Dyadic Martingale Hardy-amalgam spaces: Embeddings and Duality
aa r X i v : . [ m a t h . C A ] F e b DYADIC MARTINGALE HARDY-AMALGAM SPACES:EMBEDDINGS AND DUALITY
JUSTICE SAM BANSAH AND BENOˆIT F. SEHBA
Abstract.
We present in this paper some embeddings of variousdyadic martingale Hardy-amalgam spaces H Sp,q , H sp,q , H ∗ p,q , Q p,q and P p,q of the real line. In the same settings, we characterize thedual of H sp,q for large p and q . We also introduce a Garsia-typespace G p,q and characterize its dual space. Introduction
Our setting is a quadruplet ( R , F , {F } n ≥ , P ) where F n is the sigmaalgebra generated by all dyadic intervals of R of lentgh 2 − n , P a prob-ability measure, and F stands for the sigma algebra generated by theunion of the F n s. In this work, we are interested in the study of the em-beddings relations between the recently introduced martingale Hardy-amalgam spaces in [1] and the characterization of their dual spacesfor large exponents. We also define a natural extension of the Garsiaspace to the Wiener amalgam setting for which we characterize thedual space.Martingale inequalities have proven to be very useful in various ap-plications. For instance, the justification of martingale convergencetheorems for both forward and backward convergences have been estab-lished by applying classical martingale inequalities (see [9]). In Fourieranalysis, we have the involvement of martingale inequalities in the es-tablishment of the boundedness of the maximal Fej´er operator (see[15]). Martingale inequalities also play some important roles in thestudy of properties of Brownian motions (see for example [4]).In the past few decades, there have been various studies about theembeddings of classical martingale Hardy spaces. Some of these dis-cussions can be found in [6, 13, 14]. There are many other importantinequalities involving martingales that have been proved and applied inthe literature (see for example [3, 4, 10, 11, 12, 14]). For instance in [14] Mathematics Subject Classification.
Primary: 60G42, 60G46 Secondary:42B25, 42B35 .
Key words and phrases.
Martingales, dyadic filtration, amalgam space. we can find a discussion on the Doob’s inequality, the Convexity andConcavity inequalities for martingales. We can also find a discussionon norm inequalities for operators of matrix type on martingales and aproof of Burkholder-Davis-Gundy inequality in [4, 14]. A discussion onthe weighted norm inequality similar to the Burkholder-Davis-Gundyinequality can also be found in [10] and inequalities of operators ofnon-matrix type on martingales with a weighted probability measureare discussed in [12]. It is also interesting to mention that an analogueof weighted norm inequality for the Hardy maximal function result isalso valid in the setting of martingale theory (see [11]). Some of theseclassical results will be very useful in this paper and for consistencypurposes, we will restate these classical results when needed in theappropriate section below.Let λ = ( λ n ) n ∈ N be an adapted sequence (this will be made clear inthe next section) and let d n λ = λ n − λ n − . Then λ is said to be L p -variation integrable bounded if k P n | d n λ |k L p < ∞ . It is also said tobe L p -jump bounded if sup n | d n f | ∈ L p . The space of all martingales λ that are L p -variation integrable bounded is often referred to as variationintegrable space (see [13]). It is also referred to as the Garcia spaceand it is denoted G p (see [14]). Also the space all martingales λ thatare L p -jump bounded is simply referred to as the Jump bounded space(see [13]). This is the space denoted as BD p in [14]. The Garcia spaceis shown to be a component of the Davis decompositions of martingalesin the classical martingale Hardy spaces (see [13]). It is also establishedthat the dual space of the Garcia space is the Jump bounded space (see[13, 14]).We recall (see [1]) that if T is either the quadratic variation ( S ),the conditional quadratic variation ( s ) or the maximal function ( M f : f ∗ ), then the corresponding martingale Hardy-amalgam space H Tp,q isthe space of all martingales f such that T ( f ) belongs to the Wieneramalgam space L p,q , 0 < p, q ≤ ∞ . The amalgam space of predictivemartingales P p,q and the amalgam space of martingales with predictivequadratic variation Q p,q are defined as in the classical case (see [14])by just replacing the Lebesgue space in the definition by the amalgamspace L p,q .In this paper, we extend the embeddings of the classical martin-gale Hardy spaces, (see [14, Theorem 2.11]), to the martingale Hardy-amalgam spaces. We will also extend the Doob’s inequality and theBurkholder-Davis-Gundy inequality of the classical martingale Hardyspaces to the martingale Hardy-amalgam spaces.We shall also intro-duce the space of L p,q -variation integrable bounded martingales G p,q MBEDDINGS AND DUALITY 3 and the space L p,q -jump bounded martingales BD p,q , and we shall alsorefer to these spaces as the Garcia (or the variation integrable) spaceand Jump bounded space respectively. We will then establish that thedual of G p,q is BD p ′ ,q ′ where ( p, p ′ ) and ( q, q ′ ) are conjugate pairs. In theclassical case, the space G p played an important role in establishing thefamous Fefferman’s inequality (see for example [13]). This motivatedus to introduce G p,q and study one property of this space, which is itsduality.It was established in [1] that the dual of H sp,q when 0 < p ≤ q ≤ < p, q < ∞ was left open.We prove in this paper that for 1 < q ≤ p ≤ ≤ p ≤ q < ∞ , theduality of H sp,q identifies with H sp ′ ,q ′ .We note that the setting of [1] is more general. In a previous versionof [1], we have tried to solve embeddings and duality problem butthis quickly appeared to be a difficult task. A key argument in thedyadic setting is Lemma 4.2 which allows us to obtain the results ofthis paper. For the embeddings, we combine this lemma with knownclassical results. The proofs of duality results are more demanding butagain Lemma 4.2 is relevant here as it is used in the proof of the Doob’sinequality which is in its turn used in our proofs.The outline of this paper is as follows. In Section 2, we will getfamiliar with notations and recall the various definitions appropriatefor this work. The main results in this work are presented in Section3. In Section 4, we provide proof for the first result of this work whichis the extension of the classical martingale Hardy spaces embeddingsto the martingale Hardy-amalgam spaces. We also provide proofs forthe extension of the Doob’s inequality and Burkholder-Davis-Gundyinequalities establishing the second and third result of this work. InSection 5, we will characterize the dual space of H sp,q for 1 < p ≤ q < ∞ and also identify the dual space of G p,q . We will close the paper with aconclusion where we will make some other observations.2. Notations and Basic Definitions
In this section, we introduce the necessary definitions and recall thevarious martingale Hardy-amalgam spaces we shall consider in thispaper. We will also state some important classical results, such asDoob’s inequality and the Burkholder-Davis-Gundy inequality that wewill need later in this work.
JUSTICE SAM BANSAH AND BENOˆIT F. SEHBA
Let R be the set of real numbers and consider the following dyadicintervals of R ; I n,k = (cid:20) k n , k + 12 n (cid:19) n ∈ N , k ∈ Z . Let D n = { I n,k , k ∈ Z } , n ∈ N . Let F n = σ ( D n ) be the σ − algebragenerated by D n . Then {F n } n ∈ N is a (dyadic) filtration. With this, wedefine the probability space as Ps := ( R , F , {F n } n ∈ N , P )where P is the probability measure and F n ⊆ F for all n ∈ N . Thus allthe martingales defined in this paper are with respect to this probabilityspace with the underlying filtration {F n } n ∈ N . Let J k,n,j ∈ D n be thedyadic interval defined as J k,n,j = (cid:20) k + j n n , k + 1 + j n n (cid:19) . Then A j = [ j, j + 1) = n − [ k =0 J k,n,j (1)Therefore, A j ∈ F n for all n. Note that the A j ’s are dyadic intervals. Also, A i ∩ A j = ∅ for i = j and S j A j = R . Let L p denote the classical Lebesgue space and let ℓ q denote thesequence space. For f ∈ L p , we will be using the notation k f k p := k f k L p := (cid:18)Z R | f | p d P (cid:19) /p . The amalgam space of L p and ℓ q is defined as the space L p,q ( R ) = ( f : X j k f A j k qp < ∞ ) equipped with the (quasi)-norm k f k L p,q ( R ) = X j ∈ Z (cid:18)Z R | f | p A j d P (cid:19) qp ! q for 0 < p, q < ∞ and k f k L p, ∞ ( R ) = sup j ∈ Z (cid:18)Z R | f | p A j d P (cid:19) p MBEDDINGS AND DUALITY 5 for 0 < p < ∞ . It is interesting to note that L p,p ( R ) = L p ( R ) . We referthe interested reader to ([2, 5, 7, 8]) for more on amalgam spaces.Let M be the spaces of all martingales defined on Ps relative tothe underlying filtration F n . For f = ( f n ) n ∈ N ∈ M , we define themartingale difference as d n f = f n − f n − and we will agree that f = 0and d f = 0 . Let k · k p denote the usual L p − norm for 0 < p ≤ ∞ . Then f = ( f n ) n ∈ N ∈ M is said to be L p bounded if k f k p := sup n ∈ N k f n k p < ∞ and we define the maximal function, f ∗ , or M ( f ) of f as M ( f ) = f ∗ := sup n ∈ N | f n | . Let E and E n be the expectation and the conditional expectation op-erators respectively. Then the following measurable functions are welldefined (see for example [4]); S ( f ) = X n ∈ N | d n f | ! and s ( f ) = X n ∈ N E n − | d n f | ! and we shall agree that S n ( f ) = n X i =0 | d i f | ! and s n ( f ) = n X i =0 E i − | d i f | ! . Let ρ be the space of all sequences ̺ = ( ̺ n ) n ≥ of adapted (that is for all n ∈ Z , ̺ n is F n -measurable), non-decreasing, non-negative functionsand define ̺ ∞ := lim n →∞ ̺ n . We are now in the position to define the martingale Hardy-amalgamspaces. These spaces were originally introduce in [1]. Let 0 < p, q ≤ ∞ .Theni. H Sp,q ( R ) is the space of all f ∈ M such that S ( f ) ∈ L p,q ( R )with (quasi)-norm k f k H Sp,q ( R ) := k S ( f ) k L p,q ( R ) . ii. H sp,q ( R ) is the space of all f ∈ M such that s ( f ) ∈ L p,q ( R ) with(quasi)-norm k f k H sp,q ( R ) := k s ( f ) k L p,q ( R ) . JUSTICE SAM BANSAH AND BENOˆIT F. SEHBA iii. H ∗ p,q ( R ) is the space of all f ∈ M such that f ∗ ∈ L p,q ( R ) with(quasi)-norm k f k H ∗ p,q ( R ) := k f ∗ k L p,q ( R ) . iv. Q p,q ( R ) is the space of all f ∈ M for which there is a sequenceof functions ̺ = ( ̺ n ) n ≥ ∈ ρ such that S n ( f ) ≤ ̺ n − and k ̺ ∞ k L p,q ( R ) < ∞ with (quasi)-norm k f k Q p,q ( R ) := inf ̺ ∈ ρ k ̺ ∞ k L p,q ( R ) . v. P p,q ( R ) is the space of all f ∈ M for which there is a se-quence of functions ̺ = ( ̺ n ) n ≥ ∈ ρ such that | f n | ≤ ̺ n − and k ̺ ∞ k L p,q ( R ) < ∞ with (quasi)-norm k f k P p,q ( R ) := inf ̺ ∈ ρ k ̺ ∞ k L p,q ( R ) . We also introduce here the spaces G p,q ( R ) and BD p,q ( R ) which we shallrefer to as variation integrable space and Jump bounded space respec-tively. G p,q ( R ) := ( f ∈ M : ∞ X n =0 | d n f | ∈ L p,q ( R ) ) endowed with the norm k f k G p,q ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =0 | d n f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p,q ( R ) for 1 ≤ p ≤ q < ∞ and BD p,q ( R ) := (cid:26) f ∈ M : sup n ∈ N | d n f | ∈ L p,q ( R ) (cid:27) endowed with the norm k f k BD p,q ( R ) = (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N | d n f | (cid:13)(cid:13)(cid:13)(cid:13) L p,q ( R ) for 1 ≤ p ≤ q ≤ ∞ . For the sake of presentation, we sometimes write H Sp,q ( R ) , H sp,q ( R ) , H ∗ p,q ( R ) , Q p,q ( R ) , P p,q ( R ) , G p,q ( R ) , BD p,q ( R ) as H Sp,q , H sp,q , H ∗ p,q , Q p,q , P p,q , G p,q , BD p,q respectively and k · k L p,q ( R ) as k · k p,q . We say that the stochastic basis, F n , is regular if there exists R > f n ≤ Rf n − . MBEDDINGS AND DUALITY 7 Presentation of Results
In this section, we present the main results of this paper. We startwith the presentation of the inequalities that relate the first five spacesdefined above. To be specific, we have the following theorem.
Theorem 3.1 (Martingale Embeddings) . Let < q ≤ ∞ . Then (i) k f k H ∗ p,q ( R ) ≤ C k f k H sp,q ( R ) , k f k H Sp,q ( R ) ≤ C k f k H sp,q ( R ) (0
Theorem 3.2.
Let < q ≤ ∞ and < p < ∞ . For every non-negative L p,q − bounded submartingale ( f n , n ∈ N ) , we have that (2) (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N f n (cid:13)(cid:13)(cid:13)(cid:13) p,q ≤ pp − n ∈ N k f n k p,q . Theorem 3.3.
The spaces H Sp,q ( R ) and H ∗ p,q ( R ) are equivalent for ≤ p, q ≤ ∞ , namely, c p k f k H Sp,q ( R ) ≤ k f k H ∗ p,q ( R ) ≤ C p k f k H Sp,q ( R ) (1 ≤ p, q < ∞ ) and c p k f k H Sp, ∞ ( R ) ≤ k f k H ∗ p, ∞ ( R ) ≤ C p k f k H Sp, ∞ ( R ) (1 ≤ p < ∞ ) . In Section 5, we provide the proof of the following dual characteri-zations of the spaces H sp,q ( R ) and G p,q ( R ) . Theorem 3.4.
If either < q ≤ p ≤ or ≤ p ≤ q < ∞ , then thedual space of H sp,q ( R ) identifies with H sp ′ ,q ′ ( R ) where p + p ′ = q + q ′ = 1 . Theorem 3.5.
Let < p, q < ∞ . Then the dual space of G p,q ( R ) is BD p ′ ,q ′ ( R ) where p + p ′ = 1 and q + q ′ = 1 JUSTICE SAM BANSAH AND BENOˆIT F. SEHBA Martingale Embeddings
In this section we will discuss the various inclusions of the martin-gale Hardy-amalgam spaces H Sp,q ( R ) , H sp,q ( R ) , H ∗ p,q ( R ) , Q p,q ( R ) and P p,q ( R ) as described in Theorem 3.1 above. Ferenc ([14]) has discussedthe classical cases including the Doob’s maximal inequality for p > A j ) j ∈ Z is such that A j ∈ F n for all j ∈ Z and all n ≥ Proposition 4.1.
For any f ∈ M , the following hold. (i) k f k H ∗ p ( R ) ≤ C p k f k H sp ( R ) , k f k H Sp ( R ) ≤ C p k f k H sp ( R ) (0 < p ≤ k f k H sp ( R ) ≤ C p k f k H ∗ p ( R ) , k f k H sp ( R ) ≤ C p k f k H Sp ( R ) (2 ≤ p < ∞ )(iii) k f k H ∗ p ( R ) ≤ C p k f k P p ( R ) , k f k H Sp ( R ) ≤ C p k f k Q p ( R ) (0 < p < ∞ )(iv) k f k H ∗ p ( R ) ≤ C p k f k Q p ( R ) , k f k H Sp ( R ) ≤ C p k f k P p ( R ) (0 < p < ∞ )(v) k f k H sp ( R ) ≤ C p k f k P p ( R ) , k f k H sp ( R ) ≤ C p k f k Q p ( R ) (0 < p < ∞ ) .Moreover, if the ( F ) n ≥ is regular, the above five spaces are equivalent. We observe the following.
Lemma 4.2.
Assume that A ∈ F n for all n ≥ . Then if f ∈ M , then f A = ( f n A ) n ≥ is also a martingale in M . Moreover, if T is any ofthe operators s , S and M (the maximal operator), then T ( f A ) = T ( f ) A . Combining the above lemma with ([14, Lemma 2.20]), we obtain thefollowing.
Lemma 4.3.
Assume that A ∈ F n for all n ≥ . Then for any mar-tingale f ∈ M and < p < ∞ , we have E (cid:20) sup n E n − ( | f n | p A ) (cid:21) ≤ E (( f ∗ ) p A ) and E (cid:20) sup n E n − ( | S n ( f ) p A ) (cid:21) ≤ E ( S ( f ) p A ) . With the help of Lemma 4.2 and Lemma 4.3 we now present theproof to the martingale inequalities that relate the five sets.
MBEDDINGS AND DUALITY 9
Proof of Theorem 3.1.
Let T be any of the operators s , S and M , and H Tp ( R ) , H Tp,q ( R ) the corresponding martingale spaces, then since A j ∈F n for all j ∈ Z and all n ≥
1, by Lemma 4.2, we have that for anymartingale f ∈ M and any j ∈ Z , Z R T ( f ) p A j d P = Z R T ( f A j ) p d P = k f A j k pH Tp ( R ) , and consequently,(3) k T ( f ) k qH Tp,q ( R ) = X j k T ( f ) A j k qL p ( R ) = X j k f A j k qH Tp ( R ) . The two first assertions of the theorem then follow from (3) and Propo-sition 4.1.To obtain the other assertions, following (3) and Proposition 4.1, weonly need to prove that(4) X j k f A j k q Q p ( R ) ≤ C k f k q Q p,q ( R ) and(5) X j k f A j k q P p ( R ) ≤ C k f k q P p,q ( R ) . We only prove (4) as (5) follows similarly.Let ( ̺ n ) n ≥ be an arbitrary nonnegative nondecreasing adapted se-quence such that S n ( f ) ≤ ̺ n − , and k ̺ ∞ k L p,q ( R ) < ∞ . We have that the sequence ( γ jn ) n ≥ = ( ̺ n A j ) n ≥ is also nonnegativenondecreasing and adapted, and S n ( f A j ) = S n ( f ) A j ≤ ̺ n − A j = γ jn − and k γ j ∞ k L p ( R ) = k ̺ ∞ A j k L p ( R ) ≤ k ̺ ∞ k L p,q ( R ) < ∞ . It follows that X j k f A j k q Q p ( R ) ≤ X j k γ j ∞ k qL p ( R ) = X j k ̺ ∞ A j k qL p ( R ) = k ̺ ∞ k qL p,q ( R ) . As the sequence ( ̺ n ) n ≥ was chosen arbitrarily, we conclude that X j k f A j k q Q p ( R ) ≤ inf ̺ ∈ ρ k ̺ ∞ k qL p,q ( R ) = k f k q Q p,q ( R ) . Let us now assume that the ( F n ) n ≥ is regular. To prove the equivalencebetween the five spaces, we only need to prove that(6) k f k Q p,q ( R ) ≤ C k f k H Sp,q ( R ) and(7) k f k P p,q ( R ) ≤ C k f k H ∗ p,q ( R ) . We only prove the (6) since the proof of (7) use similar arguments.Let f = ( f n ) n ≥ be a martingale in H Sp,q ( R ). Then using the defini-tion of the regularity, one obtain that(8) S n ( f ) ≤ (cid:2) C p (cid:0) S pn − ( f ) + E n − ( S pn ( f )) (cid:1)(cid:3) p (see [14, p. 39]). Define the sequence ̺ = ( ̺ n ) n ≥ by ̺ n = (cid:2) C p (cid:0) S pn ( f ) + E n ( S pn +1 ( f )) (cid:1)(cid:3) p . Then ̺ ∈ ρ and by (8), S n ( f ) ≤ ̺ n − . Also, we have that ̺ ∞ = sup n ̺ n = (cid:20) C p (cid:18) S p ( f ) + sup n E n ( S pn +1 ( f )) (cid:19)(cid:21) p . Then using Lemma 4.2 and Lemma 4.3, we obtain for any j ∈ Z , k ̺ ∞ A j k L p ( R ) ≤ C p k S p ( f ) A j k L p ( R ) . Hence k f k Q p,q ( R ) ≤ k ̺ ∞ k L p,q ( R ) . k S ( f ) k L p,q ( R ) = k f k H Sp,q ( R ) . The proof is complete. (cid:3)
We finish this section with the proofs of martingale inequalities, thegeneralization of Doob’s inequality and Burkholder-Davis-Gundy in-equality. For this, we recall the following classical Doob’s inequality,that can be found in [14].
Proposition 4.4.
Let p > . For every non-negative L p − bounded sub-martingale ( f n , n ∈ N ) , we have that (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N f n (cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) ≤ pp − n ∈ N k f n k L p ( R ) . We start with the proof of the extension of Doob’s inequality.
MBEDDINGS AND DUALITY 11
Proof of Theorem 3.2.
Let A j be defined as equation (1). Let g n,j = f n A j . Since g n,j is a martingale, Proposition 4.4 implies that (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N g n,j (cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) ≤ pp − n ∈ N k g n,j k L p ( R ) . Therefore by definition, and for 0 < q < ∞ , k sup n ∈ N f n k qL p,q ( R ) = X j (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N f n A j (cid:13)(cid:13)(cid:13)(cid:13) qp = X j (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N g n,j (cid:13)(cid:13)(cid:13)(cid:13) qp ≤ (cid:18) pp − (cid:19) q X j (cid:18) sup n ∈ N k g n,j k p (cid:19) q = (cid:18) pp − (cid:19) q X j sup n ∈ N k g n,j k qp = (cid:18) pp − (cid:19) q sup n ∈ N X j k g n,j k qp = (cid:18) pp − (cid:19) q sup n ∈ N X j k f n A j k qp Thus k sup n ∈ N f n k qL p,q ( R ) ≤ (cid:18) pp − (cid:19) q sup n ∈ N X j k f n A j k qp = (cid:18) pp − (cid:19) q sup n ∈ N k f k qL p,q ( R ) . The case q = ∞ follows similarly. The proof is complete. (cid:3) We also obtain the proof of the extension of Burkholder-Davis-Gundy’sinequality below.
Proof of Theorem 3.3.
The proof follows from Lemma 4.2 and the clas-sical Burkholder-Davis-Gundy’s inequality (see [14, Theorem 2.12]).We note that for the second equivalence, we simply replace the sum-mation with the supremum and the result follows. (cid:3) Dual Characterizations
In this Section, we focus our attention on the characterization of thethe dual of the spaces H sp,q ( R ) when 1 < p, q < ∞ and G p,q ( R ) . Let usstart by identifying the dual space of H sp,q ( R ) . Dual of H sp,q ( R ) . As noted in the introduction, the dual of H sp,q ( R )when 0 < p ≤ q ≤ H sp,q ( R ) when 1 < p ≤ q < ∞ .Let us start with the justification of the following important result. Lemma 5.2.
Let ≤ p ≤ q < ∞ . Then the space H sp,q ( R ) is uniformlyconvex. Proof.
We recall that a Banach space H is uniformly convex if for any ǫ >
0, there exists δ > x, y ∈ H with k x k H ≤ k y k H ≤ k x − y k H ≥ ǫ , then k x + y k H ≤ − δ ).We recall that for 1 ≤ r < ∞ and for a, b > a + b ) r ≤ r − ( a r + b r ) and a r + b r ≤ ( a + b ) r . Let ǫ >
0, and assume that f, g ∈ H sp,q with k f k H sp,q ≤ k g k H sp,q ≤ k f − g k H sp,q ≥ ǫ . We start by observing that s ( f + g ) + s ( f − g ) = 2( s ( f ) + s ( g )) . We then obtain (cid:0) s ( f + g ) (cid:1) p + (cid:0) s ( f − g ) (cid:1) p ≤ (cid:0) s ( f + g ) + s ( f − g ) (cid:1) p ≤ p − [ s p ( f ) + s p ( g )] . Hence for any j ∈ Z , k s ( f + g ) A j k pp + k s ( f + g ) A j k pp ≤ p − (cid:0) k s ( f ) A j k pp + k s ( g ) A j k pp (cid:1) . Raising both members of the last inequality to the power qp ≥
1, weobtain k s ( f + g ) A j k qp + k s ( f + g ) A j k qp ≤ (cid:0) k s ( f + g ) A j k pp + k s ( f + g ) A j k pp (cid:1) qp ≤ qp ( p − (cid:0) k s ( f ) A j k pp + k s ( g ) A j k pp (cid:1) qp ≤ qp ( p − qp − (cid:0) k s ( f ) A j k qp + k s ( g ) A j k qp (cid:1) . Hence taking the sum over j ∈ Z , we obtain k s ( f + g ) k qp,q + k s ( f − g ) k qp,q ≤ q − (cid:0) k s ( f ) k qp,q + k s ( g ) k qp,q (cid:1) and so k s ( f + g ) k qp,q ≤ q − (cid:0) k s ( f ) k qp,q + k s ( g ) k qp,q (cid:1) − k s ( f − g ) k qp,q ≤ q − ǫ q . Thus k s ( f + g ) k p,q ≤ − δ )where δ = 1 − (cid:18) − ǫ q q (cid:19) q and the proof is complete. (cid:3) From the above Lemma and Milman’s Theorem (see [16, p.127]), wededuce the following.
Corollary 5.3.
Let ≤ p ≤ q < ∞ . Then the space H sp,q ( R ) isreflexive. MBEDDINGS AND DUALITY 13
Proof of Theorem 3.4.
It follows from Corollary 5.3 above that we onlyneed to prove for the case 1 < q ≤ p ≤ . Let g ∈ H sp ′ ,q ′ ( R ) and κ g ( f ) := E ∞ X n =0 d n f d n g ! (cid:0) f ∈ H sp,q ( R ) (cid:1) . Hence by Schwarz’s inequality, we have that | κ g ( f ) | ≤ Z R ∞ X n =0 E n − | d n f || d n g | d P = X j ∈ Z Z A j ∞ X n =0 E n − | d n f || d n g | d P ≤ X j ∈ Z Z A j ∞ X n =0 (cid:0) E n − | d n f | (cid:1) (cid:0) E n − | d n g | (cid:1) d P ≤ X j ∈ Z Z A j ∞ X n =0 E n − | d n f | ! ∞ X n =0 E n − | d n g | ! d P = X j ∈ Z Z A j s ( f ) s ( g )d P . Applying the H¨older’s inequality to the right hand of the last inequality,we obtain | κ g ( f ) | ≤ X j ∈ Z k s ( f ) A j k p k s ( g ) A j k p ′ ≤ X j ∈ Z k s ( f ) A j k qp ! q X j ∈ Z k s ( g ) A j k q ′ p ′ ! q ′ = k f k H sp,q ( R ) k g k H sp ′ ,q ′ ( R ) . Thus κ g ∈ (cid:0) H sp,q (cid:1) ′ and k κ g k ≤ k g k H sp ′ ,q ′ ( R ) . Conversely, let κ be a continuous linear functional on H sp,q ( R ) . Thenas H sp,q ( R ) embeds continuously into H sp ( R ) (since q < p ), we haveby the Hahn-Banach theorem that κ can be extended to a continuouslinear functional ˜ κ on H sp ( R ) having the same operator norm as κ . Itfollows from [14, Theorem 2.26] that there exists some g ∈ H sp ′ ( R ) suchthat ˜ κ ( f ) = E ( f g ) (cid:0) ∀ f ∈ H sp ( R ) (cid:1) . In particular(9) κ ( f ) = ˜ κ ( f ) = E ( f g ) (cid:0) ∀ f ∈ H sp,q ( R ) (cid:1) . Let us prove that(10) k g k H sp ′ ,q ′ ( R ) . sup f ∈ H sp,q ( R ) , k f k Hsp,q ( R ) ≤ | κ ( f ) | < ∞ . Obviously, this holds if k g k H sp ′ ,q ′ ( R ) = 0. Hence we assume that k g k H sp ′ ,q ′ ( R ) =0. We recall that, A j ∈ F n for all j ∈ Z and n ≥
1. Set µ n = X j ∈ Z s p ′ − n ( g ) A j k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A j k p ′ − q ′ p ′ . (11)Since the A j ’s are pairwise disjoint, we have that µ n = X j ∈ Z s p ′ − n ( g ) A j k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A j k p ′ − q ′ ) p ′ . From the definition of s ( · ) , we have that µ n is F n − -measurable. Wedefine h as the martingale transform of g by µ n . That is d n h = µ n d n g. (12)We then obtain ∞ X n =0 E n − | d n h | = ∞ X n =0 µ n E n − | d n g | or equivalently s ( h ) = ∞ X n =0 X j ∈ Z s p ′ − n ( g ) A j k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A j k p ′ − q ′ ) p ′ E n − | d n g | . Therefore s ( h ) = X j ∈ Z A j k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A j k p ′ − q ′ ) p ′ ∞ X n =0 s p ′ − n ( g ) E n − | d n g | = X j ∈ Z A j k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A j k p ′ − q ′ ) p ′ ∞ X n =0 s p ′ − n ( g )( s n ( g ) − s n − ( g ))= 1 k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z A j k s ( g ) A j k p ′ − q ′ ) p ′ ∞ X n =0 [ s p ′ − n ( g ) − s p ′ − n ( g ) s n − ( g )] . MBEDDINGS AND DUALITY 15
It follows that s ( h ) ≤ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z A j k s ( g ) A j k p ′ − q ′ ) p ′ ∞ X n =0 [ s p ′ − n ( g ) − s p ′ − n − ( g )]= 1 k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z s p ′ − ( g ) A j k s ( g ) A j k p ′ − q ′ ) p ′ . Thus, by disjointness of the A j ’s, s ( h ) ≤ s p ′ − ( g ) k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z A j k s ( g ) A j k p ′ − q ′ p ′ . (13)We also have that for any k ∈ Z , s ( h ) A k ≤ X j ∈ Z s p ′ − ( g ) k s ( g ) k q ′ − p ′ ,q ′ A j k s ( g ) A j k p ′ − q ′ p ′ A k = s p ′ − ( g ) k s ( g ) k q ′ − p ′ ,q ′ A k k s ( g ) A k k p ′ − q ′ p ′ . Therefore k s ( h ) A k k p ≤ k s p ′ − ( g ) A k k p k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A k k p ′ − q ′ p ′ = k s ( g ) A k k p ′ − p ′ k s ( g ) k q ′ − p ′ ,q ′ k s ( g ) A k k p ′ − q ′ p ′ = k s ( g ) A k k q ′ − p ′ k s ( g ) k q ′ − p ′ ,q ′ . Hence X k ∈ Z k s ( h ) A k k qp ≤ X k ∈ Z k s ( g ) A k k q ( q ′ − p ′ k s ( g ) k q ( q ′ − p ′ ,q ′ = X k ∈ Z k s ( g ) A k k q ′ p ′ k s ( g ) k q ′ p ′ ,q ′ = k s ( g ) k q ′ p ′ ,q ′ k s ( g ) k q ′ p ′ ,q ′ = 1 . That is k h k H sp,q ( R ) ≤ . We now test (10) with the martingale h above. First proceeding as in[14, p.37] (this is why we need p to be smaller than 2), we obtain | κ ( h ) | = E ∞ X n =0 d n hd n g ! = E ∞ X n =0 µ n | d n g | ! = 1 k s ( g ) k q ′ − p ′ ,q ′ E ∞ X n =0 X j ∈ Z s p ′ − n ( g ) A j k s ( g ) A j k p ′ − q ′ p ′ E n − | d n g | ! = 1 k s ( g ) k q ′ − p ′ ,q ′ E ∞ X n =0 X j ∈ Z s p ′ − n ( g ) A j k s ( g ) A j k p ′ − q ′ p ′ ( s n ( g ) − s n − ( g )) ! It follows that | κ ( h ) | ≥ p ′ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z k s ( g ) A j k p ′ − q ′ p ′ E A j ∞ X n =0 s p ′ n ( g ) − s p ′ n − ( g ) ! = 2 p ′ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z k s ( g ) A j k p ′ − q ′ p ′ E (cid:16) A j s p ′ ( g ) (cid:17) = 2 p ′ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z k s ( g ) A j k p ′ − q ′ p ′ Z R A j s p ′ ( g )d P = 2 p ′ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z k s ( g ) A j k p ′ − q ′ p ′ k s ( g ) A j k p ′ p ′ = 2 p ′ k s ( g ) k q ′ − p ′ ,q ′ X j ∈ Z k s ( g ) A j k q ′ p ′ = 2 p ′ k s ( g ) k q ′ p ′ ,q ′ k s ( g ) k q ′ − p ′ ,q ′ = 2 p ′ k s ( g ) k L p ′ ,q ′ ( R ) . The proof is complete. (cid:3)
Dual of G p,q . This part is devoted to the characterization of thedual of the variation integrable space. We begin our characterizationof the dual of G p,q with the introduction of the following larger space. Definition 5.5.
Let n ∈ N and let 1 ≤ p, q, r < ∞ . We define thespace K ( L p,q , ℓ r ) by K ( L p,q , ℓ r ) = (cid:8) measurable process ǫ = ( ǫ n ) n ≥ : k ǫ k K ( L p,q ,ℓ r ) < ∞ (cid:9) where k ǫ k K ( L p,q ,ℓ r ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ≥ | ǫ n | r ! r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p,q ( R ) . We observe that G p,q ⊆ K ( L p,q , ℓ ) . Indeed let f be a martingale.Then it is measurable with respect to the underlining filtration henceits increment, d n f, is also measurable. Thus we can take ǫ n = d n f andthe inclusion then follows by setting r = 1 . In the same way, we obtainthat BD p,q ⊆ K ( L p,q , ℓ ∞ ) . We also observe that since L p,p ( R ) = L p ( R ) , then K ( L p,p , ℓ r ) is thespace defined in [14, Definition 2.8]. The following lemma is part ofthe proof of Proposition 5.7 that follows, but for the sake of the pre-sentation, we isolate it. MBEDDINGS AND DUALITY 17
Lemma 5.6.
Let < p, q < ∞ , ≤ r < ∞ and let ( p, p ′ ) , ( q, q ′ ) , ( r, r ′ ) be their respective conjugate pairs. Let η ∈ K ( L p ′ ,q ′ , ℓ r ′ ) . Consider thesequence h = ( h k ) k ≥ defined as follows h k = P i ≥ | η k | r ′ η k k η k p ′− r ′ ℓr ′ k η k q ′− K ( Lp ′ ,q ′ ,ℓr ′ ) Ai k k η k ℓr ′ Ai k p ′− q ′ Lp ′ , η k = 00 , otherwiseif r > , and h k = P i ≥ η k )2 k +1 k η k p ′− ℓr ′ k η k q ′− K ( Lp ′ ,q ′ ,ℓr ′ ) Ai k k η k ℓr ′ Ai k p ′− q ′ Lp ′ , η k = 00 , otherwisefor r = 1 . Then h has a unit norm in K ( L p,q , ℓ r ) . Consequently h ∈K ( L p,q , ℓ r ) . Proof.
By definition, k h k K ( L p,q ,ℓ r ) = X j ≥ Z R X k | h k | r ! pr A j d P qp q = X j ≥ (cid:13)(cid:13) k h k ℓ r A j (cid:13)(cid:13) qL p ! q . Now for r >
1, we obtain | h k | r = | η k | r ′ k η k ( p ′ − r ′ ) rℓ r ′ k η k ( q ′ − r K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ r so that X k | h k | r = X k | η k | r ′ k η k ( p ′ − r ′ ) rℓ r ′ k η k ( q ′ − r K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ r and hence k h k rℓ r = k η k r ′ ℓ r ′ k η k ( p ′ − r ′ ) rℓ r ′ k η k ( q ′ − r K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ r = k η k r ′ +( p ′ − r ′ ) rℓ r ′ k η k ( q ′ − r K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ r = k η k ( p ′ − rℓ r ′ k η k ( q ′ − r K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ r and then(14) k h k ℓ r = k η k p ′ − ℓ r ′ k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ . One can easily check that (14) also holds for r = 1.As the A k s are disjoint, we obtain k h k ℓ r A j = k η k p ′ − ℓ r ′ k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) X i ≥ A i (cid:13)(cid:13) k η k ℓ r ′ A i (cid:13)(cid:13) p ′ − q ′ L p ′ A j = k η k p ′ − ℓ r ′ k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ . MBEDDINGS AND DUALITY 19
We now take the L p ( R )-norm of both sides. Z R k h k pℓ r A j d P = Z R k η k ( p ′ − pℓ r ′ k η k ( q ′ − p K ( L p ′ ,q ′ ,ℓ r ′ ) A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( p ′ − q ′ ) pL p ′ d P = 1 k η k ( q ′ − p K ( L p ′ ,q ′ ,ℓ r ′ ) Z R k η k ( p ′ − pℓ r ′ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( p ′ − q ′ ) pL p ′ d P = 1 k η k ( q ′ − p K ( L p ′ ,q ′ ,ℓ r ′ ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( p ′ − q ′ ) pL p ′ Z R k η k p ′ ℓ r ′ A j d P = 1 k η k ( q ′ − p K ( L p ′ ,q ′ ,ℓ r ′ ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ L p ′ (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( p ′ − q ′ ) pL p ′ = 1 k η k ( q ′ − p K ( L p ′ ,q ′ ,ℓ r ′ ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( q ′ − pL p ′ . Therefore (cid:13)(cid:13) k h k ℓ r A j (cid:13)(cid:13) L p = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) q ′ − L p ′ . Hence X j ≥ (cid:13)(cid:13) k h k ℓ r A j (cid:13)(cid:13) qL p = X j ≥ k η k ( q ′ − q K ( L p ′ ,q ′ ,ℓ r ′ ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) ( q ′ − qL p ′ = 1 k η k q ′ K ( L p ′ ,q ′ ,ℓ r ′ ) X j ≥ (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) q ′ L p ′ and then X j ≥ (cid:13)(cid:13) k h k ℓ r A j (cid:13)(cid:13) qL p = 1 k η k q ′ K ( L p ′ ,q ′ ,ℓ r ′ ) k η k q ′ K ( L p ′ ,q ′ ,ℓ r ′ ) = 1 . Therefore k h k K ( L p,q ,ℓ r ) = X j ≥ (cid:13)(cid:13) k h k ℓ r A j (cid:13)(cid:13) qL p ! q = 1 . Thus h = ( h k ) k ≥ ∈ K ( L p,q , ℓ r ) since h = ( h k ) k ≥ is measurable. (cid:3) The following Proposition characterizes the dual of K ( L p,q , ℓ r ) . Proposition 5.7.
For < p, q < ∞ and ≤ r < ∞ , the dual space, K ( L p,q , ℓ r ) ∗ , of K ( L p,q , ℓ r ) is K ( L p ′ ,q ′ , ℓ r ′ ) where p + 1 p ′ = 1 , q + 1 q ′ = 1 , r + 1 r ′ = 1 . Proof.
Let η = ( η k ) k ≥ ∈ K ( L p ′ ,q ′ , ℓ r ′ ) and ǫ = ( ǫ k ) k ≥ ∈ K ( L p,q , ℓ r ) . Let h· , ·i be the usual inner product, that is, h η, ǫ i = X k η k ǫ k . and define the functional, Λ , byΛ η ( ǫ ) = E h η, ǫ i = Z R X k ≥ ǫ k η k d P = X j ≥ Z A j X k ≥ ǫ k η k d P for all η = ( η k ) k ≥ ∈ K ( L p ′ ,q ′ , ℓ r ′ ) measurable and ǫ = ( ǫ k ) k ≥ ∈K ( L p,q , ℓ r ) . Then by H¨older inequality, | Λ η ( ǫ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ≥ Z A j X k ≥ ǫ k η k d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ǫ k K ( L p,q ,ℓ r ) k η k K ( L p ′ ,q ′ ,ℓ r ′ ) . (15)and since Λ η ( · ) is linear and bounded, it is a continuous linear func-tional on K ( L p,q , ℓ r ) . From inequality (15), we deduce that Λ η ∈ ( K ( L p,q , ℓ r )) ′ and k Λ η k ≤ k η k K ( L p ′ ,q ′ ,ℓ r ′ ) . (16)For the converse, we can suppose that q < p . Let Λ be a continuouslinear functional on K ( L p,q , ℓ r ) . Then as K ( L p,q , ℓ r ) embeds continu-ously into K ( L p , ℓ r ) (since q < p ), we have by Hahn-Banach Theoremthat Λ can be extended to a continuous linear functional ˜Λ on K ( L p , ℓ r )having the same operator norm as Λ . It follows from ([14, Lemma 2.9])that there exists some η ∈ K ( L p ′ , ℓ r ′ ) such that˜Λ η ( ǫ ) = E h η, ǫ i for all ǫ ∈ K ( L p , ℓ r ) . In particularΛ η ( ǫ ) = ˜Λ η ( ǫ ) = E h η, ǫ i for all ǫ ∈ K ( L p,q , ℓ r ) . Let us now show that k η k K ( L p ′ ,q ′ ,ℓ r ′ ) . sup ǫ ∈K ( L p,q ,ℓ r ) , k ǫ k K ( Lp,q,ℓr ) ≤ | Λ η ( ǫ ) | . Set h = ( h k ) k ≥ to be the sequence defined in Lemma 5.6. Since h =( h k ) k ≥ ∈ K ( L p,q , ℓ r ) with a unit norm, by linearity of the expectation MBEDDINGS AND DUALITY 21 operator, we have that for 1 < r < ∞ , k Λ k ≥ | Λ η ( h ) | = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) E X k ≥ | η k | r ′ ! p ′ r ′ X j ≥ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) E k η k p ′ ℓ r ′ X j ≥ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) X j ≥ E k η k p ′ ℓ r ′ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) X j ≥ E ( k η k p ′ ℓ r ′ A j ) (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) X j ∈ Z (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ L p ′ (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) "X j ∈ Z (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) q ′ L p ′ . Therefore k Λ k ≥ k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) k η k q ′ K ( L p ′ ,q ′ ,ℓ r ′ ) = k η k K ( L p ′ ,q ′ ,ℓ r ′ ) . Thus k Λ k ≥ k η k K ( L p ′ ,q ′ ,ℓ r ′ ) (17)In the case of r = 1, we note that there is and integer k such that12 k η k ℓ ∞ ≤ | η k | . Thus using the test function h defined in Lemma 5.6 for r = 1, andfollowing the steps above, we obtain k Λ k ≥ | Λ η ( h ) | = 1 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) E X k ≥ | η k | k +1 ! X j ≥ k η k p ′ − ℓ r ′ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ ≥ k +2 k η k q ′ − K ( L p ′ ,q ′ ,ℓ r ′ ) E k η k p ′ ℓ r ′ X j ≥ A j (cid:13)(cid:13) k η k ℓ r ′ A j (cid:13)(cid:13) p ′ − q ′ L p ′ = 12 k +2 k η k K ( L p ′ ,q ′ ,ℓ r ′ ) . The proof is complete. (cid:3)
We then see that ( K ( L p,q , ℓ )) ∗ = K ( L p ′ ,q ′ , ℓ ∞ ) and thus it is nowevident that the dual of the variation integrable space is the jumpbounded space. More rigorously, we prove Theorem 3.5. Proof of Theorem 3.5.
Let g ∈ BD p ′ ,q ′ ( R ) and set κ g ( f ) = ∞ X k =1 E [ d k f d k g ] for f ∈ G p,q ( R ) . We obtain | κ g ( f ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 E [ d k f d k g ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X k =1 E [ | d k f || d k g | ] ≤ E ∞ X k =1 [ | d k f | sup k ∈ N | d k g | ]= Z R ∞ X k =1 | d k f | sup k ∈ N | d k g | d P = X j ∈ Z Z A j ∞ X k =1 | d k f | sup k ∈ N | d k g | d P ≤ X j ∈ Z "Z A j ∞ X k =1 | d k f | ! p d P p "Z A j sup k ∈ N | d k g | p ′ d P p ′ ≤ X j ∈ Z "Z A j ∞ X k =1 | d k f | ! p d P qp q X j ∈ Z "Z A j sup k ∈ N | d k g | p ′ d P q ′ p ′ q ′ . MBEDDINGS AND DUALITY 23
That is | κ g ( f ) | ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =0 | d n f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,q (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N | d n g | (cid:13)(cid:13)(cid:13)(cid:13) p ′ ,q ′ = k f k G p,q ( R ) k g k BD p ′ ,q ′ ( R ) . Therefore κ g ∈ ( G p,q ( R )) ′ and k κ g k ≤ k g k BD p ′ ,q ′ ( R ) . To prove the converse, we first assume that τ is an arbitrary elementin the dual of G p,q ( R ) then we show that there exists g ∈ BD p ′ ,q ′ ( R )such that τ = κ g and k g k BD p ′ ,q ′ ≤ C k τ k for some constant C. By setting ǫ k = d k f for f ∈ M we saw earlier that G p,q ( R ) ⊆K ( L p,q , ℓ ) . We also recall that the dual space of K ( L p,q , ℓ ) is K ( L p ′ ,q ′ , ℓ ∞ )and τ is a continuous linear functional on G p,q ( R ) ⊆ K ( L p,q , ℓ ) . ByHahn-Banach Theorem, τ can be extended to a continuous linear func-tional on K ( L p,q , ℓ ) having the same operator norm as τ. Let Λ bethis extension of τ. Then we have by Proposition 5.7 that there exists η ∈ K ( L p ′ ,q ′ , ℓ ∞ ) such that k Λ k = k τ k ≈ k η k K ( L p ′ ,q ′ ,ℓ ∞ ) and Λ η ( ǫ ) = X k ≥ E ( ǫ k η k )for ǫ ∈ K ( L p,q , ℓ ) . Hence τ ( f n ) = n X k =1 E [( d k f ) η k ] = n X k =1 E [( d k f )( E k η k − E k − η k )](18)is well defined (we agree for a moment to work with f n as we will showthat f n → f in G p,q as n → ∞ ). We shall now set g n := (cid:26) P nk =1 [ E k η k − E k − η k ] if n = 0 g = 0and show that g n is a bounded martingale. Indeed we observe that as k → ∞ , E k η k − E k − η k → . Thus g n is finite (i.e. sup n | g n | < ∞ ) andalso E n − ( g n − g n − ) = E n − ( E n η n − E n − η n ) = E n − η n − E n − η n = 0 . This means that g n is a martingale. Consequently, sincesup k ∈ N | d k g | = sup k ∈ N | E k η k − E k − η k | ≤ sup k ∈ N ( | E k | η k | + E k − | η k | ) ≤ n ∈ N E n (cid:18) sup k ∈ N | η k | (cid:19) , we can invoke inequality (2) to obtain k sup k ∈ N | d k g |k p ′ ,q ′ ≤ (cid:13)(cid:13)(cid:13)(cid:13) sup n ∈ N E n (cid:18) sup k ∈ N | η k | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) p ′ ,q ′ ≤ p ′ p ′ − k ∈ N (cid:13)(cid:13)(cid:13)(cid:13) E n (cid:18) sup k ∈ N | η k | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) p ′ ,q ′ ≤ p ′ p ′ − (cid:13)(cid:13)(cid:13)(cid:13) sup k ∈ N | η k | (cid:13)(cid:13)(cid:13)(cid:13) p ′ ,q ′ = 2 p ′ p ′ − k η k K ( L p ′ ,q ′ ,ℓ ∞ ) . Hence g ∈ BD p ′ ,q ′ ( R ) and k g k BD p ′ ,q ′ ( R ) ≤ C k τ k since k τ k = k η k K ( L p ′ ,q ′ ,ℓ ∞ ) . We now show that f n → f in G p,q as n → ∞ . We observe that since f = P ∞ k =0 f k − f k − , we have that f n − f = f n − ∞ X k =0 f k − f k − = f n − n X k =0 f k − f k − − ∞ X k = n +1 f k − f k − = − ∞ X k = n +1 f k − f k − . Hence for n → ∞ , f n − f → . Also X k ≥ | d k ( f n − f ) | = X k ≥ | d k f n − d k f | ≤ X k ≥ || d k f n | + | d k f || ≤ X k ≥ | d k f | < ∞ since f ∈ G p,q and thus by the Dominated Convergence Theorem k f n − f k G p,q ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ≥ | d k ( f n − f ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p,q ( R ) → n → ∞ . That is f n → f in G p,q ( R ) . Consequently, from equation (18), as n →∞ , we have that, by setting η k = d k g,τ ( f n ) = n X k =1 E [( d k f ) η k ] → τ ( f ) = ∞ X k =1 E [( d k f ) η k ] = κ g ( f ) . Hence k g k BD p ′ ,q ′ ( R ) ≤ C k τ k = C k κ k and the poof is complete. (cid:3) Further discussions and conclusion
In this work, we were able to to characterize the dual of H sp,q for1 < p ≤ q < ∞ . We were also able to extend the Doob’s inequality andthe Burkholder-Davis-Gundy inequality from the classical Lebesguespace to the amalgam space. We have discussed the various relationsbetween the martingale Hardy-amalgam spaces and showed that uponthe condition of regular basis, the spaces are all equivalent. We havealso introduced the variation integrable space relative to our settingand characterized its dual space.
MBEDDINGS AND DUALITY 25
We observe that the Davis’ decomposition is also valid in the mar-tingale Hardy-amalgam spaces. Indeed, we have the following decom-position which can be proved as in the case p = q (see [14, Lemma2.13]) Theorem 6.1.
Let f = { f n } n ∈ N ∈ H Sp,q (1 ≤ p < ∞ , < q < ∞ ) . Then there exists h = { h n } n ∈ N ∈ G p,q and g = { g n } n ∈ N ∈ Q p,q suchthat f = { f n = h n + g n } n ∈ N for all n ∈ N and k h k G p,q ≤ (2 + 2 C ) k f k H Sp,q and k g k Q p,q ≤ (7 + 2 C ) k f k H Sp,q . Combining the above theorem with the second of the inequalities (v)in Theorem 3.1, we derive the following.
Corollary 6.2.
Let f = { f n } n ∈ N ∈ H Sp,q (1 ≤ p < ∞ , < q < ∞ ) . Then there exists h = { h n } n ∈ N ∈ G p,q and g = { g n } n ∈ N ∈ H sp,q suchthat f = { f n = h n + g n } n ∈ N for all n ∈ N and k h k G p,q ≤ (2 + 2 C ) k f k H Sp,q and k g k H sp,q ≤ (7 + 2 C ) k f k H Sp,q . The following Davis’ decomposition of H ∗ p,q follows also as in [14,Lemma 2.14]. Theorem 6.3.
Let f = { f n } n ∈ N ∈ H ∗ p,q (1 ≤ p < ∞ , < q < ∞ ) . Then there exists h = { h n } n ∈ N ∈ G p,q and g = { g n } n ∈ N ∈ P p,q such that f = { f n = h n + g n } n ∈ N for all n ∈ N and k h k G p,q ≤ (4 + 4 C ) k f k H ∗ p,q and k g k P p,q ≤ (13 + 4 C ) k f k H ∗ p,q . Combining the above theorem with inequality (v) in Theorem 3.1,we derive the following.
Corollary 6.4.
Let f = { f n } n ∈ N ∈ H ∗ p,q (1 ≤ p < ∞ , < q < ∞ ) . Then there exists h = { h n } n ∈ N ∈ G p,q and g = { g n } n ∈ N ∈ H sp,q suchthat f = { f n = h n + g n } n ∈ N for all n ∈ N and k h k G p,q ≤ (4 + 4 C ) k f k H ∗ p,q and k g k H sp,q ≤ (13 + 4 C ) k f k H ∗ p,q . We close this work with the proof of the following duality result thatextend the one of [14, Theorem 2.34].
Theorem 6.5.
The dual space of H ∗ p,q (1 < q ≤ p ≤ can be givenwith the norm k φ k := k φ k H sp ′ ,q ′ + k φ k BD p ′ q ′ where p + p ′ = q + q ′ = 1 . Proof.
Let φ ∈ H sp ′ ,q ′ ∩ BD p ′ ,q ′ . Then φ ∈ L since L is dense in H sp ′ ,q ′ (from its atomic decomposition). Define the functional κ φ by κ φ ( f ) = E ( f φ ) , ( f ∈ L ) . (19)We will show that (19) is a bounded linear functional on H ∗ p,q (1 < p ≤ q ≤ . Linearity follows since the expectation operator is linear. Also as L is dense in H ∗ p,q , we have that κ φ is well defined. As f n → f in L norm(as n → ∞ ), we have that κ φ ( f ) := E ( f φ ) = lim n →∞ E ( f n φ ) . Now for f ∈ H ∗ p,q , we know from Davis’ decomposition (Corollary 6.4)that f n = h n + g n where h = { h n } n ∈ N and g = { g n } n ∈ N are martingalessuch that k h k G p,q . k f k H ∗ p,q (20)and k g k H sp,q . k f k H ∗ p,q . (21)Hence we have by linearity of E that | E ( f n φ ) | = | E ( h n φ + g n φ ) | ≤ | E ( g n φ ) | + | E ( h n φ ) | (22)From (21) we have that g ∈ H sp,q since f ∈ H ∗ p,q . Hence since φ ∈ H sp ′ ,q ′ , it follows from Theorem 3.4 for ( q ≤ p ) that | E ( g n φ ) | ≤ k g n k H sp,q k φ k H sp ′ ,q ′ . (23)Similarly, since h ∈ G p,q and φ ∈ BD p ′ ,q ′ , we have by Theorem 4.13that | E ( h n φ ) | ≤ k h n k G p,q k φ k BD p ′ ,q ′ . (24)Therefore (22) becomes | E ( f n φ ) | ≤ k g n k H sp,q k φ k H sp ′ ,q ′ + k h n k G p,q k φ k BD p ′ ,q ′ and thus | E ( f φ ) | ≤ k g k H sp,q k φ k H sp ′ ,q ′ + k h k G p,q k φ k BD p ′ ,q ′ . MBEDDINGS AND DUALITY 27
It then follows from (20) and (21) that | E ( f φ ) | ≤ k f k H ∗ p,q k φ k H sp ′ ,q ′ + k f k H ∗ p,q k φ k BD p ′ ,q ′ . In other words, | E ( f φ ) | ≤ k f k H ∗ p,q ( k φ k H sp ′ ,q ′ + k φ k BD p ′ ,q ′ ) . (25)Thus the functional κ φ is continuous linear functional on H ∗ p,q . Conversely, assume that κ φ is an arbitrary continuous linear on H ∗ p,q . Then as H ∗ p,q embeds continuously in H ∗ p ( q < p ) , we have by Hahn-Banach theorem that κ φ can be extended to a continuous linear func-tional ˜ κ φ on H ∗ p having the same operator norm as κ φ . It follows from[14, Theorem 2.34] that there exists some φ ∈ L such that˜ κ φ ( f ) = E ( f φ ) , ( f ∈ L ) . In particular κ φ ( f ) = ˜ κ φ ( f ) = E ( f φ ) , ( f ∈ H ∗ p,q ) . We observe that since f ∗ := sup n ∈ N | f n | = sup n ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 d k f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup n ∈ N n X k =1 | d k f | ≤ sup n ∈ N ∞ X k =1 | d k f | = ∞ X k =1 | d k f | , we have that k f ∗ k p,q ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | d k f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,q (in other words , k f k H ∗ p,q ≤ k f k G p,q ) . Thus G p,q ⊆ H ∗ p,q . Therefore κ φ is also a continuous linear functionalon G p,q . It follows from Theorem 3.5 that k φ k BD p ′ ,q ′ ≤ C k κ φ k . (26)We also recall that from Theorem 3.1, we have k f k H ∗ p,q ≤ C k f k H sp,q (1 ≤ p, q ≤
2) Therefore κ φ is also a continuous linear functional on H sp,q . Hence by Theorem 3.4, for ( q ≤ p ), k φ k H sp ′ ,q ′ ≤ C k κ φ k . (27)From (26) and (27), we obtain that k φ k BD p ′ ,q ′ + k φ k H sp ′ ,q ′ ≤ C k κ φ k and the proof is complete. (cid:3) Finally, we note that even in the dyadic case, the characterization ofthe dual of H sp,q for ( p, q ) / ∈ { ( p, q ) ∈ [1 , ∞ ) : 1 < q ≤ p ≤ ≤ p ≤ q < ∞} is still open. 7. Declarations
The authors declare that they have no conflict of interest regardingthis work.
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Department of Mathematics, University of Ghana, P. O. Box LG 62Legon, Accra, Ghana
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