An abstract Coifman-Rochberg-Weiss commutator theorem
aa r X i v : . [ m a t h . F A ] M a y AN ABSTRACT COIFMAN-ROCHBERG-WEISS COMMUTATORTHEOREM
JOAQUIM MARTIN AND MARIO MILMAN
It is a special pleasure for us to dedicate this paper to you, our dear friend Dan Waterman onthe occasion of your 80th birthday. But that is not all.The things we discuss here are intimatelyrelated to important work by another dear friend, and so to you too Richard Rochberg, warmestgreetings on the occasion of your 65th birthday. We wish both of you many many morewonderful and creative years.
Abstract.
We formulate and prove a version of the celebrated Coifman-Rochberg-Weiss commutator theorem for the real method of interpolation Introduction
Commutator estimates play an important role in analysis (cf. [20]). Our startingpoint in this paper is the celebrated commutator theorem of Coifman-Rochberg-Weiss [5]. Let K be a Calder´on-Zygmund operator, and let b ∈ BM O ( R n ) . Denoteby M b the operator “multiplication by b ”, then (cf. [5])(1.1) k [ K, M b ] f k p ≤ c k b k BMO k f k p , < p < ∞ , where [ K, M b ] f = K ( bf ) − bK ( f ) . Since each of the operators f → K ( bf ) and f → bK ( f ) is unbounded on L p , the cancellation that results of taking their differenceis essential for the validity of (1.1).The Coifman-Rochberg-Weiss commutator theorem has found many applicationsin the study of PDEs, Jacobians, Harmonic Analysis, and was also the starting pointof the Rochberg-Weiss [19] abstract theory of commutators in the setting of scalesof interpolation spaces, which itself has had many applications (cf. [12], [13], [18],and the references therein).It is instructive to review informally one of the proofs of (1.1) provided in [5].Suppose that b ∈ BM O, and fix p > . Then it is well known that we can find ε > < α < ε, e αb and e − αb ∈ A p (here A p is the class ofMuckenhoupt weights). Let K be a CZ operator, then K is bounded on the weightedspaces L p ( e αb ) , | α | < ε. In other words, the family of operators f → e αb K ( e − αb f )is uniformly bounded on L p for | α | < ε. It follows readily that one can extendedthese operators to an analytic family of operators T ( z ) f = e zb K ( e − zb f ) , for | z | < ε, and then show that, ddz T ( z ) f (cid:12)(cid:12) z =0 = [ K, M b ] f is also a bounded operator on L p . In particular, it follows that, in the statement of the theorem, we can replace CZoperators by operators T with the same weighted norm inequalities, i.e. the resultholds for any operator T, such that for all weights in the A p class of Muckenhoupt, T : L p ( w ) → L p ( w ) , < p < ∞ , boundedly. The previous argument was the starting point of the Rochberg-Weiss [19] theoryof abstract commutator estimates for the complex method of interpolation, laterextended to the real method by these authors jointly with Jawerth (cf. [11]). Thesubject has been intensively developed in the last 30 years (cf. the recent surveyby Rochberg [18] and the references therein).While the Rochberg-Weiss theory, when suitably specialized to weighted L p spaces, can be used to re-prove the Coifman-Rochberg-Weiss commutator theo-rem, in this paper we consider a different problem: we give an abstract formulationof the Coifman-Rochberg-Weiss commutator theorem which is valid for interpola-tion scales themselves. Since we work with the real method, the cancellations willbe exploited via integration by parts and a suitable re-interpretation of the relevant BM O condition . Before we formulate our main result let us recall some basic definitions associatedwith the real method of interpolation (cf. [3] for more details). Let ¯ X = ( X , X )be a compatible pair of Banach spaces. To define the real interpolation spaces ( X , X ) θ,q we start by considering on X ∩ X the family of norms J ( t, x ; ¯ X ) = max {k x k X , t k x k X } , t > . Let θ ∈ (0 , , ≤ q ≤ ∞ . We consider the elements f ∈ X + X , that can berepresented by f = Z ∞ u ( s ) dss (crucially here the convergence of the integral is in the X + X sense),where u : (0 , ∞ ) → X ∩ X . We letΦ θ,q ( g ) = (cid:26)Z ∞ (cid:0) s − θ | g ( s ) | (cid:1) q dss (cid:27) /q , ¯ X θ,q = { f = Z ∞ u ( s ) dss in X + X : Φ θ,q ( J ( s, u ( s ); ¯ X )) < ∞} , k f k ¯ X θ,q = inf { Φ θ,q ( J ( s, u ( s ); ¯ X )) : f = Z ∞ u ( s ) dss in X + X } . Likewise, if w is a positive function on (0 , ∞ ) , we define the corresponding spaces¯ X θ,q,w by means of the use of the function normΦ θ,q,w ( g ) = Φ θ,q ( wg ) . In this setting we consider the nonlinear operator f → u f : (0 , ∞ ) → X ∩ X , where u f has been selected so that(1.2) f = Z ∞ u f ( s ) dss in X + X , and Φ θ,q ( J ( s, u f ( s ); ¯ X )) ≤ k f k ¯ X θ,q . We then define(1.3) Ω f = Ω ¯ X f = Z ∞ u f ( s ) log s dss . we shall only consider the J − method in this note. we use 2 for definitiness, obviously can replace 2 by 1 + ε. OIFMAN-ROCHBERG-WEISS COMMUTATOR 3
The commutator theorem in this context (cf. [11]) states that if T : ¯ X → ¯ Y is abounded linear operator, then the nonlinear operator[ T, Ω] f = T (Ω ¯ X f ) − Ω ¯ Y ( T f )= Z ∞ ( T ( u f ( s )) − u T f ( s )) log s dss (1.4)is bounded, k [ T, Ω] f k ¯ Y θ,q ≤ c k T k ¯ X → ¯ Y k f k ¯ X θ,q . One possible interpretation of the appearance of the logarithm in formula (1.3)(and hence (1.4)) can be given if we try to imitate the arguments of Coifman-Rochberg-Weiss and bring into the argument analytic functions with suitable can-cellations. Indeed, if we represent the elements of ¯ X θ ,q using the normalization u θ f ( s ) = s θ u f ( s ) , then the elements in ¯ X θ ,q can be represented by analytic func-tions (with appropriate control), F ( z ) = Z ∞ s ( z − θ ) ( u θ f ( s )) dss , F ( θ ) = f. In this setting we have F ′ ( θ ) = Ω f. The crucial point of the cancellation argument is that, while operators repre-sented by derivatives of analytic functions can be unbounded (since we may losecontrol of the norm estimates) , the canonical representation of [ T, Ω] G ′ ( θ ) = [ T, Ω] f, with G ( z ) = Z ∞ s ( z − θ ) ( T u θ f ( s ) − u θ T f ( s )) dss , exhibits the crucial cancellation G ( θ ) = Z ∞ ( T u θ f ( s ) − u θ T f ( s )) dss = T f − T f = 0 , (1.5)which allows us to control the norm of G ′ ( θ ).It is, of course, possible to eliminate all references to analytic functions, andformulate the results in terms of representations that exhibit cancellations. Fromthis point of view the “badness” of the commutators is expressed by the fact thattheir canonical representations have an extra unbounded log factor (cf. (1.4)) whichwould lead to the weaker estimate[ T, Ω] : ¯ X θ,q → ¯ Y θ,q, | log s | ) , (note that ¯ Y θ,q $ ¯ Y θ,q, | log s | ) ) . Here is where the cancellation (1.5), now expressed without reference to analyticfunctions, simply as an integral equal to zero, comes to our rescue and allows us tointegrate by parts to find the “better” representation,(1.6) [ T, Ω] f = Z ∞ ( Z t ( T u θ f ( s ) − u θ T f ( s )) dss ) dss , which leads to the correct estimate[ T, Ω] : ¯ X θ,q → ¯ Y θ,q . JOAQUIM MARTIN AND MARIO MILMAN
This point of view was developed in [15].To formulate the Coifman-Rochberg-Weiss theorem in our setting we give a dif-ferent interpretation to the logarithm that appears in the formulae. First, for agiven weight w we introduce the (possibly non linear) operators Ω w , defined byΩ w ( f ) = Z ∞ u f ( s ) w ( s ) dss . It follows that for w ∈ L ∞ (0 , ∞ ) , the corresponding Ω w is (trivially) a boundedoperator, k Ω w ( f ) k ¯ X θ,q ≤ c k w k L ∞ k f k ¯ X θ,q , and therefore the corresponding commutators [ T, Ω w ] are also bounded. On theother hand, for the mildly unbounded function w ( s ) = log( s ) , we have Ω w = Ω , which is not bounded on ¯ X θ,q , but for which cancellations imply the boundednessof commutators of the form [ T, Ω] . Now, as is well known, the logarithm is a typicalexample of a function with
BM O behavior. Therefore we now ask more generally:for which weights w can we assert that for all bounded linear operators T : ¯ X → ¯ Y , we have that [ T, Ω w ] is a bounded operator as well? The answer to this question iswhat we shall call “the abstract Coifman-Rochberg-Weiss theorem.”Not surprisingly the answer is given in terms of a suitable BM O type spacewhich allows us to control the oscillations of w. Let
P w ( t ) = t R t w ( s ) ds and define w ( t ) = P w ( t ) − w ( t ) = 1 t Z t w ( s ) ds − w ( t ) = 1 t Z t ( w ( s ) − w ( t )) ds. Then we consider the following analog of BM O ( R + ) introduced in [16]: W = { w : w ( t ) ∈ L ∞ (0 , ∞ ) } , with k w k W = k P w − w k L ∞ . There is a direct connection between W and the space L ( ∞ , ∞ ) of Bennett-DeVore-Sharpley [4]: w ∈ L ( ∞ , ∞ ) ⇔ w ∗ ∈ W, where w ∗ denotes the non-increasing rearrangement of w. In particular, we notethat, as expected, the log has bounded oscillation since(log t ) = 1 t Z t log sds − log t = − . It will turn out that W is the correct way to measure oscillation in our context.In particular, we will show below that, when dealing with the commutators [ T, Ω w ] , the corresponding “good representation” (cf. (1.6)) is given by[ T, Ω w ] f = Z ∞ ( Z t ( T u f ( s ) − u T f ( s )) dss ) w ( s ) dss . The purpose of this note is to prove the following abstract analog of the Coifman-Rochberg-Weiss commutator theorem For martingales it can be explicitly shown, by means of selecting appropriate sigma fields(cf. [10]), that W is a BMO martingale space. W has also appeared before in several papers oninterpolation theory (cf. [9], [2]). OIFMAN-ROCHBERG-WEISS COMMUTATOR 5
Theorem 1.
Suppose that w ∈ W, and let ¯ X, ¯ Y , be Banach pairs. Then, for anybounded linear operator T : ¯ X → ¯ Y , the commutator [ T, Ω w ] is bounded, [ T, Ω w ] :¯ X θ,q → ¯ Y θ,q , < θ < , ≤ q ≤ ∞ , and, moreover, k [ T, Ω w ] f k ¯ Y θ,q ≤ c k T k ¯ X → ¯ Y k w k W k f k ¯ X θ,q . We will also prove higher order versions of this result (cf. [15] and the referencestherein). Using the strong form of the fundamental lemma (cf. [6]) one can connectthe results above with those obtained in [1] for the K − method, and, moreover, giveexplicit instances of these operators.2. Representation Theorems
As we have indicated in the Introduction, commutator theorems can be formu-lated as results about special representations of certain elements in interpolationscales. To develop our program explicitly it will be necessary to integrate by partsoften, so we start by collecting some elementary calculations that will be useful forthat purpose.
Lemma 1.
The operator P is bounded on W. Proof. ( P w ) ( t ) = 1 t Z t P w ( s ) ds − P w ( t )= 1 t Z t ( P w ( s ) − w ( s )) ds + P w ( t ) − P w ( t ) . Therefore, (cid:12)(cid:12) ( P w ) ( t ) (cid:12)(cid:12) ≤ k w k W . (cid:3) Lemma 2.
Let w ∈ W , and let < θ < . Then lim t → t θ w ( t ) = lim t →∞ t − θ w ( t ) = 0 . Proof.
Write
P w = w + w, then, since w is bounded, lim t → t θ w ( t ) = lim t →∞ t − θ w ( t ) =0 , and we see that it is enough to show that lim t → t θ P w ( t ) = lim t →∞ t − θ P w ( t ) = 0 . Now,from tP w ( t ) = R t w ( s ) ds, we get ( P w ) ′ ( t ) = − P w ( t ) − w ( t ) t . Therefore, | P w ( t ) | ≤ | P w (1) | + (cid:12)(cid:12)(cid:12)(cid:12)Z t w ( s ) dss (cid:12)(cid:12)(cid:12)(cid:12) ≤ k w k W (1 + | log t | ) . and the result follows. (cid:3) Although we shall not make use of the next result in this section it is convenientto state it here to stress the
BM O characteristics of the space W. Lemma 3. (i) (cf. [1] ) Let Qf ( t ) = R t f ( s ) dss then W = L ∞ + Q ( L ∞ ) . (ii) Let W = (cid:26) w : sup s | sw ′ ( s ) | < ∞ (cid:27) . Then, W ⊂ W. JOAQUIM MARTIN AND MARIO MILMAN (iii) W = L ∞ + W . Proof. (i) see [1].(ii) Suppose that w ∈ W . Integrating by parts1 x Z x sw ′ ( s ) ds = 1 x sw ( s ) | s = xs =0 − x Z x w ( s ) ds. It is easy to see (cf. Lemma 2) that lim x → sw ( s ) = 0 , hence (cid:12)(cid:12) w ( x ) (cid:12)(cid:12) = | P ( sw ′ ( s ))( x ) | . Consequently, since P is bounded on L ∞ , it follows that w ∈ L ∞ and therefore w ∈ W. (iii) Suppose that w ∈ W. Since (cid:12)(cid:12) t ( P w ) ′ (cid:12)(cid:12) = (cid:12)(cid:12) w ( t ) (cid:12)(cid:12) , it follows that P w ∈ W . The desired decomposition is therefore given by w = ( w − P w ) | {z } L ∞ + P w |{z} W . (cid:3) The next result gives the representation theorem that we need to prove Theorem1.
Theorem 2.
Let H = ( H , H ) be a Banach pair, and suppose that w ∈ W. Supposethat an element f ∈ H + H can be represented as f = Z ∞ u ( s ) w ( s ) dss , with Z ∞ u ( s ) dss = 0 , Φ θ,q ( J ( t, u ( t ); H )) < ∞ . Then, f ∈ H θ,q , and, moreover, k f k H θ,q ≤ c θ,q k w k W Φ θ,q ( J ( t, u ( t ); H )) . Proof.
Write f = Z ∞ u ( s ) w ( s ) dss = Z ∞ u ( s )( w ( s ) − P w ( s )) dss + Z ∞ u ( s ) P w ( s ) dss = I + I . It is plain that k I k H θ,q ≤ k w k W Φ θ,q ( J ( t, u ( t ); H )) . It remains to estimate I . We integrate by parts: I = P w ( t ) Z t u ( s ) dss (cid:21) ∞ − Z ∞ (cid:18)Z t u ( s ) dss (cid:19) ( w ( t ) − P w ( t )) dtt . The integrated term vanishes. Suppose first that q > . We can write
OIFMAN-ROCHBERG-WEISS COMMUTATOR 7 (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ Z t J ( s, u ( s )) dss ≤ (cid:18)Z t (cid:18) J ( s, u ( s )) s θ (cid:19) q dss (cid:19) /q (cid:18)Z t s θq ′ dss (cid:19) /q ′ (2.1) ≤ c θ,q Φ θ,q ( J ( t, u ( t ); H )) t θ . By Lemma 1
P w ∈ W and therefore we may apply Lemma 2 to conclude thatlim t → | P w ( t ) | (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ c θ,q lim t → Φ θ,q ( J ( t, u ( t ); H )) t θ | P w ( t ) | = 0 . Likewise, using the cancelation condition(2.2) Z t u ( s ) dss = − Z ∞ t u ( s ) dss , we have that (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ c Φ θ,q ( J ( t, u ( t ); H )) t − θ , and once again we can apply Lemma 2 and find thatlim t →∞ | P w ( t ) | (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H = 0 . The case q = 1 is simpler. For example, instead of using Holder’s inequality in(2.1) we write Z t J ( s, u ( s )) dss = t θ t θ Z t J ( s, u ( s )) dss ≤ t θ Z t J ( s, u ( s )) s θ dss . It remains to estimate the H θ,q norm of I = R ∞ (cid:16)R t u ( s ) dss (cid:17) ( w ( t ) − P w ( t )) dtt . By definition,(2.3) k I k H θ,q ≤ Φ θ,q ( J ( t )) , where J ( t ) = J ( t, (cid:18)Z t u ( s ) dss (cid:19) ( w ( t ) − P w ( t )); H ) ≤ k w k W (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H + t (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ! . The first term on the right hand side can be estimated directly by Minkowski’sinequality (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ Z t J ( s, u ( s ); H ) dss , while for the second we argue that, by (2.2), t (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H = t (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t u ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ t Z ∞ t J ( s, u ( s ); H ) dss . JOAQUIM MARTIN AND MARIO MILMAN
Altogether, we arrive at J ( t ) ≤ k w k W (cid:18)Z t J ( s, u ( s ); H ) dss + t Z ∞ t J ( s, u ( s ); H ) dss (cid:19) . Therefore, applying the Φ θ,q norm on both sides of the previous inequality and thenusing Hardy’s inequalities to estimate the right hand side, we get(2.4) Φ θ,q ( J ( t )) ≤ c θ,q k w k W Φ θ,q (cid:0) J ( t, u ( t ); H ) (cid:1) . Combining (2.4) and (2.3) k I k H θ,q ≤ c θ,q k w k W Φ θ,q ( J ( t )) , and collecting the estimates for I and I we finally obtain k f k H θ,q ≤ c θ,q k w k W Φ θ,q ( J ( t, u ( t ); H ))as we wished to show. (cid:3) We are now ready for the proof of Theorem 1.
Proof.
Suppose that T is a given bounded linear operator T : ¯ X → ¯ Y , and let w ∈ W. Let ˜ u ( t ) = (( u T f ( t ) − T ( u f ( t )) . Then[ T, Ω w ] f = Z ∞ ˜ u ( t ) w ( t ) dtt with Φ θ,q ( J ( t, ˜ u ( t ); ¯ Y ) ≤ c k T k ¯ X → ¯ Y k f k X θ,q . Since, moreover, Z ∞ ˜ u ( t ) dtt = 0 , we can apply theorem 2 to conclude that k [ T, Ω w ] f k Y θ,q ≤ c k w k W k T k ¯ X → ¯ Y k f k X θ,q , as we wished to show. (cid:3) Higher order cancellations
We adapt the analysis of [15] to handle higher order cancellations. The corre-sponding higher order commutator theorems that follow will be stated and provedin the next section.
Theorem 3.
Let H be a Banach pair, and let w ∈ W. Suppose that f admits arepresentation f = Z ∞ u ( s ) ( P w ( s )) dss , with Z ∞ u ( s ) dss = 0 , Z ∞ u ( s ) P w ( s ) dss = 0; Φ θ,q ( J ( t, u ( t ); H )) < ∞ then, f ∈ H θ,q , and, moreover, k f k H θ,q ≤ c k w k W Φ θ,q ( J ( t, u ( t ); H )) . OIFMAN-ROCHBERG-WEISS COMMUTATOR 9
Proof.
We will integrate by parts repeatedly. We start writing f = Z ∞ u ( t ) ( P w ( t )) dtt = Z ∞ P w ( t ) d (cid:18)Z t u ( s ) P w ( s ) dss (cid:19) . Then, f = P w ( t ) Z t u ( s ) P w ( s ) dss (cid:21) ∞ − Z ∞ (cid:18)Z t u ( s ) P w ( s ) dss (cid:19) ( w ( t ) − P w ( t )) dtt , we will show below that the integrated term vanishes, then(3.1) f = − Z ∞ (cid:18)Z t u ( s ) P w ( s ) dss (cid:19) ( w ( t ) − P w ( t )) dtt . Now we consider the inner integral and integrate by parts Z t u ( s ) P w ( s ) dss = Z t P w ( s ) d (cid:18)Z s u ( r ) drr (cid:19) , using the fact that (cf. the proof of Theorem 2) lim t → P w ( t ) R t u ( s ) dss = 0 , we get Z t u ( s ) P w ( s ) dss = P w ( t ) Z t u ( s ) dss − Z t (cid:18)Z r u ( s ) dss (cid:19) ( w ( r ) − P w ( r )) drr . Inserting this result back in (3.1) we find that f = − Z ∞ (cid:18)Z t u ( s ) P w ( s ) dss (cid:19) ( w ( t ) − P w ( t )) dtt = Z ∞ (cid:18) P w ( t ) Z t u ( s ) dss (cid:19) w ( t ) dtt + Z ∞ (cid:18)Z t (cid:18)Z r u ( s ) dss (cid:19) w ( r ) drr (cid:19) w ( t ) dtt = I + I . Integrating by parts I we get I = P w ( t ) Z t (cid:18) w ( r ) Z r u ( s ) dss (cid:19) drr (cid:12)(cid:12)(cid:12)(cid:12) ∞ + Z ∞ (cid:18)Z t (cid:18)Z r u ( s ) dss (cid:19) w ( r ) drr (cid:19) w ( t ) dtt , where once again the integrated term vanishes. Hence, I = I . Therefore, if we let U ( t ) = 2 (cid:16)R t (cid:0)R r u ( s ) dss (cid:1) w ( r ) drr (cid:17) w ( t ) , f can be representedby f = Z ∞ U ( t ) dtt . Now we estimate the corresponding J − functional, J ( t ) = J ( t, U ( t ); ¯ H ) , by2 k w k W (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:18)Z r u ( s ) dss (cid:19) w ( r ) drr (cid:13)(cid:13)(cid:13)(cid:13) H + t (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:18)Z r u ( s ) dss (cid:19) w ( r ) drr (cid:13)(cid:13)(cid:13)(cid:13) H ! = 2 k w k W ( C + tC ) . We readily see that C is majorized by C ≤ k w k W Z t (cid:18)Z r J ( s, u ( s ); H ) dss (cid:19) drr = k w k W Z t J ( r, u ( r ); H ) ln tr drr . To handle C we work with the integral inside the norm H by first using R r u ( s ) dss = − R ∞ r u ( s ) dss and then changing the order of integration. We find that C = (cid:13)(cid:13)(cid:13) lim α → C ( α ) (cid:13)(cid:13)(cid:13) H , where C ( α ) = R t R sα w ( r ) drr u ( s ) dss + R ∞ t R tα w ( r ) drr u ( s ) dss . We compute C ( α )using the formula ( P w ) ′ ( t ) = − w ( t ) t , and we get C ( α ) = P w ( α ) Z t u ( s ) dss − Z t P w ( s ) u ( s ) dss + P w ( α ) Z ∞ t u ( s ) dss − P w ( t ) Z ∞ t u ( s ) dss . Now by the cancellation conditions: Z ∞ u ( s ) dss = 0 = ⇒ P w ( α ) Z t u ( s ) dss = − P w ( α ) Z ∞ t u ( s ) dss , and Z ∞ u ( s ) P w ( s ) dss = 0 = ⇒ Z t u ( s ) P w ( s ) dss = − Z ∞ t u ( s ) P w ( s ) dss , we have C ( α ) = Z ∞ t u ( s )[ P w ( s ) − P w ( t )] dss = Z ∞ t u ( s ) Z st w ( r ) drr dss . All in all it follows that, C ≤ k w k W Z ∞ t k u ( s ) k H ln st dss ≤ k w k W Z ∞ t J ( s, u ( s ); H ) ln st dss . Summarizing, J ( t ) ≤ k w k W (cid:18)Z t J ( r, u ( r ); H ) ln tr drr + t Z ∞ t J ( r, u ( r ); H ) ln rt drr (cid:19) . Applying the Φ θ,q norm and Hardy’s inequalities (twice) we finally obtain k f k H θ,q ≤ c Φ θ,q (cid:0) J ( t, u ( t ); H ) (cid:1) ≤ c k w k W Φ θ,q (cid:0) J ( t, u ( t ); H ) (cid:1) . To conclude the proof it remains to verify that the integrated terms we have col-lected along the way effectively vanish. More precisely, it remains to prove that(3.2) lim t → ξ P w ( t ) Z t u ( s ) P w ( s ) dss = 0, for ξ = 0 , ∞ , and(3.3) lim t → ξ P w ( t ) Z t (cid:18) ( w ( r ) − P w ( r )) Z r u ( s ) dss (cid:19) drr = 0, for ξ = 0 , ∞ . OIFMAN-ROCHBERG-WEISS COMMUTATOR 11
To handle these limits we shall assume that q >
1, the case q = 1 is easier (cf. theproof of Theorem 2 above). We start with (3.2): (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s ) P w ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ Z t J ( s, u ( s ); H ) | P w ( s ) | dss ≤ Z t (cid:18) J ( s, u ( s ); H ) s θ (cid:19) q dss ! /q (cid:18)Z t (cid:0) s θ | P w ( s ) | (cid:1) q ′ dss (cid:19) /q ′ ≤ (cid:0) Φ θ,q ( J ( s, u ( s ); H )) (cid:1) c (cid:18)Z t (cid:0) s θ | w ( s ) | (cid:1) q ′ dss (cid:19) /q ′ (by Hardy’s inequality)Let e θ > θ − e θ > . Since w ∈ W ⇒ P w ∈ W (cf. Lemma 1) , therefore, by Lemma 2, we have (cid:12)(cid:12)(cid:12) t e θ P w ( t ) (cid:12)(cid:12)(cid:12) ≤ t suff. close to 0).Thus, for small t, (cid:18)Z t (cid:0) s θ | P w ( s ) | (cid:1) q ′ dss (cid:19) /q ′ ≤ (cid:18)Z t (cid:16) s θ − e θ (cid:17) q ′ dss (cid:19) /q ′ ≤ ct θ − e θ , and lim t → (cid:13)(cid:13)(cid:13)(cid:13) P w ( t ) Z t u ( s ) w ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H ≤ lim t → ct θ − e θ | P w ( t ) | = 0 . The corresponding limit when t → ∞ can be handled by the same argument if wefirst use the cancellation property R t u ( s ) P w ( s ) dss = − R ∞ t u ( s ) P w ( s ) dss and thenapply the H norm.To see (3.3) we note that | P w ( t ) | (cid:13)(cid:13)(cid:13)(cid:13)Z t ( w ( r ) − P w ( r )) Z r u ( s ) dss drr (cid:13)(cid:13)(cid:13)(cid:13) H ≤ k w k W | P w ( t ) | Z t J ( s, u ( s ); H ) ln ts dss ≤ k w k W | P w ( t ) | t θ (cid:0) Φ θ,q ( J ( s, u ( s ); H )) (cid:1) t − θ Z t (cid:18) s θ ln ts (cid:19) q ′ dss ! /q ′ . Now, the term on the right hand side converges to zero when t → t − θ (cid:16)R t (cid:0) s θ ln ts (cid:1) q ′ dss (cid:17) /q ′ ≤ t − θ (cid:16)R t (cid:0) s θ st (cid:1) q ′ dss (cid:17) /q ′ ≤ ct − θ t − t θ . Again the case t → ∞ is reduced to the case t → (cid:3) Corollary 1.
Let H be a Banach pair, and let w ∈ W. Suppose that f = Z ∞ u ( s ) ( w ( s )) dss , with (3.4) Z ∞ u ( s ) dss = 0 , Z ∞ u ( s ) w ( s ) dss = 0 , Z ∞ u ( s ) P w ( s ) dss = 0; and Φ θ,q ( J ( t, u ( t ); H )) < ∞ . Then, f ∈ H θ,q , and, moreover, k f k H θ,q ≤ c k w k W Φ θ,q ( J ( t, u ( t ); H )) . Proof.
Write Z ∞ u ( s ) ( w ( s )) dss = Z ∞ u ( s ) ( w ( s ) − P w ( s )) w ( s ) dss + Z ∞ u ( s ) w ( s ) P w ( s ) dss . Since w ( t ) P w ( t ) = ( w ( t )) +( P w ( t )) − ( w ( t ) − P w ( t )) , we have Z ∞ u ( s ) ( w ( s )) dss = 2 Z ∞ u ( s ) ( w ( s ) − P w ( s )) w ( s ) dss − Z ∞ u ( s ) ( w ( s ) − P w ( s )) dss + Z ∞ u ( s ) ( P w ( s )) dss We now show how to control each of these terms. Let ˜ u ( s ) = u ( s ) ( w ( s ) − P w ( s )) , by the cancellation conditions (3.4) it follows that R ∞ ˜ u ( s ) dss = 0 . Therefore we canapply Theorem 2 to conclude that R ∞ ˜ u ( s ) w ( s ) dss ∈ H θ,q . It follows that (cid:13)(cid:13)(cid:13)(cid:13) Z ∞ u ( s ) ( w ( s ) − P w ( s )) w ( s ) dss (cid:13)(cid:13)(cid:13)(cid:13) H θ,q ≤ c k w k W Φ θ,q ( J ( t, u ( t ); H )) . The second term is also under control since ( w ( s ) − P w ( s )) is bounded. Finallywe may apply Theorem 3 to control the remaining term. (cid:3) Theorem 4.
Let H be a Banach pair, and let w , w ∈ W. Suppose that f = Z ∞ u ( s ) w ( s ) w ( s ) dss , with Z ∞ u ( s ) dss = 0 , Z ∞ u ( s ) w j ( s ) dss = 0 , Z ∞ u ( s ) P w j ( s ) dss = 0 ( j = 0 , and Φ θ,q ( J ( t, u ( t ); H )) < ∞ . Then, f ∈ H θ,q and, moreover, k f k H θ,q ≤ c max {k w k W , k w k W } Φ θ,q ( J ( t, u ( t ); H )) . Proof.
Write w ( s ) w ( s ) = ( w ( s ) + w ( s )) − w ( s ) − w ( s ) (cid:3) For n >
OIFMAN-ROCHBERG-WEISS COMMUTATOR 13
Theorem 5.
Let H be a Banach pair, and let w ∈ W .(i) Suppose that f = Z ∞ u ( s ) ( P w ( s )) n dss , with Z ∞ u ( s ) w ( s ) k dss = 0 , Z ∞ u ( s ) P w ( s ) k dss = 0 , ( k = 0 , · · · , n − and Φ θ,q ( J ( t, u ( t ); H )) < ∞ . Then, f ∈ H θ,q , and, moreover, k f k H θ,q ≤ c Φ θ,q ( J ( t, u ( t ); H )) . (ii) If f = Z ∞ u ( s ) ( w ( s )) n dss , with Z ∞ u ( s ) w ( s ) k dss = 0 , Z ∞ u ( s ) P w ( s ) k dss = 0 , Z ∞ u ( s ) w ( s ) n − k P w ( s ) k dss = 0 , , ( k = 0 , · · · , n − and Φ θ,q ( J ( t, u ( t ); H )) < ∞ , then, f ∈ H θ,q , and, moreover, k a k H θ,q ≤ c k w k nW Φ θ,q ( J ( t, u ( t ); H )) . Remark 1.
In the classical case (cf. [15] , theorem 3) w ( t ) = ln t, and therefore P w ( t ) = ln t − . Consequently the conditions Z ∞ u ( s ) P w ( s ) k dss = 0 , Z ∞ u ( s ) w ( s ) n − k P w ( s ) k dss = 0 , ( k = 0 , · · · , n − actually follow from Z ∞ u ( s ) ( w ( s )) k dss = 0 , ( k = 0 , · · · , n − . Higher order commutators
We consider higher order commutators defined as follows (cf. [15], [1], [18]). Let¯ X and ¯ Y be Banach pairs, and let T : ¯ X → ¯ Y be a bounded linear operator. Givena nearly optimal representation (cf. 1.2 above) f = Z ∞ u f ( s ) dss we let Ω n,w f = 1 n ! Z ∞ u f ( s )( w ( s )) n dss , n = 0 , , ... and form the commutators C n,w f = T f , n = 0[ T, Ω ,w ] f , n = 1[ T, Ω ,w ] f − Ω ,w ( C ,w f ) , n = 2 ............ [ T, Ω n,w ] f − Ω ,w ( C n − ,w f ) − · · · Ω n − ,w ( C ,w f )Observe that the commutators [ T, Ω n,w ] alone are not bounded and we need toform more complicated expressions like C n,w in order to produce the necessarycancellations. Moreover, since the operations Ω j,w are not linear, simple mindediterations of the form Ω ,w [ T, Ω ,w ] − [ T, Ω ,w ] Ω ,w , etc, cannot be treated directlyusing Theorem 1. Theorem 6.
Suppose that w ∈ W. Then the commutators C n,w are bounded, C n,w : ¯ X θ,q → ¯ Y θ,q , < θ < , ≤ q ≤ ∞ , and, moreover, for each instance g = w, or g = P w, we have k C n,w f k ¯ Y θ,q ≤ c k T k ¯ X → ¯ Y k w k nW k f k ¯ X θ,q . Proof.
We only consider in detail the case n = 2 . Writing w = ( w − P w ) +
P w, wesee that we only need to deal with the commutator C ,P w . Let u ( s ) = T ( u f ( s )) − u T ( f ) ( s )then C ,P w ( T f ) = 12 Z ∞ u ( t )( P w ( t )) dtt − Z ∞ e u ( t ) P w ( t ) dtt , with Z ∞ e u ( t ) dtt = Z ∞ u ( t ) P w ( t ) dtt ; Z ∞ u ( t ) dtt = 0 , and Φ θ,q ( J ( t, e u ( t ) , X )) ≤ c k w k W k f k ¯ X θ,q Φ θ,q ( J ( t, u ( t ) , X )) ≤ c k w k W k f k ¯ X θ,q Since 12 Z ∞ u ( t )( P w ( t )) dtt = 12 Z ∞ ( P w ( t )) d (cid:18)Z t u ( s ) dss (cid:19) = Z ∞ (cid:18)Z t u ( s ) dss (cid:19) P w ( t ) w ( t ) dtt , it follows that if we let v ( t ) = ( Z t u ( s ) dss ) w ( t )then C ,P w ( T f ) = Z ∞ ( v ( t ) − e u ( t )) P w ( t ) dtt , and Z ∞ ( v ( t ) − e u ( t )) dtt = 0 . OIFMAN-ROCHBERG-WEISS COMMUTATOR 15 then theorem 2 implies that k C ,P w ( T f ) k ¯ Y θ,q ≤ c k w k W Φ θ,q ( J ( t, u ( t ); ¯ X )) + c Φ θ,q ( J ( t, e u ( t ); ¯ X )) ≤ c k w k W k f k ¯ X θ,q . as we wished to show. (cid:3) Comparison with earlier results and some questions
This paper was originally conceived in 1999-2000, when the first named authorspent one year in the Tropics. So publication was delayed somewhat and in themean time several papers on the subject have appeared. In particular, [17] hassimilar statements framed in terms of weights of the form(5.1) w ( t ) = φ (log t ) , with φ Lipchitz.One recognizes that these weights are included in our theory since for w of the form(5.1) we have (cf. Lemma 3 above) k w k W = sup | tw ′ ( t ) | = k φ ′ k ∞ < ∞ . There is also a connection with [1] (a longer version of this paper was originally cir-culated in 1996 (cf. [2])). These papers emphasize the connection between weightednorm inequalities, commutators and BMO type conditions using the K − method,and BM O conditions are formulated in terms of properties of weights. Recall thatfor the K − method of interpolation we define the corresponding Ω operations byΩ K f = Z x ( t ) dtt − Z ∞ x ( t ) dtt , or, more generally, byΩ Kw f = Z x ( t ) w ( t ) dtt − Z ∞ x ( t ) w ( t ) dtt , where f = x ( t ) + x ( t ) , and k x ( t ) k H + t k x ( t ) k H ≤ cK ( t, f ; ¯ H ) . Using the strong form of the fundamental lemma of interpolation theory (cf. [6])we can arrange to have f = R ∞ u f ( s ) dss , and x ( t ) = Z t u f ( s ) dss , x ( t ) = Z ∞ t u f ( s ) dss . It formally follows that Ω Kw f = − Ω Gw f, where Gw ( s ) = Z s w ( r ) drr . In particular, if w = 1 , then Gw ( s ) = log s . Also note thatsup s | s ( Gw ) ′ ( s ) | = k w k ∞ . Now a brief attempt to informally connect our work with Dan Waterman’s clas-sical Fourier analysis. One source of inspiration for the formulation of some of theresults in this paper comes from the Littlewood-Paley theory, framed in terms ofsemigroups, e.g. as developed in Stein [21]. In the abstract theory of Stein [21] (cf. [21] pag 121) the relevant semigroups are represented, using the spectral theorem,by T t = Z ∞ e − λt dE ( λ ) , and one considers (multiplier) operators of the form T w f = Z ∞ e − λt w ( t ) dE ( λ ) f, with w ∈ L ∞ . The conclusion is that the operator T ( Lw ) ′ is bounded on L p ,
V M O. ..T5. In connection with T3 and T4 it would be of interest to study compactness(weak compactness) in the abstract setting of [14] using the ideas in this paper.
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Department of Mathematics, Universidad Autonoma de Barcelona
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