An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces
aa r X i v : . [ m a t h . F A ] A ug AN OPERATOR-VALUED T (1) THEOREM FOR SYMMETRICSINGULAR INTEGRALS IN UMD SPACES
TUOMAS HYTÖNEN
Abstract.
The natural BMO (bounded mean oscillation) conditions sug-gested by scalar-valued results are known to be insufficient for the boundednessof operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical“ T (1) ∈ BMO ” assumptions that are not easily checkable. Recently, Hong, Liuand Mei (J. Funct. Anal. 2020) observed that the situation improves remark-ably for singular integrals with a symmetry assumption, so that a classical T (1) criterion still guarantees their L -boundedness on Hilbert space -valuedfunctions. Here, these results are extended to general UMD (unconditionalmartingale differences) spaces with the same natural BMO condition for sym-metrised paraproducts, and requiring in addition only the usual replacementof uniform bounds by R -bounds in the case of general singular integrals. Inparticular, under these assumptions, we obtain boundedness results on non-commutative L p spaces for all < p < ∞ , without the need to replace thedomain or the target by a related non-commutative Hardy space as in theresults of Hong et al. for p = 2 . Introduction
Paraproducts are central and widespread in modern harmonic analysis. Theirsystematic introduction is due to J.-M. Bony [5] in the context of symbolic calculusfor non-linear partial differential equations but, as argued by [1], “the first versionof a paraproduct is [already] implicit in A. P. Calderón’s work on commutators [7]”.We refer the reader to [1] for a friendly introduction to the variety of objects thatnow go under the generic name “paraproduct”, and some further applications. Ourpresent interest is in the dyadic version of paraproducts Π b , and in their role in theboundedness of general singular integral operators, a connection revealed by thecelebrated T (1) and T ( b ) theorems of G. David, J.-L. Journé and S. Semmes [8, 9].More specifically, we are interested in the matrix/operator-valued versions ofthese objects and results. Matrix-valued paraproducts (disguised as matrix-valuedCarleson embeddings) appeared in the work of S. Treil and A. Volberg [29], mo-tivated by questions in multivariate stationary processes. Independently, theseauthors with F. Nazarov [25] on the one hand, and N. H. Katz [22] on the otherhand, obtained the dimensional dependence in the key inequality k b k BMO d so ( R ; L ( H )) ≤ k Π b k L ( L ( R ; H )) ≤ c (1 + log dim H ) k b k BMO d so ( R ; L ( H )) , (1.1) Mathematics Subject Classification.
Key words and phrases.
Calderón–Zygmund operator, T (1) theorem, operator-valued, UMD.The author was supported by the Academy of Finland through project Nos. 307333 (Centreof Excellence in Analysis and Dynamics Research) and 314829 (Frontiers of singular integrals). where H is a finite-dimensional Hilbert space, L ( H ) is the space of bounded linearoperators acting on this space, Π b is the dyadic paraproduct associated with the L ( H ) -valued function b , and k b k BMO d so ( R ; L ( H )) := sup x ∈ H k x k≤ sup I ⊂ R interval | I | ˆ I k ( b ( t ) − h b i I ) x k H d t (1.2)is the strong-operator (dyadic) bounded mean oscillation norm. Some time later,Nazarov, Pisier, Treil and Volberg [24] proved the sharpness of log dim H in (1.1);the necessity of some dimensional growth was already contained in [25]. (In (1.2),we have followed the notation “ BMO so ” as used e.g. in [24], but this is not univer-sal; some other related papers also incorporate the norm of the pointwise adjointfunction t b ( t ) ∗ in this notation.)For a while, there were hopes in the air of achieving a dimension-free bound byreplacing BMO so on the right of (1.1) by the larger uniform-operator BMO norm k b k BMO d ( R ; L ( H )) := sup I ⊂ R interval | I | ˆ I k b ( t ) − h b i I k L ( H ) d t but also this was ruled out by T. Mei [23] by showing that, even in the bound k Π b k L ( L ( R ; H )) ≤ φ (dim H ) k b k L ∞ ( R ; L ( H )) , with the smaller function space L ∞ in place of BMO d on the right, the dimensionaldependence cannot be better than φ ( d ) ≥ c (1 + log d ) .In particular, this severely sets back the hopes of developing such estimates ininfinite-dimensional Hilbert spaces, not to mention more general Banach spaces.This failure in infinite dimensions of classical bounds between various BMO-typequantities on the one hand, and the norms of related transformations on the otherhand, has been further elaborated by Blasco and Pott [3, 4] and, for analogousquestions dealing with the (complex-)analytic BMOA and related operators, quiterecently by Rydhe [28].As mentioned, one of the major applications of paraproducts is their role in thecharacterisation of boundedness of general (non-convolution type) singular integraloperators via the T (1) and T ( b ) theorems of David, Journé and Semmes [8, 9],as well as their many extensions. The above-discussed problems of describing theboundedness of paraproducts in the infinite-dimensional setting, in terms of acces-sible function space norms, have been a major obstacle on the way of achievinga fully satisfactory analogue of the general theory of singular integrals in infinite-dimensional Banach spaces; the available versions of the operator-valued T (1) and T ( b ) theorems – [13, 15, 19] and their extensions – suffer from complicated and noteasily verifiable variants of BMO conditions that are only distant cousins of theirsimple classical predecessors.In contrast, verifiable conditions for the boundedness of operator-valued singularintegrals of convolution type , T f ( x ) = ˆ R d K ( x − y ) f ( y ) d y, are well understood since the work of L. Weis [30] via their equivalent descriptionas operator-valued Fourier multipliers c T f = ˆ K ˆ f (but see also [20] for a singularintegral point of view to the same operators). Results on the boundedness of these SYMMETRIC T (1) THEOREM 3 operator-valued convolution-type singular integrals have profound applications toregularity problems for autonomous evolution equations; see again [30] and themany works citing this influential paper.Analogous questions for non-autonomous equations give rise to singular integralsof non-convolution type,
T f ( x ) = ˆ R d K ( x, y ) f ( y ) d y, which in principle should belong to the scope of the (operator-valued) T (1) and T ( b ) theorems [13, 15, 19]. However, the complicated form of their conditions has so farhindered such applications. In a recent paper [10, p. 535], the authors explicitlywrite: “At the moment we do not know whether the T -theorem and T b -theoremcan be applied to study maximal L p -regularity for the time dependent problems weconsider.” Nevertheless, the authors of [10] manage to obtain the L p -boundednessfor a special class of operator-valued non-convolution singular integrals suitablefor their needs. This indicates a continuing demand for checkable criteria for theboundedness of at least special classes of non-convolution operators, as long as thefull analogue of the scalar-valued T (1) and T ( b ) theorems seems out of reach.In a recent work [14], Hong, Liu and Mei achieve such a result, in the very styleof a T (1) theorem, for operator-valued singular integrals with a certain symmetry assumption, satisfied in particular by all even operators. Under natural assump-tions, their result gives the L p ( R d ; X ) -boundedness of these operators when p = 2 and X = H is a Hilbert space, and a weaker substitute result (replacing eitherthe domain or the target with a suitable non-commutative Hardy space) when p ∈ (1 , ∞ ) \ { } and X is non-commutative L p -space. (Incidentally, a symmetrycondition was also key to another recent advance on vector-valued singular inte-grals concerning the possible linear dependence of singular integral and martingaletransform norms, a problem that was solved for even singular integrals by Pott andStoica [27] but remains open in general.)In this paper, we obtain an extension of the Hong–Liu–Mei [14] result to all Ba-nach spaces X with the unconditionality property of martingale differences (UMD).Our result is a pure L p estimate for all p ∈ (1 , ∞ ) , without the need of substi-tute Hardy spaces, and it is obtained in the maximal generality of Banach spaces(namely, UMD spaces) in which such results could be hoped for. Indeed, theBeurling–Ahlfors transform T f ( z ) = − p . v . π ˆ C f ( y )( z − y ) d y, where the integration is with respect to the two-dimensional Lebesgue measure on C h R , is an even singular integral operator in the scope of Theorem 1.3 below,whose boundedness on L p ( R ; X ) is equivalent to X being a UMD space by [11].Our main result can be roughly stated as follows; see Theorem 4.7 for a detailedformulation of the various technical assumptions appearing in the statement.1.3. Theorem (Symmetric T (1) theorem) . Let X be a UMD space, L ( X ) the spaceof bounded linear operators on X , and p ∈ (1 , ∞ ) . Let T f ( x ) = ˆ R d K ( x, y ) f ( y ) d y TUOMAS HYTÖNEN be a Calderón–Zygmund operator with an L ( X ) -valued kernel K , acting on X -valued test functions f . Suppose that T and K satisfy R -bounded versions of theCalderón–Zygmund kernel estimates and the weak boundedness property as definedin Section 4. Finally, suppose that T T ∗ ∗ ∈ BMO( R d ; L ( X )) , (1.4) Then T extends to a bounded linear operator on L p ( R d ; X ) . We stress that all other assumptions of Theorem 1.3 are essentially the same asin any other operator-valued T (1) theorem in the literature (like [13, 15, 19]), andthe key novelty is the condition (1.4) inspired by [14]. This improves on all previousresults on the level of general UMD spaces by means of replacing their more com-plicated BMO-type spaces by the plain BMO( R d ; L ( X )) , which is just the classicalBMO space with absolute values replaced the norm in L ( X ) ; it achieves this atthe cost of requiring the additional symmetry imposed by the equality in (1.4).Unfortunately, Theorem 1.3 still lacks a key feature of the classical T (1) theorems:a characterisation of L p -boundedness. The condition that T ∈ BMO( R d ; L ( X )) is a relatively checkable sufficient condition, but it is still not necessary .As with the other assumptions, we refer the reader to Section 4 for a preciseinterpretation of the condition (1.4); however, the following formal explanationmay be helpful at this point: In (1.4), the (formal) action of T on the constantscalar function is an L ( X, Y ) -valued function, and part of the assumption (1.4)is to require that this function has bounded mean oscillation. Moreover, T ∗ refersto the formal adjoint operator T ∗ g ( x ) = ˆ R d K ( y, x ) ∗ g ( y ) d y, whose kernel K ( y, x ) ∗ takes values in L ( X ∗ ) and acts on test functions g with val-ues in X ∗ . The formal action of T ∗ on the constant scalar function is an L ( X ∗ ) -valued function T ∗ , and ( T ∗ ∗ refers to its pointwise adjoint, an L ( X ∗∗ ) = L ( X ) -valued function. (Note that UMD spaces are reflexive, see [17, Theorem4.3.3.].) Thus T and ( T ∗ ∗ are (at least formally) functions of the same type,and another part of the assumption (1.4) is to require that they are equal.A main ingredient of Theorem 1.3, related to the key condition (1.4), is thefollowing bound of independent interest for the symmetrised paraproduct Λ b := Π b + Π ∗ b ∗ ; the precise definition of these operators and a slightly more general statement willbe given in Section 3.1.5. Theorem (Boundedness of symmetrised paraproducts) . Let X be a UMDspace and b ∈ BMO d ( R d ; L ( X )) . Then the symmetrised paraproduct Λ b extends toa bounded linear operator on L p ( R d ; X ) with the bound k Λ b k L ( L p ( R d ; X )) ≤ c d β p,X k b k BMO( R d ; L ( X )) , where β p,X is the UMD constant of X and c d is dimensional. We note that the case when p = 2 and X = H is a Hilbert space is alreadydue to Blasco and Pott [3, Theorem 2.6]. Hong, Liu and Mei [14, Proposition2.2] provide a version with p ∈ (1 , ∞ ) , where X is a non-commutative L p space(with the same p ), and either the domain or the target needs to be replaced by an SYMMETRIC T (1) THEOREM 5 appropriate non-commutative Hardy space in place of L p ( R d ; X ) when p = 2 . This[14, Proposition 2.2] plays a similar role in their T (1) theorem, as Theorem 1.5 inour Theorem 1.3, and motivates our approach to both results.Theorem 1.3 is actually a relatively quick corollary of Theorem 1.5 and theintermediate results in essentially any existing proof of the operator-valued T (1) theorem based on the dyadic approach. Namely, these proofs typically decomposethe operator into a sum of two paraproducts Π b + Π ∗ b ∗ , estimated with the help ofsome BMO type assumptions, and the cancellative part, which is handled by usingthe Calderón–Zygmund kernel estimates and the weak boundedness property. Forthe cancellative part, we can simply borrow the estimates that were already carriedout in one of the previous works. For the paraproduct part, under our symmetryassumption (1.4), we have b = b = b , and hence this part reduces to Π b + Π ∗ b ∗ = Π b + Π ∗ b ∗ = Λ b , which is precisely the operator estimated in Theorem 1.5. Such a decompositioninto the paraproduct part and the cancellative part is at least implicitly behindessentially all known proofs of the T (1) theorem, but it is particularly clean in therecent dyadic representation theorems that originate from the resolution of the A conjecture on sharp weighted norm inequalities [21]. For our purposes, we use the operator-valued dyadic representation theorem from [13]. (Hong, Liu and Mei [14]adapt the approach of [16] instead; this would also have been a relevant alternativehere.)sThe rest of this paper is structured as follows. In Section 2, we collect somenecessary preliminaries on the vector-valued dyadic Hardy space and BMO on theone hand, and on projective tensor products and their duality on the other hand.The latter provide a key substitute in our considerations for some of the non-commutative tools used by [14]. In Section 3, we provide the definitions relatedto paraproducts, and give the proof of Theorem 1.5. Up to this point, all consid-erations are purely dyadic, and we only turn to continuous singular integrals andrelated objects in the remaining two sections. In Section 4, we provide all necessarydefinitions to give a precise formulation of our main Theorem 1.3. This theoremis then proved in the final Section 5 via the operator-valued dyadic representationtheorem of [13], which we recall there.2. Preliminaries A system of dyadic cubes in R d is a family D := [ k ∈ Z D k of (axes-parallel, left-closed, right-open) cubes Q such that, for each k ∈ Z , • D k is a partition of R d consisting of cubes Q of side-length ℓ ( Q ) = 2 − k ; • D k +1 is a refinement of D k .We consider one such dyadic system fixed for the moment. However, in our approachto the T (1) theorem below, it will be important that all estimates hold uniformlywith respect to the choice of the dyadic systems, as the later considerations willinvolve a random choice. TUOMAS HYTÖNEN
A dyadic system induces the averaging (or conditional expectation) operators E k f := X Q ∈ D k Q h f i Q , h f i Q := Q f := 1 | Q | ˆ Q f ( t ) d t and the martingale difference operators D k f := E k +1 f − E k f, both well defined for f ∈ L ( R d ; E ) , where E is any Banach space.For (say) f ∈ L p ( R d ; E ) with < p < ∞ , we have E k f → f as k → ∞ ,and E k f → as k → −∞ , both in the norm of L p ( R d ; E ) and pointwise almosteverywhere (see e.g. [17, Theorems 3.3.2 and 3.3.5]) and hence f = X k ∈ Z D k f ; (2.1)in particular, finite truncations of sums on the right are dense in L p ( R d ; E ) . If E isa UMD space, the convergence of (2.1) is unconditional. In particular, the modifiedsums P k ∈ Z ǫ k D k f with ǫ k ∈ {− , +1 } also converge, and (cid:13)(cid:13)(cid:13) X k ∈ Z ǫ k D k f (cid:13)(cid:13)(cid:13) L p ( R d ; E ) ≤ β p,E (cid:13)(cid:13)(cid:13) X k ∈ Z D k f (cid:13)(cid:13)(cid:13) L p ( R d ; E ) = β p,E k f k L p ( R d ; E ) , where β p,E is the UMD constant of E . See [17, Chapter 4] for these results andmore on UMD spaces.2.A. Dyadic H and BMO . The dyadic Hardy space H d ( R d ; E ) is defined withthe help of the cancellative dyadic maximal function M d h := sup k ∈ Z k E k h k E = sup Q ∈ D Q kh h i Q k E . It is essential that the norm is taken outside and not inside the average. Then H d ( R d ; E ) := n h ∈ L ( R d ; E ) : k h k H d := k M d h k < ∞ o . Note that the choice of L ( R d ; E ) as the ambient space does not impose any essentialrestriction. Even if we only demanded that h ∈ L ( R d ; E ) (an essentially minimalcondition to be able to make sense of M d h ) it follows from Lebesgue’s differentiationtheorem that k h ( · ) k E ≤ M d h a.e., and hence h ∈ L ( R d ; E ) if M d h ∈ L ( R d ) .The dyadic BMO space BMO d ( R d ; F ) (with valued in another Banach space F )is defined as BMO d ( R d ; F ) := n b ∈ L ( R d ; F ) : k b k BMO d := sup Q ∈ D Q k b − h b i Q k F < ∞ o . Both the dyadic H and BMO are special cases of martingale H and BMO (withrespect to a regular filtration). When F = E ∗ , there is a fundamental dualitybetween these spaces. It is essential that the following key estimate of this dualityis valid for an arbitrary Banach space E (see [6, Theorem 12]): |h b, h i| : = (cid:12)(cid:12)(cid:12) lim N →∞ ˆ R d min (cid:8) , N k b ( x ) k E ∗ (cid:9) h b ( x ) , h ( x ) i d x (cid:12)(cid:12)(cid:12) . k b k BMO d ( R d ; E ∗ ) k h k H d ( R d ; E ) , (2.2)where the implied constant depends only on the dimension d (more generally, onthe underlying filtration, which in our case is the dyadic filtration of R d ). Note SYMMETRIC T (1) THEOREM 7 that x
7→ h b ( x ) , h ( x ) i need not be integrable under these assumptions, so that thepairing needs to be defined via such a limiting process in general. Of course, if x
7→ h b ( x ) , h ( x ) i is integrable, then dominated convergence shows that the limit issimply the integral of this function.This identifies BMO d ( R d ; E ∗ ) with a subspace of ( H d ( R d ; E )) ∗ , and it is notdifficult check that this identification is isomorphic, although we only need the one-sided inequality in (2.2). (It is also known that BMO d ( R d ; E ∗ ) exhausts the entiredual of H d ( R d ; E ) , if and only if E ∗ has the Radon–Nikodým property , cf. [2], wehave no need for such considerations in the present context.)2.B.
Projective tensor product and duality.
We will also need some basicfacts about the projective tensor product E ˆ ⊗ F of Banach spaces E and F . Areference for this material is [26, Sec. 0.b]. The algebraic tensor product E ⊗ F consists of finite sums of the form v = K X k =1 e k ⊗ f k , e k ∈ E, f k ∈ F. On this space, we define the norm k v k ∧ := inf K X k =1 k e k k E k f k k F , where the infimum runs over all expansions of v of this form. Then E ˆ ⊗ F is thecompletion of E ⊗ F with respect to this norm.Let B ( E × F ) stand for the space of bounded bilinear forms on E × F . This canbe identified with either of the two spaces of bounded linear operators L ( E, F ∗ ) or L ( F, E ∗ ) . For any φ ∈ B ( E × F ) and v = P Kk =1 e k ⊗ f k ∈ E ⊗ F , the pairing h φ, v i := K X k =1 φ ( e k , f k ) is well-defined, i.e., independent of the particular representation of v . It is thenclear that |h φ, v i| ≤ k φ k B ( E,F ) k v k ∧ , and hence, by continuity, φ induces an element of ( E ˆ ⊗ F ) ∗ . Conversely, given λ ∈ ( E ˆ ⊗ F ) ∗ the formula φ ( e, f ) := λ ( e ⊗ f ) defines a bilinear form φ ∈ B ( E × F ) ,which induces λ in the above sense. This gives rise to the isometric identification ( E ˆ ⊗ F ) ∗ ≃ B ( E, F ) ≃ L ( E, F ∗ ) . In combination with the H -BMO duality, this shows that |h b, h i| . k b k BMO( R d ; L ( E,F ∗ )) k h k H ( R d ; E ˆ ⊗ F ) for all functions b and h in the indicated spaces. Our main interest lies in the casewhen F = Y ∗ is a dual space. Via the usual identification Y ⊆ Y ∗∗ , we have L ( E, Y ) ⊆ L ( E, Y ∗∗ ) , and hence in particular |h b, h i| . k b k BMO( R d ; L ( E,Y )) k h k H ( R d ; E ˆ ⊗ Y ∗ ) for all functions b and h in the indicated spaces. Note that even if E and Y arevery nice spaces (as they will be in our main application), the spaces E ˆ ⊗ Y ∗ and L ( E, Y ) are not, and hence it is quite essential for our purposes that we only usethe part of the H - BMO duality that is valid in general Banach spaces.
TUOMAS HYTÖNEN Paraproducts
The paraproduct of two functions b and f is the formal series Π b f = X k ∈ Z D k b E k − f = X k ∈ Z D k ( b E k − f ) . Given two Banach space X and Y , the individual terms of this series are well-definedfor b ∈ L ( R d ; L ( X, Y )) and f ∈ L p ( R d ; X ) , producing D k b E k − f ∈ L ∞ loc ( R d ; Y ) .The series can then be paired agains any g ∈ L p ′ c ( R d ; Y ∗ ) (here and below, thesubscript c refers to compact support) with a finitely-nonzero martingale differenceexpansion g = P k ∈ Z D k g by h Π b f, g i = X k ∈ Z h b E k − f, D k g i = X k ∈ Z h b, E k − f ⊗ D k g i = D b, X k ∈ Z E k − f ⊗ D k g E . All pairings involving the sum over k ∈ Z are the integral pairings h F, G i = ´ h F ( x ) , G ( x ) i d x where the pointwise pairing h F ( x ) , G ( x ) i is between Y and Y ∗ inthe first sum, and between L ( X, Y ) and X ⊗ Y ∗ ⊂ X ˆ ⊗ Y ∗ in the second and thefinal ones.Similarly, for b as before, f ∈ L pc ( R d ; X ) and g ∈ L p ′ ( R d ; Y ∗ ) , we have h f, Π b ∗ g i = X k ∈ Z h D k f, b ∗ E k − g i = X k ∈ Z h b D k f, E k − g i = D b, X k ∈ Z D k f ⊗ E k − g E , where the ultimate right is again an integral pairing of the same type as before.Hence, if b ∈ L ( R d ; L ( X, Y )) and both f ∈ L pc ( R d ; X ) and g ∈ L p ′ c ( R d ; Y ∗ ) have finite martingale difference expansions, then h Λ b f, g i := h (Π b + Π ∗ b ∗ ) f, g i = D b, X k ∈ Z ( E k − f ⊗ D k g + D k f ⊗ E k − g ) E , (3.1)where the sum is finite, and we have an integral pairing between a function takingvalues in L ( X, Y ) and another one with values in X ⊗ Y ∗ ⊂ X ˆ ⊗ Y ∗ . We can nowelaborate and prove Theorem 1.5 as follows:3.2. Theorem.
Let X and Y be UMD spaces and b ∈ BMO d ( R d ; L ( X, Y )) . Thenthe symmetrised paraproduct Λ b defined by (3.1) extends to a bounded linear oper-ator from L p ( R d ; X ) to L p ( R d ; Y ) with the norm estimate k Λ b k L ( L p ( R d ; X ) ,L p ( R d ; Y )) ≤ c d ( pp ′ + β p,X β p,Y ) k b k BMO( R d ; L ( X,Y )) ≤ c ′ d · β p,X β p,Y k b k BMO( R d ; L ( X,Y )) where the first c d is the constant in the H - BMO duality (2.2) .Proof.
The second inequality follows from the fact that, for any Banach space E , β p,E ≥ β p, R = max( p, p ′ ) − ≥
12 max( p, p ′ ); see [17, Proposition 4.2.17(3) and Theorem 4.5.7] for the first and second steps inthe above computation. Thus we concentrate on the first inequality in the statementof the theorem.By standard density and duality results concerning the L p ( R d ; X ) spaces, it isenough to prove that |h Λ b f, g i| ≤ c d ( pp ′ + β p,X β p,Y ) k b k BMO( R d ; L ( X,Y )) , SYMMETRIC T (1) THEOREM 9 for all f ∈ L pc ( R d ; X ) and g ∈ L p ′ c ( R d ; Y ∗ ) with finite martingale difference expan-sions and norm one in L p ( R d ; X ) and L p ′ ( R d ; Y ∗ ) , respectively.By (3.1) and the H - BMO duality (2.2) for E = X ⊗ Y ∗ and E ∗ ⊇ L ( X, Y ) ,we have |h Λ b f, g i| . k b k BMO d ( R d ; L ( X,Y )) (cid:13)(cid:13)(cid:13) X k ∈ Z ( E k − f ⊗ D k g + D k f ⊗ E k − g ) (cid:13)(cid:13)(cid:13) H d ( R d ; X ˆ ⊗ Y ∗ ) = k b k BMO d ( R d ; L ( X,Y )) (cid:13)(cid:13)(cid:13) sup K ∈ Z k X k ≤ K ( E k − f ⊗ D k g + D k f ⊗ E k − g ) k X ˆ ⊗ Y ∗ (cid:13)(cid:13)(cid:13) L ( R d ) , and the task is reduced to estimating the H d norm on the right.Since E k f ⊗ E k g − E k − f ⊗ E k − g = ( E k − + D k ) f ⊗ ( E k − + D k ) g − E k − f ⊗ E k − g = E k − f ⊗ D k g + D k f ⊗ E k − g + D k f ⊗ D k g, we find by telescoping that X k ≤ K ( E k − f ⊗ D k g + D k f ⊗ E k − g ) = E K f ⊗ E K g − X k ≤ K D k f ⊗ D k g. Let us keep in mind that, by the assumptions on f and g , all these sums are finite,and hence these functions take their values in the algebraic tensor product X ⊗ Y ∗ .By the triangle inequality, we then have (cid:13)(cid:13)(cid:13) X k ∈ Z ( E k − f ⊗ D k g + D k f ⊗ E k − g ) (cid:13)(cid:13)(cid:13) H ( R d ; X ˆ ⊗ Y ∗ ) ≤ (cid:13)(cid:13)(cid:13) sup K ∈ Z k E K f ⊗ E K g k X ˆ ⊗ Y ∗ (cid:13)(cid:13)(cid:13) L ( R d ) + (cid:13)(cid:13)(cid:13) sup K ∈ Z k X k ≤ K D k f ⊗ D k g k X ˆ ⊗ Y ∗ (cid:13)(cid:13)(cid:13) L ( R d ) =: I + II.
It is immediate that I ≤ (cid:13)(cid:13)(cid:13) sup K ∈ Z k E K f ⊗ E K g k X ˆ ⊗ Y ∗ (cid:13)(cid:13)(cid:13) L ( R d ) ≤ (cid:13)(cid:13)(cid:13) sup K ∈ Z k E K f k X sup K ∈ Z k E K g k Y ∗ (cid:13)(cid:13)(cid:13) L ( R d ) = k M d f M d g k L ( R d ) ≤ k M d f k L p ( R d ) k M d g k L p ′ ( R d ) ≤ p ′ k f k L p ( R d ; X ) · p k g k L p ′ ( R d ; Y ∗ ) by Doob’s maximal inequality (see [17, Theorem 3.2.2]) in the last step.On the other hand, introducing independent unbiased random signs ε k on someprobability space (Ω , F , P ) with expectation E ε = ´ Ω ( · ) d P , we have E ε ( ε k ε j ) = δ k,j and hence (cid:13)(cid:13)(cid:13) X k ≤ K D k f ⊗ D k g (cid:13)(cid:13)(cid:13) X ˆ ⊗ Y ∗ = (cid:13)(cid:13)(cid:13) E ε (cid:16) X k ≤ K ε k D k f (cid:17) ⊗ (cid:16) X j ≤ K ε j D j g (cid:17)(cid:13)(cid:13)(cid:13) X ˆ ⊗ Y ∗ ≤ E ε (cid:13)(cid:13)(cid:13) X k ≤ K ε k D k f (cid:13)(cid:13)(cid:13) X (cid:13)(cid:13)(cid:13) X j ≤ K ε j D j g (cid:13)(cid:13)(cid:13) Y ∗ ≤ (cid:13)(cid:13)(cid:13) X k ≤ K ε k D k f (cid:13)(cid:13)(cid:13) L p (Ω; X ) (cid:13)(cid:13)(cid:13) X j ≤ K ε j D j g (cid:13)(cid:13)(cid:13) L p ′ (Ω; Y ∗ ) ≤ (cid:13)(cid:13)(cid:13) X k ∈ Z ε k D k f (cid:13)(cid:13)(cid:13) L p (Ω; X ) (cid:13)(cid:13)(cid:13) X j ∈ Z ε j D j g (cid:13)(cid:13)(cid:13) L p ′ (Ω; Y ∗ ) by Kahane’s contraction principle (see [17, Proposition 3.2.10]) for such randomsums in the last step. Thus II ≤ (cid:13)(cid:13)(cid:13) k X k ∈ Z ε k D k f k L p (Ω; X ) k X j ∈ Z ε j D j g k L p ′ (Ω; Y ∗ ) (cid:13)(cid:13)(cid:13) L ( R d ) ≤ (cid:13)(cid:13)(cid:13) X k ∈ Z ε k D k f (cid:13)(cid:13)(cid:13) L p ( R d × Ω; X ) (cid:13)(cid:13)(cid:13) X j ∈ Z ε j D j g (cid:13)(cid:13)(cid:13) L p ′ ( R d × Ω; Y ∗ ) ≤ β p,X k f k L p ( R d ; X ) · β p ′ ,Y ∗ k g k L p ′ ( R d × Ω; Y ∗ ) by the UMD property of X and Y ∗ in the last step. We note that β p ′ ,Y ∗ = β p,Y (see [17, Proposition 4.2.17(2)]).Putting the pieces together, we have completed the proof of Theorem 3.2. (cid:3) Remark.
As discussed in the Introduction, the previous result extends [14,Proposition 2.2], which contains the case that p = 2 and X = Y is a Hilbertspace, and a weaker statement in the case that p ∈ (1 , ∪ (2 , ∞ ) and X = Y is anoncommutative L p space.Our method of proof is also analogous to, and inspired by, that of [14, Proposition2.2]. The main new ingredients consist of using the projective tensor product asa replacement of the product in the noncommutative L p spaces, and the randomsigns as a replacement of some other noncommutative constructions.4. Set-up for the T (1) theorem We now turn to a discussion and precise definition of the notions appearingin the statement of our main Theorem 1.3. Let Q ( R d ; X ) be the space of finitelinear combinations of functions of the form Q x , where Q ⊂ R d is an axes-parallelleft-closed, right-open cube and x ∈ X .Let t be a bilinear form on Q ( R d ; X ) × Q ( R d ; X ∗ ) associated with a kernel K ∈ C ( ˙ R d ; L ( X )) , where ˙ R d = { ( x, y ) ∈ R d × R d : x = y } , in the sense that t ( f, g ) = ¨ h K ( x, y ) f ( y ) , g ( x ) i d y d x (4.1)whenever f ∈ Q ( R d ; X ) and g ∈ Q ( R d ; X ∗ ) have disjoint supports. (We implicitlyassume that the integral on the right makes sense for all such f, g ; this will followfrom the assumptions that will be imposed on K next.) If we can show that | t ( f, g ) | . k f k L p ( R d ; X ) k g k L p ′ ( R d ; X ∗ ) for all ( f, g ) ∈ Q ( R d ; X ) × Q ( R d ; X ∗ ) , then one obtains the existence of a unique T ∈ L ( L p ( R d ; X ) , ( L p ′ ( R d ; X ∗ )) ∗ ) such that h T f, g i = t ( f, g ) for all ( f, g ) ∈Q ( R d ; X ) × Q ( R d ; X ∗ ) . When X = X ∗∗ is reflexive (in particular, when X isa UMD space; see [17, Theorem 4.3.3]), it has the so-called Radon–Nikodým prop-erty, and ( L p ′ ( R d ; X ∗ )) ∗ can be identified with L p ( R d ; X ∗∗ ) = L p ( R d ; X ) (see [17,Theorems 1.3.10 and 1.3.21]).We recall that a family of operators T ⊂ L ( X ) is called R -bounded if (cid:13)(cid:13)(cid:13) K X k =1 ε k T k x k (cid:13)(cid:13)(cid:13) L p (Ω; X ) ≤ C (cid:13)(cid:13)(cid:13) K X k =1 ε k x k (cid:13)(cid:13)(cid:13) L p (Ω; Y ) (4.2)for some (equivalently, by the Khintchine–Kahane inequality [17, Theorem 3.2.23],for all) p ∈ [1 , ∞ ) , for all K ∈ Z + , all x k ∈ X and all T k ∈ T , where ε k are SYMMETRIC T (1) THEOREM 11 (as before) unbiased random signs on the probability space (Ω , F , P ) , and C < ∞ depends at most on p . The least constant C admissible in (4.2) is denoted by R p ( T ) , and R ( T ) := R ( T ) is called the R -bound of T . See [18, Chapter 8] foran extensive treatment of this notion.We say that K satisfies the R -bounded Calderón–Zygmund estimates , if the set {| x − y | d K ( x, y ) : ( x, y ) ∈ ˙ R d } ⊂ L ( X ) , (4.3)as well as the sets n | x − y | d + δ [ K ( x, y ) − K ( z, y )] | x − z | δ : x, y, z ∈ R d , | x − y | > | x − z | o ⊂ L ( X ) (4.4)and n | x − y | d + δ [ K ( y, x ) − K ( y, z )] | x − z | δ : x, y, z ∈ R d , | x − y | > | x − z | o ⊂ L ( X ) , (4.5)are R -bounded, where δ ∈ (0 , .We also define the action of t : Q ( R d ) × Q ( R d ) → B ( X, X ∗ ) ≃ L ( X, X ∗∗ ) by t ( φ, ψ )( x, x ∗ ) := B ( φ ⊗ x, ψ ⊗ x ∗ ) , and we say that t satisfies the weak R -boundedness property if n t (1 Q , Q ) | Q | : Q ⊂ R d cube o ⊂ L ( X, X ∗∗ ) (4.6)is R -bounded.Finally, we define t (1 , · ) by its action on Q ( R d ) := { ψ ∈ Q ( R d ) : ´ ψ = 0 } by t (1 , ψ ) := t ( χ, ψ ) + ¨ h [ K ( x, y ) − K ( z, y )](1 − χ ( y )) , ψ ( x ) i d x d y, where z ∈ supp ψ and χ ∈ Q ( R d ) is any function that is identically in a neigh-bourhood of supp ψ . One routinely checks the convergence of the integral for anysuch z and χ , and the independence of this definition from their particular choice.We say that t (1 , · ) = b ∈ L ( R d ; L ( X )) if t (1 , ψ ) = ˆ R d b ( x ) ψ ( x ) d x for all ψ ∈ Q ( R d ) . Analogously, we define t ( · , and the meaning of t ( · ,
1) = b ∈ L ( R d ; L ( X )) .We can now restate our main Theorem 1.3 more precisely as follows: (Note inparticular that the t (1 , · ) and t ( · , defined above provide a rigorous meaning forthe heuristic notions of “ T ” and “ ( T ∗ ∗ ” featuring in Theorem 1.3.)4.7. Theorem.
Let X be a UMD spaces and p ∈ (1 , ∞ ) . Let t be a bilinear form on Q ( R d ; X ) × Q ( R d ; X ∗ ) associated with an R -bounded Calderón–Zygmund kernel K .If moreover that t satisfies the weak R -boundedness property and the “symmetric T (1) assumption” (with the usual, non-dyadic BMO space!) t (1 , · ) = t ( · ,
1) = b ∈ BMO( R d ; L ( X )) , (4.8) then there is a unique T ∈ L ( L p ( R d ; X )) such that t ( f, g ) = h T f, g i and |h T f, g i| . β p,X (cid:16) C T + k b k BMO( R d ; L ( X )) (cid:17) k f k p k g k q (4.9) for all ( f, g ) ∈ Q ( R d ; X ) × Q ( R d ; Y ∗ ) . Here C T is the sum of the R -bounds ofthe collections in (4.3) through (4.6) , and the implicit constant in (4.9) depends atmost on d , p , and δ . Proof of the T (1) theorem Theorem 4.7 is actually a relatively quick corollary of Theorem 3.2 and a recentapproach to the operator-valued T (1) theorem from [13]. In fact, most other dyadicapproaches to this theorem would work essentially equally well, but the formulationof some intermediate steps in [13] is perhaps most convenient for quoting as a blackbox, avoiding the need of repeating the considerations already covered in previousworks, although, unfortunately, it is still necessary to make some technical remarkson the applicability of the results to the setting at hand.We quote the following result from [13]:5.1. Theorem (Operator-valued Dyadic Representation; [13], Theorems 1 and 2) . Let the assumptions of Theorem 4.7 be in force, except that (4.8) is replaced by t (1 , · ) = b ∈ BMO( R d ; L ( X )) , t ( · ,
1) = b ∈ BMO( R d ; L ( X )) . (5.2) For any ǫ ∈ (0 , and ( f, g ) ∈ Q ( R d ; X ) × Q ( R d ; X ∗ ) , we have the representation t ( f, g ) = E ω (cid:16) C T ∞ X i,j =0 /ǫ − (1 − ǫ ) α max { i,j } D S ij D ω f, g E + D (Π D ω b + (Π D ω b ∗ ) ∗ ) f, g E(cid:17) , where • E ω is the expectation over a random selection of the dyadic system D ω ; • C T is the sum of the R -bounds of the collections in (4.3) through (4.6) ; • each S ij D ω is an operator with the bound k S ij D ω k L ( L p ( R d ; X )) . (1 + max { i, j } ) β p,X ; (5.3) • Π D ω b and Π D ω b ∗ are dyadic paraproducts related to the dyadic system D ω . We note that the operators S ij D ω are so-called operator-valued dyadic shifts of complexity type ( i, j ) associated with the dyadic system D ω , and (5.3) states thebound for these operators contained in [13, Theorem 1]; for the present purposes,the precise form of the S ij D ω is irrelevant, and we only care about this bound.In [13, Theorem 2], the Dyadic Representation Theorem 5.1 is stated for adifferent class of test functions (namely, ( f, g ) ∈ ( C c ( R d ) ⊗ X, C c ( R d ) ⊗ X ∗ ) )and under the qualitative a priori assumption that t ( f, g ) = h T f, g i for some T ∈ L ( L p ( R d ; X )) . However, these are only technical auxiliary assumptions usedto legitimate the formal manipulations leading to the desired representation. Thesame representation formula can also be achieved without the a priori L p ( R d ; X ) -boundedness by using our test functions, and a simple finitary variant of the un-derlying randomisation process explained in [12, Section 2.1].We are now ready for: Proof of Theorem 4.7.
The assumptions of Theorem 4.7 correspond to those of theDyadic Representation Theorem 5.1 with b = b = b ; thus Π D ω b + (Π D ω b ∗ ) ∗ = Π D ω b + (Π D ω b ∗ ) ∗ = Λ D ω b SYMMETRIC T (1) THEOREM 13 is an operator of the form considered in Theorem 3.2. The conclusion of the DyadicRepresentation Theorem 5.1 then takes the form t ( f, g ) = E ω (cid:16) C T ∞ X i,j =0 /ǫ − (1 − ǫ ) α max { i,j } D S ij D ω f, g E + D Λ D ω b f, g E(cid:17) . Using the estimate of Theorem 5.1 for S ij D ω and Theorem 3.2 for Λ D ω b , we have | t ( f, g ) | . E ω (cid:16) C T ∞ X i,j =0 /ǫ − (1 − ǫ ) α max { i,j } (1 + max { i, j } ) β p,X + β p,X k b k BMO( R d ; L ( X )) (cid:17) k f k L p ( R d ; X ) k g k L p ′ ( R d ; X ∗ ) . β p,X (cid:16) C T + k b k BMO( R d ; L ( X )) (cid:17) k f k L p ( R d ; X ) k g k L p ′ ( R d ; X ∗ ) by summing up a convergent series in the last step.This is the asserted bound and completes the proof. (cid:3) References [1] A. Bényi, D. Maldonado, and V. Naibo. What is . . . a paraproduct?
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