P k ( e − t | i − j | − e − t | k − j | e − t | i − k | ) a ki a kj and (cid:0) P t | T ( A ) | − | P t T ( A ) | (cid:1) ij = X k ( e − t | i − j | − e − t | k − j | e − t | i − k | ) sgn( k − i ) sgn( k − j ) a ki a kj . Since sgn( k − i )sgn( k − j ) = 1 iff e − t | i − k | e − t | k − j | = e − t | i − j | , we get P t | A | − | P t A | = P t | T ( A ) | − | P t T ( A ) | . The last identity implies that T is an isometry on the Poisson BMO space. (cid:3) Quantum Euclidean spaces. Given an integer n ≥ 1, fix an anti-symmetric R -valued n × n matrix Θ. We define A Θ as the universal C*-algebra generated bya family u ( s ) , u ( s ) , · · · , u n ( s ) of one-parameter unitary groups in s ∈ R n whichare strongly continuous and satisfy the following Θ-commutation relations u j ( s ) u k ( t ) = e πi Θ jk st u k ( t ) u j ( s ) . If Θ = 0, by Stone’s theorem we can take u j ( s ) = exp(2 πis h e j , ·i ) and A Θ is thespace of bounded continuous functions on R n . In general, given ξ ∈ R n , definethe unitaries λ Θ ( ξ ) = u ( ξ ) u ( ξ ) · · · u n ( ξ n ). Let E Θ be the closure in A Θ of λ Θ ( L ( R n )) with f = Z R n ˇ f Θ ( ξ ) λ Θ ( ξ ) dξ. If Θ = 0, E Θ = C ( R n ). Define τ Θ ( f ) = τ Θ (cid:18)Z R n ˇ f Θ ( ξ ) λ Θ ( ξ ) dξ (cid:19) = ˇ f Θ (0)for ˇ f Θ : R n → C integrable and smooth. τ Θ extends to a normal faithful semifinitetrace on E Θ . Let R Θ = A ′′ Θ = E ′′ Θ be the von Neumann algebra generated by E Θ in the GNS representation of τ Θ . Note that if Θ = 0, R Θ = L ∞ ( R n ). In generalwe call R Θ a quantum Euclidean space. There are two maps which play importantroles while doing analysis over quantum Euclidean spaces. The first one is thecorepresentation map σ Θ : R Θ → L ∞ ( R n ) ¯ ⊗R Θ , given by λ Θ ( ξ ) exp ξ ⊗ λ Θ ( ξ )where exp ξ stands for the Fourier character exp(2 πi h ξ, ·i ). Note that σ Θ is a normalinjective ∗ -homomorphism. The second map is π Θ : exp ξ λ Θ ( ξ ) ⊗ λ Θ ( ξ ) ∗ , whichextends to a normal ∗ -homomorphism from L ∞ ( R n ) to R Θ ¯ ⊗R opΘ , where R opΘ isthe apposite algebra of R Θ , which is obtained by preserving the linear and adjointstructures but reversing the product. We refer the readers to [24] for more detailedinformation of quantum Euclidean spaces and these two maps. BMO and Markov metric. Our first goal is to construct a natural Markov metricfor quantum Euclidean spaces. Let us recall the heat semigroup on R n acting on ϕ : R n → C admits the following form H t ϕ ( x ) = Z R n b ϕ ( ξ ) e − t | ξ | exp ξ ( x ) dξ. This induces a semigroup on R Θ determined by σ Θ ◦ S Θ ,t = ( H t ⊗ id R Θ ) ◦ σ Θ . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 39 S Θ ,t gives a Markov semigroup on R Θ which formally acts as(4.2) S Θ ,t ( f ) = Z R n ˇ f Θ ( ξ ) e − t | ξ | λ Θ ( ξ ) dξ. The corresponding semigroup column BMO norm is given by k f k BMO c ( R Θ ) = sup t> (cid:13)(cid:13)(cid:13)(cid:16) S Θ ,t ( | f | ) − | S Θ ,t ( f ) | (cid:17) (cid:13)(cid:13)(cid:13) R Θ ≈ sup B ball in R n (cid:13)(cid:13)(cid:13)(cid:16) − Z B | σ Θ ( f ) − σ Θ ( f ) B | dµ (cid:17) (cid:13)(cid:13)(cid:13) R Θ = k σ Θ ( f ) k BMO c ( R n ; R Θ ) . According to Remark 1.5, the semicommutative extension H t ⊗ id R Θ of the heatsemigroup, together with the extension of the corresponding Markov metric fromParagraph 2.4 still satisfies the Hilbert module majorization(4.3) h ξ, ξ i H t ⊗ id R Θ ≤ X j ≥ σ ∗ j,t h ξ, ξ i R j,t ⊗ id R Θ σ j,t as well as the integrability condition, where σ j,t ≡ e p j n /πe − j , γ j,t ≡ j n and R j,t f ( x ) is the average of f over B √ jt ( x ). Then we can easily produce a Markovmetric on R Θ . Let B j,t be the Euclidean ball in R n centered at the origin withradius √ jt and consider the projections q j,t = χ B j,t ⊗ R Θ . Define the cpu maps R Θ ,j,t ( f ) = 1 | B j,t | Z B j,t σ Θ ( f )( x ) dx = 1 | B j,t | Z R n b χ B j,t ( ξ ) ˇ f Θ ( ξ ) λ Θ ( ξ ) dξ. It is easy to check that(4.4) σ Θ ◦ R Θ ,j,t = ( R j,t ⊗ id R Θ ) ◦ σ Θ . The Hilbert module majorization h ξ, ξ i S Θ ,t ≤ X j ≥ σ ∗ j,t h ξ, ξ i R Θ ,j,t σ j,t for ξ ∈ R Θ ¯ ⊗ S Θ ,t R Θ is equivalent to the same inequality after composing with the ∗ -homomorphism σ Θ , which follows in turn by the intertwining identities (4.2) and(4.4), together with the majorization (4.3). Therefore, we obtain a Markov metricon R Θ associated to S Θ Q Θ = (cid:8) ( R Θ ,j,t , σ j,t , γ j,t ) | ( j, t ) ∈ Z + × R + (cid:9) . The algebraic structure. We start with the kernel representation of our CZOsover the (fully noncommutative) von Neumann algebra R Θ . Given a kernel k affiliated to R Θ ¯ ⊗R opΘ , the linear map associated to it is formally given by T k f = ( id R Θ ⊗ τ Θ ) (cid:0) k ( R Θ ⊗ f ) (cid:1) = ( id R Θ ⊗ τ Θ ) (cid:0) ( R Θ ⊗ f ) k (cid:1) . The reader is referred to [24] for more details. Our goal is to provide sufficientconditions for the L ∞ → BMO boundedness of T k . Consider the ∗ -homomorphism σ Θ : R Θ → L ∞ ( R n ) ¯ ⊗R Θ . In the case of quantum Euclidean spaces, we need thefull algebraic skeleton introduced in Section 2. In Table 1 there is a little dictionaryto identify the main objects. Next, note that σ Θ ◦ T k ( f ) = ( id R n ⊗ id R Θ ⊗ τ Θ ) (cid:0) k σ ( R n ⊗ R Θ ⊗ f ) (cid:1) , Generic algebraic objects Quantum Euclidean spaces M R Θ N ρ L ∞ ( R n ) ¯ ⊗R Θ ρ = ⊗ · , ρ = σ Θ E ρ = Lebesgue integral N π R Θ ¯ ⊗R opΘ π = ⊗ · , π = · ⊗ E π = τ Θ ⊗ id R opΘ N σ L ∞ ( R n ) ¯ ⊗ L ∞ ( R n ) ¯ ⊗R Θ N σ = ( δ ⊗ id R Θ )( N ρ ) δϕ ( x, y ) = ϕ ( x ) − ϕ ( y ) Table 1. Algebraic skeleton for R Θ .where k σ = ( σ Θ ⊗ id R opΘ )( k ). Denote σ Θ ◦ T k by T k σ . Define b T k : R Θ ¯ ⊗R Θ ∋ f ⊗ a T k σ ( f )( R n ⊗ a ) ∈ L ∞ ( R n ) ¯ ⊗R Θ . Then it is clear that the compatibility condition (2.3) holds since b T k ◦ π = σ Θ ◦ T k . Lemma 4.6. If T k is bounded on L ( R Θ ) , then (cid:13)(cid:13) b T k : L c ( R Θ ) ¯ ⊗R Θ → L c ( R n ) ¯ ⊗R Θ (cid:13)(cid:13) ≤ (cid:13)(cid:13) T k : L ( R Θ ) → L ( R Θ ) (cid:13)(cid:13) . Proof. We need to introduce two maps: j Θ : L ( R n ) ∋ Z R n ϕ ( ξ ) exp ξ dξ Z R n ϕ ( ξ ) λ Θ ( ξ ) dξ ∈ L ( R Θ ) ,W : L c ( R n ) ¯ ⊗R Θ ∋ Z R n exp ξ ⊗ a ( ξ ) dξ Z R n exp ξ ⊗ λ Θ ( ξ ) a ( ξ ) dξ ∈ L c ( R n ) ¯ ⊗R Θ . It is straightforward to show that W extends to an isometry. Moreover, j Θ is alsoan L -isometry, we refer the reader to [24, Section 1.3.2] for the proof. Observethat σ Θ ( f )( R n ⊗ a ) = Z R n ˇ f Θ ( ξ ) exp ξ ⊗ λ Θ ( ξ ) a dξ = W ( Z R n ˇ f Θ ( ξ ) exp ξ ⊗ a dξ ) = W ◦ ( j ∗ Θ ⊗ id R Θ )( f ⊗ a ) . Letting f = T k g , we get b T k ( g ⊗ a ) = W ( j ∗ Θ T k ⊗ id R Θ )( g ⊗ a ) . The properties of the maps j Θ and W readily imply the assertion. (cid:3) Now let us introduce a weak- ∗ dense subalgebra of R Θ , which is the analogue ofthe classical Schwartz class. Let S ( R n ) denote the classical Schwartz class in theEuclidean space R n and define S Θ = (cid:8) f ∈ R Θ : ˇ f Θ ∈ S ( R n ) (cid:9) . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 41 Let S ′ Θ denote the space of continuous linear functionals on S Θ , which is thequantum space of tempered distributions. Consider a continuous linear operator T ∈ L ( S Θ , S ′ Θ ). By using the unitary map j Θ : S ( R n ) → S Θ as defined in the proof of Lemma 4.6, we get j ∗ Θ T j Θ ∈ L ( S ( R n ) , S ( R n ) ′ ). By aresult of Schwartz, there exists a unique kernel K ∈ S ′ ( R n ) = ( S ( R n ) ⊗ π S ( R n )) ′ such that T admits the kernel k = ( j Θ ⊗ j Θ )( K ) ∈ ( S Θ ⊗ π S Θ ) ′ . Actually, thekernel representations T k satisfying the Calder´on-Zygmund type conditions in thefollowing theorem belong to L ( S Θ , S ′ Θ ). It provides sufficient conditions for the L ∞ ( R Θ ) → BMO c ( R Θ ) boundedness of CZO operators associated to kernels in( S Θ ⊗ π S Θ ) ′ . We shall use the quantum analogue of the bands around the diagonal a B = π Θ ( χ ) = Z R b χ ( ξ ) λ Θ ( ξ ) ⊗ λ Θ ( ξ ) ∗ dξ. Theorem 4.7. Let T k ∈ L ( S Θ , S ′ Θ ) and assume i) Cancellation (cid:13)(cid:13) T k : L ( R Θ ) → L ( R Θ ) (cid:13)(cid:13) < ∞ . ii) For any f ∈ R Θ and any Euclidean ball B centered at the origin − Z B × B (cid:12)(cid:12) Σ Θ ,k,f, B ( y ) − Σ Θ ,k,f, B ( y ) (cid:12)(cid:12) dy dy . k f k R Θ , where Σ Θ ,k,f, B = ( id R n ⊗ id R Θ ⊗ τ Θ ) (cid:2) k σ ( R n ⊗ R Θ ⊗ f )( R n ⊗ a ⊥ B ) (cid:3) .Then, the Calder´on-Zygmund operator T k is bounded from L ∞ ( R Θ ) to BMO c ( R Θ ) . Proof. By Theorem 1.4, it suffices to prove T k : L ∞ ( R Θ ) → BMO c Q Θ . Arguing as in Paragraph 1.4, the Markov metric BMO norm takes the simpler form k f k BMO c Q Θ = sup t> sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) γ − j,t (cid:2) R Θ ,j,t ( | f | ) − | R Θ ,j,t ( f ) | (cid:3) γ − j,t (cid:17) (cid:13)(cid:13)(cid:13) R Θ . In other words, the extra term in the definition of BMO is dominated by theabove expression as in (1.3). As noticed in Remark 2.5, the size kernel condition isthen superfluous. This also reduces the analytic conditions and the smooth kernelconditions to be checked. In summary, according to the proof of Theorem 2.1, theassertion will follow if we can justify:C0) Initial condition T k : A Θ → R Θ for A Θ ⊂ R Θ weak- ∗ dense.Al1) Q Θ -monotonicity of E ρ E ρ ( q j,t | ξ | q j,t ) ≤ E ρ ( | ξ | ) . Al2) Right modularity of b T k b T k ( ηπ ( b )) = b T k ( η ) ρ ( b ) . An1) Mean differences b R Θ ,j,t ( ξ ∗ ξ ) − b R Θ ,j,t ( ξ ) ∗ b R Θ ,j,t ( ξ ) ≤ Φ j,t (cid:0) δ ( ξ ) ∗ δ ( ξ )) (cid:1) for some cpu Φ j,t . An2) Metric/measure growth ≤ π ρ − E ρ ( q j,t ) − E π ( a ∗ j,t a j,t ) π ρ − E ρ ( q j,t ) − . π ρ − ( γ j,t ) . CZ1) L c -boundedness condition b T k : L c ∞ ( N π ; E π ) → L c ∞ ( N ρ ; E ρ ) . CZ2) Kernel smoothness conditionΦ j,t (cid:16)(cid:12)(cid:12) δ (cid:0) b T k ( π ( f )( − a j,t )) (cid:1)(cid:12)(cid:12) (cid:17) . γ j,t k f k ∞ . The initial condition trivially holds for good kernels k ∈ S Θ ⊗ alg S Θ . In [24] itwas required to extend the main result from this class of kernels to general onesin S ′ Θ ⊕ Θ , by reproving certain auxiliary results in the context of distributions. Inour case, this is much simpler. Indeed, when dealing with general kernels, we justnote that T k ( f ) ∈ L ( R Θ ) for all f ∈ S Θ by assumption. Given the form of R Θ ,j,t ,it trivially follows that R Θ ,j,t ( | T k f | ) and R Θ ,j,t T k f are well-defined operators in L ( R Θ ) and L ( R Θ ) respectively. In particular, the proof of Theorem 2.1 followsexactly as it was written there under this more flexible assumption. Therefore, theinitial condition can be relaxed to the condition T k : S Θ → L ( R Θ ) . In fact, according to [24, Proposition 2.17], every algebraic column CZO is normal.Thus, it suffices —as we did in Theorem 2.1— to justify that T k : S Θ → BMO c Q Θ is bounded, as we shall do by justifying the remaining conditions.Al1 holds trivially since q j,t = χ B j,t ⊗ R Θ lives in the center of N ρ . On the otherhand, according to the definition of ρ , π from Table 1, the algebraic condition Al2can be rewritten as follows b T k (cid:0) η ( R Θ ⊗ b ) (cid:1) = b T k ( η )( R n ⊗ b ) . This is clear from the definition of b T k . Next, condition An1 reads as − Z B j,t | ξ | dµ − (cid:12)(cid:12)(cid:12) − Z B j,t ξdµ (cid:12)(cid:12)(cid:12) ≤ − Z B j,t × B j,t (cid:12)(cid:12) ξ ( x ) − ξ ( y ) (cid:12)(cid:12) dµ ( x ) dµ ( y )for R Θ -valued functions, when Φ j,t is chosen to be the average over B j,t × B j,t .As in (2.4), this is a consequence of the operator-valued Jensen’s inequality. Nextrecalling that a j,t = π Θ ( χ j,t ), condition An2 takes the form | B j,t | R Θ ≤ ( τ Θ ⊗ id R opΘ )( π Θ ( χ j,t )) . j n | B j,t | R Θ . To verify it we note that( τ Θ ⊗ id R opΘ )( π Θ ( ϕ )) = ( τ Θ ⊗ id R opΘ ) (cid:16) Z R n b ϕ ( ξ ) λ Θ ( ξ ) ⊗ λ Θ ( ξ ) ∗ dξ (cid:17) = b ϕ (0) R Θ . Then we get ( τ Θ ⊗ id R opΘ )( π Θ ( χ j,t )) = 5 | B j,t | R Θ . Condition CZ1 reduces to our L -boundedness assumption by Lemma 4.6. Finally, the smoothness condition ii)in the statement readily implies condition CZ2 for all values of j, t . (cid:3) LGEBRAIC CALDER ´ON-ZYGMUND THEORY 43 The smoothness condition in Theorem 4.7 is of H¨ormander type, while the onein the main result of [24] is a gradient condition. As expected, we shall show thatour condition in this paper is more flexible than that of [24, Theorem 2.6]. We use • for the product in M ¯ ⊗M op , so that( a ⊗ b ) • ( a ′ ⊗ b ′ ) = ( aa ′ ) ⊗ ( b ′ b ) . The quantum analogue of the metric is defined byd Θ = π Θ ( | · | )for the Euclidean norm | · | . Moreover, we also introduce the Θ-deformation of thefree gradient. Let L ( F n ) denote the group von Neumann algebra associated to thefree group with n generators F n . It is well-known from (say) [64] that L ( F n ) isgenerated by n semicircular random variables s , s , . . . , s n . Note that there existderivations ∂ j Θ in S Θ which are determined by ∂ j Θ ( λ Θ ( ξ )) = 2 πiξ j λ Θ ( ξ )for 1 ≤ j ≤ n . Define the Θ-deformed free gradient as ∇ Θ = n X j =1 s j ⊗ ∂ j Θ : S Θ → L ( F n ) ¯ ⊗R Θ . If ∇ denotes the free gradient for Θ = 0, it is easy to check that( id L ( F n ) ⊗ σ Θ ) ◦ ∇ Θ = n X j =1 s j ⊗ ( σ Θ ◦ ∂ j Θ )(4.5) = n X j =1 s j ⊗ ( ∂ j ◦ σ Θ ) = ( ∇ ⊗ id R Θ ) ◦ σ Θ . For the convenience of the reader, we cite Theorem 2.6 from [24] below. Theorem 4.8. Let T k ∈ L ( S Θ , S ′ Θ ) and assume: i) Cancellation k T k : L ( R Θ ) → L ( R Θ ) k ≤ A . ii) Gradient condition. There exists α < n < β < n satisfying the gradient conditions below for ρ = α, β (cid:12)(cid:12)(cid:12) d ρ Θ • ( ∇ Θ ⊗ id R opΘ )( k ) • d n +1 − ρ Θ (cid:12)(cid:12)(cid:12) ≤ A . Then, we find the following L ∞ → BMO c estimate (cid:13)(cid:13) T k : L ∞ ( R Θ ) → BMO c ( R Θ ) (cid:13)(cid:13) ≤ C n ( α, β )( A + A ) . To simplify notation, we shall write in what follows Σ for Σ Θ ,k,f, B . According tothe semicommutative Poincar´e type inequality introduced in [24, Proposition 1.6]we obtain (cid:13)(cid:13)(cid:13) − Z B × B (cid:12)(cid:12) δ (Σ) (cid:12)(cid:12) dµ × µ (cid:13)(cid:13)(cid:13) R Θ ≤ (cid:13)(cid:13)(cid:13) ( ⊗ χ B ⊗ )( ∇ ⊗ id R Θ )(Σ) (cid:13)(cid:13)(cid:13) L ( F n ) ¯ ⊗ L ∞ ( R n ) ¯ ⊗R Θ for R = radius of B. By (4.5), we may rewrite( ⊗ χ B ⊗ )( ∇ ⊗ id R Θ )(Σ)= ( id ⊗ ⊗ τ Θ ) (cid:16) ( ⊗ χ B ⊗ ⊗ )( ∇ ⊗ id ⊗ )( k σ )( ⊗ ⊗ f )( ⊗ ⊗ a ⊥ j,t ) (cid:17) = ( id ⊗ ⊗ τ Θ ) (cid:16) ( ⊗ χ B ⊗ ⊗ )( id ⊗ σ Θ ⊗ id )( ∇ Θ ⊗ id )( k )( ⊗ ⊗ f )( ⊗ ⊗ a ⊥ j,t ) (cid:17) = ( id ⊗ ⊗ τ Θ ) (cid:0) K • ( ⊗ ⊗ f ) (cid:1) with K = ( ⊗ χ B ⊗ ⊗ )( id ⊗ σ Θ ⊗ id )( ∇ Θ ⊗ id )( k ) • ( ⊗ ⊗ a ⊥ j,t )in L ( F n ) ¯ ⊗ ( S ( R n ) ⊗ π S Θ ⊗ π S Θ ) ′ . Thus, ( ⊗ χ B ⊗ )( ∇ ⊗ id R Θ )(Σ) = T K ( f ).We turn to the proofs of Theorem 2.6, Proposition 2.15 and Remark 2.16 (as thegeneralizations of Theorem 2.6) in [24], they show that the condition ii) in Theorem4.8 implies (cid:13)(cid:13) T K ( f ) (cid:13)(cid:13) L ( F n ) ¯ ⊗ L ∞ ( R n ) ¯ ⊗R Θ ≤ C n ( α, β ) A R k f k R Θ , which is inequality (2.2) in [24]. Combining the calculations above, we deducethat condition ii) in Theorem 4.8 is stronger than condition ii) in Theorem 4.7. Inconclusion, the Calder´on-Zygmund extrapolation on R Θ that we obtain by applyingTheorem 2.1 improves the corresponding result in [24].4.4. Quantum Fourier multipliers. We now refine our abstract result for locallycompact quantum groups. We shall need some basic notions from the theory ofquantum groups, details can be found in Kustermans/Vaes’ papers [38, 39]. Letus consider a von Neumann algebra N equipped with a comultiplication map, anormal injective unital ∗ -morphism ∆ : N → N ¯ ⊗N satisfying the coassociativitylaw ( id N ⊗ ∆)∆ = (∆ ⊗ id N )∆ . Assume also the existence of two n.s.f weights ψ and ϕ on N such that( id N ⊗ ψ )∆( a ) = ψ ( a ) N and ( ϕ ⊗ id N )∆( a ) = ϕ ( a ) N for a ∈ N + . We call ψ and ϕ the left-invariant Haar weight and the right-invariant Haar weight on N respectively. Then the quadruple G = ( N , ∆ , ψ, ϕ ) is called a (von Neumannalgebraic) locally compact quantum group and we write L ∞ ( G ) for the quantumgroup von Neumann algebra N . Using the Haar weights, one can construct anantipode S on N which is a densely defined anti-automorphism on N satisfying theidentity ( id N ⊗ ψ ) (cid:0) ( N ⊗ a ∗ )∆( b ) (cid:1) = S (cid:0) ( id N ⊗ ψ ) (cid:0) ∆( a ∗ )( N ⊗ b ) (cid:1)(cid:1) . The comultiplication map ∆ determines a multiplication on the predual L ( G )given by convolution ϕ ⋆ ϕ ( a ) = ( ϕ ⊗ ϕ )∆( a ). The pair ( L ( G ) , ⋆ ) forms aBanach algebra. In what follows, if not specified otherwise, the quantum groups G we shall work with admit a tracial left-invariant Haar weight ψ . The simplestmodel of noncommutative quantum groups are group von Neumann algebras L (G)associated to discrete groups. If λ is the left regular representation of G, thecomultiplication is determined by ∆( λ ( g )) = λ ( g ) ⊗ λ ( g ). Its isometric naturefollows from Fell’s absorption principle and the convolution is abelian. The standardtrace on L (G) is a left and right-invariant Haar weight. Moreover, in this case, theantipode is bounded and S ( λ ( g )) = λ ( g − ). LGEBRAIC CALDER ´ON-ZYGMUND THEORY 45 A convolution semigroup of states is a family ( φ t ) t ≥ of normal states on L ∞ ( G )such that φ t ⋆ φ t = φ t + t . The corresponding semigroup of completely positivemaps is given by S ∆ ,t ( a ) = ( φ t ⊗ id G ) ◦ ∆( a ) . When S ∆ = ( S ∆ ,t ) t ≥ is a Markov semigroup, we call it a convolution semigroup . Lemma 4.9. Let G be a locally compact quantum group equipped with a convolutionsemigroup of states ( φ t ) t ≥ . Then, S ∆ = (( φ t ⊗ id G ) ◦ ∆) t ≥ is a Markov semigroupon L ∞ ( G ) whenever i) φ t ◦ S = φ t for all t ≥ , ii) S ∆ ,t ( a ) → a as t → + in the weak- ∗ topology of L ∞ ( G ) . Proof. Let us begin with the self-adjointness ψ (cid:0) a ∗ S ∆ ,t ( b ) (cid:1) = ψ (cid:0) a ∗ ( φ t ⊗ id G )∆( b ) (cid:1) = φ t ⊗ ψ (cid:0) ( G ⊗ a ∗ )∆( b ) (cid:1) = φ t (cid:16) ( id G ⊗ ψ ) (cid:0) ( G ⊗ a ∗ )∆( b ) (cid:1) = φ t (cid:16) S ( id G ⊗ ψ ) (cid:0) ∆( a ∗ )( G ⊗ b ) (cid:1)| {z } ρ (cid:17) . This means that ψ (cid:0) a ∗ S ∆ ,t ( b ) (cid:1) = φ t ( S ( ρ )) = φ t ( ρ ) and we get ψ (cid:0) a ∗ S ∆ ,t ( b ) (cid:1) = φ t ⊗ ψ (cid:0) ∆( a ∗ )( G ⊗ b ) (cid:1) = ψ (cid:0) ( φ t ⊗ id G ) ◦ ∆( a ∗ ) b (cid:1) = ψ (cid:0) S ∆ ,t ( a ) ∗ b (cid:1) . The remainder properties are straightforward. Indeed, identity S ∆ ,t ( G ) = G isobvious. The weak- ∗ convergence of the S ∆ ,t ( a )’s as t → + is assumed and thecomplete positivity is clear. The normality follows from the weak- ∗ continuity of φ t and ∆. Finally, the semigroup law easily follows from coassociativity. (cid:3) In what follows, we shall assume that the hypotheses of Lemma 4.9 hold. Letus fix a quantum group G = ( N , ∆ , ψ, ϕ ) and consider a convolution semigroup S ∆ associated to it. A Markov metric Q = (cid:8) ( R j,t , σ j,t , γ j,t ) : j, t ∈ Z + × R + (cid:9) in L ∞ ( G ) = N associated to S ∆ will be called an intrinsic Markov metric whenthere exists an increasing family of projections p j,t in L ∞ ( G ) such that the cpumaps take the form(4.6) R j,t f = 1 ψ ( p j,t ) ( ψ ⊗ id G ) (cid:0) ( p j,t ⊗ G )∆( f ) (cid:1) . In other words, we use the algebraic skeleton (cid:0) N ρ = N π , ρ , ρ , E ρ , q j,t (cid:1) = (cid:0) L ∞ ( G ) ¯ ⊗ L ∞ ( G ) , ⊗ · , ∆ , ψ ⊗ id G , p j,t ⊗ G (cid:1) . Remark 4.10. Assume that γ j,t ∈ R + and γ j,t ≥ ψ ( p j,t ) ψ ( p ,t ) . Then, the term | R j,t f − M t f | in the metric BMO norm satisfies for M t = R ,t that (cid:12)(cid:12) R j,t f − M t f (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ψ ( p ,t ) ( ψ ⊗ id G ) (cid:0) (∆( f ) − ⊗ R j,t f )( p ,t ⊗ ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ψ ( p ,t ) ( ψ ⊗ id G ) h(cid:12)(cid:12) (∆( f ) − ⊗ R j,t f )( p ,t ⊗ ) (cid:12)(cid:12) i = ψ ( p j,t ) ψ ( p ,t ) (cid:0) R j,t | f | − | R j,t f | (cid:1) ≤ γ j,t (cid:0) R j,t | f | − | R j,t f | (cid:1) γ j,t . According to Theorem 1.4, this yields(4.7) k f k BMO c S ∆ . k f k BMO c Q . sup t> sup j ≥ (cid:13)(cid:13)(cid:13)(cid:16) R j,t | f | − | R j,t f | (cid:17) (cid:13)(cid:13)(cid:13) L ∞ ( G ) . Additionally, we may consider transferred Markov metrics in other von Neumannalgebras. Consider a convolution semigroup of states ( φ t ) t ≥ on a locally compactquantum group L ∞ ( G ). A corepresentation π : M → L ∞ ( G ) ¯ ⊗M is a normalinjective ∗ -representation satisfying the identity( id G ⊗ π ) ◦ π = (∆ ⊗ id M ) ◦ π. Every such π yields a transferred convolution semigroup S π = ( S π,t ) t ≥ with S π,t : M → M ,S π,t f = ( φ t ⊗ id M ) ◦ π ( f ) . Lemma 4.11. Assume that • τ ( S π,t ( f ) ∗ f ) = τ ( f ∗ S π,t ( f )) , • S π,t f → f as t → + in the weak- ∗ topology of M .Then S π defines a Markov semigroup on M such that π ◦ S π,t = ( S ∆ ,t ⊗ id M ) ◦ π . Proof. It is easy to check that S π,t is cpu and the normality follows from the weak- ∗ continuity of φ t and π . Hence, it remains to show the identity π ◦ S π,t = ( S t ⊗ id G ) ◦ π and the semigroup law. We first observe that π ( φ t ⊗ id M ) = ( φ t ⊗ id G ⊗ id M )( id G ⊗ π )as maps on L ∞ ( G ) ¯ ⊗M . Indeed, by weak- ∗ continuity, it suffices to test the identityon elementary tensors n ⊗ m , for which the identity is trivial. Therefore, we have( S ∆ ,t ⊗ id M ) π = ( φ t ⊗ id G ⊗ id M )(∆ ⊗ id M ) π = ( φ t ⊗ id G ⊗ id M )( id G ⊗ π ) π = π ( φ t ⊗ id M ) π = πS π,t . For the semigroup law we note that S π,t S π,t = ( φ t ⊗ id M )( φ t ⊗ id G ⊗ id M )( id G ⊗ π ) π = ( φ t ⊗ id M )( φ t ⊗ id G ⊗ id M )(∆ ⊗ id M ) π = ( φ t ⊗ φ t ⊗ id M )(∆ ⊗ id M ) π = ( φ t ⋆ φ t ⊗ id M ) π = S π,t + t . (cid:3) In the sequel, we shall assume that the assumptions in Lemma 4.11 hold. IntrinsicMarkov metrics on L ∞ ( G ) yield transferred Markov metrics on M associated to thetransferred convolution semigroup S π . Indeed, given any intrinsic Markov metric LGEBRAIC CALDER ´ON-ZYGMUND THEORY 47 Q = { ( R j,t , σ j,t , γ j,t ) } in G with cpu maps R j,t given by (4.6), the transferred cpumaps R π,j,t are given by R π,j,t f = 1 ψ ( p j,t ) ( ψ ⊗ id M ) (cid:0) ( p j,t ⊗ ) π ( f ) (cid:1) . It is easy to check that π ◦ R π,j,t = ( R j,t ⊗ id M ) ◦ π . Assume in addition that σ j,t ∈ R + . Then, arguing as we did before Corollary 4.5 for the corepresentation u of R in B ( ℓ ), we get a Markov metric in MQ π = (cid:8) ( R π,j,t , σ j,t , γ j,t ) : j ∈ Z + , t ∈ R + (cid:9) . Let α : N → N be a strictly increasing function with α ( j ) > j . This Markov metricis called α -doubling if there exists some constant c α such that ψ ( q α ( j ) ,t ) ≤ c α ψ ( q j,t ) . Remark 4.12. In what follows, we impose our Markov metrics to be α -doublingfor some function α : N → N , to satisfy σ j,t ∈ R + as well as the condition inRemark 4.10. Altogether, this allows to eliminate the size CZ condition and reducethe number of analytic and smoothness CZ conditions to be checked for both theintrinsic Markov metric and the transferred one.Observe that the transferred formulation above includes the intrinsic formulationby taking ( M , π ) = ( G , ∆). Let us now state the corresponding Calder´on-Zygmundtheory. Given A M a weakly dense ∗ -subalgebra of M , let T be a (not necessarilybounded) operator T : A M → M . We say T is a transferred map if there exists anamplification map b T : D ⊂ L ∞ ( G ) ¯ ⊗M → L ∞ ( G ) ¯ ⊗M satisfying the identity(4.8) π ◦ T = b T ◦ π | AM . Again, D is a weakly dense ∗ -subalgebra for which π ( A M ) ⊂ D . In the case( M , π ) = ( G , ∆), we can always take the amplification T ⊗ id M and condition abovejust imposes that T is a quantum Fourier multiplier. In the following theorem, weprovide sufficient conditions on the amplification map to make a given transferredCZO T bounded from A M to BMO c S π . Theorem 4.13. Let π : M → L ∞ ( G ) ¯ ⊗M be a corepresentation of a locally compact quantum group G in a semifinite vonNeumann algebra ( M , τ ) . Assume that L ∞ ( G ) comes equipped with an α -doublingintrinsic Markov metric Q determined by an increasing family of projections p j,t asabove. Then, a transferred map T ( with amplification for which (4.8) holds ) will bebounded from A M to BMO c S π provided :i) b T : L c ( G ) ¯ ⊗M → L c ( G ) ¯ ⊗M is bounded, ii) ( ψ ⊗ ψ ⊗ id M ) ψ ( p j,t ) (cid:16) ( p j,t ⊗ p j,t ⊗ M ) (cid:12)(cid:12) δ G (cid:0) b T ( π ( f ) p ⊥ α ( j ) ,t ) (cid:1)(cid:12)(cid:12) (cid:17) . k f k M . Proof. We use the algebraic skeleton (cid:0) M , N ρ = N π , ρ , ρ , E ρ , q j,t (cid:1) = (cid:0) M , L ∞ ( G ) ¯ ⊗M , ⊗ · , π, ψ ⊗ id G , p j,t ⊗ G (cid:1) . Identity (4.8) is the compatibility condition (2.3). Let us justify the algebraicconditions. The second one is trivial since both E ρ ( q j,t ) and ρ ( γ j,t ) belong to R +8 JUNGE, MEI, PARCET AND XIA in this case. For the first one, consider the product • in L ∞ ( G ) ¯ ⊗M op . Then, wejust observe that E ρ ( q j,t | ξ | q j,t ) = ( ψ ⊗ id M ) (cid:0) ( p j,t ⊗ ) ξ ∗ ξ (cid:1) = ( ψ ⊗ id M ) (cid:0) ξ • ( p j,t ⊗ ) • ξ ∗ (cid:1) ≤ ( ψ ⊗ id M )( ξ • ξ ∗ ) = ( ψ ⊗ id M )( ξ ∗ ξ ) = E ρ ( | ξ | ) . Define the amplifications b R π,j,t : L ∞ ( G ) ¯ ⊗M ∋ ξ ψ ( p j,t ) ( ψ ⊗ id M ) (cid:0) ( p j,t ⊗ M ) ξ (cid:1) ∈ M . Consider also the cpu mapsΦ j,t : L ∞ ( G ) ¯ ⊗ L ∞ ( G ) ¯ ⊗M → M , Φ j,t ( η ) = 1 ψ ( p j,t ) ( ψ ⊗ ψ ⊗ id M ) (cid:0) ( p j,t ⊗ p j,t ⊗ M ) η (cid:1) . Recalling that δ G ( x ) = x ⊗ − ⊗ x , the identityΦ j,t ( | δ G ( ξ ) | ) = 2 b R π,j,t ( | ξ | ) − (cid:12)(cid:12) b R π,j,t ( ξ ) (cid:12)(cid:12) . is straightforward. This readily implies the first analytic condition. On the otherhand, since the auxiliary Markov metric is α -doubling, the second analytic conditionreduces to note that( ψ ⊗ id M )( q α ( j ) ,t ⊗ M ) ≤ c α ( ψ ⊗ id M )( q j,t ⊗ M ) . Thus, according to inequalitty (4.7), the assertion follows from Theorem 2.1. (cid:3) Remark 4.14. As noticed, the main particular case of Theorem 4.13 arises for( M , π ) = ( G , ∆) with amplification T ⊗ id G . Condition (4.8) becomes the identity∆ ◦ T = ( T ⊗ id G ) ◦ ∆ . In other words, these are translation invariant CZ operators. We also call themquantum Fourier multipliers in this paper and it can be checked, as expected, thatthese maps are of convolution type in the sense that there exists a kernel k affiliatedto L ∞ ( G ) so that T f = k ⋆ f = ( id G ⊗ ψ ) (cid:0) ∆( k )( G ⊗ Sf ) (cid:1) . In this particular case, it is not difficult to prove that our conditions in Theorem4.13 reduce to those in Theorem B2 from the Introduction. Of course, Theorem4.13 also applies as well for nonconvolution CZ operators on quantum groups, oreven for transferred forms of them to other von Neumann algebras M . Remark 4.15. One may consider twisted convolution CZO’s on quantum groupsapplying Theorem 4.13. As an illustration, assume that G y L ∞ ( G ) by a tracepreserving action α and that G is a quantum group satisfying ( α g ⊗ α g )∆ = ∆ α g forall g ∈ G. This property is quite natural in the commutative case, where quantumgroups come from locally compact groups and α is typically implemented by ameasure preserving map β . Note that the underlying Haar measure is translationinvariant and the condition above just imposes that β is an homomorphism. Letus see what we get for a map X g f g λ ( g ) X g T g ( f g ) λ ( g ) , LGEBRAIC CALDER ´ON-ZYGMUND THEORY 49 where the T g ’s are normal convolution maps on L ∞ ( G ). Assume L ∞ ( G ) comesequipped with a convolution G-equivariant semigroup S ∆ which admits a η -doublingintrinsic Markov metric. Then, we get a bounded map L ∞ ( G ) ⋊ . G → BMO c S ⋊ whenthe following conditions hold:i) We have a bounded map L c ( G ) ¯ ⊗B ( ℓ (G)) ∋ ξ (cid:0) T gh − (cid:1) • ξ ∈ L c ( G ) ¯ ⊗B ( ℓ (G)) , where • stands once more for the generalized Schur product of matrices.ii) Letting R = L ∞ ( G ) ¯ ⊗B ( ℓ (G)) and Ψ( ξ ) = P g,h ( α g − ) • ( T gh − ) • ( α g ) • ξ, ( ψ ⊗ ψ ⊗ id B ( ℓ (G)) ) ψ ( p j,t ) (cid:16) ( p j,t ⊗ p j,t ⊗ ) (cid:12)(cid:12) δ G (cid:0) Ψ( ξq ⊥ η ( j ) ,t (cid:1)(cid:12)(cid:12) (cid:17) . k ξ k R . Remark 4.16. All our results in this paragraph impose the additional assumptionthat our quantum groups admit a tracial Haar weight. We believe however that ourresults can be extended to the general non-tracial case. We leave this generalizationopen to the interested reader.5. Noncommutative transference Originally motivated by Cotlar’s paper [15] and the method of rotations, Calder´ondeveloped a circle of ideas [5] which was called the transference method after thesystematic study of Coifman/Weiss in their monograph [13]. The fundamental workof K. de Leeuw [18] also had a big impact in this line of research. Let us consideran amenable locally compact group G with left Haar measure µ , a σ -finite measurespace (Ω , ν ) and a uniformly bounded representation β : G → B ( L p (Ω)). Roughly,Calder´on’s transference is a technique which allows to transfer the L p boundednessof a convolution operator f k ⋆ f on L p (G) to the corresponding transferredoperator on L p (Ω) Vf ( w ) = Z G k ( g ) β g − f ( w ) dµ ( g ) , for some compactly supported kernel k in L (G). A case by case limiting procedurealso allows to consider more general (singular) kernels. In the rest of this sectionwe shall develop a noncommutative form of Calder´on-Coifman-Weiss technique.Our first task is to clarify what we mean by ‘representation’ and ‘amenable’ in thecontext of quantum groups. Using the commutative locally compact quantum group L ∞ (G) as above, a representation β : G → Aut( M ) induces a ∗ -representation π β : M → L ∞ (G; M ) by π β f ( g ) = β g − f. Note that we have( id G ⊗ π β )( π β f )( g, h ) = π β ( β g − f )( h ) = β h − β g − f = β ( gh ) − f = (∆ G ⊗ id M )( π β f )( g, h ) . Given a semifinite von Neumann algebra ( M , τ ) and a locally compact quantumgroup G , this leads us to consider corepresentations π : M 7→ L ∞ ( G ) ¯ ⊗M satisfying( id G ⊗ π ) ◦ π = (∆ ⊗ id M ) ◦ π . Note that comultiplication is a corepresentation bycoassociativity. To show what we mean by ‘uniformly bounded’, let us go back to our motivating example β : G → Aut( M ), where we take M = L ∞ (Ω) for some σ -finite measure space (Ω , ν ). In the classical case k β g f k p ∼ k f k p for all g ∈ Gup to an absolute constant independent of f, g . We say that a corepresentation π : M → L ∞ ( G ) ¯ ⊗M is uniformly bounded in L p ( M ) if for any f ∈ M ∩ L p ( M )we have 1 c π k f k pL p ( M ) ≤ ( id G ⊗ τ ) (cid:0) | π ( f ) | p (cid:1) ≤ c π k f k pL p ( M ) for some absolute constant c π independent of f . Note that our notion again reducesto the classical one on L ∞ (G). Note also that, since | π ( f ) | p = π ( | f | p ), our definitionreduces to the p -independent condition1 c π k f k L ( M ) ≤ ( id G ⊗ τ ) (cid:0) π ( f ) (cid:1) ≤ c π k f k L ( M ) for all f ∈ M + ∩ L ( M ) . Now we introduce what we mean by an ‘amenable’ quantum group. We say that G satisfies Følner’s condition if for every projection q ∈ L ( G ) and every ε > 0, thereexists two projections q , q ∈ L ( G ) such that∆( q )( q ⊗ q ) = q ⊗ q and ψ ( q ) ≤ (1 + ε ) ψ ( q ) . In the standard example for a locally compact group G, where ( L ∞ ( G ) , ψ ) is L ∞ (G)equipped with the left Haar measure µ and ∆ is given by ∆ G ( ξ )( g, h ) = ξ ( gh ) theclassical comultiplication, it turns out that G is amenable iff G is an amenablegroup. Indeed, our notion can be rephrased in this case by saying that for anycompact set K in G and any ε > 0, there exists a neighborhood of the identity Wof finite measure such that µ (KW) ≤ (1 + ε ) µ (W) , which corresponds to ( q, q , q ) = ( χ K , χ KW , χ W ) in our formulation. This is exactlythe classical characterization of amenability, known as Følner’s condition, used byCoifman and Weiss in [13]. Given an amenable locally compact group G withleft Haar measure µ , it is clear that L ∞ (G , µ ) with its natural quantum groupstructure is amenable. On the other hand, as expected, any compact quantumgroup is amenable just by taking q = q = G .Assume that G admits a corepresentation π : M → L ∞ ( G ) ¯ ⊗M . Given A M aweakly dense ∗ -subalgebra of M , we say that a linear operator V : A M → M is a transferred convolution map if there exists Φ : D ⊂ L ∞ ( G ) ¯ ⊗M → L ∞ ( G ) ¯ ⊗M , anauxiliary convolution map such that π ◦ V = Φ ◦ π | AM . The classical transferredoperator V = Z G k ( g ) β g − f ( w ) dµ ( g )comes fromΦ( ξ )( g, w ) = Z G k ( h ) ξ ( hg, w ) dµ ( h ) = ( ϕ ⊗ id G ⊗ id Ω )(∆ G ⊗ id Ω ) . If π β f ( g ) = β g − f denotes the corresponding corepresentation, we may then applythe identities in the proof of Lemma 4.11 again to deduce the following identitiesΦ ◦ π β = ( ϕ ⊗ id G ⊗ id Ω )(∆ G ⊗ id Ω ) π β = ( ϕ ⊗ id G ⊗ id Ω )( id G ⊗ π β ) π β = π β ( ϕ ⊗ id Ω ) ◦ π β . LGEBRAIC CALDER ´ON-ZYGMUND THEORY 51 By injectivity of π β , we must have Vf ( w ) = ( ϕ ⊗ id Ω ) π β f ( w ) = Z G k ( g ) β g − f ( w ) dµ ( g )as expected. This shows how we recover the classical construction.Let us now settle the framework for our transference result. Assume that G isamenable and consider π : M → L ∞ ( G ) ¯ ⊗M a uniformly bounded corepresentationin some noncommutative measure space ( M , τ ). We say that T : L p ( G ) → L p ( G )is a convolution map with finitely supported L kernel when the map T has theform T = ( φ ⊗ id G ) ◦ ∆ for some functional φ = ψ ( d · ), with d an element in L ( G )whose left support q satisfies ψ ( q ) < ∞ . In the commutative case, this is the kind ofoperators which are transferred. Roughly, the goal is to show how a limit operator T = lim γ T γ of such maps which is bounded on L ( G ) and L ∞ ( G ) → BMO S canbe transferred under suitable conditions to a bounded map on L p ( M ). Remark 5.1. Young’s inequality extends to this setting as k d ⋆ f k p ‘=’ k ( φ ⊗ id G )∆( f ) k p ≤ k d k k f k p , where φ = ψ ( d · ) and 1 ≤ p ≤ ∞ . Indeed, when d and f are positive the inequalityholds with constant 1. This can by justified by interpolation. When p = 1 we useFubini and the left-invariance of ψ , while for p = ∞ it follows from the fact that( φ ⊗ id G )∆ is a positive map with G ψ ( d ). In the general case, we split d, f intotheir positive parts and obtain the constant 4. In fact, the same argument still holdsafter matrix amplification and we deduce that ( φ ⊗ id G )∆ is completely boundedon L p ( G ) with cb-norm 4 k d k . This is however not enough for transference, sincethe norms k d γ k might not be uniformly bounded. Theorem 5.2. Let G be an amenable quantum group and consider a uniformlybounded corepresentation π : M → L ∞ ( G ) ¯ ⊗M in some noncommutative measurespace ( M , τ ) . Let T : L ( G ) → L ( G ) be a bounded map and assume that ( T ⊗ id M ) = SOT − lim γ ( T γ ⊗ id M ) for some net T γ = ( φ γ ⊗ id G ) ◦ ∆ of convolution maps with finitely supported L kernels and such that lim γ k T γ k B ( L ( G )) ≤ k T k B ( L ( G )) . Then, the net of transferredoperators V γ = ( φ γ ⊗ id M ) ◦ π satisfies the inequalities k V γ k B ( L ( M )) ≤ c π k T γ k B ( L ( G )) . We thus find a WOT -cluster point V satisfying k V k B ( L ( M )) ≤ c π k T k B ( L ( G )) . Proof. Note that we have πV γ = ( φ γ ⊗ id )( id G ⊗ π ) π = ( φ γ ⊗ id )(∆ ⊗ id M ) π = ( T γ ⊗ id M ) π. Hence, the uniform boundedness of π yields1 c π k V γ f k ≤ ( ρ ⊗ τ ) (cid:0) | πV γ ( f ) | (cid:1) = ( ρ ⊗ τ ) (cid:0) | ( T γ ⊗ id M ) π ( f ) | (cid:1) for any state ρ on L ∞ ( G ). On the other hand, if φ γ = ψ ( d γ · ) and q γ denotes theleft support of d γ , we know from the amenability assumption that for any ε > q γ and q γ such that∆( q γ )( q γ ⊗ q γ ) = q γ ⊗ q γ and ψ ( q γ ) ≤ (1 + ε ) ψ ( q γ ) . Taking ρ = ψ ( q γ · ) /ψ ( q γ ), we obtain the inequality1 c π k V γ f k ≤ ( ψ ⊗ τ ) ψ ( q γ ) (cid:16)(cid:12)(cid:12)(cid:12) ( φ ⊗ id ) (cid:0) (∆ ⊗ id M ) π ( f )( G ⊗ q γ ⊗ M ) (cid:1)(cid:12)(cid:12)(cid:12) (cid:17) since ρ is supported by q γ . Moreover, d γ ⊗ q γ is supported on the left by q γ ⊗ q γ and amenability provides d γ ⊗ q γ = ∆( q γ )( d γ ⊗ q γ ). Once we have created∆( q γ ), we can eliminate q γ . Altogether gives1 c π k V γ f k ≤ ψ ( q γ ) (cid:13)(cid:13)(cid:13) ( T γ ⊗ id ) (cid:0) π ( f )( q γ ⊗ M ) (cid:1)(cid:13)(cid:13)(cid:13) L ( L ∞ ( G ) ¯ ⊗M ) . Now we use the L boundedness of T γ and uniform boundedness of π to conclude1 c π k V γ f k ≤ ψ ( q γ ) k T γ k B ( L ( G )) ψ (cid:16) ( q γ ⊗ M )( id G ⊗ τ ) (cid:0) | π ( f ) | (cid:1)(cid:17) ≤ c π ψ ( q γ ) k T γ k B ( L ( G )) ψ ( q γ ) k f k ≤ c π (1 + ε ) k T γ k B ( L ( G )) k f k . Letting ε → 0, we prove the inequality k V γ k B ( L ( M )) ≤ c π k T γ k B ( L ( G )) . Since T is bounded on L ( G ) and lim γ k T γ k B ( L ( G )) ≤ k T k B ( L ( G )) , the operators V γ are eventually in a ball of radius c π (1 + δ ) k T k B ( L ( G )) for any δ > 0. The closureof such ball is weak operator compact and thus we find our cluster point. (cid:3) We now study L ∞ → BMO transference and then interpolate/dualize to obtain L p -transference. This approach seems to be new even in the classical theory andwhere our semigroup formulation becomes an essential ingredient. Corollary 5.3. Let G be a compact ( hence amenable ) quantum group equippedwith a uniformly bounded corepresentation π : M → L ∞ ( G ) ¯ ⊗M . Let ( φ t ) t ≥ bea convolution semigroup of states on L ∞ ( G ) , giving rise to Markov semigroups S ∆ on ( G , ψ ) and S π on ( M , τ ) . Let T = SOT − lim γ T γ be as above and take A M = M ∩ L ( M ) . Then, if T : L ∞ ( G ) → BMO S ∆ is completely bounded, we findthat V = WOT − lim γ V γ : A M → BMO S π is completely bounded. Moreover, if T π is regular, the complete boundedness of J p V : L p ( M ) → L p ( M ) follows for every < p < ∞ by interpolation. In additionthe complete boundedness of V J p : L p ( M ) → L p ( M ) for < p < holds under thesame assumptions for T ∗ . Proof. By uniform boundedness of π we have (cid:13)(cid:13)(cid:13) ( id G ⊗ τ ) (cid:0) | π ( f ) | (cid:1) (cid:13)(cid:13)(cid:13) L ∞ ( G ) ≤ c π k f k , which implies that π : L ( M ) → L ∞ ( G ) ¯ ⊗ L c ( M ) is bounded by c π . According tothe finiteness of L ∞ ( G ), we deduce that in fact π : L ( M ) → L ( L ∞ ( G ) ¯ ⊗M ) isstill bounded with the same norm. This proves that πV = WOT − lim γ πV γ = WOT − lim γ ( T γ ⊗ id M ) π = ( T ⊗ id M ) π. LGEBRAIC CALDER ´ON-ZYGMUND THEORY 53 In particular, πV = ( T ⊗ id M ) π over A M and identity πS π,t = ( S t ⊗ id M ) π yields k V f k BMO c S π = sup t ≥ (cid:13)(cid:13)(cid:13) S π,t | V f | − | S π,t V f | (cid:13)(cid:13)(cid:13) M = sup t ≥ (cid:13)(cid:13)(cid:13) πS π,t | V f | − | πS π,t V f | (cid:13)(cid:13)(cid:13) L ∞ ( G ) ¯ ⊗M = (cid:13)(cid:13) ( T ⊗ id M ) π ( f ) (cid:13)(cid:13) BMO c S ≤ k T k cb k π ( f ) k L ∞ ( G ) ¯ ⊗M = k T k cb k f k M for f ∈ A M . Since the same inequality holds after matrix amplification, we deducethat V : A M → BMO c S π is completely bounded with cb-norm ≤ k T k cb . The rowcase is similar because πV † = ( πV ) † = ( T π ) † = T † π. The assertions on L p boundedness follow as usual from Theorem 1.3. (cid:3) Remark 5.4. Under the above assumptions, we see that for V = WOT − lim γ V γ we can find the concrete form of its amplification map Φ defined on L ∞ ( G ) ¯ ⊗M . 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(1969), 529-564.61. M. Takesaki, Theory of operator algebras I. Springer-Verlag, New York, 1979.62. X. Tolsa, BMO, H , and Calder´on-Zygmund operators for non doubling measures. Math.Ann. (2001), 89-149.63. X. Tolsa, Littlewood-Paley theory and the T (1) theorem with non-doubling measures. Adv.Math. (2001), no. 1, 57-116.64. D. V. Voiculescu, K. J. Dykema and A. Nica, Free random variables. CRM Monograph Series, . American Mathematical Society, Providence, RI, 1992.65. X. Xiong, Q. Xu and Z. Yin. Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori.Mem. Amer. Math. Soc. (2018), 1203. Marius Junge Department of MathematicsUniversity of Illinois at Urbana-Champaign1409 W. Green St. Urbana, IL 61891. USA [email protected] Tao Mei Department of MathematicsBaylor University1301 S University Parks Dr, Waco, TX 76798, USA Javier Parcet Instituto de Ciencias Matem´aticasConsejo Superior de Investigaciones Cient´ıficasC/ Nicol´as Cabrera 13-15. 28049, Madrid. Spain [email protected] Runlian Xia