COMMUTATORS IN REAL INTERPOLATION WITH QUASI-POWER PARAMETERS

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.


Introduction
A current topic in interpolation theory is to estimate the nonlinear commutators in different situations.In the eighties, Rochberg, Weiss, and Jawerth initiated the study of the second order and abstract commutator theorems for scales of the classical complex and real interpolation spaces.The applications in many important differential and integral expressions were found ever since (see [6,7,10,17] for details).The higher order commutator theorem was proved by Rochberg for the complex interpolation [16], and by Milman for the real interpolation [14].Recently, these results were extended to the real interpolation with the Calder ón weights [1].In addition, Milman and Rochberg presented a comparison between the commutator results of the real and complex interpolation methods, and emphasized the role of internal cancellation in these results [15].
In the present paper, we formulate the basic higher order commutator theorem for a wide family of the real interpolation methods, the Kand J-methods due to Brudnyȋ and Krugljak associated with the quasi-power parameters.This is motivated, in a large part, by the observation that the Hardy inequalities for the L p -spaces (1 < p < ∞) play an important role in the proof of the commutator theorem for the classical interpolation methods.Furthermore, the quasilogarithmic operators in our commutator theorem is more general than the original ones.These operators, which are inspired by Kalton's work [12,13], are determined by a much wide choice of Lipschitz functions.It therefore seems natural to us to carry over the commutator theorem to the real interpolation methods associated with the function spaces on which Hardy inequalities are valid.This unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the general theory of commutator estimates for the real interpolation methods.
This paper is organized as follows: in Section 2, we review the real interpolation methods with quasi-power parameters.In Section 3, we give definitions and results about quasilogarithmic operators corresponding to the above mentioned real interpolation methods.In Section 4, the main results about cancellation and commutators are formulated.Some useful remarks are included in Section 5.
Conventions.The notations ⊆ and = between Banach spaces stand for continuous inclusion and isomorphic equivalence, respectively.The notation Ꮾ(X, Y ), respectively, Ꮾ( X, Ȳ ), stands for the Banach space of all bounded linear operators from Banach space X to Banach space Y , respectively, from Banach couple X to Banach couple Ȳ .We simply write Ꮾ(X) = Ꮾ(X, X) and Ꮾ( X) = Ꮾ( X, X).

Real interpolation with quasi-power parameters
We suppose that the reader is familiar with the basic notations and definitions of interpolation theory, and we refer to [3,4] for background information.We start by giving a brief review of the real interpolation methods with the quasi-power parameters.
Assume that X = (X 0 ,X 1 ) is a Banach couple with ∆ X = X 0 ∩ X 1 and Σ X = X 0 + X 1 .For t > 0, recall that the Jand K-functionals on ∆ X and Σ X, respectively, are given by J t, x; In particular, J(1,x; X) = x ∆ X and K(1,x; X) = x Σ X .For simplicity, we usually write J(t, x) = J(t, x; X) and K(t, x) = K(t, x; X) when it does not lead to ambiguity.We now introduce Brudnyȋ-Krugljak's Kand J-methods as follows.
Definition 2.1.Let Φ be a Banach function space over (R + ,dt/t) such that 1 ∧ t ∈ Φ and and define J Φ ( X) as the space of all x ∈ Σ X, which permits a canonical representation x = ∞ 0 u(t) dt/t for a strongly measurable function u : R + → ∆ X, with Ming Fan 241 the norm Let (Ω,µ) be a completely σ-finite measurable space and let w be a weight on (Ω,µ).We denote by L 0 (Ω,dµ) the space of all measurable functions on (Ω,µ) with f L 0 = µ(supp f ) < ∞, and denote by the weighted L p -space for 1 ≤ p < ∞, with the usual change for p = ∞.For the measurable function ϕ : R + → R + , we write In particular, we write L p θ = L p ϕ θ for the power function ϕ θ (t) = t θ .Now let us define the Hardy operators P, Q and the Calder ón operator S on L 0 (R + ,dt/t) by (2.6) The Banach function space Φ is said to be a quasi-power parameter for the real interpolation if S ∈ Ꮾ(Φ).Observe that S ∈ Ꮾ(Φ) if and only if P, Q ∈ Ꮾ(Φ).In this case, we have the equivalence J Φ ( X) = K Φ ( X) with the norm estimate for all Banach couples X [4, Corollary 3.5.35].A function ϕ : R + → R + is said to be quasi-power if Sϕ is equivalent to ϕ.The function space L p ϕ is a quasi-power parameter for the real interpolation if ϕ is a quasi-power function.We simply write J p ϕ ( X) = J L p ϕ ( X) and The following examples show that several interpolation methods in the literature can be formulated in terms of Brudnyȋ-Krugljak's Kand J-methods with the quasi-power parameters for the real interpolation.
Example 2.2 (real methods associated with Calder ón weights [1]).A weight w on (R + ,dt) is said to be a Calderón weight if the Calder ón type operator S ∈ Ꮾ(L p (w dt)), where For a Banach couple X, we define Xp,w,K to be the space of all those x ∈ Σ X, for which K(t, x)/t ∈ L p (w dt), with the norm x Xp,w,K = K(t, x)/t L p (w dt) , and define Xp,w,J to be the space of all those x ∈ Σ X, for which there exists a strongly measurable function u : R + → ∆ X satisfying x = ∞ 0 u(t) dt/t and J(t, u(t))/t ∈ L p (w dt), with the norm x Xp,w,J = inf u { J(t, u(t))/t L p (w dt) }.If we set ϕ(t) = t 1−1/p w(t) −1/ p , then the space L p ϕ is a quasi-power parameter for the real interpolation.By substituting Φ with L p ϕ in (2.7), we obtain that Xp,w,J = J p ϕ X = K p ϕ X = Xp,w,K . (2.9) Example 2.3 (E-and E α -methods [4,10]).Let r > 0 and let x ∈ Σ X.We first consider the E-functional which is given by For a quasi-power parameter Φ for the real interpolation, we set and define the space E Ψ ( X) consisting of all x ∈ Σ X such that (2.12) According to [10, Lemma 2.1], for each x ∈ Σ X, the right continuous inverse of the function E α (r, x) is K(t, x)/t α .For p ≥ 1 and 0 < θ < 1, we choose η = 1/(θ − α), q = p(α − θ) and define the space Xθ,p,E α consisting of all x ∈ Σ X for which (2.15)

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By the change of variable r = K(t, x)/t α and integrating by parts, we have (2.16)

Quasilogarithmic operators
Now consider the relevant definitions and results about quasilogarithmic operators in the real interpolation, which came originally from [10].In this section, however, those operators will be considered in a more general sense.The main result here is to present the connection between the quasilogarithmic operators in different versions of the real interpolation methods.In this and the next section, we always assume that Φ is a quasi-power parameter for the real interpolation.
Let X be a Banach couple, and let c > 1 be a constant which is fixed during the discussion in this section.For x ∈ Σ X, the decomposition x = x 0 (t) + x 1 (t), t > 0, is (c-)almost optimal for the K-methods if A (c-)almost optimal projection for the K-methods is a (usually nonlinear) operator D K (t) : Σ X → X 0 defined by for some almost optimal decomposition.It is always possible to choose x j (t) ( j = 0, 1) to be continuous.Now we consider the J Φ methods.Given x ∈ Σ X, we say that defines a (c-)almost optimal decomposition for the Here and throughout, we assume that ψ : R → R is a Lipschitz function satisfying where γ ψ is a constant depending on ψ.For n = 1, 2,..., we introduce the following concept.
Definition 3.1.The quasilogarithmic operators Ω J ψ,n and Ω K ψ,n of order n are defined on Σ X in the following way In particular, if ψ(t) = t, then Ω J ψ,n = Ω n,J and Ω K ψ,n = −Ω n,K as given in [15].Remark 3.2.For the E α -methods, respectively, E-methods, we can also introduce the almost optimal decomposition x = x 0 (r) + x 1 (r), r > 0, and the almost optimal projection D E α (r)x = x 0 (r), respectively, D E (r)x = x 0 (r), where for α > 1, and respectively, The corresponding quasilogarithmic operator Ω α ψ,n = Ω E α ψ,n , respectively, Ω E ψ,n , can be similarly defined by ) (3.10) Before proceeding, recall some basic facts about rearrangement invariant function spaces, which can be found in [2].Let (Ω,µ) be a completely σ-finite measurable space as before.For f ∈ L 0 (Ω,dµ), we denote the rank function of f by the distribution function of f by and the measure-preserving rearrangement of Two functions in L 0 (Ω,dµ) are said to be equimeasurable if they have the same distribution function.A Banach function space X over (Ω,dµ) is said to be rearrangement invariant (r.i. in short) if f X = g X for every pair of equimeasurable functions f and g in L 0 (Ω,dµ).We refer to [2, Definitions III.5.10 and III.5.12] for the Boyd indices of r.i.spaces.
The main tools to be used in the course of the proof for the main theorems in this and the next section are integrating by parts and internal cancellation.The following lemma is crucial for this reason.We refer to [5, Lemma 3.2] for the classical case.Lemma 3.4.Let Φ be a quasi-power parameter for the real interpolation, let ψ be a Lipschitz function satisfying (3.4), and let u : R + → ∆ X, for which Proof.We begin with the special case ψ(t) = t.Let Φ * and Φ be given as in (3.14) and (3.15).Then the Boyd indices α, β of Φ * satisfy 0 < α ≤ β < 1 by Lemma 3.3.
It is reasonable to extend [6, Theorem 2.8] as follows: Theorem 3.5.Let x ∈ Σ X0 , the closure of ∆ X in Σ X.Then there is a decomposition x = ∞ 0 u(t) dt/t for the J Φ methods, which is almost optimal simultaneously for all choices of quasi-power parameters Φ for the real interpolation.Furthermore, there exists an almost optimal projection of x for the K Φ methods such that As in the proof of [6, Theorem 2.8], we can find a decomposition where That is, D K (t)x = x 0 (t) is an almost optimal projection for the K-method.By the fundamental inequality for the real interpolation [11] and by the estimate in (2.7), we have This implies that D J (t)x = u(t) is an almost optimal decomposition for the J Φ methods for all choices of quasi-power parameters Φ.Further calculation gives that (3.33) Here we use Lemma 3.4 integration by parts.
Remark 3.6.For the operator Ω α ψ,n with α ≥ 1, as in Remark 3.2, we consider by Remark 3.2 and a similar argument as in Theorem 3.5.

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In the following examples, we consider some Banach couples of function spaces, with weights w 0 , w 1 sometimes, over the completely σ-finite measurable space (Ω,µ), and determine the corresponding quasilogarithmic operators.

.43)
Example 3.9 (On (L p 0 ,L p 1 ), where L p j = L p j (Ω,dµ) with 1 ≤ p 0 < p 1 ≤ ∞).For f ∈ L p 0 + L p 1 , let r f , λ f , and f * be given as before.It is known that λ f and f * are right continuous and nonincreasing with f * (t) = λ −1 f (t) for each t at which f * is continuous.Moreover, lim s→0 λ f (s) = 0 and r f (ω [10,Section 4.3].We can now calculate

and let
(3.45) By dealing with two different cases, we obtain that (3.47) Remark 3.10.In Examples 3.8 and 3.9, on the couple (L 1 (dµ),L ∞ (dµ)), we have These operators were first studied by Kalton in [12, Section 3], and were used by him to determine the commutators of trace-class operators [13].In those papers, Kalton developed a general commutator theorem for the rearrangement invariant function spaces with the nontrivial Boyd indices.It is an interesting problem how to put Kalton's work in the context of real interpolation [7, Section VIII].

Cancellation and commutator estimates
As a preparation for the commutator theorem, we include now the following cancellation result.The arguments on this topic have been used to deal with the classical real interpolation in [15] but are of a much more general nature, so we take them up once more.
Theorem 4.1.Let Φ be a quasi-power parameter for the real interpolation, let ψ be a Lipschitz function satisfying (3.4), and let y ∈ Σ X. (i) If where Ming Fan 251 where x j : R + → ∆ X ( j = 0, 1), for which x 0 (t) + x 1 (t) = 0, and ) Proof.(i) For n = 1, we have in terms of integrating-by-parts and Lemma 3.4.Observe that according to the proof of [15,Theorem 3].Therefore, Since Φ is a quasi-power parameter for the real interpolation, we obtain that and hence y J Φ ≤ γ ψ S Ꮾ(Φ) J(t, u(t)) Φ .
Proof.We mention first that all estimates are trivial for n = 0 since C 0 = T.For (4.22), we have C 1 (T) = [T, Ω 1 ] and hence Furthermore, we have This, together with Lemma 3.4, implies that (4.32)By Theorem 4.1(i) again, we obtain According to Theorem 4.1(ii), we have (4.37) The case n > 1 and the estimates in (4.24) and (4.25) can be proved by making a simple substitution or by induction as we did above.We leave all details to the reader.

Final remarks
Remark 5.1.According to Theorem 4.3, C n can be considered as a bounded (nonlinear) operator on Ꮾ( X, Ȳ ).This uniform boundedness plays an important role in future studies.
The equivalence X = K Φ (L 1 ,L ∞ ) can be proved as in [8, Section 1].With this construction, we may apply Theorem 4.3 on the r.i.space X with the nontrivial Boyd indices, and obtain the commutator theorem for operators Ω K ψ,1 , which are given in Remark 3.10, on X.This is the case considered by Kalton

Remark 5 . 3 .
If we replace the natural number n by any positive real number η in Definitions 3.1 and 4.2, then we can obtain the quasilogarithmic operators Ω J ψ,η [12, Corollary 3.2].By applying Theorem 4.3 on the noncommutative L p -spaces, the Schatten classes C p , one can immediately get other results by Kalton [13, Theorems 4.2 and 4.3].The methods used by Kalton to prove those results are totally different.