Multilinear fractional Calderón-Zygmund operators on weighted Hardy spaces
aa r X i v : . [ m a t h . C A ] M a r MULTILINEAR FRACTIONAL CALDER ´ON-ZYGMUNDOPERATORS ON WEIGHTED HARDY SPACES
DAVID CRUZ-URIBE, OFS, KABE MOEN, AND HANH VAN NGUYEN
Abstract.
We prove norm estimates for multilinear fractional integrals acting onweighted and variable Hardy spaces. In the weighted case we develop ideas weused for multilinear singular integrals [7]. For the variable exponent case, a keyelement of our proof is a new multilinear, off-diagonal version of the Rubio deFrancia extrapolation theorem. Introduction
The purpose of this paper is to continue the study of multilinear operators on Hardyspaces begun in [7, 10]. In those papers we considered multilinear Calder´on-Zygmundoperators and multipliers. Here we consider the multilinear fractional Calder´on-Zygmund operators introduced by Lin and Lu [15]. In the linear case, fractionalCalder´on-Zygmund operators have been studied by a number of authors. See, forinstance, the recent papers [17, 18].Given positive integers m, n and a real number 0 < γ < mn , let K γ be a functiondefined in R ( m +1) n away from the diagonal x = y = · · · = y m that satisfies the sizecondition(1.1) | K γ ( x, y , . . . , y m ) | . (cid:16) | x − y | + · · · + | x − y m | (cid:17) γ − mn , and the smoothness condition(1.2) m X i =1 X | β | = N | ∂ βi K γ ( x, y , . . . , y m ) | . (cid:16) | x − y | + · · · + | x − y m | (cid:17) γ − mn − N for some sufficiently large integer N . We define the multilinear fractional Calder´on-Zygmund operator T γ by T γ ( f , . . . , f m )( x ) = Z R mn K γ ( x, y , . . . , y m ) f ( y ) · · · f m ( y m ) d~y. Date : February 28, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Muckenhoupt weights, weighted Hardy spaces, variable Hardy spaces,multilinear fractional operators, Rubio de Francia extrapolation.The first author is supported by research funds from the Dean of the College of Arts & Sciences,the University of Alabama. The second author is supported by the Simons Foundation.
The simplest example of such an operator is the multilinear fractional integralintroduced by Kenig and Stein [14]: I γ ( f , . . . , f m )( x ) = Z R mn f ( y ) · · · f m ( y m )( | x − y | + · · · + | x − y m | ) mn − γ d~y. They proved that for 1 < p , . . . , p m ≤ ∞ and q such that q = p + · · · + p m − γn > k I γ ( f , . . . , f m ) k L q . C k f k L p · · · k f m k L pm . Moreover, if p i = 1 for some i , then the above inequality is replaced by the corre-sponding weak-type estimate.Lin and Lu [15] proved Hardy space estimates for multilinear fractional Calder´on-Zygmund operators, generalizing the results of Grafakos and Kalton [13] for multi-linear singular integrals and the results in the linear case for fractional integrals dueto Str¨omberg and Wheeden [20] and Gatto, et al. [12]. More precisely, they provedthat if 0 < p , . . . , p m , q ≤
1, then k T γ ( f , . . . , f m ) k L q . C k f k H p · · · k f m k H pm . However, they had to make the restrictive assumption that 0 < γ < n .Our first theorem is a generalization of the result of Lin and Lu to weighted Hardyspaces. To state it, we first recall some basic definitions from the theory of Muck-enhoupt weights. By a weight we mean a non-negative, locally integrable function.Given a weight w and 1 < p < ∞ , we say w is in the Muckenhoupt class A p , denotedby w ∈ A p , if for every cube Q , − Z Q w dx (cid:18) − Z Q w − p ′ dx (cid:19) p − ≤ C < ∞ . The smallest such constant C is denoted by [ w ] A p . The A p classes are nested: A p ⊂ A q if p < q . Hence we can define A ∞ as the union of all the A p classes, and define r w = inf { p : w ∈ A p } . For s >
1, we say that w satisfies a reverse H¨older inequalitywith exponent s , denoted by w ∈ RH s , if for every cube Q , (cid:18) − Z Q w s dx (cid:19) s ≤ C − Z Q w dx. The infimum of all the constants for which this is true is denoted by [ w ] RH s . A weightis in A p for some p > RH s for some s > Theorem 1.1.
Let < γ < mn . Given < p , . . . , p m < ∞ , define < p < ∞ by (1.3) 1 p = 1 p + · · · + 1 p m > γn , and define < q < ∞ by (1.4) 1 q = 1 p − γn . ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 3
Suppose that ( w , . . . , w m ) is a vector of weights satisfying w i ∈ RH qp . If K γ satisfies (1.1) and (1.2) for some positive integer N > max { mn ( r wi p i − , ≤ i ≤ m } , then (1.5) k T γ ( f , . . . , f m ) k L q ( w ) . m Y i =1 k f i k H pi ( w i ) , where w = m Y i =1 w qpi i . Remark 1.2.
Even in the unweighted case Theorem 1.1 is a more general result thanthat of Lin and Lu, since we extend the values of γ to the full range 0 < γ < mn .Below, we will prove Theorem 1.1 as a special case of a more general result, The-orem 3.1. It is more complicated to state, since it requires the existence of certain q i > p i such that w i ∈ RH qipi . However, this result has the advantage that it respectsthe product structure in the multilinear setting, in that we do not have to assumean identical condition on each weight w i . This phenomenon does not appear in thediagonal case for multilinear singular integrals considered in [7], but it does play arole in the conditions for multilinear multipliers given in [10].As an application of our weighted estimates we extend our results to the variableexponent setting. Variable exponent spaces are generalizations of the classical L p and H p spaces where the constant exponent p is replaced by an exponent function p ( · ). Intuitively, L p ( · ) consists of all functions such that Z R n | f ( x ) | p ( x ) dx < ∞ . Harmonic analysis has been extensively studied on these spaces: see [1] for the historyand detailed references. The theory of variable exponent Hardy spaces H p ( · ) wasintroduced in [11]. Our second main result is Theorem 1.3. For brevity, we defersome technical definitions to Section 4. Theorem 1.3.
Given < γ < mn , let p i ( · ) ∈ P , ≤ i ≤ m , be log-H¨oldercontinuous exponent functions such that < [ p i ( · )] − ≤ [ p i ( · )] + < ∞ and p ( · )] + + · · · + 1[ p m ( · )] + > γn . Define q ( · ) by q ( · ) = 1 p ( · ) + · · · + 1 p m ( · ) − γn ; then k T γ ( f , . . . , f m ) k L q ( · ) . m Y i =1 k f i k H pi ( · ) . CRUZ-URIBE, MOEN, AND NGUYEN
The key tool in the proof Theorem 1.3 is a multilinear, off-diagonal version ofRubio de Francia extrapolation, Theorem 4.1, which is of interest in its own right.This result generalizes earlier multilinear extrapolation theorems into the scale ofvariable exponent spaces [7, 8] and also the multilinear extrapolation theory in [3].The remainder of this paper is organized as follows. In Section 2 we gather somedefinitions and preliminary results needed in the proof of Theorem 1.1. In Section 3we prove Theorem 1.1. Our proof draws upon ideas from [7, 10], but significantmodifications were required to handle the fractional nature of the kernel. Finally, inSection 4 we give the necessary definitions and prove Theorems 4.1 and 1.3.Throughout this paper, we will use n to denote the dimension of the underlyingspace, R n , and will use m to denote the “dimension” of our multilinear operators. Bya cube Q we will always mean a cube whose sides are parallel to the coordinate axes,and for τ > τ Q denote the cube with same center such that ℓ ( τ Q ) = τ ℓ ( Q ). Inparticular, let Q ∗ = 2 √ nQ and Q ∗∗ = ( Q ∗ ) ∗ . By C , c , etc. we will mean constantsthat may depend on the underlying parameters in the problem. The values of theseconstants may change from line to line. If we write A . B , we mean that A ≤ cB for some constant c . 2. Preliminary results
For 0 ≤ γ < n , the fractional maximal function M γ is defined by M γ f ( x ) = sup Q ℓ ( Q ) γ (cid:16) − Z Q | f ( y ) | dy (cid:17) χ Q ( x ) . When γ = 0 we get the Hardy-Littlewood maximal operator and write M f insteadof M f .For 1 < p < ∞ and 1 < r < ∞ , given w ∈ A p we have the Fefferman-Steininequality:(2.1) (cid:13)(cid:13)(cid:13)(cid:16) X k M ( f k ) r (cid:17) r (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13)(cid:16) X k | f k | r (cid:17) r (cid:13)(cid:13)(cid:13) L p ( w ) A similar result holds for M γ . Given 0 < γ < n, < p < nγ and q = p − γn , we say w ∈ A p,q if for all cubes Q , (cid:18) − Z Q w q dx (cid:19) q (cid:18) − Z Q w − p ′ dx (cid:19) p ′ ≤ C < ∞ . Muckenhoupt and Wheeden [16] showed that if w ∈ A p,q , then k M γ f k L q ( w q ) . k f k L p ( w p ) . As a consequence of the off-diagonal Rubio de Francia extrapolation [4, Theorem 3.23],we have that for 1 < r < ∞ and w ∈ A p,q ,(2.2) (cid:13)(cid:13)(cid:13)(cid:16) X k M γ ( f k ) r (cid:17) r (cid:13)(cid:13)(cid:13) L q ( w q ) . (cid:13)(cid:13)(cid:13)(cid:16) X k | f k | r (cid:17) r (cid:13)(cid:13)(cid:13) L p ( w p ) . ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 5
For γ >
0, we have that(2.3) ℓ ( Q ) γ χ Q ∗ . M γδ ( χ Q ) δ for all 0 < δ ≤
1. If we combine this estimate with (2.2) we get the followingvector-valued estimate.
Lemma 2.1.
Given < γ < ∞ and < p < nγ , define q > by q = p − γn . Thenfor any w ∈ RH qp , (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) γ χ Q ∗ j (cid:13)(cid:13)(cid:13) L q ( w qp ) . (cid:13)(cid:13) X j λ j χ Q j (cid:13)(cid:13) L p ( w ) , where λ j > and { Q j } j is any sequence of cubes. Remark 2.2.
Lemma 2.1 was first proved in [20] when 1 < p < n/γ in a two weightsetting. Our proof is much simpler. For another proof that also uses extrapolationbut avoids the vector-valued inequality see [6, Lemma 4.9].
Proof.
For each δ ∈ (0 , p ), set q δ = q/δ , p δ = p/δ and u δ = w pδ . Since w ∈ RH qp ,there exists δ > u q δ δ = w qp ∈ A qδ ( pδ ) ′ . Therefore, it follows from the definitions that u δ ∈ A p δ ,q δ . Then we can apply in-equalities (2.3) and (2.2) to get that (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) γ χ Q ∗ j (cid:13)(cid:13)(cid:13) L q ( w qp ) . (cid:13)(cid:13)(cid:13) X j λ j M γδ ( χ Q j ) δ (cid:13)(cid:13)(cid:13) L q ( w qp ) = (cid:13)(cid:13)(cid:13)(cid:16) X j λ j M γδ ( χ Q j ) δ (cid:17) δ (cid:13)(cid:13)(cid:13) δ L qδ ( w qδpδ ) = (cid:13)(cid:13)(cid:13)(cid:16) X j λ j M γδ ( χ Q j ) δ (cid:17) δ (cid:13)(cid:13)(cid:13) δ L qδ ( u qδδ ) . (cid:13)(cid:13)(cid:13)(cid:16) X j λ j χ Q j (cid:17) δ (cid:13)(cid:13)(cid:13) δ L pδ ( u pδδ ) = (cid:13)(cid:13)(cid:13) X j λ j χ Q j (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:3) Lemma 2.3.
Let γ, p , and q be real numbers as in Lemma 2.1 and suppose w ∈ RH qp ∩ A r for some r > . Then for ǫ > max( nrp , n ) and any sequence { Q j } j ofcubes, (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) ǫ χ ( Q ∗ j ) c | · − c j | ǫ − γ (cid:13)(cid:13)(cid:13) L q ( w qp ) . (cid:13)(cid:13)(cid:13) X j λ j χ Q j (cid:13)(cid:13)(cid:13) L p ( w ) , where λ j > and c j is the center of the cube Q j . CRUZ-URIBE, MOEN, AND NGUYEN
Proof.
For each j , we first decompose R n \ Q ∗ j into annuli and then into non-overlappingcubes R klj such that R n \ Q ∗ j = ∞ [ l =0 3 n − [ k =1 R klj and such that | x − c j | ≈ l ℓ ( Q j ) ≈ ℓ ( R klj ) for all x ∈ R klj and all 1 ≤ h ≤ n − s > | x − c j | − s χ ( Q ∗ j ) c ( x ) ≈ ∞ X l =0 3 n − X k =1 (cid:0) l ℓ ( Q j ) (cid:1) − s χ R klj ( x ) . Then by Lemma 2.1 and the equivalence (2.4) we have that (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) ǫ χ ( Q ∗ j ) c | · − c j | ǫ − γ (cid:13)(cid:13)(cid:13) L q ( w qp ) . (cid:13)(cid:13)(cid:13) X j ∞ X l =0 3 n − X k =1 λ j − ǫl ℓ ( R klj ) γ χ R klj (cid:13)(cid:13)(cid:13) L q ( w qp ) . (cid:13)(cid:13)(cid:13) X j ∞ X l =0 3 n − X k =1 λ j − ǫl χ R klj (cid:13)(cid:13)(cid:13) L p ( w ) = (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) ǫ ∞ X l =0 3 n − X k =1 (3 l ℓ ( Q j )) − ǫ χ R klj (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13) X j λ j ℓ ( Q j ) ǫ | · − c j | − ǫ χ ( Q ∗ j ) c (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13) X j λ j M ( χ Q j ) ǫn (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13)(cid:16) X j λ j M ( χ Q j ) ǫn (cid:17) nǫ (cid:13)(cid:13)(cid:13) ǫn L ǫpn ( w ) . (cid:13)(cid:13)(cid:13) X j λ j χ Q j (cid:13)(cid:13)(cid:13) L p ( w ) . The last inequality holds because by our choice of ǫ , ǫpn > r , so w ∈ A ǫpn and we canapply inequality (2.1). (cid:3) The proof of Theorem 1.1
In this section, we prove Theorem 1.1, and as we said in the introduction, weactually prove a more general result.
Theorem 3.1.
Given < γ < mn and < p , . . . , p m < ∞ , define p as in (1.3) and q as in (1.4) . Suppose that q i are such that p i < q i < ∞ , ≤ i ≤ m , and q + · · · + q m = q , and ( w , . . . , w m ) are weights such that w i ∈ RH qipi . If K γ satisfies ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 7 (1.1) and (1.2) for some positive integer
N > max { mn ( r wi p i − , ≤ i ≤ m } , then k T γ ( f , . . . , f m ) k L q ( w ) . m Y i =1 k f i k H pi ( w i ) , where w = m Y i =1 w qpi i . Remark 3.2.
Theorem 1.1 follows from Theorem 3.1 by taking q i = qp p i . Proof.
Recall that for w ∈ A ∞ , H p ( w ) is defined as the set of all distributions f suchthat M N f ∈ L p ( w ); here N is some large constant whose precise value does notmatter, though it will be implicit in our constants. For more information, see [19].Let N be the positive integer as in the hypotheses of Theorem 1.1. Define O N = (cid:8) f ∈ C ∞ : Z R n x β f ( x ) dx = 0 , ≤ | β | ≤ N (cid:9) . Then E i = O N ∩ H p i ( w i ) is dense in H p i ( w i ) for all 1 ≤ i ≤ m (see [7, 19]). Asproved in [7, Theorem 2.6] for each f i ∈ E i , 1 ≤ i ≤ m , we have a finite atomicdecomposition:(3.1) f i = X k i λ i,k i a i,k i , where λ i,k i > , | a i | ≤ χ Q i,ki for some cube Q i,k i , R x α a i,k i dx = 0 for all | α | ≤ N , and(3.2) (cid:13)(cid:13) X k i λ i,k i χ Q i,ki (cid:13)(cid:13) L pi ( w i ) . k f i k H pi ( w i ) . By a standard density argument, it will suffice to show that inequality (1.5) holdsfor f i of the form (3.1). Define 0 < γ i < ∞ by γ i n = 1 p i − q i , ≤ i ≤ m. From the hypothesis (1.4) we get(3.3) m X i =1 γ i = γ, < γ i < np i , ≤ i ≤ m. By the multilinearity of T γ we get that T γ ( f , . . . , f m )( x ) = X k ,...,k m λ ,k · · · λ m,k m T γ ( a ,k , . . . , a m,k m )( x ) . Given a cube Q and ~k = ( k , . . . , k m ), define E ~k = ∩ mi =1 Q ∗ i,k i , F ~k = R n \ E ~k . CRUZ-URIBE, MOEN, AND NGUYEN
Then we can decompose T γ ( f , . . . , f m ) = G + G where G = X k ,...,k m λ ,k · · · λ m,k m T γ ( a ,k , . . . , a m,k m ) χ E ~k ,G = X k ,...,k m λ ,k · · · λ m,k m T γ ( a ,k , . . . , a m,k m ) χ F ~k . To estimate the first term, we may assume that E ~k is not empty. With γ i as definedby (3.3) we can estimate T γ ( a ,k , . . . , a m,k m )( x ) for all x ∈ E ~k as follows: | T γ ( a ,k , . . . , a m,k m )( x ) | ≤ Z R mn χ Q ,k ( y ) · · · χ Q m,km ( y m ) d~y (cid:0) | x − y | + · · · + | x − y m | (cid:1) mn − γ ≤ m Y i =1 (cid:16) Z Q i,ki | x − y i | γ i − n dy i (cid:17) χ Q ∗ i,ki ( x ) . m Y i =1 ℓ ( Q i,k i ) γ i χ Q ∗ i,ki ( x ) . (3.4)We can now estimate the quasi-norm of G as follows: k G k L q ( w ) = (cid:13)(cid:13)(cid:13) X k ,...,k m λ ,k · · · λ m,k m T γ ( a ,k , . . . , a m,k m ) χ E ~k (cid:13)(cid:13)(cid:13) L q ( w ) . (cid:13)(cid:13)(cid:13) X k ,...,k m λ ,k · · · λ m,k m m Y i =1 ℓ ( Q i,k i ) γ i χ Q ∗ i,ki (cid:13)(cid:13)(cid:13) L q ( w ) = (cid:13)(cid:13)(cid:13) m Y i =1 (cid:16) X k i λ i,k i ℓ ( Q i,k i ) γ i χ Q ∗ i,ki (cid:17)(cid:13)(cid:13)(cid:13) L q ( w ) ≤ m Y i =1 (cid:13)(cid:13)(cid:13) X k i λ i,k i ℓ ( Q i,k i ) γ i χ Q ∗ i,ki (cid:13)(cid:13)(cid:13) L qi ( w qi/pii ) (3.5) . m Y i =1 (cid:13)(cid:13)(cid:13) X k i λ i,k i χ Q i,ki (cid:13)(cid:13)(cid:13) L pi ( w i ) ;(3.6)for inequality (3.5) we used H¨older’s inequality, and for (3.6) we used Lemma 2.1.To estimate G , fix x ∈ F ~k . Then there exists a non-empty subset Λ of { , . . . , m } such that x / ∈ Q ∗ i,k i , for all i ∈ Λ and x ∈ Q ∗ j,k j for all 1 ≤ j ≤ m, j Λ. Let Q i ,k i ,for some i ∈ Λ, be the cube with smallest length among Q i,k i , i ∈ Λ and let c i,k i bethe center of the cube Q i,k i . Note that since x / ∈ Q ∗ i ,k i , | x − y | . | x − c i ,k i | for all y i ∈ Q i ,k i . Let P N ( x, y , . . . , , c i ,k i , . . . , y m ) = X | β |
If 0 < γ < ( m − l ) n for some 1 ≤ l < m , then we can allow at most l exponents among the { p , . . . , p m } to be infinite and the conclusion of Theorem 3.1is still true, replacing H p i ( w i ) with L ∞ . To see this, first note that we may assumethat p m − l +1 = · · · = p m = ∞ . Then we can integrate in y m − l +1 , . . . , y m to estimate(3.7) as follows: | T γ ( a ,k , . . . , a m − l,k m − l , f m − l +1 , . . . , f m )( x ) |≤ Z R mn χ Q ,k ( y ) · · · χ Q m − l,km − l ( y m − l ) f m − l +1 ( y m − l +1 ) · · · f m ( y m ) d~y (cid:0) | x − y | + · · · + | x − y l | (cid:1) nl − γ ≤ Z R nl χ Q ,k ( y ) · · · χ Q m − l,km − l ( y m − l ) dy · · · dy l (cid:0) | x − y | + · · · + | x − y l | (cid:1) nl − γ m Y i = l +1 k f i k L ∞ ≤ l Y i =1 (cid:16) Z Q i,ki | x − y i | γ i − n dy i (cid:17) χ Q ∗ i,ki ( x ) m Y i = l +1 k f i k L ∞ . l Y i =1 ℓ ( Q i,k i ) γ i χ Q ∗ i,ki ( x ) m Y i = l +1 k f i k L ∞ . If we now repeat the argument in the proof of Theorem 1.1, we get k T γ ( f , . . . , f m ) k L q ( w ) . l Y i =1 k f i k H pi ( w i ) m Y i = l +1 k f i k L ∞ . In their work, Lin and Lu [15, Theorem 2.1] assumed an unweighted estimatesimilar to this one when l = m −
1. This helps to explain the restriction 0 < γ < n in their results. 4.
Boundedness on Variable Hardy Spaces
In this section, we state and prove the analogue of Theorem 1.1 on the variableexponent Hardy spaces, H p ( · ) . To do so, we first recall some basic facts about thevariable exponent Lebesgue and Hardy spaces. For complete background informationwe refer the reader to [1].Let P ( R n ) be the set of all measurable functions p ( · ) : R n → (0 , ∞ ). Define[ p ( · )] − = ess inf x ∈ R n p ( x ) , [ p ( · )] + = ess sup x ∈ R n p ( x ) . ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 11
Given p ( · ) ∈ P ( R n ) define L p ( · ) = L p ( · ) ( R n ) to be the set of all measurable functions f such that for some λ > ρ ( f /λ ) = Z R n (cid:18) | f ( x ) | λ (cid:19) p ( x ) dx < ∞ . This becomes a quasi-Banach space with the quasi-norm k f k L p ( · ) = inf { λ > ρ ( f /λ ) ≤ } . If [ p ( · )] − ≥
1, then k · k L p ( · ) is a norm and L p ( · ) is a Banach space. For all p >
0, if p ( · ) = p a constant, then L p ( · ) = L p with equality of norms, so the variable exponentLebesgue spaces are a generalization of the classical L p spaces.Let B denote the collection of exponents p ( · ) such that the Hardy-Littlewood max-imal operator is bounded on L p ( · ) . A sufficient (but not necessary) condition for p ( · ) ∈ B is that 1 < [ p ( · )] − ≤ [ p ( · )] + < ∞ and p ( · ) is log-H¨older continuous: thereexist constants C , C ∞ and p ∞ such that | p ( x ) − p ( y ) | ≤ C − log( | x − y | ) , < | x − y | < , and | p ( x ) − p ∞ | ≤ C ∞ log( e + | x | ) . Given p ( · ) ∈ P ( R n ), the variable Hardy space H p ( · ) is defined to be the set ofall distributions f such that M N f ∈ L p ( · ) . Again, we here assume N > H p ( · ) . For further details, see [7, 11].We will prove norm inequalities in the variable exponent Lebesgue and Hardyspaces from the corresponding weighted norm inequalities by applying the theoryof Rubio de Francia extrapolation. In the linear case this approach was introducedin [1, 2, 5], and multilinear versions of extrapolation into the variable Lebesgue spaceswere proved in [7, 8]. Here we need a generalization of these results similar to themultilinear extrapolation theorem proved in [3]. We state our extrapolation resultsin terms of extrapolation ( m + 1)-tuples; for more on this approach, see [5, 8]. Theorem 4.1.
Let F = (cid:8) ( f , . . . , f m , F ) (cid:9) be a family of ( m + 1) -tuples of non-negative, measurable functions on R n . Given < γ < mn and exponents < p i < ∞ , ≤ i ≤ m , such that (1.3) holds, define q > by (1.4) . Suppose that for all exponents p i < q i < ∞ such that q = 1 q + · · · + 1 q m , and for all weights w i ∈ RH q i /p i , with w = Q mi =1 w q/p i i , we have that (4.1) k F k L q ( w ) . m Y i =1 k f i k L pi ( w i ) for all ( f , . . . , f m , F ) ∈ F such that F ∈ L q ( w ) , where the implicit constant dependsonly on n , p i , and [ w i ] RH qi/pi . Let p i ( · ) ∈ P , ≤ i ≤ m , be such that each p i ( · ) is log-H¨older continuous, p i < [ p i ( · )] − ≤ [ p i ( · )] + < ∞ , and (4.2) 1[ p ( · )] + + · · · + 1[ p m ( · )] + > γn . Define q ( · ) by (4.3) 1 q ( · ) = 1 p ( · ) + · · · + 1 p m ( · ) − γn . Then for all ( f , . . . , f m , F ) ∈ F such that k F k L q ( · ) < ∞ , (4.4) k F k L q ( · ) . m Y i =1 k f i k L pi ( · ) . The implicit constant only depends on n , [ p i ( · )] − , [ p i ( · )] + , and the log-H¨older con-stants of p i ( · ) . Remark 4.2.
It will be clear from the proof that we can weaken the hypothesis thateach p i ( · ) is log-H¨older continuous, and instead assume that the maximal operator isbounded on a certain family of variable exponent Lebesgue spaces. Details are leftto the interested reader. Remark 4.3.
Theorem 4.1 is stated so that its hypotheses coincide with the weightedresults in Theorem 3.1. We can also prove an extrapolation theorem starting fromthe weaker conclusion given in Theorem 1.1. The proof below can be modified, butwe need to assume that the exponents p i ( · ) have bounded oscillation: more precisely,that p i < [ p i ( · )] − ≤ [ p i ( · )] + < qp i q − p . Details of this result are left to the interested reader. The fact that we can removethis upper bound on [ p i ( · )] + is another reason for proving the stronger result inTheorem 3.1. Proof.
For our proof we need a family of Rubio de Francia iteration algorithms. Toconstruct them, we will define some exponent functions and show that the maximaloperator is bounded on the associated variable Lebesgue space. By (4.2), for each i ,1 ≤ i ≤ m , we can fix γ i > γ = n X i =1 γ i , and [ p i ( · )] + < n/γ i . Define q i > p i − q i = γ i n and define the variable exponents q i ( · ) by1 p i ( · ) − q i ( · ) = γ i n . ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 13
But then by (4.3) we have that1 q ( · ) = 1 q ( · ) + · · · + 1 q m ( · ) , and so 1[ q ( · )] − ≤ m X i =1 p i ( · )] − − γn < m X i =1 p i − γn = 1 q . Therefore, if we define q ( · ) = q ( · ) /q , then [ q ( · )] − >
1. Similarly, if we define p i ( · ) = p i ( · ) /p i , then [ p i ( · )] − > σ i ( · ) = p i q i p ′ i ( · ). We claim that [ σ i ( · )] − >
1. However, this inequalityfollows from some standard estimates for dual exponents in the variable exponentLebesgue spaces [1, p. 14]: this inequality is equivalent to [ p ′ i ( · )] − > q i p i , which in turnis equivalent to [ p i ( · )] ′ + > q i p i , and this in turn is equivalent to[ p i ( · )] + < p i (cid:18) q i p i (cid:19) ′ = nγ i , which we know to hold.We also have that each σ i ( · ) is log-H¨older continuous since each p i ( · ) is. Therefore,the Hardy-Littlewood maximal operator is bounded on L σ i ( · ) . Hence, we can definethe iteration operator R i , acting on non-negative functions h , by R i h ( x ) = ∞ X j =0 M j h ( x )2 j k M k jL σi ( · ) , where M j h = M ◦ · · ·◦ M h is j iterates of the maximal operator, and M h = h . Thenby a standard argument [1, p. 210] and a rescaling property of A ∩ RH s weights [9],we have the following:(1) h ( x ) ≤ R i h ( x );(2) kR i h k L σi ( · ) ≤ k h k L σi ( · ) ;(3) R i h ∈ A , and [ R i h ] A ≤ k M k L σi ( · ) ;(4) ( R i h ) p i /q i ∈ A ∩ RH q i /p i , and [( R i h ) p i /q i ] RH qi/pi depends only on [ R i h ] A .Define a family of auxiliary exponents θ i , 1 ≤ i ≤ m , by θ i ( · ) = qq ′ ( · ) p i p ′ i ( · ) . Then(4.5) m X i =1 θ i ( · ) = qq ′ ( · ) m X i =1 p i p ′ i ( · ) = qq ′ ( · ) m X i =1 p i (cid:18) − p i p i ( · ) (cid:19) = qq ′ ( · ) m X i =1 (cid:18) p i − p i ( · ) (cid:19) = q ′ ( · ) − q ′ ( · ) q ( · ) = 1 . We can now prove the desired inequality. Fix ( f , . . . , f m , F ) ∈ F such that F ∈ L q ( · ) . Since q ( · ) >
1, by rescaling and the associate norm in variable exponent
Lebesgue spaces [1, Prop. 2.18, Thm. 2.34], there exists h ∈ L q ′ ( · ) , k h k L q ′ ( · ) = 1, suchthat(4.6) k F k qL q ( · ) = k F q k L q ( · ) . Z R n F q h dx = Z R n F q m Y i =1 h θ i ( · ) dx . Z R n F q m Y i =1 (cid:20) R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi (cid:21) qpi dx. By construction, we have that for each i R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi ∈ A ∩ RH q i /p i . Assume for the moment that the last term in the above inequality is finite. If it is,then we can apply our hypothesis (4.1) and the generalized H¨older’s inequality in thescale of variable Lebesgue spaces [1, Theorem. 2.26] to get Z R n F q m Y i =1 (cid:20) R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi (cid:21) qpi dx . m Y i =1 (cid:18)Z R n f p i i R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi dx (cid:19) qpi . m Y i =1 k f p i i k qpi L pi ( · ) kR i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi k qpi L p ′ i ( · ) . Again by rescaling we have that k f p i i k qpi L pi ( · ) = k f i k qL pi ( · ) . Thus to complete the proofof inequality (4.4) we will show that(4.7) kR i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi k L p ′ i ( · ) = kR i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) k piqi L σi ( · ) . . By the properties of the iteration operator and rescaling, kR i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) k piqi L σi ( · ) . k h q ′ ( · ) qip ′ i ( · ) pi k piqi L σi ( · ) = k h q ′ ( · ) p ′ i ( · ) k L p ′ i ( · ) . By the relationship between the norm and the modular in variable exponent spaces[1, Prop. 2.21], since k h k L q ′ ( · ) = 1,1 = Z R n h ( x ) q ′ ( x ) dx = Z R n (cid:18) h ( x ) q ′ ( x ) p ′ i ( x ) (cid:19) p ′ i ( x ) dx, and this in turn implies that k h q ′ ( · ) p ′ i ( · ) k L p ′ i ( · ) = 1 . Finally, to complete the proof we need to justify our assumption that the last termin (4.6) is finite. If we divide the second and last terms of the identity (4.5) by q ′ ( · ),we get 1 = 1 q ( · ) + m X i =1 qp i p ′ i ( · ) = 1 q ( · ) . ULTILINEAR FRACTIONAL OPERATORS ON WEIGHTED HARDY SPACES 15
Hence, by the multi-term generalized H¨older’s inequality in variable exponent Lebesguespaces [1, Cor. 2.30], Z R n F q m Y i =1 (cid:20) R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi (cid:21) qpi dx . k F q k L q ( · ) m Y i =1 k (cid:20) R i (cid:0) h q ′ ( · ) qip ′ i ( · ) pi (cid:1) piqi (cid:21) qpi k L pip ′ i ( · ) /q . By rescaling, the first term becomes k F k qL q ( · ) which is finite, and by rescaling and (4.7)we see that the remaining terms are all uniformly bounded. (cid:3) Theorem 1.3 follows directly from Theorem 4.1 and a careful density argument.
Proof of Theorem 1.3.
From condition (4.2) we can find p i > p i < [ p i ( · )] − and 1 p + · · · + 1 p m > γn . Therefore, by Theorem 3.1, given q i > p i such that w i ∈ RH q i /p i , inequality (1.5)holds. We can use this to apply Theorem 1.3 if we can define the appropriate family F of extrapolation ( m + 1)-tuples.Since each p i ( · ) is log-H¨older continuous and [ p i ( · )] − >
0, 1 ≤ i ≤ m , there existsan N depending only on the p i ( · ) and on n such that functions of the form f = M X j =1 λ j a j , where each a j is an ( N, ∞ ) atom, are dense in H p i ( · ) [11, Theorem. 6.3]. All suchfunctions are also contained in H p ( w ), for any p > w ∈ A ∞ . Denote the setof such functions by A . Define the family of ( m + 1)-tuples F = { ( f , . . . , f m , F R ) } ,where f i = M N g i , g i ∈ A , R >
0, and F R = min (cid:0) | T γ ( g , . . . , g m ) | , R (cid:1) χ B (0 ,R ) . Since k χ B (0 ,R ) k L q ( · ) < ∞ [1, Lemma 2.39], we have that k F R k L q ( · ) < ∞ . Further,given any weights w i ∈ RH q i /p i , and w = Q mi =1 w q/p i i , by H¨older’s inequality withexponents q i /q we have that k F R k L q ( w ) ≤ R Z B (0 ,R ) m Y i =1 w q/p i i dx ! /q ≤ R m Y i =1 (cid:18)Z B (0 ,R ) w q i /p i i dx (cid:19) /q i < ∞ . But then by (1.5) we have that given any ( m + 1)-tuple in F , k F R k L p ( w ) . m Y i =1 k g i k H pi ( w i ) = m Y i =1 k f i k L pi ( w i ) , which gives us (4.1). Therefore, by Theorem 4.1, k F R k L q ( · ) . m Y i =1 k f i k L pi ( · ) = m Y i =1 k g i k H pi ( · ) . By Fatou’s lemma in the variable exponent Lebesgue spaces [1, Theorem. 2.61], k T ( g , . . . , g m ) k L q ( · ) ≤ lim inf R →∞ k F R k L q ( · ) . m Y i =1 k g i k H qi ( · ) . This establishes the desired norm inequality of T for a dense family of functions, andTheorem 1.3 follows by a standard approximation argument. (cid:3) References [1] D. Cruz-Uribe and A. Fiorenza.
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Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487
E-mail address : [email protected] Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487
E-mail address : [email protected] Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487
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