aa r X i v : . [ m a t h . F A ] M a r OPERATOR-VALUED LOCAL HARDY SPACES
RUNLIAN XIA AND XIAO XIONG
Abstract.
This paper gives a systematic study of operator-valued local Hardy spaces. Thesespaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties withMei’s Hardy spaces. We prove the h -bmo duality, as well as the h p -h q duality for any conjugatepair ( p, q ) when 1 < p < ∞ . We show that h ( R d , M ) and bmo( R d , M ) are also good endpointsof L p ( L ∞ ( R d ) ⊗M ) for interpolation. We obtain the local version of Calder´on-Zygmund theory,and then deduce that the Poisson kernel in our definition of the local Hardy norms can bereplaced by any reasonable test function. Finally, we establish the atomic decomposition of thelocal Hardy space h c ( R d , M ). Contents
0. Introduction and Preliminaries 10.1. Notation 20.2. Noncommutative L p -spaces 30.3. Operator-valued Hardy spaces 31. Operator-valued local Hardy spaces 61.1. Operator-valued local Hardy spaces 61.2. Operator-valued bmo spaces 72. The dual space of h cp for 1 ≤ p < cq q = bmo q p ( R d , M ) and H p ( R d , M ) 376. The atomic decomposition 39References 410. Introduction and Preliminaries
This paper is devoted to the study of operator-valued local Hardy spaces. It follows the currentline of investigation of noncommutative harmonic analysis. This field arose from the noncommuta-tive integration theory developed by Murray and von Neumann, in order to provide a mathematicalfoundation for quantum mechanics. The objective was to construct and study a linear functional onan operator algebra which plays the role of the classical integral. In [37], Pisier and Xu developed apioneering work on noncommutative martingale theory; since then, many classical results have beensuccessfully transferred to the noncommutative setting, see for instance, [18, 19, 21, 22, 39, 34, 40].Inspired by the above mentioned developments and the Littlewood-Paley-Stein theory of quan-tum Markov semigroups (cf. [20, 25, 24]), Mei [30] studied operator-valued Hardy spaces, which
Primary: 46L52, 42B30. Secondary: 46L07, 47L65.
Key words:
Noncommutative L p -spaces, operator-valued Hardy spaces, operator-valued bmo spaces, duality,interpolation, Calder´on-Zygmund theory, characterization, atomic decomposition. are defined by the Littlewood-Paley g -function and Lusin area integral function associated to thePoisson kernel. These spaces are shown to be very useful for many aspects of noncommutativeharmonic analysis. In [51], we obtain general characterizations of Mei’s Hardy spaces, which statethat the Poisson kernel can be replaced by any reasonable test function. This is done mainly byusing the operator-valued Calder´on-Zygmund theory.In the classical setting, the theory of Hardy spaces is one of the most important topics in harmoicanalysis. The local Hardy spaces h p ( R d ) were first introduced by Goldberg [12]. These spaces areviewed as local or inhomogeneous counterparts of the classical real Hardy spaces H p ( R d ). Gold-berg’s motivation of introducing these local spaces was the study of pseudo-differential operators.It is known that pseudo-differential operators are not necessarily bounded on the classical Hardyspace H ( R d ), but bounded on h ( R d ) under some appropriate assumptions. Afterwards, manyother inhomogeneous spaces have also been studied. Our references for the classical theory are[12, 46, 9]. However, they have not been investigated so far in the operator-valued case.Motivated by [52, 51, 30], we provide a localization of Mei’s operator-valued Hardy spaces on R d in this paper. The norms of these spaces are partly given by the truncated versions of theLittlewood-Paley g -function and Lusin area integral function. Some techniques that we use todeal with our local Hardy spaces are modelled after those of [51]; however, some highly non-trivialmodifications are needed. Since with the truncation, we only know the L p -norms of the Poissonintegrals of functions on the strip R d × (0 , cp ( R d , M ) and bmo c ( R d , M ). The first major result of thispart is the h cp -bmo cq duality for 1 ≤ p <
2, where q denotes the conjugate index of p . In particular,when p = 1, we obtain the operator-valued local analogue of the classical Fefferman-Stein theorem.The pattern of the proof of this theorem is similar to that of Mei’s non-local case. We also showthat h cq ( R d , M ) = bmo cq ( R d , M ) for 2 < q < ∞ like in the martingale and non-local settings. Thusthe dual of h cp ( R d , M ) agrees with h cq ( R d , M ) when 1 < p ≤ (cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p = h cp ( R d , M )for 1 < p < ∞ . We reduce this interpolation problem to the corresponding one on the non-localHardy spaces in order to use Mei’s interpolation result in [30]. This proof is quite simple.The third major result concerns the Calder´on-Zygmund theory. The usual M -valued Calder´on-Zygmund operators which satisfy the H¨ormander condition are in general not bounded on inho-mogeneous spaces. Thus, in order to guarantee the boundedness of a Calder´on-Zygmund operatoron h cp ( R d , M ), we need to impose an extra decay at infinity to the kernel.The Calder´on-Zygmund theory mentioned above will be applied to the general characterization ofh cp ( R d , M ) with the Poisson kernel replaced by any reasonable test function. This characterizationwill play an important role in our recent study of (inhomogeneous) Triebel-Lizorkin spaces on R d ,see [49].0.1. Notation.
In the following,we collect some notation which will be frequently used in thispaper. Throughout, we will use the notation A . B , which is an inequality up to a constant: A ≤ cB for some constant c >
0. The relevant constants in all such inequalities may depend onthe dimension d , the test function Φ or p , etc, but never on the function f in consideration. Theequivalence A ≈ B will mean A . B and B . A simultaneously.The Bessel and Riesz potentials are J α = (1 − (2 π ) − ∆) α and I α = ( − (2 π ) − ∆) α , respectively.If α = 1, we will abbreviate J as J and I as I . We denote also J α ( ξ ) = (1 + | ξ | ) α on R d and perator-valued local Hardy spaces 3 I α ( ξ ) = | ξ | α on R d \ { } . Then J α and I α are the symbols of the Fourier multipliers J α and I α ,respectively.We denote by H σ ( R d ) the potential Sobolev space, consisting of all tempered distributions f such that J σ ( f ) ∈ L ( R d ). If σ > d , the elements in H σ ( R d ) will serve as important convolutionkernels in the sequel.0.2. Noncommutative L p -spaces. We also recall some preliminaries on noncommutative L p -spaces and operator-valued Hardy spaces. We start with a brief introduction of noncommutative L p -spaces. Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ and S + M be the set of all positive elements x in M with τ ( s ( x )) < ∞ , where s ( x ) denotes thesupport of x , i.e., the smallest projection e such that exe = x . Let S M be the linear span of S + M .Then every x ∈ S M has finite trace, and S M is a w*-dense ∗ -subalgebra of M .Let 1 ≤ p < ∞ . For any x ∈ S M , the operator | x | p belongs to S + M (recalling | x | = ( x ∗ x ) ). Wedefine k x k p = (cid:0) τ ( | x | p ) (cid:1) p . One can prove that k · k p is a norm on S M . The completion of ( S M , k · k p ) is denoted by L p ( M ),which is the usual noncommutative L p -space associated to ( M , τ ). In this paper, the norm of L p ( M ) will be often denoted simply by k · k p if there is no confusion. But if different L p -spacesappear in a same context, we will precise their norms in order to avoid possible ambiguity. Werefer the reader to [54] and [38] for further information on noncommutative L p -spaces.Now we introduce noncommutative Hilbert space-valued L p -spaces L p ( M ; H c ) and L p ( M ; H r ),which are studied at length in [20]. Let H be a Hilbert space and v ∈ H with k v k = 1, and p v be the orthogonal projection onto the one-dimensional subspace generated by v . Then define thefollowing row and column noncommutative L p -spaces: L p ( M ; H r ) = ( p v ⊗ M ) L p ( B ( H ) ⊗M ) and L p ( M ; H c ) = L p ( B ( H ) ⊗M )( p v ⊗ M ) , where the tensor product B ( H ) ⊗M is equipped with the tensor trace while B ( H ) is equipped withthe usual trace, and where 1 M denotes the unit of M . For f ∈ L p ( M ; H c ), k f k L p ( M ; H c ) = k ( f ∗ f ) k p . A similar formula holds for the row space by passing to adjoint: f ∈ L p ( M ; H r ) if and only if f ∗ ∈ L p ( M ; H c ), and k f k L p ( M ; H r ) = k f ∗ k L p ( M ; H c ) . It is clear that L p ( M ; H c ) and L p ( M ; H r )are 1-complemented subspaces of L p ( B ( H ) ⊗M ) for any p .0.3. Operator-valued Hardy spaces.
Throughout the remainder of the paper, unless explicitlystated otherwise, ( M , τ ) will be fixed as before and N = L ∞ ( R d ) ⊗M , equipped with the tensortrace. In this subsection, we introduce Mei’s operator-valued Hardy spaces. Contrary to thecustom, we will use letters s, t to denote variables of R d since letters x, y are reserved for operatorsin noncommutative L p -spaces. Accordingly, a generic element of the upper half-space R d +1+ will bedenoted by ( s, ε ) with ε >
0, where R d +1+ = { ( s, ε ) : s ∈ R d , ε > } .Let P be the Poisson kernel on R d :P( s ) = c d | s | + 1) d +12 with c d the usual normalizing constant and | s | the Euclidean norm of s . LetP ε ( s ) = 1 ε d P( sε ) = c d ε ( | s | + ε ) d +12 . For any function f on R d with values in L ( M ) + M , its Poisson integral, whenever it exists, willbe denoted by P ε ( f ): P ε ( f )( s ) = Z R d P ε ( s − t ) f ( t ) dt, ( s, ε ) ∈ R d +1+ . Note that the Poisson integral of f exists if f ∈ L (cid:0) M ; L c ( R d , dt | t | d +1 ) (cid:1) + L ∞ (cid:0) M ; L c ( R d , dt | t | d +1 ) (cid:1) . R. Xia and X. Xiong
This space is the right space in which all functions considered in this paper live as far as onlycolumn spaces are involved. As it will appear frequently later, to simplify notation, we will denotethe Hilbert space L ( R d , dt | t | d +1 ) by R d :(0.1) R d = L ( R d , dt | t | d +1 ) . The Lusin area square function of f is defined by(0.2) S c ( f )( s ) = (cid:16) Z Γ (cid:12)(cid:12) ∂∂ε P ε ( f )( s + t ) (cid:12)(cid:12) dt dεε d − (cid:17) , s ∈ R d , where Γ is the cone { ( t, ε ) ∈ R d +1+ : | t | < ε } . For 1 ≤ p < ∞ define the column Hardy space H cp ( R d , M ) to be H cp ( R d , M ) = (cid:8) f : k f k H cp = k S c ( f ) k p < ∞ (cid:9) . Note that [30] uses the gradient of P ε ( f ) instead of the sole radial derivative in the definition of S c above, but this does not affect H cp ( R d , M ) (up to equivalent norms). At the same time, it isproved in [30] that H cp ( R d , M ) can be equally defined by the Littlewood-Paley g -function:(0.3) G c ( f )( s ) = (cid:16) Z ∞ ε (cid:12)(cid:12) ∂∂ε P ε ( f )( s ) (cid:12)(cid:12) dε (cid:17) , s ∈ R d . Thus k f k H cp ≈ k G c ( f ) k p , f ∈ H cp ( R d , M ) . The row Hardy space H rp ( R d , M ) is the space of all f such that f ∗ ∈ H cp ( R d , M ), equipped withthe norm k f k H rp = k f ∗ k H cp . Finally, we define the mixture space H p ( R d , M ) as H p ( R d , M ) = H cp ( R d , M ) + H rp ( R d , M ) for 1 ≤ p ≤ k f k H p = inf (cid:8) k f k H cp + k f k H rp : f = f + f (cid:9) , and H p ( R d , M ) = H cp ( R d , M ) ∩ H rp ( R d , M ) for 2 < p < ∞ equipped with the intersection norm k f k H p = max (cid:0) k f k H cp , k f k H rp (cid:1) . Observe that H c ( R d , M ) = H r ( R d , M ) = L ( N ) with equivalent norms.It is proved in [30] that for 1 < p < ∞H p ( R d , M ) = L p ( N ) with equivalent norms.The operator-valued BMO spaces are also studied in [30]. Let Q be a cube in R d (with sidesparallel to the axes) and | Q | its volume. For a function f with values in M , f Q denotes its meanover Q : f Q = 1 | Q | Z Q f ( t ) dt. The column BMO norm of f is defined to be(0.4) k f k BMO c = sup Q ⊂ R d (cid:13)(cid:13)(cid:13) | Q | Z Q (cid:12)(cid:12) f ( t ) − f Q (cid:12)(cid:12) dt (cid:13)(cid:13)(cid:13) M . Then BMO c ( R d , M ) = (cid:8) f ∈ L ∞ (cid:0) M ; R cd (cid:1) : k f k BMO c < ∞ (cid:9) . Similarly, we define the row space BMO r ( R d , M ) as the space of f such that f ∗ lies in BMO c ( R d , M ),and BMO( R d , M ) = BMO c ( R d , M ) ∩ BMO r ( R d , M ) with the intersection norm.In [30], it is showed that the dual of H c ( R d , M ) can be naturally identified with BMO c ( R d , M ).This is the operator-valued analogue of the celebrated Fefferman-Stein H -BMO duality theorem.On the other hand, one of the main results of [51] asserts that the Poisson kernel in the definitionof Hardy spaces can be replaced by more general test functions. perator-valued local Hardy spaces 5 Take any Schwartz function Φ with vanishing mean. We will assume that Φ is nondegeneratein the following sense:(0.5) ∀ ξ ∈ R d \ { } ∃ ε > , s.t. b Φ( εξ ) = 0 . Set Φ ε ( s ) = ε − d Φ( sε ) for ε >
0. The radial and conic square functions of f associated to Φ aredefined by replacing the partial derivative of the Poisson kernel P in S c ( f ) and G c ( f ) by Φ :(0.6) S c Φ ( f )( s ) = (cid:16) Z Γ | Φ ε ∗ f ( s + t ) | dtdεε d +1 (cid:17) , s ∈ R d and(0.7) G c Φ ( f )( s ) = (cid:16) Z ∞ | Φ ε ∗ f ( s ) | dεε (cid:17) . The following two lemmas are taken from [51]. The first one says that the two square functionsabove define equivalent norms in H cp ( R d , M ): Lemma 0.1.
Let ≤ p < ∞ and f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) . Then f ∈ H cp ( R d , M ) if and onlyif G c Φ ( f ) ∈ L p ( N ) if and only if S c Φ ( f ) ∈ L p ( N ) . If this is the case, then k G c Φ ( f ) k p ≈ k S c Φ ( f ) k p ≈ k f k H cp with the relevant constants depending only on p, d and Φ . The above square functions G c Φ and S c Φ can be discretized as follows: G c,D Φ ( f )( s ) = (cid:16) ∞ X j = −∞ | Φ − j ∗ f ( s ) | (cid:17) S c,D Φ ( f )( s ) = (cid:16) ∞ X j = −∞ dj Z B ( s, − j ) | Φ − j ∗ f ( t ) | dt (cid:17) . (0.8)Here B ( s, r ) denotes the ball of R d with center s and radius r . To prove that these discrete squarefunctions also describe our Hardy spaces, we need to impose the following condition on the previousSchwartz function Φ, which is stronger than (0.5):(0.9) ∀ ξ ∈ R d \ { } ∃ < a ≤ b < ∞ s.t. b Φ( εξ ) = 0 , ∀ ε ∈ ( a, b ] . The following is the discrete version of Lemma 0.1:
Lemma 0.2.
Let ≤ p < ∞ and f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) . Then f ∈ H cp ( R d , M ) if and onlyif G c,D Φ ( f ) ∈ L p ( N ) if and only if S c,D Φ ( f ) ∈ L p ( N ) . Moreover, k G c,D Φ ( f ) k p ≈ k S c,D Φ ( f ) k p ≈ k f k H cp with the relevant constants depending only on p, d and Φ . Finally, let us give some easy facts on operator-valued functions. The first one is the followingCauchy-Schwarz type inequality for the operator-valued square function,(0.10) (cid:12)(cid:12) Z R d φ ( s ) f ( s ) ds (cid:12)(cid:12) ≤ Z R d | φ ( s ) | ds Z R d | f ( s ) | ds, where φ : R d → C and f : R d → L ( M ) + M are functions such that all integrations of the aboveinequality make sense. We also require the operator-valued version of the Plancherel formula. Forsufficiently nice functions f : R d → L ( M ) + M , for example, for f ∈ L ( R d ) ⊗ L ( M ), we have(0.11) Z R d | f ( s ) | ds = Z R d | b f ( ξ ) | dξ. Given two nice functions f and g , the polarized version of the above equality is(0.12) Z R d f ( s ) g ∗ ( s ) ds = Z R d b f ( ξ ) b g ( ξ ) ∗ dξ. The paper is organized as follows. In the next section, we give the definitions of operator-valuedlocal Hardy and bmo spaces. Section 2 is devoted to the proofs of duality results, including the
R. Xia and X. Xiong h -bmo duality and the h p -h q duality for 1 < p < p + q = 1. Section 3 gives the resultsof interpolation. In section 4, we develop Calder´on-Zymund theory that is suitable for our localversion of Hardy spaces. In section 5, we prove general characterizations of h cp ( R d , M ), and thenconnect the local Hardy spaces h cp ( R d , M ) with Mei’s non-local Hardy spaces H cp ( R d , M ). In thelast section of this paper, we give the atomic decomposition of h c ( R d , M ).1. Operator-valued local Hardy spaces
Operator-valued local Hardy spaces.
In this subsection, we give the definition of operator-valued local Hardy spaces as well as some basic facts of them. Let f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd )(recalling that the Hilbert space R d is defined by (0.1)). Then the Poisson integral of f is well-defined and takes values in L ( M ) + M . Now we define the local analogue of the Lusin area squarefunction of f by s c ( f )( s ) = (cid:16) Z e Γ (cid:12)(cid:12) ∂∂ε P ε ( f )( s + t ) (cid:12)(cid:12) dtdεε d − (cid:17) , s ∈ R d , where e Γ is the truncated cone { ( t, ε ) ∈ R d +1+ : | t | < ε < } . It is the intersection of the cone { ( t, ε ) ∈ R d +1+ : | t | < ε } and the strip S ⊂ R d +1+ defined by: S = { ( s, ε ) : s ∈ R d , < ε < } . For 1 ≤ p < ∞ define the column local Hardy space h cp ( R d , M ) to beh cp ( R d , M ) = { f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) : k f k h cp < ∞} , where the h cp ( R d , M )-norm of f is defined by k f k h cp = k s c ( f ) k L p ( N ) + k P ∗ f k L p ( N ) . The row local Hardy space h rp ( R d , M ) is the space of all f such that f ∗ ∈ h cp ( R d , M ), equippedwith the norm k f k h rp = k f ∗ k h cp . Moreover, define the mixture space h p ( R d , M ) as follows:h p ( R d , M ) = h cp ( R d , M ) + h rp ( R d , M ) for 1 ≤ p ≤ k f k h p = inf {k g k h cp + k h k h rp : f = g + h, g ∈ h cp ( R d , M ) , h ∈ h rp ( R d , M ) } , and h p ( R d , M ) = h cp ( R d , M ) ∩ h rp ( R d , M ) for 2 < p < ∞ equipped with the intersection norm k f k h p = max {k f k h cp , k f k h rp } . The local analogue of the Littlewood-Paley g -function of f is defined by g c ( f )( s ) = (cid:16) Z | ∂∂ε P ε ( f )( s ) | εdε (cid:17) , s ∈ R d . We will see in section 5 that k s c ( f ) k p + k P ∗ f k p ≈ k g c ( f ) k p + k P ∗ f k p for all 1 ≤ p < ∞ .In the following, we give some easy facts that will be frequently used later. Firstly, we have(1.1) k s c ( f ) k + k P ∗ f k ≈ k f k . Indeed, by (0.11), we have Z R d (cid:12)(cid:12) ∂∂ε P ε ( f )( s ) (cid:12)(cid:12) ds = Z R d (cid:12)(cid:12) \ ∂∂ε P ε ( ξ ) (cid:12)(cid:12) | b f ( ξ ) | dξ = Z R d π | ξ | | b f ( ξ ) | e − πε | ξ | dξ. Then Z R d Z (cid:12)(cid:12) ∂∂ε P ε ( f )( s ) (cid:12)(cid:12) εdεds = 14 Z R d (1 − e − π | ξ | − π | ξ | e − π | ξ | ) | b f ( ξ ) | dξ. perator-valued local Hardy spaces 7 Therefore k s c ( f ) k = τ Z R d Z e Γ (cid:12)(cid:12) ∂∂ε P ε ( f )( s + t ) (cid:12)(cid:12) dεdtε d − ds = τ Z R d Z Z B ( s,ε ) (cid:12)(cid:12) ∂∂ε P ε ( f )( t ) (cid:12)(cid:12) dεdtε d − ds = c d τ Z R d Z (cid:12)(cid:12) ∂∂ε P ε ( f )( s ) (cid:12)(cid:12) εdεds = c d τ Z R d (1 − e − π | ξ | − π | ξ | e − π | ξ | ) | b f ( ξ ) | dξ, where c d is the volume of the unit ball in R d . Meanwhile, k P ∗ f k = τ Z R d e − π | ξ | | b f ( ξ ) | dξ. Then we deduce (1.1) from the equality4 c d k s c ( f ) k + k P ∗ f k = τ Z R d (1 − π | ξ | e − π | ξ | ) | b f ( ξ ) | dξ and the fact that 0 ≤ π | ξ | e − π | ξ | ≤ e for every ξ ∈ R d . Passing to adjoint, (1.1) also tells us that k f k h r ( R d , M ) ≈ k f ∗ k = k f k . Then we have(1.2) h c ( R d , M ) = h r ( R d , M ) = L ( N )with equivalent norms.Next, if we apply (0.12) instead of (0.11) in the above proof, we get the following polarizedversion of (1.1), Z R d f ( s ) g ∗ ( s ) ds = 4 Z R d Z ∂∂ε P ε ( f )( s ) ∂∂ε P ε ( g ) ∗ ( s ) ε dεds + Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds + 4 π Z R d P ∗ f ( s )( I (P) ∗ g ( s )) ∗ ds. = 4 c d Z R d Z Z e Γ ∂∂ε P ε ( f )( s + t ) ∂∂ε P ε ( g ) ∗ ( s + t ) dtdεε d − ds + Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds + 4 π Z R d P ∗ f ( s )( I (P) ∗ g ( s )) ∗ ds (1.3)for nice f , g ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) (recalling that I is the Riesz potential of order 1).1.2. Operator-valued bmo spaces.
Now we introduce the noncommutative analogue of bmospaces defined in [12]. For any cube Q ⊂ R d , we denote its center by c Q , its side length by l ( Q ), and its volume by | Q | . Let f ∈ L ∞ ( M ; R cd ). The mean value of f over Q is denoted by f Q := | Q | R Q f ( s ) ds . We set(1.4) k f k bmo c = max n sup | Q | < (cid:13)(cid:13) ( 1 | Q | Z Q | f − f Q | dt ) (cid:13)(cid:13) M , sup | Q | =1 (cid:13)(cid:13) ( Z Q | f | dt ) (cid:13)(cid:13) M o . Then we define bmo c ( R d , M ) = { f ∈ L ∞ ( M ; R cd ) : k f k bmo c < ∞} . Respectively, define bmo r ( R d , M ) to be the space of all f ∈ L ∞ ( M ; R rd ) such that k f ∗ k bmo c < ∞ with the norm k f k bmo r = k f ∗ k bmo c . And bmo( R d , M ) is defined as the intersection of these twospaces bmo( R d , M ) = bmo c ( R d , M ) ∩ bmo r ( R d , M )equipped with the norm k f k bmo = max {k f k bmo c , k f k bmo r } . R. Xia and X. Xiong
Remark 1.1.
Let Q be a cube with volume k d ≤ | Q | < ( k + 1) d for some positive integer k . Then Q can be covered by at most ( k + 1) d cubes with volume 1, say Q j ’s. Evidently,1 | Q | Z Q | f | dt ≤ k − d Z Q | f | dt ≤ k − d ( k +1) d X j =1 Z Q j | f | dt. Hence, sup | Q |≥ (cid:13)(cid:13) ( 1 | Q | Z Q | f | dt ) (cid:13)(cid:13) M ≤ d sup | Q | =1 (cid:13)(cid:13) ( Z Q | f | dt ) (cid:13)(cid:13) M . Thus, if we replace the second supremum in (1.4) over all cubes of volume one by that over allcubes of volume not less than one, we get an equivalent norm of bmo c ( R d , M ). Proposition 1.2.
Let f ∈ bmo c ( R d , M ) . Then k f k L ∞ ( M ;R cd ) . k f k bmo c . Moreover, bmo( R d , M ) , bmo c ( R d , M ) and bmo r ( R d , M ) are Banach spaces.Proof. Let Q be the cube centered at the origin with side length 1 and Q m = Q + m for each m ∈ Z d . For f ∈ L ∞ ( M ; R cd ), k f k L ∞ ( M ;R cd ) = (cid:13)(cid:13)(cid:13) Z R d | f ( t ) | | t | d +1 dt (cid:13)(cid:13)(cid:13) M ≤ X m ∈ Z d (cid:13)(cid:13)(cid:13) Z Q m | f ( t ) | | t | d +1 dt (cid:13)(cid:13)(cid:13) M . X m ∈ Z d (cid:13)(cid:13)(cid:13)
11 + | m | d +1 Z Q m | f ( t ) | dt (cid:13)(cid:13)(cid:13) M . k f k c X m ∈ Z d
11 + | m | d +1 . k f k c . It is then easy to check that bmo c ( R d , M ) is a Banach space. (cid:3) Proposition 1.3.
We have the inclusion bmo c ( R d , M ) ⊂ BMO c ( R d , M ) . More precisely, thereexists a constant C depending only on the dimension d , such that for any f ∈ bmo c ( R d , M ) , (1.5) k f k BMO c ≤ C k f k bmo c . Proof.
By virtue of Remark 1.1, it suffices to compare the term (cid:13)(cid:13)(cid:13) ( | Q | R Q | f | dt ) (cid:13)(cid:13)(cid:13) M and the term (cid:13)(cid:13)(cid:13) ( | Q | R Q | f − f Q | dt ) (cid:13)(cid:13)(cid:13) M for | Q | ≥
1. By the triangle inequality and (0.10), we have (cid:13)(cid:13)(cid:13) ( 1 | Q | Z Q | f − f Q | dt ) (cid:13)(cid:13)(cid:13) M ≤ (cid:13)(cid:13)(cid:13) ( 1 | Q | Z Q | f | dt ) (cid:13)(cid:13)(cid:13) M + k f Q k M ≤ (cid:13)(cid:13)(cid:13) ( 1 | Q | Z Q | f | dt ) (cid:13)(cid:13)(cid:13) M , which leads immediately to (1.5). (cid:3) Classically, BMO functions are related to Carleson measures (see [11]). A similar relation stillholds in the present noncommutative local setting. We say that an M -valued measure dλ on thestrip S = R d × (0 ,
1) is a Carleson measure if N ( λ ) = sup | Q | < { | Q | (cid:13)(cid:13) Z T ( Q ) dλ (cid:13)(cid:13) M : Q ⊂ R d cube } < ∞ , where T ( Q ) = Q × (0 , l ( Q )]. Lemma 1.4.
Let g ∈ bmo c ( R d , M ) . Then dλ g = | ∂∂ε P ε ( g )( s ) | ε dsdε is an M -valued Carlesonmeasure on the strip S and max { N ( λ g ) , k P ∗ g k L ∞ ( N ) } . k g k bmo c . perator-valued local Hardy spaces 9 Proof.
Given a cube Q with | Q | <
1, denote by 2 Q the cube with the same center and twice the sidelength of Q . We decompose g = g + g + g , where g = ( g − g Q ) Q and g = ( g − g Q ) R d \ Q .Since R ∂∂ε P ε ( s ) ds = 0 for any ε >
0, we have ∂∂ε P ε ( g ) = ∂∂ε P ε ( g ) + ∂∂ε P ε ( g ). By (0.10), N ( λ g ) ≤ N ( λ g ) + N ( λ g )) . We first deal with N ( λ g ). By (0.11) and (1.5), we have Z T ( Q ) (cid:12)(cid:12) ∂∂ε P ε ( g )( s ) (cid:12)(cid:12) εdsdε ≤ Z R d Z ∞ (cid:12)(cid:12) ∂∂ε P ε ( g )( s ) (cid:12)(cid:12) εdsdε = Z R d Z ∞ (cid:12)(cid:12)(cid:12) \ ∂∂ε P ε ( ξ ) (cid:12)(cid:12)(cid:12) | b g ( ξ ) | εdεds . Z R d | g ( s ) | ds = Z Q | g − g Q | ds . | Q | k g k c . Thus, N ( λ g ) . k g k c . Since (cid:12)(cid:12) ∂∂ε P ε ( s ) (cid:12)(cid:12) . ε + | s | ) d +1 , applying (0.10), we obtain (cid:12)(cid:12) ∂∂ε P ε ( g )( s ) (cid:12)(cid:12) . ε Z R d \ Q | g ( t ) − g Q | ( ε + | s − t | ) d +1 dt. The integral on the right hand side of the above inequality can be treated by a standard argumentas follows: for any ( s, ε ) ∈ T ( Q ), Z R d \ Q | g ( t ) − g Q | ( ε + | s − t | ) d +1 dt . Z R d \ Q | g ( t ) − g Q | | t − c Q | d +1 dt . X k ≥ Z k +1 Q \ k Q | g ( t ) − g Q | | t − c Q | d +1 dt . l ( Q ) X k ≥ − k | k +1 Q | Z k +1 Q | g ( t ) − g Q | dt . l ( Q ) k g k c , where c Q is the center of Q . Then, it follows that N ( λ g ) . k g k c .Now we deal with the term k P ∗ g ( s ) k M . Let Q m = Q + m be the translate of the cube withvolume one centered at the origin, so R d = ∪ m ∈ Z d Q m . By (0.10), for any s ∈ R d , we have k P ∗ g ( s ) k M = (cid:13)(cid:13) X m Z Q m P( t ) g ( s − t ) dt (cid:13)(cid:13) M ≤ X m (cid:0) Z Q m | P( t ) | dt ) · sup m ∈ Z d k ( Z Q m | g ( s − t ) | dt (cid:1) k M . sup | Q | =1 (cid:13)(cid:13)(cid:13) ( 1 | Q | Z Q | g ( t ) | dt ) (cid:13)(cid:13)(cid:13) M . k g k bmo c . Thus, k P ∗ g k L ∞ ( N ) = sup s ∈ R d k P ∗ g ( s ) k M . k g k bmo c , which completes the proof. (cid:3) Reexamining the above proof, we find that the facts used to prove N ( λ g ) . k g k bmo c are • R R d ε ∂∂ε P ε ( s ) ds = 0 for ∀ ε < • sup ξ ∈ R d R ∞ (cid:12)(cid:12)(cid:12) \ ε ∂∂ε P ε ( ξ ) (cid:12)(cid:12)(cid:12) dεε < ∞ ; • (cid:12)(cid:12) ε ∂∂ε P ε ( s ) (cid:12)(cid:12) . ε ( ε + | s | ) d +1 .We can easily check that if we replace ε ∂∂ε P ε above by Ψ ε = ε d Ψ( · ε ), where Ψ is a Schwartzfunction such that b Ψ(0) = 0, the corresponding three conditions still hold. On the other hand, theonly fact used for proving the inequality k P ∗ g k L ∞ ( N ) . k g k bmo c is that X m ( Z Q m | P( t ) | dt ) < ∞ . Recall that H σ ( R d ) denotes the potential Sobolev space, consisting of distributions f such that J σ ( f ) ∈ L ( R d ). It is equipped with the norm k f k H σ ( R d ) = k J σ f k L ( R d ) . If ψ is a function on R d such that b ψ ∈ H σ ( R d ) for some σ > d , we have X m (cid:0) Z Q m | ψ ( s ) | dt (cid:1) . (cid:0) X m | m | ) σ (cid:1) (cid:0) Z R d (1 + | s | ) σ | ψ ( s ) | ds (cid:1) . k b ψ k H σ . Based on the above observation, we have the following generalization of Lemma 1.4:
Lemma 1.5.
Let ψ be the (inverse) Fourier transform of a function in H σ ( R d ) , and Ψ be aSchwartz function such that b Ψ(0) = 0 . If g ∈ bmo c ( R d , M ) , then dµ g = | Ψ ε ∗ g ( s ) | dεdsε is an M -valued Carleson measure on the strip S and (1.6) max { N ( µ g ) , k ψ ∗ g k L ∞ ( N ) } . k g k bmo c . In particular, (1.7) max { N ( µ g ) , k J (P) ∗ g k L ∞ ( N ) } . k g k bmo c . Proof. (1.6) follows from the above discussion; (1.7) is ensured by (1.6) and the fact that (1 + | ξ | ) e − π | ξ | ∈ H σ ( R d ), which can be checked by a direct computation. (cid:3) Remark 1.6.
We will see in the next section that the converse inequality of (1.7) also holds.2.
The dual space of h cp for ≤ p < cp ( R d , M ) for 1 ≤ p < cq ( R d , M ) (with q the conjugate index of p ). The argument used here is modelled onthe one used in [12] when studying the duality between H cp ( R d , M ) and BMO cq ( R d , M ). However,due to the truncation of the square functions, some highly non-trivial modifications are needed.2.1. Definition of bmo cq . Let 2 < q ≤ ∞ . We define bmo cq ( R d , M ) to be the space of all f ∈ L q ( M ; R cd ) such that k f k bmo cq = (cid:16)(cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z Q | f ( t ) − f Q | dt (cid:13)(cid:13)(cid:13) q q + (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | =1 | Q | Z Q | f ( t ) | dt (cid:13)(cid:13)(cid:13) q q (cid:17) q < ∞ . If q = ∞ , bmo cq ( R d , M ) coincides with the space bmo c ( R d , M ) introduced in the previous section.Note that the norm k sup + i a i k q is just an intuitive notation since the pointwise supremumdoes not make any sense in the noncommutative setting. This is the norm of the Banach space L q ( N ; ℓ ∞ ); we refer to [36, 18, 22] for more information.If 1 ≤ p < ∞ and ( a i ) i ∈ Z is a sequence of positive elements in L p ( N ), it has been proved byJunge (see [18], Remark 3.7) that(2.1) k sup i + a i k p = sup (cid:8) X i ∈ Z τ ( a i b i ) : b i ∈ L q ( N ) , b i ≥ , k X i ∈ Z b i k q ≤ (cid:9) . It is also known that a positive sequence ( x i ) i belongs to L p ( N ; ℓ ∞ ) if and only if there is an a ∈ L p ( N ) such that x i ≤ a for all i , and moreover, k ( x i ) k L p ( N ; ℓ ∞ ) = inf {k a k p : a ∈ L p ( N ) , x i ≤ a, ∀ i } . Then we get the following fact (which can be taken as an equivalent definition): f ∈ bmo cq ( R d , M )if and only if(2.2) ∃ a ∈ L q ( N ) s.t. 1 | Q | Z Q | f ( t ) − f Q | dt ≤ a ( s ) , ∀ s ∈ Q and ∀ Q ⊂ R d with | Q | < ∃ b ∈ L q ( N ) s.t. 1 | Q | Z Q | f ( t ) | dt ≤ b ( s ) , ∀ s ∈ Q and ∀ Q ⊂ R d with | Q | = 1 . If this is the case, then k f k bmo cq = inf (cid:8)(cid:0) k a k q q + k b k q q (cid:1) q : a, b as in (2.2) and (2.3) respectively (cid:9) . perator-valued local Hardy spaces 11 In fact, the cubes considered in the definition of bmo cq ( R d , M ) can be reduced to cubes withdyadic lengths. Let Q ks denote the cube centered at s and with side length 2 − k , k ∈ Z . Set f k ( s ) = 1 | Q ks | Z Q ks (cid:12)(cid:12) f ( t ) − f Q ks (cid:12)(cid:12) dt and f ( s ) = 1 | Q s | Z Q s (cid:12)(cid:12) f ( t ) (cid:12)(cid:12) dt. Lemma 2.1. If q > , then (cid:16) k sup k ≥ f k k q q + k f k q q (cid:17) q gives an equivalent norm in bmo cq ( R d , M ) .Proof. It is obvious from the definition that k sup k ≥ f k k q ≤ k f k bmo cq and k f k q ≤ k f k bmo cq . We notice that for any cube Q with | Q | < s ∈ Q , there exists k ≥ − Q ⊂ Q ks and | Q ks | ≤ d | Q | . Thus14 d (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z Q | f ( t ) − f Q | dt (cid:13)(cid:13)(cid:13) q . k sup k ≥− f k k q . d k sup k ≥ f k k q . Similarly, 14 d (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | =1 | Q | Z Q | f ( t ) | dt (cid:13)(cid:13)(cid:13) q ≤ d k f k q . Thus the lemma is proved. (cid:3)
From the proofs of Proposition 1.2 and Lemma 1.4, we can easily see that their q -analogues stillhold in the present setting. We leave the proofs to the reader. Proposition 2.2.
Let q > and f ∈ bmo cq ( R d , M ) . Then k f k L q ( M ;R cd ) . k f k bmo cq . Lemma 2.3.
Let f ∈ bmo cq ( R d , M ) and assume that the operators a and b satisfy (2.2) and (2.3) respectively. Then dλ f is a q -Carleson measure in the following sense: | Q | Z T ( Q ) | ∂∂ε P ε ( f )( t ) | εdtdε . a ( s ) , ∀ s ∈ Q and ∀ Q ⊂ R d with | Q | < . Moreover, | ψ ∗ f ( s ) | . b ( s ) for any s ∈ R d , if ψ is the (inverse) Fourier transform of a functionin H σ ( R d ) . A bounded map.
In the sequel, we equip the truncated cone e Γ = { ( s, ε ) ∈ R d +1+ : | s | <ε < } with the measure dtdεε d +1 . For any 1 ≤ p < ∞ , we will embed h cp ( R d , M ) into a larger space L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ). Here L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) is the ℓ p -direct sum of the Banach spaces L p (cid:0) N ; L c ( e Γ) (cid:1) and L p ( N ), equipped with the norm k ( f, g ) k = (cid:16) k f k pL p (cid:0) N ; L c ( e Γ) (cid:1) + k g k pL p ( N ) (cid:17) p for f ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) and g ∈ L p ( N ), with the usual modification for p = ∞ . Definition 2.4.
We define a map E from h cp ( R d , M ) to L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) by E ( f )( s, t, ε ) = (cid:0) ε ∂∂ε P ε ( f )( s + t ) , P ∗ f ( s ) (cid:1) , and a map F for sufficiently nice h = ( h ′ , h ′′ ) ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) by F ( h )( u ) = Z R d h c d Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( s + t − u ) dtdεε d + h ′′ ( s )(P + 4 πI (P))( s − u ) i ds . By definition, the map E embeds h cp ( R d , M ) isometrically into L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ). Thefollowing results, Theorems 2.8 and 2.18 show that by identifying h cp ( R d , M ) as a subspace of L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) via E , h cp ( R d , M ) is complemented in L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) for every1 < p < ∞ by virtue of the map F . Proposition 2.5.
Let ≤ p < ∞ . Then for any nice f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) , we have F ( E ( f )) = f. Proof.
Applying (1.3), we get, for any nice function g , Z R d F ( E ( f ))( u ) g ( u ) du = Z R d h c d Z Z e Γ ∂∂ε P ε ( f )( s + t ) ∂∂ε P ε ( s + t − u ) dtdεε d − g ( u ) du + P ∗ f ( s ) Z (P( s − u ) + 4 πI (P)( s − u )) g ( u ) du i ds = Z R d h c d Z Z e Γ ∂∂ε P ε ( f )( s + t ) ∂∂ε P ε ( g )( s + t ) dtdεε d − + P ∗ f ( s )(P ∗ g + 4 πI (P) ∗ g )( s ) i ds = Z R d f ( u ) g ( u ) du , which completes the proof. (cid:3) The following dyadic covering lemma is known. Tao Mei [30] proved this lemma for the d -torusand also for the real line. For the case R d with d >
1, we refer the interested readers to [6, 16] formore details. In the following, we give a sketch of the way how we choose the dyadic covering.
Lemma 2.6.
There exist a constant
C > , depending only on d , and d + 1 dyadic increasingfiltrations D i = {D ij } j ∈ Z of σ -algebras on R d for ≤ i ≤ d , such that for any cube Q ⊂ R d , thereis a cube D im,j ∈ D ij satisfying Q ⊂ D im,j and | D im,j | ≤ C | Q | .Proof. Let { α i } di =0 be a sequence in the interval (0 ,
1) such thatmin i = i ′ | α i − α i ′ | > . Then we define(2.4) α ij = α i , j ≥ ,α i + (2 − j − , j < − j even ,α i − (2 − j + 1) , j < − j odd . The σ -algebra D ij is generated by the cubes D im,j = ( α ij + m − j , α ij + ( m + 1)2 − j ] × · · · × ( α ij + m d − j , α ij + ( m d + 1)2 − j ] , for all m = ( m , · · · , m d ) ∈ Z d .For any cube Q ⊂ R d , there exist a constant C , depending only on { α i } di =0 and d , and a dyadiccube D im,j such that Q ⊂ D im,j and | D im,j | ≤ C | Q | . (cid:3) To show the boundedness of the map F , we need the following assertion by Mei, see [30, Propo-sition 3.2]; we include a proof for this lemma, since the one in [30] is the one dimensional case. Let1 ≤ p < ∞ , and f ∈ L p ( N ) be a positive function. Let Q be a cube centered at the origin, anddenote Q t = t + Q . Then we define f Q ( t ) = 1 | Q | Z Q t f ( s ) ds. Lemma 2.7.
Let ≤ p < ∞ and let ( f k ) k ∈ Z be a positive sequence in L p ( N ) and ( Q k ) k ∈ Z be asequence of cubes centered at the origin. Then k X k ∈ Z ( f k ) Q k k p . k X k ∈ Z f k k p . perator-valued local Hardy spaces 13 Proof.
Similarly to the proof of [30, Proposition 3.2], we are going to apply [18, Theorem 0.1] fornoncommutative martingales. By Lemma 2.6, we can cover every Q k by some D im ′ ,j k , and thus bysome D im,j k − , which has twice the side length of D im ′ ,j k . Moreover, | D im,j k − | ≤ C | Q k | . Obviously, t + Q k is still covered by t + D im,j k − , but the later is not necessary a dyadic cube in D ij k − . Letus adjust the translation vector t = ( t , ..., t d ) as follows. Write Q k = ( − a, a ] × ... × ( − a, a ] and D im,j k − = ( b , b ] × ... × ( b , b ], then either b − a ≥ − j k or − a − b ≥ − j k . Without loss ofgenerality, we can assume b − a ≥ − j k . Now set e t = ( e t , ..., e t d ) with e t j the largest real number inthe set 2 − j k Z less than t j . Then we can check that t + Q k is covered by e t + D im,j k − and that thelater is a dyadic cube. Thus, ( f k ) Q k ≤ C X ≤ i ≤ d E ( f k |D ij k ) , where E ( ·|D ij ) denotes the conditional expectation with respect to D ij . Then the lemma followsfrom [18, Theorem 0.1]. (cid:3) Theorem 2.8.
For < p ≤ ∞ , the map F extends to a bounded map from L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) to bmo cp ( R d , M ) .Proof. We have to show that for any h = ( h ′ , h ′′ ) ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ), k F ( h ) k bmo cp . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) . Fix h = ( h ′ , h ′′ ) ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) and set ϕ = F ( h ). We will apply Lemma 2.1 to estimatethe bmo cp -norm of F ( h ). For v ∈ R d and k ∈ N , denote by Q kv the cube centered at v and with sidelength 2 − k , then we have Q kv = v + Q k . We set h ′ ( s, t, ε ) = h ′ ( s, t, ε ) Q k − v ( s ) , h ′ ( s, t, ε ) = h ′ ( s, t, ε ) ( Q k − v ) c ( s )and ϕ k ( v ) = 1 | Q kv | Z Q kv (cid:12)(cid:12) ϕ ( u ) − ϕ Q kv (cid:12)(cid:12) du. Let B Q k ( v ) = Z R d Z Z e Γ ( ∂∂ε P ε ) Q k ( s, t, v ) h ′ ( s, t, ε ) dtdεε d ds with ( ∂∂ε P ε ) Q k ( s, t, v ) = | Q kv | R Q kv ∂∂ε P ε ( s + t − u ) du . Then, we have ϕ k ( v ) . | Q kv | Z Q kv | ϕ ( u ) − B Q k ( v ) | du . | Q kv | Z Q kv (cid:12)(cid:12)(cid:12) Z ( Q k − v ) c Z Z e Γ h ′ ( s, t, ε ) (cid:2) ∂∂ε P ε ( s + t − u ) − ( ∂∂ε P ε ) Q k ( s, t, v ) (cid:3) dtdεε d ds (cid:12)(cid:12)(cid:12) du + 1 | Q kv | Z Q kv (cid:12)(cid:12)(cid:12) Z Q k − v Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( s + t − u ) dtdεε d ds (cid:12)(cid:12)(cid:12) du + 1 | Q kv | Z Q kv (cid:12)(cid:12)(cid:12) Z R d h ′′ ( s )[P( s − u ) + 4 πI (P)( s − u )] ds (cid:12)(cid:12)(cid:12) du. When s ∈ ( Q k − v ) c , u ∈ Q kv and ( t, ε ) ∈ e Γ, we have | s + t − u | + ε ≈ | s − v | + ε with uniformconstants. Then, Z Z e Γ (cid:12)(cid:12)(cid:12) ∂∂ε P ε ( s + t − u ) − ( ∂∂ε P ε ) Q k ( s, t, v ) (cid:12)(cid:12)(cid:12) dtdεε d − . Z Z e Γ (cid:16) − k ( | s + t − u | + ε ) d +2 (cid:17) dtdεε d − . Z Z B (0 ,ε ) − k ( | s − v | + ε ) d +2 dt dεε d − = c d Z − k ε ( | s − v | + ε ) d +2 dε . − k | s − v | d +2 . Let ( a k ) k ∈ N be a positive sequence such that k P k ≥ a k k ( p ) ′ ≤
1, where r ′ denotes the conjugateindex of r . LetA = X k ≥ τ Z R d Z ( Q k − v ) c − k | s − v | d +1 ds · Z ( Q k − v ) c | s − v | d +1 Z Z e Γ | h ′ ( s, t, ε ) | dtdεε d +1 ds · a k ( v ) dv B = X k ≥ τ Z R d | Q kv | Z Q kv (cid:12)(cid:12) Z Q k − v Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( s + t − u ) dtdεε d +1 ds (cid:12)(cid:12) du · a k ( v ) dv C = X k ≥ τ Z R d | Q kv | Z Q kv (cid:12)(cid:12) Z R d h ′′ ( s )[P( s − u ) + 4 πI (P)( s − u )] ds (cid:12)(cid:12) du · a k ( v ) dv. Then, X k ≥ τ Z ϕ k ( v ) a k ( v ) dv . A + B + C . First, we estimate the term A. Applying the Fubini theorem and the H¨older inequality, we arriveat A . X k ≥ τ Z R d − k Z ( Q k − s ) c | v − s | − d − Z Z e Γ | h ′ ( s, t, ε ) | dtdεε d +1 ds a k ( v ) dv ≤ (cid:13)(cid:13)(cid:13) Z Z e Γ | h ′ ( · , t, ε ) | dtdεε d +1 (cid:13)(cid:13)(cid:13) p · (cid:13)(cid:13)(cid:13) X k ≥ − k Z ( Q k − s ) c | v − s | − d − a k ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ . k h ′ k L p ( N ; L c ( e Γ)) · (cid:13)(cid:13)(cid:13) X k ≥ − k X j ≤ k Z Q j − s \ Q j − s ( j − d +1) a k ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ . Here and in the context below, k · k ( p ) ′ is the norm of L ( p ) ′ ( N ) with respect to the variable s ∈ R d .Now we apply Lemma 2.7 to estimate the second factor of the last term: (cid:13)(cid:13)(cid:13) X k ≥ X j ≤ k ( j − d Z Q j − s \ Q j − s j − k − a k ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ . (cid:13)(cid:13)(cid:13) X j ∈ Z X k ≥ jk ≥ j − k − a k (cid:13)(cid:13)(cid:13) ( p ) ′ . (cid:13)(cid:13) X k ≥ a k (cid:13)(cid:13) ( p ) ′ ≤ . Then we move to the estimate of B:B ≤ X k ≥ Z R d kd τ Z R d (cid:12)(cid:12)(cid:12) Z Q k − v Z Z e Γ h ′ ( s, t, ε ) a k ( v ) ∂∂ε P ε ( s + t − u ) dtdεε d ds (cid:12)(cid:12)(cid:12) dudv ≤ X k ≥ Z R d kd sup k f k =1 (cid:12)(cid:12)(cid:12) τ Z Q k − v Z Z e Γ h ′ ( s, t, ε ) a k ( v ) ∂∂ε P ε ( f ) ∗ ( s + t ) dtdεε d ds (cid:12)(cid:12)(cid:12) dv. Since h c ( R d , M ) = L ( N ) with equivalent norms, by the Cauchy-Schwarz inequality and Lemma2.7, we get B ≤ X k ≥ Z R d kd τ Z Q k − v Z Z e Γ | h ′ ( s, t, ε ) | dtdεε d +1 ds a k ( v ) dv · k f k h c . X k ≥ τ Z R d Z Z e Γ | h ′ ( s, t, ε ) | dtdεε d +1 kd Z Q k − s a k ( v ) dvds ≤ k h ′ k L p (cid:0) N ; L c ( e Γ) (cid:1)(cid:13)(cid:13)(cid:13) X k ≥ kd Z Q k − s a k ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ ≤ d k h ′ k L p (cid:0) N ; L c ( e Γ) (cid:1)(cid:13)(cid:13) X k ≥ a k (cid:13)(cid:13) ( p ) ′ ≤ d k h ′ k L p (cid:0) N ; L c ( e Γ) (cid:1) . perator-valued local Hardy spaces 15 The techniques used to estimate the term C are similar to that of B:C = X k ≥ τ Z R d kd Z Q k − v (cid:12)(cid:12) Z R d h ′′ ( s )[P( s − u ) + 4 πI (P)( s − u )] ds (cid:12)(cid:12) a k ( v ) dvdu ≤ (cid:13)(cid:13)(cid:13) X k ≥ kd Z Q k − s a k ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Z R d h ′′ ( s )[P( s − · ) + 4 πI (P)( s − · )] ds (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) p . (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Z R d h ′′ ( s )[P( s − · ) + 4 πI (P)( s − · )] ds (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) p , Take f ∈ L p ′ ( N ) with norm one such that (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Z R d h ′′ ( s )[P( s − u ) + 4 πI (P)( s − u )] ds (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) p = (cid:12)(cid:12)(cid:12) τ Z R d h ′′ ( s )[P ∗ f ( s ) + 4 πI (P) ∗ f ( s )] ds (cid:12)(cid:12)(cid:12) . Then (cid:12)(cid:12)(cid:12) τ Z R d h ′′ ( s )[P ∗ f ( s ) + 4 πI (P) ∗ f ( s )] ds (cid:12)(cid:12)(cid:12) ≤ k h ′′ k p k P ∗ f + 4 πI (P) ∗ f k p ′ . k h ′′ k p k f k p ′ = k h ′′ k p . Combining the estimates of A , B and C with (2.1), we obtain k sup k ≥ ϕ k k p . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) . It remains to establish the L p -norm of ϕ ( s ) = | Q s | R Q s (cid:12)(cid:12) ϕ ( t ) (cid:12)(cid:12) dt , which is relatively easy. Forany positive operator a such that k a k L ( p ′ ( N ) ≤
1, we have τ Z ϕ ( v ) a ( v ) dv . τ Z R d Z Q v (cid:12)(cid:12) Z R d Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( s + t − u ) dtdεε d +1 ds (cid:12)(cid:12) du · a ( v ) dv + τ Z R d Z Q v (cid:12)(cid:12) Z R d h ′′ ( s )[P( s − u ) + 4 πI (P)( s − u )] ds (cid:12)(cid:12) du · a ( v ) dv def = B ′ + C ′ . The terms B ′ and C ′ are treated in the same way as B and C respectively. The results areB ′ ≤ τ Z R d Z Z e Γ | h ′ ( s, t, ε ) | dtdεε d +1 Z Q s a ( v ) dvds ≤ k h ′ k L p (cid:0) N ; L c ( e Γ) (cid:1) k a k ( p ) ′ , C ′ ≤ (cid:13)(cid:13)(cid:13) Z Q s a ( v ) dv (cid:13)(cid:13)(cid:13) ( p ) ′ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Z R d h ′′ ( s )[P( s − · ) + 4 πI (P)( s − · )] ds (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) p . k h ′′ k p . So we obtain k ϕ k p . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) . Thus, Lemma 2.1 ensures that k F ( h ) k bmo cp . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) , which proves the theorem. (cid:3) Corollary 2.9.
Let ≤ p < . For any f ∈ L p (cid:0) M ; L c ( R d , (1 + | t | d +1 ) dt ) (cid:1) , we have k f k h cp . k f k L p (cid:0) M ; L c ( R d , (1+ | t | d +1 ) dt ) (cid:1) . Proof.
To simplify notation, we denote L (cid:0) R d , (1 + | t | d +1 ) dt (cid:1) by W d . Let q be the conjugate indexof p . By duality, we can choose h = ( h ′ , h ′′ ) ∈ L q ( N ; L c ) ⊕ q L q ( N ) with norm one such that k s c ( f ) k p + k P ∗ f k p = (cid:12)(cid:12)(cid:12) τ Z R d Z Z e Γ ∂∂ε P ε ( f )( s + t ) h ′∗ ( s, t, ε ) dtdεε d ds + τ Z R d P ∗ f ( s ) h ′′∗ ( s ) ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) τ Z f ( u ) e F ( h ) ∗ ( u ) du (cid:12)(cid:12) , where(2.5) e F ( h )( u ) = Z R d h Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( s + t − u ) dtdεε d + h ′′ ( s )P( s − u ) i ds . Following the proof of Theorem 2.8, we can easily check that e F is also bounded from L q (cid:0) N ; L c ( e Γ) (cid:1) ⊕ q L q ( N ) to bmo cq ( R d , M ). They applying Proposition 2.2 and Theorem 2.8, we have (cid:12)(cid:12) τ Z f ( s ) e F ( h ) ∗ ( s ) ds (cid:12)(cid:12) . sup k ϕ k bmo cq ( R d, M ) ≤ (cid:12)(cid:12) τ Z f ( s ) ϕ ∗ ( s ) ds (cid:12)(cid:12) . sup k ϕ k Lq ( M ;R cd ) ≤ (cid:12)(cid:12)(cid:12) τ Z (1 + | s | d +1 ) f ( s ) ϕ ∗ ( s ) ds | s | d +1 (cid:12)(cid:12)(cid:12) = k (1 + | s | d +1 ) f k L p ( M ;R cd ) = k f k L p ( M ;W cd ) . Thus we obtain the desired assertion. (cid:3)
Duality.
Now we are going to present the h cp -bmo cq duality for 1 ≤ p <
2. We begin thissubsection by a lemma which will be very useful in the sequel.
Lemma 2.10.
Let ≤ p ≤ and q be its conjugate index. For f ∈ h cp ( R d , M ) ∩ L ( N ) and g ∈ bmo cq ( R d , M ) , (cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12) . k f k h cp k g k bmo cq . Proof.
It suffices to prove the lemma for compactly supported (relative to the variable of R d ) f ∈ h cp ( R d , M ). We assume that f is sufficiently nice that all calculations below are legitimate.We need two auxiliary square functions. For s ∈ R d and ε ∈ [0 , s c ( f )( s, ε ) = (cid:16) Z ε Z B ( s,r − ε ) (cid:12)(cid:12) ∂∂r P r ( f )( t ) (cid:12)(cid:12) dtdrr d − (cid:17) , (2.7) s c ( f )( s, ε ) = (cid:16) Z ε Z B ( s, r ) (cid:12)(cid:12) ∂∂r P r ( f )( t ) (cid:12)(cid:12) dtdrr d − (cid:17) . Both s c ( f )( s, ε ) and s c ( f )( s, ε ) are decreasing in ε and s c ( f )( s,
0) = s c ( f )( s ). In addition, it isclear that s c ( f )( s, ε ) ≤ s c ( f )( s, ε ). Let ( e i ) i ∈ I be an increasing family of τ -finite projections of M such that e i converges to 1 M in the strong operator topology. Then we can approximate s c ( f )( s, ε )by s c ( e i f e i )( s, ε ). Thus we can assume that τ is finite; under this finiteness assumption, for anysmall δ > s c ( f )( s, ε ) + δ M instead of s c ( f )( s, ε ), we can assume that s c ( f )( s, ε ) is invertible in M for every ( s, ε ) ∈ S . By (1.3) and theFubini theorem, we have (cid:12)(cid:12) τ Z f ( s ) g ∗ ( s ) ds (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) τ Z R d Z ∂∂ε P ε ( f )( s ) ∂∂ε P ε ( g ) ∗ ( s ) ε dεds (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) τ Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds + τ Z R d P ∗ f ( s )( I (P) ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) d c d τ Z R d Z Z B ( s, ε ) ∂∂ε P ε ( f )( t ) ∂∂ε P ε ( g ) ∗ ( t ) dεdtε d − ds (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) τ Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds + τ Z R d P ∗ f ( s )( I (P) ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) . perator-valued local Hardy spaces 17 Then, (cid:12)(cid:12) τ Z f ( s ) g ∗ ( s ) ds (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) d c d τ Z R d Z Z B ( s, ε ) ∂∂ε P ε ( f )( t ) s c ( f )( s, ε ) p − s c ( f )( s, ε ) − p ∂∂ε P ε ( g ) ∗ ( t ) dεdtε d − ds (cid:12)(cid:12)(cid:12) + (cid:16) (cid:12)(cid:12)(cid:12) τ Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) τ Z R d P ∗ f ( s )( I (P) ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) (cid:17) def = I + II . The term II is easy to deal with. By the H¨older inequality and (1.7), we getII ≤ k P ∗ f k p k P ∗ g k q + k P ∗ f k p k I (P) ∗ g k q . Then by [43, Proposition V.3 and Lemma V.3.2] we have k P ∗ g k q . k J (P) ∗ g k q , and k I (P) ∗ g k q . k J (P) ∗ g k q . Hence, by Lemma 2.3, II . k g k bmo cq k f k h cp . Now we estimate the term I. By the Cauchy-Schwarz inequality c d d I ≤ τ Z R d Z (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( f )( t ) | dtε d − (cid:17) s c ( f )( s, ε ) p − dεds · τ Z R d Z (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) s c ( f )( s, ε ) − p dεds def = A · B. Note here that s c ( f )( s, ε ) is the function of two variables defined by (2.6), which is differentiablein the w ∗ -sense. We first deal with A . Using s c ( f )( s, ε ) ≤ s c ( f )( s, ε ), we have A ≤ τ Z R d Z Z B ( s, ε ) | ∂∂ε P ε ( f )( t ) | s c ( f )( s, ε ) p − dεdtε d − ds = − τ Z R d Z (cid:0) ∂∂ε s c ( f )( s, ε ) (cid:1) s c ( f )( s, ε ) p − dεds = − τ Z R d Z s c ( f )( s, ε ) p − ∂∂ε s c ( f )( s, ε ) dεds. Since 1 ≤ p < s c ( f )( s, ε ) is decreasing in ε , s c ( f )( s, ε ) p − ≤ s c ( f )( s, p − . At the sametime, ∂∂ε s c ( f )( s, ε ) ≤
0. Therefore, A . − τ Z R d s c ( f )( s, p − Z ∂∂ε s ( f ) c ( s, ε ) dεds . τ Z R d s c ( f )( s, p ds = k f k p h cp . The estimate of B is harder. For any j ∈ N , we need to create a square net partition in R d asfollows: Q m,j = ( 1 √ d ( m − − j , √ d m − j ] × · · · × ( 1 √ d ( m d − − j , √ d m d − j ]with m = ( m , · · · , m d ) ∈ Z d . Let c m,j denote the center of Q m,j . Define(2.8) S c ( f )( s, j ) = (cid:16) Z − j Z B ( c m,j ,r ) | ∂∂r P r ( f )( t ) | dtdrr d − (cid:17) if s ∈ Q m,j . For any s ∈ R d and k ∈ N ( N being the set of nonnegative integers), we define d ( s, k ) = S c ( f )( s, k ) − p − S c ( f )( s, k − − p . Since B ( s, r − ε ) ⊂ B ( c m,j , r ) whenever s ∈ Q m,j and ε ≥ − j , we have s c ( f )( s, ε ) ≤ S c ( f )( s, j ) , ∀ s ∈ Q m,j , ε ≥ − j . It is clear that S c ( f )( s, j ) is increasing in j , so d ( s, k ) ≥
0. At the same time, d ( s, k ) is constanton Q m,k and P k ≤ j d ( s, k ) = S c ( f )( s, j ) − p . Therefore, B . τ X m ∈ Z d X j ≥ Z Q m,j Z − j +1 − j (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) S c ( f )( s, j ) − p dεds = τ Z R d X j ≥ S c ( f )( s, j ) − p Z − j +1 − j (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) dεds = τ Z R d X j ≥ X ≤ k ≤ j d ( s, k ) Z − j +1 − j (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) dεds = τ Z R d X k ≥ d ( s, k ) X j ≥ k Z − j +1 − j (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) dεds = τ X m X k ≥ d ( s, k ) Z Q m,k Z − k +1 (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) dεds . Since g ∈ bmo cq , Lemma 2.3 ensures the existence of a positive operator a ∈ L q ( N ) such that k a k q . k g k cq and1 | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε ≤ a ( s ) and for s ∈ Q and for all cubes Q with | Q | < . Let e Q m,k be the cube concentric with Q m,k and having side length 2 − k +1 . By the Fubini theoremand Lemma 1.4, we have Z Q m,k Z − k +1 (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( g )( t ) | dtε d − (cid:17) dεds ≤ d Z e Q m,k Z − k +1 | ∂∂ε P ε ( g )( s ) | εdεds = 2 d Z T ( e Q m,k ) | ∂∂ε P ε ( g )( s ) | εdεds . Z Q m,k a ( s ) ds. Then we deduce B . τ X m X k ≥ Z Q m,k d ( s, k ) a ( s ) ds = τ Z R d X k ≥ d ( s, k ) a ( s ) ds = τ Z R d S c ( f )( s, + ∞ ) − p a ( s ) ds = τ Z R d S c ( f )( s ) − p a ( s ) ds ≤ k S c ( f ) k − pp k a k q ≤ k f k − p h cp k a k q . k f k − p h cp k g k cq . Combining the estimates of A , B and II, we complete the proof. (cid:3) The following is the main theorem of this section.
Theorem 2.11.
Let ≤ p < and q be its conjugate index. We have h cp ( R d , M ) ∗ = bmo cq ( R d , M ) with equivalent norms. More precisely, every g ∈ bmo cq ( R d , M ) defines a continuous linear func-tional on h cp ( R d , M ) by ℓ g ( f ) = τ Z f ( s ) g ∗ ( s ) ds, ∀ f ∈ L p ( M ; W cd ) . perator-valued local Hardy spaces 19 Conversely, every ℓ ∈ h cp ( R d , M ) ∗ can be written as above and is associated to some g ∈ bmo cq ( R d , M ) with k ℓ k (h cp ) ∗ ≈ k g k bmo cq . Proof.
First, by Lemma 2.10, we get(2.9) | ℓ g ( f ) | . k g k bmo cq k f k h cp Now we prove the converse. Suppose that ℓ ∈ h cp ( R d , M ) ∗ . By the Hahn-Banach theorem, ℓ extends to a continuous functional on L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) with the same norm. Thus, thereexists h = ( h ′ , h ′′ ) ∈ L q (cid:0) N ; L c ( e Γ) (cid:1) ⊕ q L q ( N ) such that k h k L q (cid:0) N ; L c ( e Γ) (cid:1) ⊕ q L q ( N ) = k ℓ k (h cp ) ∗ and ℓ ( f ) = τ Z R d Z Z e Γ ∂∂ε P ε ( f )( s + t ) h ′∗ ( s, t, ε ) dtdεε d ds + τ Z R d P ∗ f ( s ) h ′′∗ ( s ) ds, = τ Z R d f ( u ) e F ( h ) ∗ ( u ) du, where e F is the map defined in (2.5).Let g = e F ( h ). Following the proof of Theorem 2.8, we have k g k bmo cq . k ℓ k (h cp ) ∗ and ℓ ( f ) = τ Z R d f ( s ) g ∗ ( s ) ds, ∀ f ∈ L p (cid:0) M ; W cd (cid:1) . Thus, we have accomplished the proof of the theorem. (cid:3)
The following corollary gives an equivalent norm of the space bmo cq . Note that it is a strength-ening of the one-sided estimates in Lemmas 1.4 and 1.5. Corollary 2.12.
Let < q ≤ ∞ . Then g ∈ bmo cq ( R d , M ) if and only if dλ g = | ∂∂ε P ε ( g )( s ) | εdsdε is an M -valued Carleson q -measure on S and k J (P) ∗ g k q < ∞ . Furthermore, k g k bmo cq ≈ (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε (cid:13)(cid:13)(cid:13) q + k J (P) ∗ g k q . Proof.
From the proof of Lemma 2.10, we can see that if dλ g = | ∂∂ε P ε ( g )( s ) | εdsdε is an M -valued Carleson q -measure on S and J (P) ∗ g ∈ L q ( N ), then g defines a continuous functional onh cp ( R d , M ): ℓ ( f ) = τ Z R d f ( s ) g ∗ ( s ) ds, and k ℓ k (h cp ) ∗ . (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε (cid:13)(cid:13)(cid:13) q + k J (P) ∗ g k q . According to Theorem 2.11, there exists a function g ′ ∈ bmo cq ( R d , M ) such that k g ′ k bmo cq . (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε (cid:13)(cid:13)(cid:13) q + k J (P) ∗ g k q and that τ Z R d f ( s ) g ∗ ( s ) ds = τ Z R d f ( s ) g ′∗ ( s ) ds, for any f ∈ h cp ( R d , M ). Thus, g = g ′ with k g k bmo cq . (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε (cid:13)(cid:13)(cid:13) q + k J (P) ∗ g k q . The inverse inequality is already contained in Lemmas 1.4 and 1.5. We obtain the desired assertion. (cid:3)
The equivalence h q = bmo q . We now show that h cq ( R d , M ) = bmo cq ( R d , M ) for 2 < q < ∞ .Thus according to the duality obtained in the last subsection, the dual of h cp ( R d , M ) agrees withh cq ( R d , M ) when 1 < p <
2. Let us begin with two lemmas concerning the comparison of s c ( f ) and g c ( f ). We require an auxiliary truncated square function. For s ∈ R d and ε ∈ [0 , ], we define:(2.10) e g c ( f )( s, ε ) = (cid:16) Z ε | ∂∂r P r ( f )( s ) | rdr (cid:17) . Lemma 2.13.
We have e g c ( f )( s, ε ) . s c ( f )( s, ε , where the relevant constant depends only on the dimension d .Proof. By translation, it suffices to prove this inequality for s = 0. Given ε ∈ [0 , ], for any r suchthat ε ≤ r ≤ , let us denote the ball centered at (0 , r ) and tangent to the boundary of the cone { ( t, u ) ∈ R d +1+ : | t | < r − ε r u } by e B r . We notice that the radius of e B r is greater than or equal to r √ . By the harmonicity of ∂∂r P r ( f ), we have ∂∂r P r ( f )(0) = 1 | e B r | Z e B r ∂∂u P u ( f )( t ) dt. Then by (0.10), we arrive at | ∂∂r P r ( f )(0) | ≤ √ d +1 c d +1 r d +1 Z e B r | ∂∂u P u ( f )( t ) | dt, where c d +1 is the volume of the unit ball of R d +1 . Integrating the above inequality, we get(2.11) Z ε | ∂∂r P r ( f )(0) | rdr ≤ Z ε √ d +1 c d +1 r d Z e B r | ∂∂u P u ( f )( t ) | dtdudr. Since ( t, u ) ∈ e B r implies √ √ u ≤ r ≤ √ √ − u and ε ≤ u ≤
1, the right hand side of (2.11) can bemajorized by √ d +1 c d +1 Z ε Z e B r | ∂∂u P u ( f )( t ) | Z u u r d drdtdu ≤ C | s c ( f )(0 , ε | , where C is a constant depending only on d . Therefore, e g c ( f )(0 , ε ) . s c ( f )(0 , ε ). (cid:3) Lemma 2.14.
Let ≤ p < ∞ . Then for any f ∈ h cp ( R d , M ) , we have k s c ( f ) k p + k P ∗ f k p . k g c ( f ) k p + k P ∗ f k p . Proof.
We first deal with the case when 1 ≤ p <
2. Let g be a function in bmo cq ( R d , M ) ( q is theconjugate index of p ). Following a similar calculation as (1.3), we can easily check that τ Z R d f ( s ) g ∗ ( s ) ds = 4 τ Z R d Z ∂∂ε P ε ( f )( s ) ∂∂ε P ε ( g ) ∗ ( s ) εdεds + (cid:16) τ Z R d P ∗ f ( s )(P ∗ g ( s )) ∗ ds + 8 π τ Z R d P ∗ f ( s )( I (P ) ∗ g ( s )) ∗ ds (cid:17) def = I + II . The term II can be treated in the same way as in the proof of Lemma 2.10:II . k P ∗ f k p · k J (P ) ∗ f k p . Applying Lemma 2.3, we have II . k P ∗ f k p · k g k bmo cq . perator-valued local Hardy spaces 21 Concerning the term I, we have | I | . τ Z R d Z | ∂∂ε P ε ( f )( s ) | e g c ( f )( s, ε ) p − εdεds · τ Z R d Z | ∂∂ε P ε ( g )( s ) | e g c ( f )( s, ε ) − p εdεds def = A ′ · B ′ . Following the argument for the estimate of A in the proof of Lemma 2.10, we deduce similarly that A ′ . k e g c ( f ) k pp . Now we deal with term B ′ . By Lemma 2.13, we have B ′ ≤ τ Z R d Z | ∂∂ε P ε ( g )( s ) | s c ( f )( s, ε εdεds. Then we can apply almost the same argument as in the estimate of B . There is only one minordifference: when ε ≥ − j and s ∈ Q m,j , we have s c ( f )( s, ε ) ≤ S c ( f )( s, j + 1). We conclude that B ′ . k g k cq k s c ( f ) k − pp . Combining the estimates of I, A ′ and B ′ with Theorem 2.11, we get k s c ( f ) k p + k P ∗ f k p . k e g c ( f ) k p + k P ∗ f k p . k g c ( f ) k p + k P ∗ f k p . The case p = 2 is obvious. For p >
2, choose a positive g ∈ L ( p ) ′ ( N ) with norm one such that, k s c ( f ) k p = (cid:13)(cid:13)(cid:13) Z Z e Γ | ∂∂ε P ε ( f )( · + t ) | dtdεε d − (cid:13)(cid:13)(cid:13) p = τ Z R d Z Z e Γ | ∂∂ε P ε ( f )( s + t ) | dtdεε d − g ( s ) ds = τ Z R d Z | ∂∂ε P ε ( f )( t ) | dtdεε d − Z B ( t,ε ) g ( s ) ds. By the noncommutative Hardy-Littlewood maximal inequality (the one dimension R case is givenby [30, Theorem 3.3], the case R d is a simple corollary of (2.1) and Lemma 2.7), there exists apositive a ∈ L ( p ) ′ ( N ) such that k a k ( p ) ′ ≤ | B ( t, − k ) | Z B ( t, − k ) g ( s ) ds ≤ a ( t ) , ∀ t ∈ R d , ∀ ε > . Therefore, k s c ( f ) k p = τ Z R d Z | ∂∂ε P ε ( f )( t ) | dtdεε d − Z B ( t,ε ) g ( s ) ds ≤ c d τ Z R d Z | ∂∂ε P ε ( f )( t ) | εa ( t ) dtdε ≤ c d (cid:13)(cid:13) Z | ∂∂ε P ε ( f )( t ) | εdtdε (cid:13)(cid:13) p k a k ( p ) ′ ≤ c d k g c ( f ) k p . Then the assertion for the case p > (cid:3)
To proceed further, we introduce the definition of tent spaces. In the noncommutative setting,these spaces were first defined and studied by Mei [31].
Definition 2.15.
For any function defined on R d × (0 ,
1) = S with values in L ( M )+ M , wheneverit exists, we define A c ( f )( s ) = (cid:16) Z e Γ | f ( t + s, ε ) | dtdεε d +1 (cid:17) , s ∈ R d . For ≤ p < ∞ , we define T cp ( R d , M ) = { f : A c ( f ) ∈ L p ( N ) } equipped with the norm k f k T cp ( R d , M ) = k A c ( f ) k p . For p = ∞ , define the operator-valued column T c ∞ norm of f as k f k T c ∞ = sup | Q |≤ (cid:13)(cid:13)(cid:13)(cid:16) | Q | Z T ( Q ) | f ( s, ε ) | dsdεε (cid:17) (cid:13)(cid:13)(cid:13) M , and the corresponding space is T c ∞ ( R d , M ) = { f : k f k T c ∞ < ∞} . Remark 2.16.
By the same arguments used in the proof of Theorem 2.11, we can prove theduality that T cp ( R d , M ) ∗ = T cq ( R d , M ) for 1 ≤ p < ∞ and p + q = 1. For the case p = 1, itsuffices to replace ∂∂ε P ε ( f )( s ) and ∂∂ε P ε ( g )( s ) in the proof of Lemma 2.10 by f ( s, ε ) and g ( s, ε )respectively. A similar argument will give us the inclusion that T c ∞ ( R d , M ) ⊂ T c ( R d , M ) ∗ . Onthe other hand, since L ∞ ( N ; L c ( e Γ)) ⊂ T c ∞ ( R d , M ), we get the reverse inclusion. For 1 < p < ∞ ,the tent space T cp ( R d , M ) we define above is a complemented subspace of the column tent spacedefined in [30]. So by Remark 4.6 in [51], we obtain the duality that T cp ( R d , M ) ∗ = T cq ( R d , M ). Theorem 2.17.
For < q < ∞ , h cq ( R d , M ) = bmo cq ( R d , M ) with equivalent norms.Proof. First, we show the inclusion h cq ( R d , M ) ⊂ bmo cq ( R d , M ). By Theorem 2.11, it suffices toshow that h cq ( R d , M ) ⊂ h cp ( R d , M ) ∗ . Applying (1.3), for any f ∈ h cq ( R d , M ) and g ∈ h cp ( R d , M ),we have τ Z R d g ( s ) f ∗ ( s ) ds = 4 c d Z R d Z Z e Γ ∂∂ε P ε ( g )( s + t ) ∂∂ε P ε ( f ) ∗ ( s + t ) dtdεε d − ds + Z R d P ∗ g ( s )(P ∗ f ( s )) ∗ ds + 4 π Z R d I (P) ∗ g ( s )(P ∗ f ( s )) ∗ ds. Then, by the H¨older inequality, (cid:12)(cid:12) τ Z R d g ( s ) f ∗ ( s ) ds (cid:12)(cid:12) ≤ (cid:13)(cid:13) ε · ∂∂ε P ε ( g ) (cid:13)(cid:13) L p (cid:0) N ; L ( e Γ) (cid:1) k ε · ∂∂ε P ε ( f ) k L q (cid:0) N ; L ( e Γ) (cid:1) + k (P + I (P)) ∗ g k p · k P ∗ f k q ≤ (cid:16)(cid:13)(cid:13) s c ( g ) k p + k (P + I (P)) ∗ g (cid:13)(cid:13) p (cid:17) k f k h cq . Now, we show that for any 1 ≤ p < g ∈ h cp ( R d , M ), we have k (P + I (P)) ∗ g k p . k g k h cp .Since 2 < q < ∞ , we have 1 < q < ∞ . Applying the noncommutative Hardy-Littlewood maximalinequality, we get k f k bmo cq . (cid:13)(cid:13)(cid:13) sup s ∈ Q ⊂ R d + | Q | Z Q | f ( t ) | dt (cid:13)(cid:13)(cid:13) q . k| f | k q = k f k q . This implies that L q ( N ) ⊂ bmo cq ( R d , M ) for any 2 < q ≤ ∞ . Then by Theorem 2.11, we geth cp ( R d , M ) ⊂ L p ( N ). Therefore we deduce that(2.12) k (P + I (P)) ∗ g k p . k g k p . k g k h cp . Thus, (cid:12)(cid:12) τ Z R d g ( s ) f ∗ ( s ) ds (cid:12)(cid:12) . k f k h cq k g k h cp . We have proved h cq ( R d , M ) ⊂ bmo cq ( R d , M ).Let us turn to the reverse inclusion bmo cq ( R d , M ) ⊂ h cq ( R d , M ). We need to make use of thetent spaces in Definition 2.15. We claim that for q >
2, every f ∈ bmo cq ( R d , M ) induces a linearfunctional on T cp ( R d , M ) ⊕ p L p ( N ). Indeed, for any h = ( h ′ , h ′′ ) ∈ T cp ( R d , M ) ⊕ p L p ( N ), we define ℓ f ( h ) = τ Z R d Z h ′ ( s, ε ) ∂∂ε P ε ( f ) ∗ ( s ) dεds + τ Z R d h ′′ ( s )[(P ∗ f ) ∗ ( s ) + 4 π ( I (P) ∗ f ) ∗ ( s )] ds. (2.13)Set A c ( h ′ )( s, ε ) = Z ε Z B ( s,r − ε ) | h ′ ( s, ε ) | dtdrr d +1 ,A c ( h ′ )( s, ε ) = Z ε Z B ( s, r ) | h ′ ( s, ε ) | dtdrr d +1 . perator-valued local Hardy spaces 23 Then by the Cauchy-Schwarz inequality, we arrive at | ℓ f ( h ) | . (cid:16) τ Z R d Z (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( f )( t ) | dtε d − (cid:17) A c ( h ′ )( s, ε ) p − dεds (cid:17) · (cid:16) τ Z R d Z (cid:16) Z B ( s, ε ) | ∂∂ε P ε ( f )( t ) | dtε d − (cid:17) A c ( h ′ )( s, ε ) − p dεds (cid:17) + (cid:12)(cid:12) τ Z R d h ′′ ( s )(P ∗ f ( s )) ∗ ds (cid:12)(cid:12) + (cid:12)(cid:12) τ Z R d h ′′ ( s )( I (P) ∗ f ( s )) ∗ ds (cid:12)(cid:12) . Following a similar argument as in the proof of Lemma 2.10, we obtain that | ℓ f ( h ) | . ( k h ′ k T cp + k h ′′ k L p ) k f k bmo cq . k h k T cp ⊕ p L p · k f k bmo cq , which implies that k ℓ f k ≤ c q k f k bmo cq . So the claim is proved.Next we show that k f k h cq ≤ C q k ℓ f k . By definition, we can regard T cp as a closed subspace of L p (cid:0) N ; L c ( e Γ) (cid:1) in the natural way. Then, ℓ f extends to a linear functional on L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ). Thus, there exists g = ( g ′ , g ′′ ) ∈ L q (cid:0) N ; L c ( e Γ) (cid:1) ⊕ q L q ( N ) such that k g k L q (cid:0) N ; L c ( e Γ) (cid:1) ⊕ q L q ( N ) ≤ k ℓ f k and for any h = ( h ′ , h ′′ ) ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ), ℓ f ( h ) = τ Z R d Z Z e Γ h ′ ( s, ε ) g ′∗ ( s, t, ε ) dtdεε d +1 ds + τ Z R d h ′′ ( s ) g ′′∗ ( s ) ds = τ Z R d Z h ′ ( s, ε ) Z B ( s,ε ) g ′∗ ( s, t, ε ) dt dsdεε d +1 + τ Z R d h ′′ ( s ) g ′′∗ ( s ) ds. Comparing the equalities above with (2.13), we get ∂∂ε P ε ( f )( s ) = 1 ε d +1 Z B (0 ,ε ) g ′ ( s, t, ε ) dt and P ∗ f + 4 πI (P) ∗ f = g ′′ . By Lemma 2.14, we have k f k h cq . (cid:13)(cid:13)(cid:13)(cid:0) Z | ∂∂ε P ε ( f ) | εdε (cid:1) (cid:13)(cid:13)(cid:13) q + k P ∗ f k q ≤ c d (cid:13)(cid:13)(cid:13)(cid:0) Z ε d +1 Z B (0 ,ε ) | g ′ ( s, t, ε ) | dtdε (cid:1) (cid:13)(cid:13)(cid:13) q + k P ∗ f k q . k g ′ k L q (cid:0) N ; L c ( e Γ) (cid:1) + k P ∗ f k q . Now let us majorize the second term k P ∗ f k q by k g ′′ k q . Indeed, consider the function G ( s ) = 2 π Z ∞ e − πε P ε ( s ) dε. We can easily check that G ∈ L ( R d ), k G k ≤ b G ( ξ ) = (1 + | ξ | ) − . This means that theoperator (1 + I ) − is a contractive Fourier multiplier on L q ( N ). Therefore, k P ∗ f k q ≤ k (P + I (P)) ∗ f k q ≤ π k g ′′ k q . Finally, we conclude that k f k h cq . k ℓ f k . k f k bmo cq and then h cq ( R d , M ) = bmo cq ( R d , M ) withequivalent norms. (cid:3) Armed with the theorem above, we are able to extend the content of Theorem 2.8.
Theorem 2.18. (1)
The map F extends to a bounded map from L ∞ (cid:0) N ; L c ( e Γ) (cid:1) ⊕ ∞ L ∞ ( N ) into bmo c ( R d , M ) and k F ( h ) k bmo c . k h k L ∞ (cid:0) N ; L c ( e Γ) (cid:1) ⊕ ∞ L ∞ ( N ) . (2) For < p < ∞ , F extends to a bounded map from L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) into h cp ( R d , M ) and k F ( h ) k h cp . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) . Proof. (1) is already contained in Theorem 2.8. When p >
2, (2) follows from Theorem 2.8 andTheorem 2.17. The case p = 2 is trivial. For the case 1 < p <
2, according to Theorem 2.11, wehave k F ( h ) k h cp . sup k f k bmo cq ≤ (cid:12)(cid:12) τ Z R d F ( h )( s ) f ∗ ( s ) ds (cid:12)(cid:12) . Then, by Theorem 2.17 and (2.12), for h = ( h ′ , h ′′ ) ∈ L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ), we havesup k f k bmo cq ≤ (cid:12)(cid:12) τ Z R d F ( h )( s ) f ∗ ( s ) ds (cid:12)(cid:12) . sup k f k h cq ≤ (cid:12)(cid:12)(cid:12) τ Z R d (cid:2) Z Z e Γ h ′ ( s, t, ε ) ∂∂ε P ε ( f ) ∗ ( s + t ) dtdε + h ′′ ( s )([P + 4 πI (P)] ∗ f ∗ ( s )) (cid:3) ds (cid:12)(cid:12)(cid:12) . k h k L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) . The desired inequality is proved. (cid:3)
The above theorem shows that, for any 1 < p < ∞ , h cp ( R d , M ) is a complemented subspace of L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ). Thus, we deduce the following duality theorem: Theorem 2.19.
We have h cp ( R d , M ) ∗ = h cq ( R d , M ) with equivalent norms for any < p < ∞ . Interpolation
In this section we study the interpolation of local Hardy and bmo spaces by transferring theproblem to that of the operator-valued Hardy and BMO spaces defined in [30]. We begin with aneasy observation on the difference between bmo cq and BMO cq norms. Lemma 3.1.
For < q ≤ ∞ , we have k g k bmo cq ≈ (cid:0) k g k q BMO cq + k J (P) ∗ g k qq (cid:1) q . Proof.
Repeating the proof of Proposition 1.3 with k · k M replaced by k · k L q ( N ; ℓ ∞ ) , we have k g k BMO cq . k g k bmo cq . By Lemma 2.3, it is also evident that k J (P) ∗ g k q . k g k bmo cq . Then we obtain (cid:0) k g k q BMO cq + k J (P) ∗ g k qq (cid:1) q . k g k bmo cq . On the other hand, by Corollary 2.12, we have k g k bmo cq . (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) | ∂∂ε P ε ( g )( t ) | εdtdε (cid:13)(cid:13)(cid:13) q + k J (P) ∗ g k q . Clearly, the first term on the right side can be estimated from above by k g k BMO qc (see [51, Theo-rem 3.4]). Therefore, k g k bmo cq . k g k BMO qc + k J (P) ∗ g k q ≈ (cid:0) k g k q BMO cq + k J (P) ∗ g k qq (cid:1) q . Thus, the lemma is proved. (cid:3)
Define F q ( N ) to be the space of all f ∈ L q ( M ; R cd ) such that k J (P) ∗ f k q < ∞ . From the abovelemma, we see that bmo cq ( R d , M ) and BMO cq ( R d , M ) ⊕ q F q ( N ) have equivalent norms. By theinterpolation between BMO cq ( R d , M ) and BMO c ( R d , M ) (see [30] for more details), we deduce thefollowing lemma: Lemma 3.2.
Let < q < ∞ and < θ < . Then (cid:0) bmo cq ( R d , M ) , bmo c ( R d , M ) (cid:1) θ ⊂ bmo c̺ ( R d , M ) with ̺ = q − θ . perator-valued local Hardy spaces 25 Proof.
By Lemma 3.1, we see thatbmo cq ( R d , M ) = BMO cq ( R d , M ) ⊕ q F q ( N ) . with equivalent norms. Define a map Υ q : F q ( N ) −→ L q ( N ) f J (P) ∗ f. Thus, Υ q defines an isometric embedding of F q ( N ) into L q ( N ). Then by the interpolation betweenBMO cq ( R d , M ) and BMO c ( R d , M ), we get (cid:0) bmo cq ( R d , M ) , bmo c ( R d , M ) (cid:1) θ = (cid:0) BMO cq ( R d , M ) ⊕ q F q ( N ) , BMO c ( R d , M ) ⊕ ∞ F ∞ ( N ) (cid:1) θ = (cid:0) BMO cq ( R d , M ) , BMO c ( R d , M ) (cid:1) θ ⊕ ̺ (cid:0) F q ( N ) , F ∞ ( N ) (cid:1) θ ⊂ BMO c̺ ( R d , M ) ⊕ ̺ F ̺ ( N ) = bmo c̺ ( R d , M ) , which completes the proof. (cid:3) Theorem 3.3.
Let < p < ∞ . We have (cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p = h cp ( R d , M ) . Proof.
Let 1 < p < p ′ = − θp + θ . Since the map E in Definition 2.4 is an isometry fromh cp ( R d , M ) to L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ), we have(3.1) (cid:0) h cp ( R d , M ) , h c ( R d , M ) (cid:1) θ ⊂ h cp ′ ( R d , M ) . By Theorem 2.19, h cp is a reflexive Banach space. Then applying [2, Corollary 4.5.2], we know thatthe dual of (cid:0) h cp ( R d , M ) , h c ( R d , M ) (cid:1) θ is (cid:0) bmo cq ( R d , M ) , bmo c ( R d , M ) (cid:1) θ . Therefore, if the inclusion(3.1) is proper, we will get the proper inclusionbmo c̺ ( R d , M ) ( (cid:0) bmo cq ( R d , M ) , bmo c ( R d , M ) (cid:1) θ , which is in contradiction with Lemma 3.2. Thus, we have(3.2) (cid:0) h cp ( R d , M ) , h c ( R d , M ) (cid:1) θ = h cp ′ ( R d , M ) . By duality and [2, Corollary 4.5.2] again, the above equality implies that for q ′ = q − θ ,(3.3) (cid:0) h cq ( R d , M ) , bmo c ( R d , M ) (cid:1) θ = h cq ′ ( R d , M ) . For the case where 1 < p , p < ∞ , the interpolation of h cp ( R d , M ) and h cp ( R d , M ) is mucheasier to handle. Indeed, by Theorem 2.18, we have, for any 1 < p < ∞ , h cp ( R d , M ) is a comple-mented subspace of L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ) via the maps E and F in Definition 2.4. This impliesthat, for any 1 < p , p < ∞ , (cid:0) h cp ( R d , M ) , h cp ( R d , M ) (cid:1) θ = h cp ( R d , M ) , with p = − θp + θp . Combining this equivalence with (3.2), (3.3), and applying Wolff’s interpolationtheorem (see [48]), we get the desired assertion. (cid:3) The following theorem is the mixed version of Theorem 3.3, which states that h ( R d , M ) andbmo( R d , M ) are also good endpoints of L p ( N ). Theorem 3.4.
Let < p < ∞ . We have (cid:0) X, Y (cid:1) p = L p ( N ) with equivalent norms, where X = bmo( R d , M ) or L ∞ ( N ) , and Y = h ( R d , M ) or L ( N ) .Proof. By the same argument as in the proof of Theorem 3.3, we have the inclusion (cid:0) bmo q ( R d , M ) , bmo( R d , M ) (cid:1) θ ⊂ bmo q ′ ( R d , M ) q ′ = qθ , which ensures by duality that (cid:0) h p ( R d , M ) , h ( R d , M ) (cid:1) θ ⊃ h p ′ ( R d , M ) = L p ′ ( N )for p ′ = − θp + θ . Then by Proposition 5.18, L p ′ ( N ) ⊂ (cid:0) h p ( R d , M ) , h ( R d , M ) (cid:1) θ = (cid:0) L p ( N ) , h ( R d , M ) (cid:1) θ . Since h ( R d , M ) ⊂ L ( N ), then (cid:0) h p ( R d , M ) , h ( R d , M ) (cid:1) θ ⊂ (cid:0) L p ( N ) , L ( N ) (cid:1) θ = L p ′ ( N ) . Combining the estimates above, we have (cid:0) h p ( R d , M ) , h ( R d , M ) (cid:1) θ = L p ′ ( N ) . Again, using duality and Wolff’s interpolation theorem, we conclude the proof by the same trickas in the proof of Theorem 3.3. (cid:3)
We end this section by some real interpolation results.
Corollary 3.5.
Let < p < ∞ . Then we have (1) (cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p ,p = h cp ( R d , M ) with equivalent norms. (2) (cid:0) X, Y (cid:1) p ,p = L p ( N ) with equivalent norms, where X = bmo( R d , M ) or L ∞ ( N ) , and Y =h ( R d , M ) or L ( N ) .Proof. Both (1) and (2) follow from [2, Theorem 4.7.2]; we only prove (1). Let 1 < p < p < p < ∞ with p = − ηp + ηp . By [2, Theorem 4.7.2], we write (cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p ,p = (cid:16)(cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p , (cid:0) bmo c ( R d , M ) , h c ( R d , M ) (cid:1) p (cid:17) η,p . (3.4)Then the assertion (1) follows from Theorem 3.4 and the facts that (cid:0) L p ( N ) , L p ( N ) (cid:1) η,p = L p ( N )and that h cp ( R d , M ) is a complemented subspace of L p (cid:0) N ; L c ( e Γ) (cid:1) ⊕ p L p ( N ). (cid:3) Calder´on-Zygmund theory
We introduce the Calder´on-Zygmund theory on operator-valued local Hardy spaces in this sec-tion. It is closely related to the similar results of [14], [26], [33] and [53]. The results in thefollowing will be used in the next section to investigate various square functions that characterizelocal Hardy spaces.Let K be an L ( M ) + M )-valued tempered distribution which coincides on R d \ { } with alocally integrable L ( M ) + M -valued function. We define the left singular integral operator K c associated to K by K c ( f )( s ) = Z R d K ( s − t ) f ( t ) dt, and the right singular integral operator K r associated to K by K r ( f )( s ) = Z R d f ( t ) K ( s − t ) dt. Both K c ( f ) and K r ( f ) are well-defined for sufficiently nice functions f with values in L ( M ) ∩ M ,for instance, for f ∈ S ⊗ ( L ( M ) ∩ M ).Let bmo c ( R d , M ) denote the subspace of bmo c ( R d , M ) consisting of compactly supported func-tions. The following lemma is an analogue of Lemma 2.1 in [51] for inhomogeneous spaces. Noticethat the usual Calder´on-Zygmund operators (the operators satisfying the condition (1) and (3) inthe following lemma) are not necessarily bounded on h c ( R d , M ). Thus, we need to impose an extradecay at infinity on the kernel K . Lemma 4.1.
Assume that (1) the Fourier transform of K is bounded: sup ξ ∈ R d k b K ( ξ ) k M < ∞ ; (2) K satisfies the size estimate at infinity: there exist C and ρ > such that k K ( s ) k M ≤ C | s | d + ρ , ∀ | s | ≥ K has the Lipschitz regularity: there exist C and γ > such that k K ( s − t ) − K ( s ) k M ≤ C | t | γ | s − t | d + γ , ∀ | s | > | t | . perator-valued local Hardy spaces 27 Then K c is bounded on h cp ( R d , M ) for ≤ p < ∞ and from bmo c ( R d , M ) to bmo c ( R d , M ) .A similar statement also holds for K r and the corresponding row spaces.Proof. First suppose that K c maps constant functions to zero. This amounts to requiring that K c ( R d ) = 0. Let Q ⊂ R d be a cube with | Q | <
1. Since the assumption of Lemma 2.1 in [51] areincluded in the ones of this lemma, we get (cid:13)(cid:13)(cid:13)(cid:0) | Q | Z Q | K c ( f ) − K c ( f ) Q | dt (cid:1) (cid:13)(cid:13)(cid:13) M . k f k BMO c . k f k bmo c . Now let us focus on the cubes with side length 1. Let Q be a cube with | Q | = 1 and e Q = 2 Q bethe cube concentric with Q and with side length 2. Decompose f as f = f + f , where f = e Q f and f = R d \ e Q f . Then K c ( f ) = K c ( f ) + K c ( f ). We have (cid:13)(cid:13)(cid:13) | Q | Z Q | K c ( f ) | ds (cid:13)(cid:13)(cid:13) M . (cid:13)(cid:13)(cid:13) | Q | Z Q | K c ( f ) | ds (cid:13)(cid:13)(cid:13) M + (cid:13)(cid:13)(cid:13) | Q | Z Q | K c ( f ) | ds (cid:13)(cid:13)(cid:13) M . The first term is easy to estimate. By assumption (1) and (0.10), (cid:13)(cid:13)(cid:13) | Q | Z Q | K c ( f ) | ds (cid:13)(cid:13)(cid:13) M ≤ (cid:13)(cid:13)(cid:13) | Q | Z R d | b K ( ξ ) b f ( ξ ) | dξ (cid:13)(cid:13)(cid:13) M . (cid:13)(cid:13)(cid:13) | Q | Z R d | b f ( ξ ) | dξ (cid:13)(cid:13)(cid:13) M = (cid:13)(cid:13)(cid:13) | Q | Z e Q | f ( s ) | ds (cid:13)(cid:13)(cid:13) M . sup | Q | =1 (cid:13)(cid:13)(cid:13) | Q | Z Q | f ( s ) | ds (cid:13)(cid:13)(cid:13) M . To estimate the second term, using assumption (2) and (0.10) again, we have | K c ( f )( s ) | = (cid:12)(cid:12)(cid:12) Z R d K ( s − t ) f ( t ) dt (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z R d \ e Q K ( s − t ) f ( t ) dt (cid:12)(cid:12)(cid:12) ≤ Z R d \ e Q k K ( s − t ) k M dt · Z R d \ e Q k K ( s − t ) k − M | K ( s − t ) f ( t ) | dt . Z R d \ e Q k K ( s − t ) k M | f ( t ) | dt . Z R d \ e Q | s − t | d + ρ | f ( t ) | dt. Set e Q m = e Q + 2 m for every m ∈ Z d . Then R d \ e Q = ∪ m =0 e Q m . Continuing the estimate of | K c ( f )( s ) | , for any s ∈ Q , we have | K c ( f )( s ) | ≤ X m =0 Z e Q m | s − t | d + ρ | f ( t ) | dt ≈ X m =0 | m | d + ρ Z e Q m | f ( t ) | dt . k f k bmo c . Combining the previous estimates, we deduce that K c is bounded from bmo c ( R d , M ) to bmo c ( R d , M ).Now we illustrate that the additional requirement K c ( R d ) = 0 is not needed. First, a similarargument as above ensures that for every compactly supported f ∈ L ∞ ( N ), k K c ( f ) k bmo c . k f k ∞ .Then we follow the argument of [10, Proposition II.5.15] to extend K c on the whole L ∞ ( N ), as K c ( f )( s ) = lim j (cid:2) K c ( f B j )( s ) − Z < | t |≤ j K ( − t ) f ( t ) dt (cid:3) , ∀ s ∈ R d , where B j is the ball centered at the origin with radius j . Let us show that the sequence on theright hand side converges pointwise in the norm k · k M and uniformly on any compact set Ω ⊂ R d . To this end, we denote by g j the j -th term of this sequence. Let l be the first natural number suchthat l ≥ s ∈ Ω | s | . Then for s ∈ Ω and j > l , we have g j ( s ) = g l ( s ) + Z l< | t |≤ j (cid:0) K ( s − t ) − K ( − t ) (cid:1) f ( t ) dt. By assumption (3), the integral on the right hand side is bounded by a bounded multiple of k f k ∞ ,uniformly on s ∈ Ω. This ensures the convergence of g j , so K c ( f ) is a well-defined function.Now we have to estimate the bmo c -norm of K c ( f ). Taking any cube Q ⊂ R d , by the uniformconvergence of g j on Q in M , we have (cid:13)(cid:13)(cid:0) Z Q | K c ( f )( s ) − ( K c ( f )) Q | ds (cid:1) (cid:13)(cid:13) M = lim j (cid:13)(cid:13)(cid:0) Z Q | g j ( s ) − ( g j ) Q | ds (cid:1) (cid:13)(cid:13) M . Similarly, (cid:13)(cid:13)(cid:0) Z Q | K c ( f )( s ) | ds (cid:1) (cid:13)(cid:13) M = lim j (cid:13)(cid:13)(cid:0) Z Q | g j ( s ) | ds (cid:1) (cid:13)(cid:13) M . Hence, by the fact that g j and K c ( f B j ) differ by a constant, we obtain k K c ( f ) k bmo c = lim j k g j k bmo c . lim sup j k K c ( f B j ) k bmo c + k f k ∞ . k f k ∞ . Therefore, K c defined above extends to a bounded operator from L ∞ ( N ) to bmo c ( R d , M ). Inparticular, K c ( R d ) determines a function in bmo c ( R d , M ). Then for f and Q as above, we have K c ( f ) = K c ( f ) + K c ( f ) + K c ( R d ) f e Q , so k K c ( f ) k bmo c ≤ k K c ( f ) k bmo c + k K c ( f ) k bmo c + k K c ( R d ) k bmo c k f e Q k M . k f k bmo c + k f e Q k M . k f k bmo c . Thus we have proved the bmo c -boundedness of K c in the general case.By duality, the boundedness of K c on h c ( R d , M ) is equivalent to that of its adjoint map ( K c ) ∗ on bmo c ( R d , M ). It is easy to see that ( K c ) ∗ is also a singular integral operator:( K c ) ∗ ( g ) = Z R d e K ( s − t ) g ( t ) dt, where e K ( s ) = K ∗ ( − s ). Obviously, e K also satisfies the same assumption as K , so ( K c ) ∗ is boundedon bmo c ( R d , M ). Thus we get the boundedness of K c on h c ( R d , M ). Then, by the interpolationbetween h c ( R d , M ) and bmo c ( R d , M ) in Theorem 3.3, we get the boundedness of K c on h cp ( R d , M )for 1 < p < ∞ . The assertion is proved. (cid:3) Remark 4.2.
Under the assumption of the above lemma, K c ( R d ) is a constant, so it is the zeroelement in BMO c ( R d , M ).A special case of Lemma 4.1 concerns the Hilbert-valued kernel K . Let H be a Hilbert space and k : R d → H be a H -valued kernel. We view the Hilbert space as the column matrices in B ( H ) withrespect to a fixed orthonormal basis. Put K ( s ) = k ( s ) ⊗ M ∈ B ( H ) ⊗M . For nice functions f : R d → L ( M )+ M , K c ( f ) takes values in the column subspace of L ( B ( H ) ⊗M )+ L ∞ ( B ( H ) ⊗M ).Consequently, k K c ( f ) k L p ( B ( H ) ⊗N ) = k K c ( f ) k L p ( N ; H c ) . Since k ( s ) ⊗ M commutes with M , K c ( f ) = K r ( f ) for f ∈ L ( N ). Let us denote this commonoperator by k c . Here the superscript c refers to the previous convention that H is identified withthe column matrices in B ( H ). Thus, Lemma 4.1 implies the following Corollary 4.3.
Assume that (1) sup ξ ∈ R d k b k ( ξ ) k H < ∞ ; (2) k k ( s ) k H . | s | d + ρ , ∀ | s | ≥ , for some ρ > ; (3) k k ( s − t ) − k ( s ) k H . | t | γ | s − t | d + γ , ∀ | s | > | t | , for some γ > .Then the operator k c is bounded (1) from bmo α ( R d , M ) to bmo α ( R d , B ( H ) ⊗M ) , where α = c , α = r or we leave out α ; (2) and from h cp ( R d , M ) to h cp ( R d , B ( H ) ⊗M ) for ≤ p < ∞ . perator-valued local Hardy spaces 29 Proof.
Since K c ( f ) = K r ( f ) on the subspace L p ( N ) ⊂ L p ( B ( H ) ⊗N ), (1) follows immediatelyfrom Lemma 4.1. Denote the column subspace of bmo c ( R d , B ( H ) ⊗M ) (resp. h cp ( R d , B ( H ) ⊗M ))by bmo c ( R d , H c ⊗M ) (resp. h cp ( R d , H c ⊗M )). Consider the adjoint operator of k c which is de-noted by ( k c ) ∗ . It admits the convolution kernel e K ( s ) = e k ( s ) ⊗ M , where e k ( s ) = k ( − s ) ∗ (so it is arow matrix). Applying Lemma 4.1 to ( k c ) ∗ , we get that ( k c ) ∗ is bounded from bmo c ( R d , H c ⊗M )to bmo c ( R d , M ). Then k c is bounded from h c ( R d , M ) to h c ( R d , H c ⊗M ), and thus boundedfrom h c ( R d , M ) into h c ( R d , B ( H ) ⊗M ). Interpolating this with the boundedness of k c frombmo c ( R d , M ) to bmo c ( R d , B ( H ) ⊗M ), we deduce the desired assertion in (2). (cid:3) Remark 4.4.
Let 1 ≤ p ≤
2. Since L ∞ ( N ) ⊆ bmo c ( R d , M ), we get h c ( R d , M ) ⊆ L ( N ). ByTheorem 3.3 and the fact that h c ( R d , M ) = L ( N ), we have h cp ( R d , M ) ⊆ L p ( N ). Then Corollary4.3 ensures that k k c ( f ) k L p ( N ; H c ) . k k c ( f ) k h cp ( R d ,B ( H ) ⊗M ) . k f k h cp ( R d , M ) for any f ∈ h cp ( R d , M ). 5. General characterizations
Applying the operator-valued Calder´on-Zygmund theory developed in the last section, we willshow that the Poisson kernel in the square functions which are used to define h cp ( R d , M ) canbe replaced by any reasonable test function. As an application, we are able to compare theoperator-valued local Hardy spaces h cp ( R d , M ) defined in this paper with the operator-valued Hardyspaces H cp ( R d , M ) in [30]. We will use multi-index notation. For m = ( m , · · · , m d ) ∈ N d and s = ( s , · · · , s d ) ∈ R d , we set s m = s m · · · s m d d . Let | m | = m + · · · + m d and D m = ∂ m ∂s m · · · ∂ md ∂s mdd .5.1. General characterizations.
Let Φ be a complex-valued infinitely differentiable functiondefined on R d \{ } . Recall that e Γ = { ( t, ε ) ∈ R d +1+ : | t | < ε < } and Φ ε ( s ) = ε − d Φ( sε ). For any f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ), we define the local versions of the conic and radial square functionsof f associated to Φ by s c Φ ( f )( s ) = (cid:16) Z Z e Γ | Φ ε ∗ f ( s + t ) | dtdεε d +1 (cid:17) , s ∈ R d ,g c Φ ( f )( s ) = (cid:16) Z | Φ ε ∗ f ( s ) | dεε (cid:17) , s ∈ R d . The function Φ that we use to characterize the operator-valued local Hardy spaces satisfies thefollowing conditions:(1) Every D m Φ with 0 ≤ | m | ≤ d makes f s cD m Φ f and f g cD m Φ f Calder´on-Zygmundsingular integral operators in Corollary 4.3;(2) There exist functions Ψ , ψ and φ such that(5.1) b φ ( ξ ) b ψ ( ξ ) + Z b Φ( εξ ) b Ψ( εξ ) dεε = 1 , ∀ ξ ∈ R d ;(3) The above Ψ and ψ make dµ g = | Ψ ε ∗ g ( s ) | dεdsε and φ ∗ g satisfy:max n(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) dµ g (cid:13)(cid:13) q , k ψ ∗ g k q o . k g k bmo cq for q > φ makes f φ ∗ f a Calder´on-Zygmund singular integral operator in Corollary 4.3.Fix the four functions Φ , Ψ , φ, ψ as above. The following is one of our main results in thissection, which states that the functions Φ , φ satisfying the above four conditions give a generalcharacterization for h cp ( R d , M ). Theorem 5.1.
Let ≤ p < ∞ and φ , Φ be as above. For any f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) , f ∈ h cp ( R d , M ) if and only if s c Φ ( f ) ∈ L p ( N ) and φ ∗ f ∈ L p ( N ) if and only if g c Φ ( f ) ∈ L p ( N ) and φ ∗ f ∈ L p ( N ) . If this is the case, then (5.2) k f k h cp ≈ k s c Φ ( f ) k p + k φ ∗ f k p ≈ k g c Φ ( f ) k p + k φ ∗ f k p with the relevant constants depending only on d, p , and the pairs (Φ , Ψ) and ( φ, ψ ) . One implication of the above theorem is an easy consequence of conditions (1) and (4) that(5.3) k s c Φ ( f ) k p + k φ ∗ f k p . k f k h cp (5.4) k g c Φ ( f ) k p + k φ ∗ f k p . k f k h cp . In order to prove the converse inequalities, we need the following lemma, which can be seen as ageneralization of Lemma 2.10.
Lemma 5.2.
Let ≤ p < , q be its conjugate index and Φ , φ be the functions satisfying the aboveassumption. For f ∈ h cp ( R d , M ) ∩ L ( N ) and g ∈ bmo cq ( R d , M ) , (cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12) . ( k s c Φ ( f ) k p + k φ ∗ f k p ) k g k bmo cq . Proof.
The proof of this Lemma is very similar to that of Lemma 2.10, we will just point out thenecessary modifications to avoid duplication. We need two auxiliary square functions associatedwith Φ. For s ∈ R d , ε ∈ [0 , s c Φ ( f )( s, ε ) = (cid:16) Z ε Z B ( s,r − ε ) | Φ r ∗ f ( t ) | dtdrr d +1 (cid:17) , (5.6) s c Φ ( f )( s, ε ) = (cid:16) Z ε Z B ( s, r ) | Φ r ∗ f ( t ) | dtdrr d +1 (cid:17) . By assumption (2) of Φ, we have τ Z R d f ( s ) g ∗ ( s ) ds = τ Z R d Z Φ ε ∗ f ( s )(Ψ ε ∗ g ( s )) ∗ dsdεε + τ Z R d φ ∗ f ( s )( ψ ∗ g ( s )) ∗ ds = 2 d c d τ Z R d Z Z B ( s, ε ) Φ ε ∗ f ( t ) s c Φ ( f )( s, ε ) p − s c Φ ( f )( s, ε ) − p (Ψ ε ∗ g ( t )) ∗ dtdεε d +1 ds + τ Z R d φ ∗ f ( s )( ψ ∗ g ( s )) ∗ ds def = I + II . Then by the Cauchy-Schwarz inequality, | I | . τ Z R d Z (cid:0) Z B ( s, ε ) | Φ ε ∗ f ( t ) | dtε d +1 (cid:1) s c Φ ( f )( s, ε ) p − dεds · τ Z R d Z (cid:0) Z B ( s, ε ) | Ψ ε ∗ g ( t ) | dtε d +1 (cid:1) s c Φ ( f )( s, ε ) − p dεds def = A · B. Replacing ε ∂∂ε P ε ( f ) and ε ∂∂ε P ε ( g ) in the proof of Lemma 2.10 by Φ ε ∗ f and Ψ ε ∗ g respectivelyand applying Lemma 5.7 and assumption (3) of Ψ and ψ , we get the estimates for the terms A and B that A . k s c Φ ( f ) k pp and B . k g k cq k s c Φ ( f ) k − pp . The term II is easy to deal with. By the H¨older inequality, Lemma 5.7 and assumption (3) again,we get (cid:12)(cid:12)(cid:12) τ Z R d φ ∗ f ( s )( ψ ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) ≤ k φ ∗ f k p k ψ ∗ g k q . k φ ∗ f k p k g k bmo cq . Combining the estimates for A , B and II, we finally get the desired inequality. (cid:3) perator-valued local Hardy spaces 31 We also need the radial version of Lemma 5.2. To this end, we need to majorize the radialsquare function by the conic one. When we consider the Poisson kernel, this result follows fromthe harmonicity of the Poisson integral (see Lemma 2.13). However, in the general case, theharmonicity is no longer available. To overcome this difficulty, a more sophisticated inequalityhas been developped in [51] to compare non-local radial and conic functions. Observe that theresult given in [51, Lemma 4.3] is a pointwise one, which also works for the local version of squarefunctions if we consider integration over the interval 0 < ε <
1. The following lemma is an obviousconsequence of [51, Lemma 4.3].
Lemma 5.3.
Let f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) . Then g c Φ ( f )( s ) . X | m | ≤ d s cD m Φ ( f )( s ) , ∀ s ∈ R d . Lemma 5.4.
Let ≤ p < . For f ∈ h cp ( R d , M ) ∩ L ( N ) and g ∈ bmo cq ( R d , M ) , (cid:12)(cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) . ( k g c Φ ( f ) k p + k φ ∗ f k p ) p k f k − p h cp k g k bmo cq . Proof.
This proof is similar to that of Lemma 5.2 and we keep the notation there. Let f ∈ h cp ( R d , M ) with compact support (relative to the variable of R d ). We assume that f is sufficientlynice so that all calculations below are legitimate. Now we need the radial version of s c Φ ( f )( s, ε ), g c Φ ( f )( s, ε ) = (cid:0) Z ε | Φ r ∗ f ( s ) | drr (cid:1) for s ∈ R d and 0 ≤ ε ≤
1. By approximation, we can assume that g c Φ ( f )( s, ε ) is invertible for every( s, ε ) ∈ S . By (5.1), (0.12) and the Fubini theorem, we have (cid:12)(cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) . τ Z R d Z | Φ ε ∗ f ( s ) | g c Φ ( f )( s, ε ) p − dεdsε · τ Z R d Z | Ψ ε ∗ g ( s ) | g c Φ ( f )( s, ε ) − p dεdsε + (cid:12)(cid:12)(cid:12) τ Z R d φ ∗ f ( s )( ψ ∗ g ( s )) ∗ ds (cid:12)(cid:12)(cid:12) = A ′ B ′ + II ′ . II ′ is treated exactly in the same way as before,II ′ . k φ ∗ f k p k ψ ∗ g k q . k φ ∗ f k pp k f k − p h cp k g k cq .A ′ is also estimated similarly as in Lemma 5.2, we have A ′ . k g c Φ ( f ) k pp .To estimate B ′ , we notice that the proof of [51, Lemma 1.3] also gives g c Φ ( f )( s, ε ) . X | m | ≤ d s cD m Φ ( f )( s, ε ) , where s cD m Φ ( f )( s, ε ) is defined by (5.5) with D m Φ instead of Φ. Then by the above inequality,Lemma 5.7 and inequality (5.3) with D m Φ instead of Φ, we obtain B ′ . X | m | ≤ d τ Z R d Z | Ψ ε ∗ g ( s ) | s cD m Φ ( f )( s, ε ) − p dεdsε . X | m | ≤ d k g k cq k s cD m Φ ( f ) k − pp . k g k cq k f k − p h cp . Therefore, | τ Z R d f ( s ) g ∗ ( s ) ds | . ( k g c Φ ( f ) k p + k φ ∗ f k p ) p k f k − p h cp k g k cq , which completes the proof. (cid:3) Proof of Theorem 5.1.
From Lemmas 5.2, 5.4 and Theorem 2.11, we conclude that if 1 ≤ p ≤ k f k h cp . k s c Φ ( f ) k p + k φ ∗ f k p , k f k h cp . k g c Φ ( f ) k p + k φ ∗ f k p . For the case 2 < p < ∞ , by Theorem 2.19, we can choose g ∈ h cq ( R d , M ) (with q the conjugateindex of p ) with norm one such that k f k h cp ≈ τ Z R d f ( s ) g ∗ ( s ) ds = τ Z R d Z Φ ε ∗ f ( s ) · (Ψ ε ∗ g ( s )) ∗ dsdεε + τ Z R d φ ∗ f ( s )( ψ ∗ g ( s )) ∗ ds. Then by the H¨older inequality and (5.4) (applied to g, Ψ and q ), k f k h cp . k g c Φ ( f ) k p k g c Ψ ( g ) k q + k φ ∗ f k p k ψ ∗ g k q . ( k g c Φ ( f ) k p + k φ ∗ f k p ) k g k h cq = k g c Φ ( f ) k p + k φ ∗ f k p . Similarly, we have k f k h cp . k s c Φ ( f ) k p + k φ ∗ f k p . Therefore, combined with (5.4) and (5.3), we have proved the assertion. (cid:3)
The rest part of this subsection is devoted to explaining how Theorem 5.1 generalizes the char-acterization of h cp ( R d , M ).Firstly and most naturally, we show how Theorem 5.1 covers the original definition of h cp ( R d , M ).Let us take Φ = − πI (P) and φ = P for example. A simple calculation shows that we can chooseΨ = − πI (P) and ψ = 4 πI (P) + P to fulfil (5.1). By the inverse Fourier transform formula, wehave − πf ∗ I (P) ε ( t ) = − π Z e π i t · ξ b f ( ξ ) | εξ | e − πε | ξ | dξ = ε ∂∂ε Z e π i t · ξ b f ( ξ ) e − πε | ξ | dξ = ε ∂∂ε P ε ( f )( t ) . So we return back to the original definition of h cp ( R d , M ). Theorem 5.1 implies that k f k h cp ≈ k s c Φ ( f ) k p + k φ ∗ f k p ≈ k g c Φ ( f ) k p + k φ ∗ f k p . In particular, we have the following equivalent norm of h cp ( R d , M ): Corollary 5.5.
Let ≤ p < ∞ . Then for any f ∈ h cp ( R d , M ) , we have k f k h cp ≈ k g c ( f ) k p + k P ∗ f k p . Secondly, consider Φ to be a Schwartz function on R d satisfying:(5.7) ( Φ is of vanishing mean;Φ is nondegenerate in the sense of (0.5) . Set Φ ε ( s ) = ε − d Φ( sε ) for ε >
0. In the sequel, we will show that every Schwartz function satisfying(5.7) fulfils the four conditions in the beginning of this subsection. So they all can be used tocharacterize h p ( R d , M ).It is a well-known elementary fact (ef. e.g. [44, p. 186]) that there exists a Schwartz function Ψof vanishing mean such that(5.8) Z ∞ b Φ( εξ ) b Ψ( εξ ) dεε = 1 , ∀ ξ ∈ R d \ { } . Lemma 5.6. R b Φ( ε · ) b Ψ( ε · ) dεε is an infinitely differentiable function on R d if we define its valueat the origin as . perator-valued local Hardy spaces 33 Proof.
To prove the assertion, it suffices to show that R b Φ( ε · ) b Ψ( ε · ) dεε is infinitely differentiable atthe origin. Given ε ∈ (0 , b Φ( ε · ) in the Taylor series at the origin b Φ( εξ ) = X | γ | ≤ N D γ b Φ(0) ε | γ | ξ γ γ ! + X | γ | = N +1 R γ ( εξ ) ξ γ , with the remainder of integral form equal to R γ ( εξ ) = ( N + 1) ε N +1 γ ! Z (1 − θ ) N D γ b Φ( θεξ ) dθ . Since b Φ(0) = 0, the above Taylor series implies that b Φ( εξ ) = X ≤| γ | ≤ N D γ b Φ(0) ε | γ | ξ γ γ ! + X | γ | = N +1 R γ ( εξ ) ξ γ . Similarly, we have b Ψ( εξ ) = X ≤| β | ≤ N D β b Ψ(0) ε | β | ξ β β ! + X | β | = N +1 R ′ β ( εξ ) ξ β , where R ′ β is the integral form remainder of b Ψ. Thus, both b Φ( εξ ) and b Ψ( εξ ) contain only powersof ε with order at least 1. Therefore, the integral R b Φ( εξ ) b Ψ( εξ ) dεε (and the integrals of arbitraryorder derivatives of b Φ( εξ ) and b Ψ( εξ )) converge uniformly for ξ ∈ R d close to the origin. We thenobtain that R b Φ( εξ ) b Ψ( εξ ) dεε is infinitely differentiable at the origin ξ = 0. (cid:3) It follows immediately from Lemma 5.6 that R ∞ b Φ( ε · ) b Ψ( ε · ) dεε is a Schwartz function if we defineits value at the origin by 1. Then we can find two other functions φ , ψ such that b φ, b ψ ∈ H σ ( R d ), b φ (0) > , b ψ (0) > b φ ( ξ ) b ψ ( ξ ) + Z b Φ( εξ ) b Ψ( εξ ) dεε = 1 , ∀ ξ ∈ R d ;Indeed, for β > | · | ) − β belongs to H σ ( R d ). On the other hand,if F ∈ S ( R d ), the function (1 + | · | ) β F is still in H σ ( R d ). Thus we obtain (5.1).Now let show that conditions (1) and (4) hold for Φ, φ satisfying (5.7). First, we deal withthe case 1 ≤ p ≤
2. Let H = L ((0 , , dεε ). Define the kernel k : R d → H by k ( s ) = Φ · ( s ) withΦ · ( s ) : ε Φ ε ( s ). Then we can check thatsup ξ ∈ R d k b Φ( εξ ) k H < ∞ , k Φ ε ( s ) k H . | s | d +1 , ∀ s ∈ R d \ { } and that k∇ Φ ε ( s ) k H . | s | d +1 , ∀ s ∈ R d \ { } . Thus, k satisfies the assumption of Corollary 4.3. By Remark 4.4, we have, for any 1 ≤ p ≤ k Φ ε ∗ f k L p ( N ; H c ) = k g c Φ ( f ) k p . k f k h cp . The treatment of s c Φ is similar. In this case, we take the Hilbert space H = L ( e Γ , dtdεε d +1 ). On theother hand, b φ ∈ H σ ( R d ) implies φ ∈ L ( R d ), then k φ ∗ f k L p ( N ) . k f k L p ( N ) . k f k h cp . Thus,combining the above estimates, we obtain k g c Φ ( f ) k p + k φ ∗ f k p . k f k h cp k s c Φ ( f ) k p + k φ ∗ f k p . k f k h cp . Then, a simple duality argument using (5.1) and Theorem 2.19 gives the above inequalities for thecase p >
2. Moreover, it is obvious that if we replace Φ by D m Φ, the above two inequalities stillhold for any 1 ≤ p < ∞ .In the end, it remains to check the condition (3) for Ψ, ψ obtained in (5.8) and (5.9). This canbe done by showing a Carleson measure characterization of bmo cq by general test functions. Theproof of the following lemma has the same pattern with that of Lemma 1.5, so is left to the reader. Lemma 5.7.
Let < q ≤ ∞ , g ∈ bmo cq ( R d , M ) and dµ g = | Ψ ε ∗ g ( s ) | dsdεε . Then dµ g is an M -valued q -Carleson measure on the strip R d × (0 , . Furthermore, let ψ be any function on R d such that (5.10) b ψ ∈ H σ ( R d ) with σ > d . We have max n(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) dµ g (cid:13)(cid:13) q , k ψ ∗ g k q o . k g k bmo cq . Remark 5.8.
It is worthwhile to note that, if Ψ and ψ are determined by (5.8) and (5.9), theopposite of the above lemma is also true. This can be deduced by a similar argument as that ofCorollary 2.12; we omit the details.By the discussion above, we deduce the following corollary from Theorem 5.1. Corollary 5.9.
Let Φ be the Schwartz function on R d satisfying (5.7) and φ be the function givenby (5.9) . Then for any ≤ p < ∞ , we have (5.11) k f k h cp ≈ k s c Φ ( f ) k p + k φ ∗ f k p ≈ k g c Φ ( f ) k p + k φ ∗ f k p with the relevant constants depending only on d, p, Φ and φ . Discrete characterizations.
In this subsection, we give a discrete characterization for operator-valued local Hardy spaces. To this end, we need some modifications of the four conditions in thebeginning of last subsection. The square functions s c Φ ( f ) and g c Φ ( f ) can be discretized as follows: g c,D Φ ( f )( s ) = (cid:16) X j ≥ | Φ j ∗ f ( s ) | (cid:17) ,s c,D Φ ( f )( s ) = (cid:16) X j ≥ dj Z B ( s, − j ) | Φ j ∗ f ( t ) | dt (cid:17) . Here Φ j is the inverse Fourier transform of Φ(2 − j · ). This time, to get a resolvent of the unit on R d , we need to assume that Φ , Ψ , φ, ψ satisfy(5.12) ∞ X j =1 b Φ(2 − j ξ ) b Ψ(2 − j ξ ) + b φ ( ξ ) b ψ ( ξ ) = 1 , ∀ ξ ∈ R d . In brief, the complex-valued infinitely differentiable function Φ considered in this subsection satis-fies:(1) Every D m Φ with 0 ≤ | m | ≤ d makes f s c,DD m Φ f and f g c,DD m Φ f Calder´on-Zygmundsingular integral operators in Corollary 4.3;(2) There exist functions Ψ , ψ and φ that fulfil (5.12);(3) The above Ψ and ψ make dµ Df = P j ≥ | Ψ j ∗ f ( s ) | ds × dδ − j ( ε ) (with δ − j ( ε ) the unit Diracmass at the point 2 − j ) and φ ∗ f satisfy:max n(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z T ( Q ) dµ Df (cid:13)(cid:13) q , k ψ ∗ f k q o . k f k bmo cq for q > φ makes f φ ∗ f a Calder´on-Zygmund singular integral operator in Corollary 4.3. Remark 5.10.
Any Schwartz function that has vanishing mean and is nondegenerate in the senseof (0.5) satisfies all the four conditions above.The following discrete version of Theorem 5.1 will play a crucial role in the study of operator-valued Triebel-Lizorkin spaces on R d in our forthcoming paper [49]. Now we fix the pairs (Φ , Ψ)and ( φ, ψ ) satisfying the above four conditions.
Theorem 5.11.
Let ≤ p < ∞ . Then for any f ∈ L ( M ; R cd ) + L ∞ ( M ; R cd ) , f ∈ h cp ( R d , M ) ifand only if s c,D Φ ( f ) ∈ L p ( N ) and φ ∗ f ∈ L p ( N ) if and only if g c,D Φ ( f ) ∈ L p ( N ) and φ ∗ f ∈ L p ( N ) .Moreover, k f k h cp ≈ k s c,D Φ ( f ) k p + k φ ∗ f k p ≈ k g c,D Φ ( f ) k p + k φ ∗ f k p perator-valued local Hardy spaces 35 with the relevant constants depending only on d, p , and the pairs (Φ , Ψ) and ( φ, ψ ) . The following paragraphs are devoted to the proof of Theorem 5.11 which is similar to thatof Theorem 5.1. We will just indicate the necessary modifications. We first prove the discretecounterparts of Lemmas 5.2 and 5.4.
Lemma 5.12.
Let ≤ p < and q be the conjugate index of p . For any f ∈ h cp ( R d , M ) ∩ L ( N ) and g ∈ bmo cq ( R d , M ) , (cid:12)(cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) . (cid:0) k s c,D Φ ( f ) k p + k φ ∗ f k p (cid:1) k g k bmo cq . Proof.
First, note that by (5.12), we have τ Z R d f ( s ) g ∗ ( s ) ds = τ Z R d X j ≥ Φ j ∗ f ( s ) (cid:0) Ψ j ∗ g ( s ) (cid:1) ∗ ds + τ Z R d φ ∗ f ( s ) (cid:0) ψ ∗ g ( s ) (cid:1) ∗ ds. The second term on the right hand side of the above formula is exactly the same as the corre-sponding term II in the proof of Lemma 5.2. Now we need the discrete versions of s c Φ and s c Φ : For j ≥ s ∈ R d , let s c,D Φ ( f )( s, j ) = (cid:16) X ≤ k ≤ j dk Z B ( s, − k − − j − ) | Φ j ∗ f ( t ) | dt (cid:17) s c,D Φ ( f )( s, j ) = (cid:16) X ≤ k ≤ j dk Z B ( s, − k − ) | Φ j ∗ f ( t ) | dt (cid:17) . Denote s c,D Φ ( f )( s, j ) and s c,D Φ ( f )( s, j ) simply by s ( s, j ) and s ( s, j ), respectively. By approximation,we may assume that s ( s, j ) and s ( s, j ) are invertible for every s ∈ R d and j ≥
1. By the Cauchy-Schwarz inequality, (cid:12)(cid:12)(cid:12) τ Z R d X j ≥ Φ j ∗ f ( s ) (cid:0) Ψ j ∗ g ( s ) (cid:1) ∗ ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) d c d τ Z R d X j dj Z B ( s, − j − ) Φ j ∗ f ( t ) (cid:0) Ψ j ∗ g ( t ) (cid:1) ∗ dt ds (cid:12)(cid:12)(cid:12) . τ Z R d X j s ( s, j ) p − (cid:16) dj Z B ( s, − j − )) | Φ j ∗ f ( t ) | dt (cid:17) ds · τ Z R d X j s ( s, j ) − p (cid:16) dj Z B ( s, − j − ) | Ψ j ∗ g ( t ) | dt (cid:17) ds def = A · B . The term A is less easy to estimate than the corresponding term A in the proof of Lemma 5.2. Todeal with it we simply set s j = s ( s, j ) and s = s ( s, + ∞ ) ≤ s c,D ( f )( s ). ThenA = τ Z R d X j ≥ s p − j ( s j − s j − ) ds ≤ τ Z R d X j s p − j ( s j − s j − ) ds = τ Z R d X j ( s j − s j − ) ds + τ Z R d X j s p − j s j − ( s j − s j − ) ds, where s = 0. Since 1 ≤ p < s p − j ≤ s p − , we have τ Z R d X j s p − j ( s j − s j − ) ds . τ Z R d s p ds ≤ k s c,D ( f ) k pp . On the other hand, τ Z R d X j s p − j s j − ( s j − s j − ) ds = τ Z R d X j s − p s p − j s j − s − p s p − ( s j − s j − ) s p − ds, since s j ≥ s j − for any j ≥
1, we have s − p s p − j s j − s − p ≤
1. Thus, by the H¨older inequality, τ Z R d X j s p − j s j − ( s j − s j − ) ds ≤ τ Z R d X j s p − ( s j − s j − ) s p − ds = τ Z R d s p ds ≤ k s c,D Φ ( f ) k pp . Combining the preceding inequalities, we get the desired estimate of A:A ≤ k s c,D Φ ( f ) k pp . The estimate of the term B is, however, almost identical to that of B in the proof of Lemma5.2. There are only two minor differences. The first one concerns the square function S c ( f )( s, j )in (2.8): it is now replaced by S c ( f )( s, j ) = (cid:16) X ≤ k ≤ j dk Z B ( c m,j , − k ) | Φ j ∗ f ( t ) | dt (cid:17) if s ∈ Q m,j . Then we have s ( s, j ) ≤ S c ( f )( s, j ). The second difference is about the Carleson characterizationof bmo cq ; we now use its discrete analogue, namely, dµ Dg . Apart from these two differences, theremainder of the argument is identical to that in the proof of Lemma 5.2. (cid:3) Lemma 5.13.
Let ≤ p < and f ∈ h cp ( R d , M ) ∩ L ( N ) , g ∈ bmo cq ( R d , M ) . Then (cid:12)(cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) . (cid:16) k g c,D Φ ( f ) k p + k φ ∗ f k p (cid:17) p k f k − p h cp k g k bmo cq . Proof.
We use the truncated version of g c,D Φ ( f ): g c,D Φ ( f )( s, j ) = (cid:16) X k ≤ j | Φ k ∗ f ( s ) | (cid:17) . The proof of [51, Lemma 4.3] is easily adapted to the present setting to ensure g c,D Φ ( f )( s, j ) . X | m | ≤ d s c,DD m Φ ( f )( s, j ) . Then (cid:12)(cid:12)(cid:12) τ Z R d f ( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) ≤ I ′ · II ′ + (cid:12)(cid:12)(cid:12) τ Z φ ∗ f ( s ) (cid:0) ψ ∗ g ( s ) (cid:1) ∗ ds (cid:12)(cid:12)(cid:12) , where I ′ = τ Z R d X j g c,D Φ ( f )( s, j ) p − | Φ j ∗ f ( s ) | ds , II ′ = τ Z R d X j g c,D Φ ( f )( s, j ) − p | Ψ j ∗ g ( s ) | ds . Both terms I ′ and II ′ are estimated exactly as before, so we haveI ′ ≤ k g c Φ ( f ) k pp and II ′ . k f k − p h cp k g k cq . This gives the announced assertion. (cid:3)
Armed with the previous two lemmas and the Calder´on-Zygmund theory in section 4, we canprove Theorem 5.11 in the same way as Theorem 5.1. Details are left to the reader.We also include a discrete Carleson measure characterization of bmo cq by general test functions.Much as the characterization in Lemma 5.7 and Remark 5.8, it is a byproduct of the proof ofTheorem 5.11. perator-valued local Hardy spaces 37 Corollary 5.14.
Let < q ≤ ∞ , ψ and Ψ be given in (5.12) . Assume further b ψ ∈ H σ ( R d ) with σ > d . Then for every g ∈ bmo cq , we have k g k bmo cq ≈ k ψ ∗ g k q + (cid:13)(cid:13)(cid:13) sup + s ∈ Q ⊂ R d | Q | < | Q | Z Q X j ≥− log ( l ( Q )) | Ψ j ∗ g ( s ) | ds (cid:13)(cid:13)(cid:13) q . The relation between h p ( R d , M ) and H p ( R d , M ) . Due to the noncommutativity, for any1 < p < ∞ and p = 2, the column operator-valued local Hardy space h cp ( R d , M ) and the columnoperator-valued Hardy space H cp ( R d , M ) are not equivalent. On the other hand, if we consider themixture spaces h p ( R d , M ) and H p ( R d , M ), then we will have the same situation as in the classicalcase.Since k P ∗ f k p . k f k p . k f k H cp for any 1 ≤ p ≤
2, we deduce the inclusion(5.13) H cp ( R d , M ) ⊂ h cp ( R d , M ) for 1 ≤ p ≤ . Then by the duality obtained in Theorem 2.19, we have(5.14) h cp ( R d , M ) ⊂ H cp ( R d , M ) for 2 < p < ∞ . However, we can see from the following proposition that we do not have the inverse inclusion of(5.13) nor (5.14).
Proposition 5.15.
Let φ be a function on R d such that b φ (0) ≥ and b φ ∈ H σ ( R d ) with σ > d .Let < p < ∞ . If for any f ∈ H cp ( R d , M ) , (5.15) k φ ∗ f k p . k f k H cp , then we must have b φ (0) = 0 .Proof. We prove the assertion by contradiction. Suppose that there exists φ such that b φ (0) > b φ ∈ H σ ( R d ) and (5.15) holds for any f ∈ H cp ( R d , M ). Since both H cp ( R d , M ) and L p ( N ) arehomogeneous spaces, we have, for any ε > k φ ∗ f ( ε · ) k p = k ( φ ε ∗ f )( ε · ) k p = ε − dp k φ ε ∗ f k p and k f ( ε · ) k H cp = ε − dp k f k H cp . This implies that(5.16) k φ ε ∗ f k p . k f k H cp , for any ε > ε . Now we consider a function f ∈ L p ( N )which takes values in S + M and such that supp b f is compact, i.e. there exists a positive real number N such that supp b f ⊂ { ξ ∈ R d : | ξ | ≤ N } . Since b φ (0) >
0, we can find ε > c > b φ ( ε ξ ) ≥ c whenever | ξ | ≤ N . Thus, in this case, k φ ε ∗ f k p ≥ c k f k p . Then by (5.16), we have k f k p . k f k H cp , which leads to a contradiction when p >
2. Therefore, b φ (0) = 0. (cid:3) By the definition of the h cp -norm and the duality in Theorem 2.19, we get the following result: Corollary 5.16.
Let ≤ p < ∞ and p = 2 . h cp ( R d , M ) and H cp ( R d , M ) are not equivalent. Although h cp ( R d , M ) and H cp ( R d , M ) do not coincide when p = 2, for those functions whoseFourier transforms vanish at the origin, their h cp -norms and H cp -norms are still equivalent. Theorem 5.17.
Let φ ∈ S such that R R d φ ( s ) ds = 1 . (1) If ≤ p ≤ and f ∈ h cp ( R d , M ) , then f − φ ∗ f ∈ H cp ( R d , M ) and k f − φ ∗ f k H cp . k f k h cp . (2) If < p < ∞ and f ∈ H cp ( R d , M ) , then f − φ ∗ f ∈ h cp ( R d , M ) and k f − φ ∗ f k h cp . k f k H cp . Proof. (1) Let f ∈ h cp ( R d , M ) and Φ be a nondegenerate Schwartz function with vanishing mean.By the general characterization of H cp ( R d , M ) in Lemma 0.1, k f − φ ∗ f k H cp ( R d , M ) ≈ k G c Φ ( f − φ ∗ f ) k p .Let us split k G c Φ ( f − φ ∗ f ) k p into two parts: k G c Φ ( f − φ ∗ f ) k p . (cid:13)(cid:13)(cid:13)(cid:0) Z | Φ ε ∗ ( f − φ ∗ f ) | dεε (cid:1) (cid:13)(cid:13)(cid:13) p + (cid:13)(cid:13)(cid:13)(cid:0) Z ∞ | Φ ε ∗ ( f − φ ∗ f ) | dεε (cid:1) (cid:13)(cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13)(cid:0) Z | Φ ε ∗ ( f − φ ∗ f ) | dεε (cid:1) (cid:13)(cid:13)(cid:13) p + (cid:13)(cid:13)(cid:13)(cid:0) Z ∞ | (Φ ε − Φ ε ∗ φ ) ∗ f | dεε (cid:1) (cid:13)(cid:13)(cid:13) p . In order to estimate the first term in the last equality, we notice that φ ∗ f ∈ h cp ( R d , M ), thus wehave f − φ ∗ f ∈ h cp ( R d , M ). Then by Theorem 5.1, this term can be majorized from above by k f k h cp .To deal with the second term, we express it as a Calder´on-Zygmund operator with Hilbert-valuedkernel. Let H = L ((1 , + ∞ ) , dεε ) and define the kernel k : R d → H by k ( s ) = Φ · ( s ) − Φ · ∗ φ ( s )(Φ · ( s ) being the function ε Φ ε ( s )). Now we prove that k satisfies the hypotheses of Corollary4.3. The condition (1) of that corollary is easy to verify. So we only check the conditions (2) and(3) there. By the fact that R R d φ ( s ) ds = 1 and the mean value theorem, we have (cid:12)(cid:12) (Φ ε − Φ ε ∗ φ )( s ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z R d [Φ ε ( s ) − Φ ε ( s − t )] φ ( t ) dt (cid:12)(cid:12)(cid:12) ≤ Z R d | t | ε d +1 sup <θ< (cid:12)(cid:12) ∇ Φ (cid:0) s − θtε (cid:1)(cid:12)(cid:12) | φ ( t ) | dt. Then we split the last integral into two parts: (cid:13)(cid:13) (Φ · − Φ · ∗ φ )( s ) (cid:13)(cid:13) H . (cid:16) Z ∞ (cid:0) Z | t | < | s | | t | ε d +1 sup <θ< (cid:12)(cid:12) ∇ Φ (cid:0) s − θtε (cid:1)(cid:12)(cid:12) | φ ( t ) | dt (cid:1) dεε (cid:17) + (cid:16) Z ∞ (cid:0) Z | t | > | s | | t | ε d +1 sup <θ< (cid:12)(cid:12) ∇ Φ (cid:0) s − θtε (cid:1)(cid:12)(cid:12) | φ ( t ) | dt (cid:1) dεε (cid:17) def = I + II . If | t | < | s | , we have | s − θt | ≥ | s | , thus |∇ Φ( s − θtε ) | . ε d + 12 | s | d + 12 for any 0 ≤ θ ≤
1. ThenI . (cid:0) Z ∞ ε dε (cid:1) | s | d + . | s | d + . When | t | > | s | , since φ ∈ S , we have R | t | > | s | | t | | φ ( t ) | dt . | s | d + 12 . HenceII . (cid:0) Z ∞ ε d +2 dεε (cid:1) · | s | d + . | s | d + . The estimates of I and II imply k (Φ ε − Φ ε ∗ φ )( s ) k H . | s | d + . In a similar way, we obtain k∇ (Φ ε − Φ ε ∗ φ )( s ) k H . | s | d +1 . Thus, it follows from Corollary 4.3 that (cid:13)(cid:13)(cid:13) ( R ∞ | (Φ ε − Φ ε ∗ φ ) ∗ f | dεε ) (cid:13)(cid:13)(cid:13) p is also majorized fromabove by k f k h cp .(2) The case p > cp and h cq (Theorem 2.19) and thatbetween H cp and H cq ( q being the conjugate index of p ). There exists g ∈ h cq ( R d , M ) with norm perator-valued local Hardy spaces 39 one such that k f − φ ∗ f k h cp = (cid:12)(cid:12)(cid:12) τ Z R d ( f − φ ∗ f )( s ) g ∗ ( s ) ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) τ Z R d f ( s )( g ∗ − φ ∗ g ∗ )( s ) ds (cid:12)(cid:12)(cid:12) ≤ k f k H cp k g − φ ∗ g k H cq . k f k H cp k g k h cq = k f k H cp , which completes the proof. (cid:3) From the interpolation result of mixture local hardy spaces in Proposition 3.4, we can deducethe equivalence between mixture local Hardy spaces and L p -spaces. Proposition 5.18.
For any < p < ∞ , h p ( R d , M ) = H p ( R d , M ) = L p ( N ) with equivalentnorms.Proof. It is known that H p ( R d , M ) = L p ( N ) with equivalent norms. One can see [30, Corollary5.4] for more details. One the other hand, since L ∞ ( N ) ⊂ bmo c ( R d , M ), by duality, we geth c ( R d , M ) ⊂ L ( N ). Combining (1.2) and the interpolation result in Theorem 3.3, we deduce thath cp ( R d , M ) ⊂ L p ( N ) for any 1 < p ≤ L p ( N ) ⊂ h cp ( R d , M ) for any 2 < p < ∞ . Similarly,we also have h rp ( R d , M ) ⊂ L p ( N ) for any 1 < p ≤ L p ( N ) ⊂ h rp ( R d , M ) for any 2 < p < ∞ .Combined with (5.13) and (5.14), we get(5.17) H p ( R d , M ) ⊂ h p ( R d , M ) ⊂ L p ( N ) for 1 < p ≤ , and(5.18) L p ( N ) ⊂ h p ( R d , M ) ⊂ H p ( R d , M ) for 2 < p < ∞ . Then (5.17), (5.18) and [30, Corollary 5.4] imply thath p ( R d , M ) = H p ( R d , M ) = L p ( N ) for 1 < p < ∞ , which completes the proof. (cid:3) The atomic decomposition
In this section, we give the atomic decomposition of h c ( R d , M ). The atomic decomposition of H c ( R d , M ) studied in [30] and the characterizations obtained in the last section will be the maintools for us. Definition 6.1.
Let Q be a cube in R d with | Q | ≤ . If | Q | = 1 , an h c -atom associated with Q isa function a ∈ L ( M ; L c ( R d )) such that • supp a ⊂ Q ; • τ (cid:0) R Q | a ( s ) | ds (cid:1) ≤ | Q | − .If | Q | < , we assume additionally: • R Q a ( s ) ds = 0 . Let h c , at ( R d , M ) be the space of all f admitting a representation of the form f = ∞ X j =1 λ j a j , where the a j ’s are h c -atoms and λ j ∈ C such that P ∞ j =1 | λ j | < ∞ . The above series converges inthe sense of distribution. We equip h c , at ( R d , M ) with the following norm: k f k h c , at = inf { ∞ X j =1 | λ j | : f = ∞ X j =1 λ j a j ; a j ’s are h c -atoms, λ j ∈ C } . Similarly, we define the row version h r , at ( R d , M ). Then we seth , at ( R d , M ) = h c , at ( R d , M ) + h r , at ( R d , M ) . Theorem 6.2.
We have h c , at ( R d , M ) = h c ( R d , M ) with equivalent norms. Proof.
First, we show the inclusion h c , at ( R d , M ) ⊂ h c ( R d , M ). To this end, it suffices to provethat for any atom a in Definition 6.1, we have(6.1) k a k h c . . Recall that the atomic decomposition of H c ( R d , M ) has been considered in [30]. An H c -atom is afunction b ∈ L ( M ; L c ( R d )) such that, for some cube Q , • supp b ⊂ Q ; • R Q b ( s ) ds = 0; • τ (cid:0) R Q | b ( s ) | ds (cid:1) ≤ | Q | − .If a is supported in Q with | Q | <
1, then a is also an H c -atom, so k a k h c . k a k H c .
1. Nowassume that the supporting cube Q of a is of side length one. We use the discrete characterizationobtained in Theorem 5.11, i.e. k a k h c ≈ (cid:13)(cid:13) ( ∞ X j =1 | Φ j ∗ a | ) (cid:13)(cid:13) + k φ ∗ a k . Apart from the assumption on Φ and φ in Theorem 5.1, we may take Φ and φ satisfyingsupp Φ , supp φ ⊂ B = { s ∈ R d : | s | ≤ } . Then supp φ ∗ a ⊂ Q and supp Φ ε ∗ a ⊂ Q for any 0 < ε < . By the Cauchy-Schwarz inequality we have k φ ∗ a k ≤ Z Q (cid:0) Z Q | φ ( t − s ) | ds (cid:1) · τ (cid:0) Z | a ( s ) | ds (cid:1) dt . . Similarly, (cid:13)(cid:13) ( ∞ X j =1 | Φ j ∗ a | ) (cid:13)(cid:13) = τ Z Q ( ∞ X j =1 | Φ j ∗ a ( s ) | ) ds . τ (cid:16) Z Q ∞ X j =1 | Φ j ∗ a ( s ) | ds (cid:17) = τ (cid:16) Z R d ∞ X j =1 | b Φ(2 − j ξ ) b a ( ξ ) | dξ (cid:17) ≤ τ (cid:0) Z | a ( s ) | ds (cid:1) ≤ . Therefore, h c , at ( R d , M ) ⊂ h c ( R d , M ).Now we turn to proving the reverse inclusion. Observe that H c -atoms are also h c -atoms. Then bythe atomic decomposition of H c ( R d , M ) and the duality between H c ( R d , M ) and BMO c ( R d , M ),every continuous functional ℓ on h c , at ( R d , M ) corresponds to a function g ∈ BMO c ( R d , M ). More-over, since for any cube Q with side length one, L (cid:0) M ; L c ( Q ) (cid:1) ⊂ h c , at ( R d , M ), ℓ induces acontinuous functional on L (cid:0) M ; L c ( Q ) (cid:1) with norm less than or equal to k ℓ k (h c , at ) ∗ . Thus, thefunction g satisfies the condition that(6.2) g ∈ BMO c ( R d , M ) and sup Q ⊂ R d | Q | =1 k g | Q k L ∞ ( M ; L c ( Q )) ≤ k ℓ k (h c , at ) ∗ . Consequently, g ∈ bmo c ( R d , M ) and ℓ ( f ) = τ Z R d f ( s ) g ∗ ( s ) ds, ∀ f ∈ h c , at ( R d , M ) . Thus, h c , at ( R d , M ) ∗ ⊂ bmo c ( R d , M ). On the other hand, by the previous result, we havebmo c ( R d , M ) ⊂ h c , at ( R d , M ) ∗ . Thus, h c , at ( R d , M ) ∗ = bmo c ( R d , M ) with equivalent norms. Sinceh c , at ( R d , M ) ⊂ h c ( R d , M ) densely, we deduce that h c , at ( R d , M ) = h c ( R d , M ) with equivalentnorms. (cid:3) perator-valued local Hardy spaces 41 Acknowledgements.
The authors are greatly indebted to Professor Quanhua Xu for havingsuggested to them the subject of this paper, for many helpful discussions and very careful readingof this paper. The authors are partially supported by the the National Natural Science Foundationof China (grant no. 11301401).
References [1] T. Bekjan, Z. Chen, M. Perrin and Z. Yin. Atomic decomposition and interpolation for Hardy spaces of non-commutative martingales.
J. Funct. Anal.
258 (2010), no. 7, 2483-2505.[2] J. Bergh and J. L¨ofstr¨om. Interpolation Spaces: An Introduction. Springer, Berlin, 1976.[3] Z. Chen, Q. Xu and Z. Yin. Harmonic analysis on quantum tori.
Comm. Math. Phys.
322 (2013), no. 3, 755-805.[4] R. Coifman, Y. Meyer and E. M. Stein. Some new function spaces and their applications to Harmonic analysis.
J. Funct. Anal.
62 (1985), no. 2, 304-335.[5] R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis.
Bull. Amer. Math. Soc.
J. Math. Anal. Appl. H p spaces of several variables. Acta Math.
129 (1972), no. 3-4, 137-193.[9] M. Frazier, R. Torres and G. Weiss. The boundedness of Caldern-Zygmund operators on the spaces ˙ F α,qp . Rev.Mat. Iberoam.
Vol. 4 (1988), 41-72.[10] J. Garca-Cuerva and J.L. Rubio de Francia. Weighted norm inequalities and related topics. North-HollandMathematics Studies, 116. Notas de Matemtica, 104. North-Holland Publishing Co., Amsterdam, 1985. x+604pp.[11] J. B. Garnett. Bounded analytic functions. Pure and Applied Mathematics, 96. Academic Press, Inc. [HarcourtBrace Jovanovich, Publishers], New York-London, 1981. xvi+467 pp.[12] D. Goldberg. A local version of real Hardy spaces.
Duke Math. J.
46 (1979), no. 1, 27-42.[13] L. Grafakos. Classical Fourier analysis. Second Edition. Springer-Verlag, New York, 2008.[14] G. Hong, L. D. L´opez-Snchez, J. M. Martell, and J. Parcet. Caldern-Zygmund operator associated to matrix-valued kernels.
Int. Math. Res.
Not. 2014, no. 5, 1221-1252.[15] G. Hong and T. Mei. John-Nirenberg inequality and atomic decomposition for noncommutative martingales.
J.Funct. Anal.
263 (2012), no. 4, 1064-1097.[16] T. Hyt¨onen. The real-variable Hardy space and BMO. Lecture notes of a course at the University of Helsinki,Winter 2010.[17] Y. Jiao, F. Sukochev, D. Zanin and D. Zhou. Johnson-Schechtman inequalities for noncommutative martingales.
J. Funct. Anal.
272 (2017), no. 3, 976-1016.[18] M. Junge. Doob’s inequality for non-commutative martingales.
J. Reine Angew. Math.
549 (2002), 149-190.[19] M. Junge and M. Musat. A noncommutative version of the John-Nirenberg theorem.
Trans. Amer. Math. Soc.
359 (2007), no. 1, 115-142.[20] M. Junge, C. Le Merdy and Q. Xu. H ∞ -functional calculus and square functions on noncommutative L p -spaces. Astrisque
No. 305 (2006), vi+138 pp.[21] M. Junge and Q. Xu. Non-commutative Burkholder/Rosenthal Inequalities.
Ann. Prob.
31 (2003), no. 2, 948-995.[22] M. Junge and Q. Xu. Noncommutative Burkholder/Rosenthal inequalities. II. Applications.
Israel J. Math.
J. Amer. Math. Soc.
20 (2007), 385-439.[24] M. Junge and T. Mei. Noncommutative Riesz transforms - a probabilistic approach.
Amer. J. Math.
132 (2010),611-681.[25] M. Junge and T. Mei. BMO spaces associated with semigroups of operators.
Math. Ann.
352 (2012), 691-743.[26] M. Junge and T. Mei and J. Parcet. Smooth Fourier multipliers on group von Neumann algebras.
GAFA. C ∗ -modules. A toolkit for operator algebraists. London Mathematical Society Lecture NoteSeries, 210. Cambridge University Press, Cambridge, 1995. x+130 pp.[28] F. Lust-Piquard and G. Pisier. Noncommutative Khintchine and Paley Inequalities. Ark. Mat.
29 (1991), 241-260.[29] T. Mei. BMO is the intersection of two translates of dyadic BMO.
C. R. Math. Acad. Sci. Paris
336 (2003),no. 12, 1003-1006.[30] T. Mei. Operator-valued Hardy Spaces.
Mem. Amer. Math. Soc.
188 (2007), no. 881, vi+64 pp.[31] T. Mei. Tent spaces associated with semigroups of operators.
J. Funct. Anal.
255 (2008), no. 12, 3356-3406.[32] M. Musat. Interpolation between non-commutative BMO and non-commutative L p -spaces. J. Funct. Anal.
J. Funct.Anal.
256 (2009), no. 2, 509-593.[34] J. Parcet and N. Randrianantoanina. Gundy’s decomposition for non-commutative martingales and applications.
Proc. London Math. Soc.
93 (2006), 227-252. [35] G. Pisier. Martingales in Banach spaces. Cambridge Studies in Advanced Mathematics, 155. Cambridge Uni-versity Press, Cambridge, 2016. xxviii+561 pp.[36] G. Pisier. Noncommutative vector valued L p spaces and completely p -summing maps. Ast´erisque.
247 (1998),vi+131 pp.[37] G. Pisier and Q. Xu. Non-commutative Martingale Inequalities.
Comm. Math. Phys.
189 (1997), 667-698.[38] G. Pisier and Q. Xu. Noncommutative L p -spaces. Handbook of the geometry of Banach spaces, Vol. 2, 1459-1517, North-Holland, Amsterdam, 2003.[39] N. Randrianantoanina. Noncommutative martingale transforms. J. Funct. Anal.
194 (2002), 181-212.[40] N. Randrianantoanina. Conditional square functions for noncommutative martingales.
Ann. Prob.
35 (2007),1039-1070.[41] N. Randrianantoanina and L. Wu. Noncommutative Burkholder/Rosenthal inequalities associated with convexfunctions.
Ann. Inst. Henri Poincar´e Probab. Stat.
53 (2017), no. 4, 1575-1605.[42] N. Randrianantoanina, L. Wu and Q. Xu. Noncommutative Davis type decompositions and applications.arXiv:1712.01374.[43] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No.30 Princeton University Press, Princeton, N.J. 1970.[44] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Math-ematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.[45] E. M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series,No. 32. Princeton University Press, Princeton, N.J., 1971.[46] R. H. Torres. Boundedness results for operators with singular kernels on distribution spaces.
Mem. Amer. Math.Soc.
90 (1991), no. 442, viii+172 pp.[47] H. Triebel. Theory of function spaces. II. Monographs in Mathematics, 84. Birkh¨auser Verlag, Basel, 1992.[48] T. Wolff. A note on interpolation spaces. Harmonic analysis (Minneapolis, Minn., 1981), pp. 199-204, LectureNotes in Math., 908, Springer, Berlin-New York, 1982.[49] R. Xia and X. Xiong. Operator-valued inhomogeneous Triebel-Lizorkin spaces. Preprint.[50] R. Xia and X. Xiong. Mapping properties of operator-valued pseudo-differential operators. Preprint.[51] R. Xia, X. Xiong and Q. Xu. Characterizations of operator-valued Hardy spaces and applications to harmonicanalysis on quantum tori.
Adv. Math.
291 (2016), 183-227.[52] X. Xiong, Q. Xu and Z. Yin. Function spaces on quantum tori.
C. R. Math. Acad. Sci. Paris
353 (2015), no.8, 729-734.[53] X. Xiong, Q. Xu and Z. Yin. Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori.
Mem. Amer. Math.Soc.
252 (2018), no. 1203, vi+118 pp.[54] Q. Xu. Noncommutative L p -spaces and martingale inequalities. Book manuscript, 2007. Laboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 25030 Besanc¸on Cedex, France, andInstituto de Ciencias Matem´aticas, 28049 Madrid, Spain
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