The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces
aa r X i v : . [ m a t h . C A ] A ug THE BOUNDEDNESS OF MULTILINEAR CALDER ´ON-ZYGMUNDOPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES
DAVID CRUZ-URIBE, OFS, KABE MOEN, AND HANH VAN NGUYEN
Abstract.
We establish the boundedness of the multilinear Calderon-Zygmund operators from aproduct of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalizeto the weighted setting results obtained by Grafakos and Kalton [20] and recent work by thethird author, Grafakos, Nakamura, and Sawano [22]. As part of our proof we provide a finiteatomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right.As a consequence of our weighted results, we prove the corresponding estimates on variable Hardyspaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the firstauthor and Naibo [11]. Introduction
In this paper we study the boundedness of multilinear Calder´on-Zygmund operators ( m -CZOs)on products of weighted and variable Hardy spaces. More precisely, we are interested in thefollowing operators. Let K ( y , y , . . . , y m ) be a kernel that is defined away from the diagonal y = y = · · · = y m in ( R n ) m +1 and satisfies the smoothness condition(1.1) (cid:12)(cid:12) ∂ α y · · · ∂ α m y m K ( y , y , . . . , y m ) (cid:12)(cid:12) ≤ A α ,...,α m (cid:16) m X k,l =0 | y k − y l | (cid:17) − ( mn + | α | + ··· + | α m | ) for all α = ( α , . . . , α m ) such that | α | = | α | + · · · + | α m | ≤ N , where N is a sufficiently largeinteger. An m -CZO is a multilinear operator T that satisfies T : L q ( R n ) × · · · × L q m ( R n ) → L q ( R n )for some 1 < q , . . . , q m < ∞ and q = q + · · · + q m , and T has the integral representation T ( f , . . . , f m )( x ) = Z ( R n ) m K ( x, y , . . . , y m ) f ( y ) · · · f ( y m ) dy · · · dy m whenever f i ∈ L ∞ c ( R n ) and x / ∈ ∩ i supp( f i ).Multilinear CZOs were introduced by Coifman and Meyer [3, 4] in the 1970s and were sys-tematically studied by Grafakos and Torres [24]. They showed that m -CZOs are bounded from L p ( R n ) × · · · × L p m ( R n ) → L p ( R n ), for any 1 < p , . . . , p m < ∞ and p defined by p = p + · · · + p m .Further, m -CZOs satisfy weak endpoint bounds when p i = 1 for some i . For Lebesgue spacebounds, it is sufficient to take N = 1 in (1.1) and in fact weaker regularity conditions are sufficient.Bounds for m -CZOs from products of Hardy spaces into Lebesgue spaces were proved by Kalton Date : July 20, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Muckenhoupt weights, weighted Hardy spaces, variable Hardy spaces, multilinearCalder´on-Zygmund operators, singular integrals.The first author is supported by NSF Grant DMS-1362425 and research funds from the Dean of the College ofArts & Sciences, the University of Alabama. The second author is supported by the Simons Foundation. and Grafakos [20] (see also Grafakos and He [19]). As in the linear case, more regularity is requiredon the operators: in this case, N ≥ s = ⌊ n ( p − ⌋ + where x + = max(0 , x ). Very recently, boundsinto Hardy spaces were proved by the third author, Grafakos, Nakamura and Sawano [22]. To mapinto Hardy spaces the kernel K must satisfy (1.1) for N > s + max (cid:26)(cid:22) mn (cid:16) p k − (cid:17)(cid:23) + : 1 ≤ k ≤ m (cid:27) + mn. Moreover, in the multilinear case the operator T must satisfy an additional cancelation condition:(1.2) Z x α T ( a , . . . , a m )( x ) dx = 0 , for | α | ≤ s and all ( p k , ∞ , N ) atoms a k . For linear CZOs of convolution type, this condition holdsautomatically: see [23, Lemma 2.1]. An example of a bilinear CZO that satisfies this cancelationcondition is T = R + R , where R i is the bilinear Riesz transform R i ( f, g )( x ) = pv Z R Z R x − y i | ( x − y , x − y ) | f ( y ) g ( y ) dy dy . Somewhat surprisingly, neither Riesz transform itself has sufficient cancellation. For more examplesof convolution-type multilinear operators that do and do not satisfy this cancelation condition,see [22, 23]. Weighted norm inequalities for multilinear operators were first considered by Grafakos and Tor-res [25]. Later, Lerner, et al. [27] characterized the weighted inequalities for m -CZOs using amultilinear generalization of the Muckenhoupt A p condition. Weighted Hardy spaces were intro-duced by Garc´ıa-Cuerva [16]. A complete treatment of weighted Hardy spaces is due to Str¨ombergand Torchinsky [33]; they proved that (linear) Calder´on-Zygmund operators whose kernels haveenough regularity map H p ( w ) into L p ( w ) or H p ( w ), for 0 < p < ∞ and for weights w ∈ A ∞ .Our goal is to generalize the results of Str¨omberg and Torchinsky to m -CZOs. To state them,we first define some notation. To do so we rely on some (hopefully) well-known concepts; completedefinitions will be given below. Given w ∈ A ∞ , we define r w = inf { r ∈ (1 , ∞ ) : w ∈ A r } and for 0 < p < ∞ we define the critical index s w of w by s w = (cid:22) n (cid:16) r w p − (cid:17)(cid:23) + . Our first result gives the boundedness of m -CZOs into weighted Lebesgue spaces. Theorem 1.1.
Given an integer m ≥ , < p , . . . , p m < ∞ , and w k ∈ A ∞ , ≤ k ≤ m , let T bean m -CZO associated to a kernel K that satisfies (1.1) for N such that (1.3) N ≥ max (cid:26) (cid:22) mn (cid:16) r w k p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + ( m − n. Then T : H p ( w ) × · · · × H p m ( w m ) → L p ( w ) , We note in passing that the results for m -CZOs in [22] are stated for convolution type operators, but as theauthors note (see Remark 3.4), their results extend to non-convolution type m -CZOs. ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 3 where w = Q mk =1 w ppk k and p = 1 p + · · · + 1 p m . Our second result gives boundedness of m -CZOs into weighted Hardy spaces. Theorem 1.2.
Given p, p , . . . , p m , w, w , . . . , w m and T as in Theorem 1.1, suppose the kernel K satisfies (1.1) for N such that (1.4) N > s w + max (cid:26) (cid:22) mn (cid:16) r w k p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + mn. Suppose further that T satisfies the cancellation condition (1.2) for all | α | ≤ s w , where for ≤ k ≤ m , a k is an ( N, ∞ ) atom: i.e., a k is supported on a cube Q k , k a k k ∞ ≤ , and (1.5) Z R n x β a k ( x ) dx = 0 for all | β | ≤ N . Then T : H p ( w ) × · · · × H p m ( w m ) → H p ( w ) . Remark 1.3.
In Theorems 1.1 and 1.2, if all the weights w k = 1, then r w k = 1, so we recapturethe unweighted results in [20, 22]. Remark 1.4. If p > w ∈ A p , then H p ( w ) = L p ( w ) (see [33]). Therefore, in Theorems 1.1and 1.2, if w k ∈ A p k , then we can replace H p k ( w k ) by L p k ( w k ) in the conclusion. Remark 1.5.
Implicit in the statement of Theorem 1.2 is the assumption that w ∈ A ∞ . However,this is always the case: see Lemma 2.1 below. Remark 1.6.
Earlier, Xue and Yan [35] proved a version of Theorem 1.1 with the additionalrestriction that 0 < p k ≤ ≤ k ≤ m . We want to thank the authors for calling ourattention to their paper, which we had overlooked.Our next pair of results are the analogs of Theorems 1.1 and 1.2 for the variable Lebesgue spaces.The variable Lebesgue spaces are a generalization of the classical L p spaces with the exponent p replaced by a measurable exponent function p ( · ) : R n → (0 , ∞ ). It consists of all measurablefunctions 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 quasi-norm k f k p ( · ) = inf { λ > ρ ( f /λ ) ≤ } . If p ( x ) ≥ L p ( · ) is a Banach space. These spaces were introducedby Orlicz [32] in 1931, and have been extensively studied by a number of authors in the past 25years. For complete details and references, see [7]. Variable Hardy spaces were introduced by thefirst author and Wang [13] and independently by Nakai and Sawano [31].In variable Lebesgue exponent spaces, harmonic analysis requires some assumption of regularityon the exponent function p ( · ). A common assumption that is sufficient for almost all applicationsis that the exponent function is log-H¨older continuous both locally and at infinity. More precisely,there exist constants C , C ∞ and p ∞ such that(1.6) | p ( x ) − p ( y ) | ≤ C − log( | x − y | ) , < | x − y | < , CRUZ-URIBE, MOEN, AND NGUYEN and(1.7) | p ( x ) − p ∞ | ≤ C ∞ log( e + | x | ) . Finally, given an exponent function p ( · ), we define p − = ess inf x ∈ R n p ( x ) , p + = ess sup x ∈ R n p ( x ) . As an immediate application of Theorems 1.1 and 1.2, and multilinear Rubio de Francia extrap-olation in the scale of variable Lebesgue spaces, we get the following two results.
Theorem 1.7.
Given an integer m ≥ , let p , . . . , p m be real numbers, and let q ( · ) , . . . , q m ( · ) belog-H¨older continuous exponent functions such that < p k < ( q k ) − ≤ ( q k ) + < ∞ . Define q ( · ) = 1 q ( · ) + · · · + 1 q m ( · ) , p = 1 p + · · · + 1 p m . Let T be an m -CZO as in Theorem 1.1 satisfying (1.1) for all | α | ≤ N with N ≥ max (cid:26) (cid:22) mn (cid:16) p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + ( m − n. Then T : H q ( · ) × · · · × H q m ( · ) → L q ( · ) . Theorem 1.8.
Given q ( · ) , q ( · ) , . . . , q m ( · ) , p, p , . . . , p m as in Theorem 1.7, let T be an m -CZO asin Theorem 1.1 satisfying (1.1) for all | α | ≤ N with N > (cid:22) n (cid:16) p − (cid:17)(cid:23) + + max (cid:26) (cid:22) mn (cid:16) p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + mn. Suppose further that T satisfies (1.2) for all | α | ≤ ⌊ n (1 /p − ⌋ + . Then T : H q ( · ) × · · · × H q m ( · ) → H q ( · ) . Remark 1.9.
As we were completing this paper we learned that a version of Theorems 1.7 and 1.8,with the additional hypothesis that ( q k ) + ≤ ≤ k ≤ m , was independently proved byTan [34]. We want to thank the author for sharing with us a preprint of his work.The remainder of this paper is organized as follows. In Section 2 we give some basic definitionsand theorems about weights that we will use in subsequent sections. In particular, we prove a finiteatomic decomposition for weighted Hardy spaces that extends the results in [13]. In Section 3 wegather together a number of technical lemmas that we need for the proofs of Theorems 1.1 and 1.2.Then in Sections 4 and 5 we prove these results. Finally, in Section 6 we give some basic facts aboutvariable exponent spaces and prove Theorems 1.7 and 1.8. In fact, we prove more general resultswhich include these theorems as special cases. Their statements, however, require additional factsabout variable exponent spaces, and so we delay their statement until the final section.Throughout this paper, we will use n to denote the dimension of the underlying space, R n , andwill use m to denote the “dimension” of our multilinear operators. By a cube Q we will alwaysmean a cube whose sides are parallel to the coordinate axes, and for τ > τ Q denote thecube with same center such that ℓ ( τ Q ) = τ ℓ ( Q ). We define the average of a function f on a cube Q by f Q = − R Q f dx = | Q | − R Q f dx . By C , c , etc. we will mean constants that may depend onthe underlying parameters in the problem. Sometimes, to emphasize that they (only) depend oncertain parameters, we will write C ( X, Y, Z, . . . ). The values of these constants may change fromline to line. If we write A . B , we mean that A ≤ cB for some constant c . ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 5 Weights and weighted Hardy spaces
Weights and weighted norm inequalities.
In this section we give some basic definitions andresults about A p weights. For complete information, we refer the reader to [14, 17, 18]. By aweight w we always mean a non-negative, locally integrable function such that 0 < w ( x ) < ∞ a.e.For 1 < p < ∞ , we say that w is in the Muckenhoupt class A p , denoted by w ∈ A p , if[ w ] A p = sup Q (cid:18) − Z Q w dx (cid:19) (cid:18) − Z Q w − p ′ dx (cid:19) p − < ∞ . When p = 1 we say that w ∈ A if there is a constant C such that for every cube Q and a.e. x ∈ Q , − Z Q w dx ≤ Cw ( x ) . The infimum over all such constants will be denoted by [ w ] A . The A p classes are nested: for1 < p < q < ∞ , A ( A p ( A q . Let A ∞ denote the union of all the A p classes, p ≥ w ∈ A ∞ , then w is a doubling measure. More precisely, if w ∈ A p for some p ≥
1, then itfollows from the definition that given any cube Q and τ > w ( τ Q ) ≤ Cτ np w ( Q ) . In the study of multilinear weighted norm inequalities, we often need the fact that the convexhull of A ∞ weights is again in A ∞ . The following result can be found, for instance, in [35] or in [21,Lemma 5]. For completeness we sketch a short proof, using a multilinear reverse H¨older inequality:if w , . . . , w m ∈ A ∞ , 1 < p , . . . , p m < ∞ , and p = p + · · · + p m , then for every cube Q , Y k (cid:18) − Z Q w k dx (cid:19) ppk . − Z Q Y k w ppk k dx. This was originally proved in the bilinear case by the first author and Neugebauer [12]; for simplerproofs in the multilinear case, see [10, 35].
Lemma 2.1.
Given m ≥ , < p , . . . , p m < ∞ , p = p + · · · + p m , if w , . . . , w m ∈ A ∞ , then w = Q mk =1 w ppk k ∈ A ∞ .Proof. Since each w k ∈ A ∞ , by choosing C sufficiently large and δ < Q and E ⊂ Q , w k ( E ) w k ( Q ) ≤ C (cid:18) | E || Q | (cid:19) δ . But then, if we apply H¨older’s inequality and the multilinear reverse H¨older’s inequality, we havethat w ( E ) w ( Q ) ≤ Q mk =1 (cid:0)R E w k dx (cid:1) ppk Q mk =1 (cid:16)R Q w k dx (cid:17) ppk ≤ C (cid:18) | E || Q | (cid:19) δ . (cid:3) There is a close connection between Muckenhoupt weights and the Hardy-Littlewood maximaloperator, defined by
M f ( x ) = sup Q − Z Q | f ( y ) | dy · χ Q ( x ) , CRUZ-URIBE, MOEN, AND NGUYEN where the supremum is taken over all cubes Q . We have that if 1 < p < ∞ , then the maximaloperator is bounded L p ( w ) if and only if w ∈ A p . Moreover, we have a weighted vector-valuedinequality that generalizes the Fefferman-Stein inequality. This was first proved by Anderson andJohn [1]; for an elementary proof via extrapolation, see [8]. Lemma 2.2.
Given < p, q < ∞ and w ∈ A p , then for any sequence { f k } in L p ( w ) , (cid:13)(cid:13)(cid:13)(cid:16) X k ( M f k ) q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13)(cid:16) X k | f k | q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) . Remark 2.3.
Below we will repeatedly apply Lemma 2.2 in the following way. Fix 0 < p < ∞ and w ∈ A ∞ . Then w ∈ A q and without loss of generality we may assume p < q . Let r = qp >
1. Givena sequence of cubes Q k , let Q ∗ k = τ Q k , τ >
1. Then χ Q ∗ k . M ( χ Q k ), and the implicit constantdepends only on n and τ . But then by Lemma 2.2, we have that for any non-negative λ k , (cid:13)(cid:13)(cid:13) X k λ k χ Q ∗ k (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13) X k M (cid:0) λ r k χ r Q k (cid:1) r (cid:13)(cid:13)(cid:13) L p ( w ) = (cid:13)(cid:13)(cid:13)(cid:16) X k M (cid:0) λ r k χ r Q k (cid:1) r (cid:17) r (cid:13)(cid:13)(cid:13) rL q ( w ) . (cid:13)(cid:13)(cid:13)(cid:16) X k λ k χ Q k (cid:17) r (cid:13)(cid:13)(cid:13) rL q ( w ) = (cid:13)(cid:13)(cid:13) X k λ k χ Q k (cid:13)(cid:13)(cid:13) L p ( w ) . Below we will need to prove a weighted norm inequality for an m -CZO. To do so, we will makeuse of some recent developments in the theory of harmonic analysis on the domination of singularintegrals by sparse operators. Here we sketch the basic definitions; for further information, see, forinstance, [6].A collection of cubes S is called a sparse family if each cube Q ∈ S contains measurable subset E Q ⊂ Q such that | E Q | ≥ | Q | and the family { E Q } Q ∈S is pairwise disjoint. Given a sparse family S we define a linear sparse operator T S f ( x ) = X Q ∈S − Z Q f ( y ) dy · χ Q ( x ) . The following estimate is proved in [9, 30].
Proposition 2.4. If < q < ∞ and w ∈ A q , then given any sparse linear operator T S , k T S f k L q ( w ) = (cid:13)(cid:13)(cid:13) X Q ∈S − Z Q f dy · χ Q (cid:13)(cid:13)(cid:13) L q ( w ) ≤ C [ w ] max(1 , q − ) A q k f k L q ( w ) . In a similar way, given a sparse family S we define the multilinear sparse operator T S ( f , . . . , f m )( x ) = X Q ∈S m Y k =1 − Z Q f k ( y k ) dy k · χ Q ( x ) . The following pointwise domination theorem was proved in [26, Theorem 13.2] (see also [5]).
Proposition 2.5.
Let T be an m -CZO whose kernel satisfies (1.1) for any N ≥ . Then givenany collection f , . . . , f m of bounded functions of compact support, there exists n sparse families S j such that | T ( f , . . . , f m )( x ) | ≤ C n X j =1 T S j ( | f | , . . . , | f m | )( x ) . ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 7
Weighted Hardy spaces.
In this section we define the weighted Hardy spaces and prove a finiteatomic decomposition theorem. In defining them we follow Str¨omberg and Torchinsky [33] and werefer the reader there for more information.Let S ( R n ) denote the Schwartz class of smooth functions. For N ∈ N to be a large valuedetermined later, define F N = { ϕ ∈ S ( R n ) : Z (1 + | x | ) N (cid:16) X | α |≤ N (cid:12)(cid:12)(cid:12) ∂ α ∂x α ϕ ( x ) (cid:12)(cid:12)(cid:12) (cid:17) dx ≤ } . Fix 0 < p < ∞ and w ∈ A ∞ ; we define the weighted Hardy space H p ( w ) to be the set ofdistributions H p ( w ) = { f ∈ S ′ ( R n ) : M N ( f ) ∈ L p ( w ) } with the quasi-norm k f k H p ( w ) = kM N ( f ) k L p ( w ) , where the grand maximal function M N ( f ) is defined by M N ( f )( x ) = sup ϕ ∈ F N sup t> (cid:12)(cid:12) ϕ t ∗ f ( x ) (cid:12)(cid:12) . Note that in this definition, N is taken to be a large positive integer, depending on n, p and w ,whose value is chosen so that the usual definitions of unweighted Hardy spaces remain equivalentin the weighted setting. Its exact value does not matter for us.Given an integer N >
0, an ( N, ∞ ) atom is a function a such that there exists a cube Q withsupp( a ) ⊂ Q , k a k ∞ ≤
1, and for | β | ≤ N , Z R n x β a ( x ) dx = 0 . In [33, Chapter VIII] it was shown that every f ∈ H p ( w ) has an atomic decomposition: for every N ≥ s w there exist a sequence of non-negative numbers { λ k } and a sequence of smooth ( N, ∞ )atoms { a k } with supp( a k ) ⊂ Q k , such that f = X k λ k a k , and the sum converges in the sense of distributions and in the H p ( w ) quasi-norm. Moreover, wehave that (cid:13)(cid:13)(cid:13) X k λ k χ Q k (cid:13)(cid:13)(cid:13) L p ( w ) . k f k H p ( w ) . Below, we want to use the atomic decomposition to estimate the norm of an m -CZO. Onetechnical obstacle, however, is that this atomic decomposition may be an infinite sum, and thereforeit is not immediate that we can exchange sum and integral in the definition of an m -CZO. Forthe argument to overcome this problem in the unweighted setting, see [19]. Our approach here isdifferent: we show that for a dense subset of H p ( w ), we can form the atomic decomposition usinga finite sequence of atoms. Our result generalizes a result in the unweighted case from [29]; in theweighted case it generalizes results proved in [13, 31].To state our result, note that for N ≥ s w , if we define O N = (cid:8) f ∈ C ∞ : Z R n x α f ( x ) dx = 0 , ≤ | α | ≤ N (cid:9) , then O N ∩ H p ( w ) is dense in H p ( w ). CRUZ-URIBE, MOEN, AND NGUYEN
Theorem 2.6.
Fix w ∈ A ∞ and < p < ∞ , and let N ≥ s w . For each f ∈ O N ∩ H p ( w ) ,there exists a finite sequence of non-negative numbers { λ k } k and a sequence { a k } of ( N, ∞ ) atoms, supp( a k ) ⊂ Q k , such that f = P k λ k a k and (2.1) (cid:13)(cid:13)(cid:13) X k λ k χ Q k (cid:13)(cid:13)(cid:13) L p ( w ) ≤ C k f k H p ( w ) , The proof of Theorem 2.6 is gotten by a close analysis of the atomic decomposition given above.To prove it, we use the following technical result. It is adapted from the corresponding resultfrom [33, Chapter VIII] (in the weighted case) and from the proof of the unweighted version ofTheorem 2.6 in [29]. (See also the construction of the atomic decomposition in [13].) Indeedweights play almost no role in the result except in (4).
Lemma 2.7.
Fix w ∈ A ∞ , < p < ∞ , and N ≥ s w , and let f ∈ O N ∩ H p ( w ) . For each k ∈ Z , let Ω k = { x ∈ R n : M N f ( x ) > k } . Then there exists a sequence { β k,i } of smooth functions with compact support and a family of cubes { Q k,i } with finite overlap that such that the following hold:(1) For each k and all i , Q k,i ⊂ Q ∗ k,i ⊂ Ω k , where Q ∗ k,i = τ Q k,i for a fix constant τ > and the Q ∗ k,i also have finite overlap.(2) The β k,i are ( N, ∞ ) atoms with supp( β k,i ) ⊂ Q ∗ k,i . In particular, P i | β k,i | . C uniformlyfor all k ∈ Z .(3) f = P k,i λ k,i β k,i , where the convergence is unconditional both pointwise and in the senseof distributions.(4) λ k,i . k for all k, i and P k,i λ k,i χ Q k,i . M N ( f ) . In particular, P k,i λ k,i β k,i also con-verges absolutely to f in L q ( w ) , whenever q > is such that w ∈ A q .Proof of Theorem 2.6. Fix f ∈ O N ∩ H p ( w ); by homogeneity we may assume without loss ofgenerality that k f k H p ( w ) = 1. Then there exists R > f ) ⊂ B (0 , R ) = B . Let B ∗ = B (0 , R ). We claim that for all x / ∈ B ∗ ,(2.2) M N f ( x ) . w ( B ) − p k f k H p ( w ) . w ( B ∗ ) p . To prove this, we argue as in [13, Lemma 7.11] (cf. inequality (7.7)). There they showed a pointwiseinequality: given any ϕ ∈ F N and t > | f ∗ ϕ t ( x ) | . inf z ∈ B ∗ M N f ( z ) , where B ∗ = B (0 , R ). Therefore, we have that | f ∗ ϕ t ( x ) | p . w ( B ∗ ) Z B ∗ M N f ( z ) p w ( z ) dz ≤ w ( B ∗ ) ;inequality (2.2) follows if we take the supremum over all ϕ ∈ F N and t >
0, and note that since w ∈ A ∞ , w ( B ∗ ) . w ( B ∗ ).Now let k be the smallest integer such that for all k > k , Ω k ⊂ B ∗ . More precisely, by (2.2)we can take k to be the largest integer such that 2 k ≤ Cw ( B ∗ ) − p .By Lemma 2.7 we can decompose f as f = X k,i λ k,i β k,i ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 9 where the β k,i are ( N, ∞ ) atoms. We will show that this sum can be rewritten as a finite sum ofatoms. Set F = X k ≤ k X i λ k,i β k,i = f − X k>k X i λ k,i β k,i . Since the β k,i are supported in Ω k ⊂ B ∗ for all k > k , the function F is also supported in B ∗ .Moreover k F k ∞ ≤ X k ≤ k (cid:13)(cid:13)(cid:13) X i λ k,i | β k,i | (cid:13)(cid:13)(cid:13) L ∞ . X k ≤ k k = C k . Further, F has vanishing moments up to order N . To see this, fix | α | ≤ N and q > w ∈ A q . Then, since supp( β k,i ) ⊂ B ∗ , (cid:13)(cid:13)(cid:13) X k ≤ k X i | x α || λ k,i β k,i | (cid:13)(cid:13)(cid:13) L ≤ (4 R ) | α | (cid:13)(cid:13)(cid:13) X k ≤ k X i | λ k,i β k,i | (cid:13)(cid:13)(cid:13) L q ( w ) w − q ′ ( B ∗ ) q ′ . (4 R ) | α | kM N f k qL ( w ) w − q ′ ( B ∗ ) q ′ . (4 R ) | α | k f k qL ( w ) w − q ′ ( B ∗ ) q ′ < ∞ . Therefore, the series on the left-hand side converges absolutely, so you can exchange the sum andintegral; since each β k,i has vanishing moments, so does F . Therefore, if we set a = C − − k F then a is an ( N, ∞ ) atom supported in B ∗ .To estimate the remaining terms, note that f is a bounded function and so there exists an integer k ∞ > k such that Ω k = ∅ for all k ≥ k ∞ . Thus the sum X k>k X i λ k,i β k,i = X k In this section we state and prove several lemmas on averaging operators and m -CZOs neededfor the proofs of Theorems 1.1 and 1.2. Averaging operators. We begin with a well-known result on the maximal operator M µ definedwith respect to a measure µ : M µ f ( x ) = sup Q µ ( Q ) Z Q | f | dµ · χ Q ( x ) . For a proof, see [17, Chapter II]. Proposition 3.1. Let µ be a doubling measure on R n . Then the maximal operator M µ satisfiesthe weak (1 , inequality (3.1) sup t> t µ ( { x ∈ R n : M µ f ( x ) > t } ) ≤ C ( µ ) Z R n | f | dµ, and for < p < ∞ the strong ( p, p ) inequality (3.2) Z R n ( M µ f ) p dµ ≤ C ( µ, p ) Z R n | f | p dµ. The next three lemmas on averaging operators are weighted extensions of results from [20]. Ourproofs, however, are different and are motivated by ideas from [33]. Lemma 3.2. Let µ be a doubling measure on R n and fix < p < . Then given any finite collection J of cubes and any set { f Q : Q ∈ J } of non-negative integrable functions with supp( f Q ) ⊂ Q , (cid:13)(cid:13)(cid:13) X Q ∈J f Q (cid:13)(cid:13)(cid:13) L p ( µ ) ≤ C ( µ, p, n ) (cid:13)(cid:13)(cid:13) X Q ∈J a µ ( Q ) χ Q (cid:13)(cid:13)(cid:13) L p ( µ ) , where a µ ( Q ) = µ ( Q ) − Z Q f Q ( x ) dµ ( x ) . Proof. Let F = P Q ∈J f Q and G = P Q ∈J a µ ( Q ) χ Q and for each t > L t = { x ∈ R n : G ( x ) > t } , U t = { y ∈ R n : M µ χ L t ( y ) > } . By (3.1) we have that µ ( U t ) ≤ C ( µ ) µ ( L t ). We can now estimate as follows: µ ( { x ∈ R n : F ( x ) > t } ) ≤ µ ( U t ) + µ ( U ct ∩ { x ∈ R n : F ( x ) > t } ) . µ ( L t ) + 1 t Z U ct F ( x ) dµ ( x ) . µ ( L t ) + 1 t X Q ∈J : Q ∩ U ct = ∅ Z Q f Q ( x ) dµ ( x ) . µ ( L t ) + 1 t X Q ∈J : Q ∩ U ct = ∅ a µ ( Q ) µ ( Q ) . ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 11 If Q ∈ J is such that Q ∩ U ct = ∅ , then M µ χ L t ( z ) ≤ for all z ∈ Q ∩ U ct . In particular, we have µ ( L t ∩ Q ) µ ( Q ) ≤ , and so µ ( Q ) ≤ µ ( L ct ∩ Q ) for all Q ∈ J . Thus we have that µ ( { x ∈ R n : F ( x ) > t } ) . µ ( L t ) + 1 t X Q ∈J a µ ( Q ) µ ( Q ∩ L ct ) . µ ( L t ) + 1 t X Q ∈J a µ ( Q ) Z L ct χ Q ( x ) dµ ( x ) . µ ( L t ) + 1 t Z L ct G ( x ) dµ ( x ) . Given this estimate, if we multiply by pt p − and integrate, by Fubini’s theorem we get k F k pL p ( µ ) = Z ∞ pt p − µ ( { x ∈ R n : F ( x ) > t } ) dt . Z ∞ pt p − µ ( { x ∈ R n : G ( x ) > t } ) dt + Z ∞ pt p − Z { x ∈ R n : G ( x ) ≤ t } G ( x ) dµ ( x )= Z R n G ( x ) p dµ ( x ) + Z R n G ( x ) Z ∞ G ( x ) pt p − dt dµ ( x ) . k G k pL p ( µ ) . (cid:3) Lemma 3.3. Let µ be a doubling measure on R n and fix ≤ p < q < ∞ . Then given anyfinite collection of cubes J and any set { f Q : Q ∈ J } of non-negative integrable functions with supp( f Q ) ⊂ Q , (3.3) (cid:13)(cid:13)(cid:13) X Q ∈J f Q (cid:13)(cid:13)(cid:13) L p ( µ ) ≤ C ( µ, p, q, n ) (cid:13)(cid:13)(cid:13) X Q ∈J a µq ( Q ) χ Q (cid:13)(cid:13)(cid:13) L p ( µ ) , where a µq ( Q ) = (cid:16) µ ( Q ) Z Q | f Q ( x ) | q dµ ( x ) (cid:17) q . Proof. First suppose that p > 1; we estimate by duality. Then there exists non-negative g ∈ L p ′ ( µ ), k g k L p ′ ( dµ ) = 1, such that (cid:13)(cid:13)(cid:13) X Q ∈J f Q (cid:13)(cid:13)(cid:13) L p ( µ ) = X Q ∈J Z Q f Q ( x ) g ( x ) dµ ( x ) ≤ X Q ∈J (cid:16) Z Q f Q ( x ) q dµ ( x ) (cid:17) q (cid:16) Z Q g ( x ) q ′ dµ ( x ) (cid:17) q ′ = X Q ∈J a µq ( Q ) µ ( Q ) h µ ( Q ) Z Q g q ′ dµ i q ′ ≤ X Q ∈J a µq ( Q ) Z Q M µ ( g q ′ )( x ) q ′ dµ ( x ) ≤ Z R n h X Q ∈J a µq ( Q ) χ Q i M µ ( g q ′ )( x ) q ′ dµ ( x ) ≤ (cid:13)(cid:13)(cid:13) X Q ∈J a µq ( Q ) χ Q (cid:13)(cid:13)(cid:13) L p ( µ ) k M µ ( g q ′ ) q ′ k L p ′ ( µ ) = (cid:13)(cid:13)(cid:13) X Q ∈J a µq ( Q ) χ Q (cid:13)(cid:13)(cid:13) L p ( dµ ) k M µ ( g q ′ ) k q ′ L p ′ q ′ ( µ ) . (cid:13)(cid:13)(cid:13) X Q ∈J a µq ( Q ) χ Q (cid:13)(cid:13)(cid:13) L p ( µ ) ;the first and third inequalities follow from H¨older’s inequality, and the last from (3.2) (since p ′ > q ′ )and the fact that k g k L p ′ ( µ ) = 1.Finally, when p = 1 the proof is essentially the same except that we use use the fact that M µ isbounded on L ∞ . This completes the proof. (cid:3) Lemma 3.4. Let w ∈ A ∞ , and fix < p < ∞ and max(1 , p ) < q < ∞ . Then given any collectionof cubes { Q k } ∞ k =1 and nonnegative integrable functions { g k } with supp( g k ) ⊂ Q k , (cid:13)(cid:13)(cid:13) ∞ X k =1 g k (cid:13)(cid:13)(cid:13) L p ( w ) ≤ C ( w, p, q, n ) (cid:13)(cid:13)(cid:13) ∞ X k =1 (cid:16) w ( Q k ) Z Q k g k ( x ) q w ( x ) dx (cid:17) q χ Q k (cid:13)(cid:13)(cid:13) L p ( w ) . Proof. Since w ∈ A ∞ , the measure µ = w ( x ) dx is doubling. If p ≥ 1, then if we fix an arbitraryinteger K and apply Lemma 3.3 to the functions { g k } Kk =1 , we immediately get (cid:13)(cid:13) K X k =1 g k (cid:13)(cid:13) L p ( w ) ≤ C ( w, p, q, n ) (cid:13)(cid:13)(cid:13) K X k =1 (cid:16) w ( Q k ) Z Q k g k ( x ) q w ( x ) dx (cid:17) q χ Q k (cid:13)(cid:13)(cid:13) L p ( w ) . The desired inequality now follows from Fatou’s lemma.When 0 < p < 1, we can apply Lemma 3.2 to get the same conclusion, using the fact that1 w ( Q k ) Z Q k g k ( x ) w ( x ) dx ≤ (cid:18) w ( Q k ) Z Q k g k ( x ) q w ( x ) dx (cid:19) q . (cid:3) Estimates for m -CZOs. In this section we prove three estimates on m -CZOs. Lemma 3.5. Let T be the operator as in Theorem 1.1 and fix w ∈ A q , q > . Then given anycollection f , . . . , f m of bounded functions of compact support, k T ( f , f , . . . , f m ) k L q ( w ) ≤ C k f k L q ( w ) k f k L ∞ · · · k f m k L ∞ . Proof. By the domination estimate in Proposition 2.5 it will suffice to prove this estimate for anymultilinear sparse operator T S and non-negative functions f , . . . , f m . By the definition of thesparse operator we have T S ( f , . . . , f m ) ≤ k f k ∞ · · · k f m k ∞ X Q ∈S − Z Q f dy · χ Q = k f k ∞ · · · k f m k ∞ T S f , where on the right-hand side we now have a linear sparse operator. But then by Proposition 2.4we have that k T S ( f , · · · , f m ) k L q ( w ) . k T S f k L q ( w ) k f k ∞ · · · k f m k ∞ . k f k L q ( w ) k f k ∞ · · · k f m k ∞ . (cid:3) ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 13 The following lemma was first prove in [22]. For completeness we include its short proof. Lemma 3.6. For ≤ k ≤ m let a k be an ( N, ∞ ) atom supported in Q k and let c k be the center of Q k . Then, given any non-empty subset Λ ⊂ { , . . . , m } , we have that for all y / ∈ ∪ k ∈ Λ Q ∗ k , (3.4) | T ( a , . . . , a m )( y ) | . min { ℓ ( Q k ) : k ∈ Λ } n + N +1 (cid:0) P k ∈ Λ | y − c k | (cid:1) n + N +1 . In particular, we always have that (3.5) | T ( a , . . . , a m ) | χ ( Q ∗ ∩ ... ∩ Q ∗ m ) c . m Y k =1 (cid:0) M ( χ Q k ) (cid:1) n + N +1 mn . Proof. Without loss of generality we may assume that Λ = { , . . . , r } for some 1 ≤ r ≤ m and that ℓ ( Q ) = min { ℓ ( Q k ) : k ∈ Λ } . Fix y / ∈ ∪ k ∈ Λ Q ∗ k ; because a has vanishing moments up to order N , we can rewrite T ( a , . . . , a m )( y ) = Z R mn K ( y, y , . . . , y m ) a ( y ) · · · a m ( y m ) d~y = Z R mn (cid:2) K ( y, y , . . . , y m ) − P N ( y, y , y , . . . , y m ) (cid:3) a ( y ) · · · a m ( y m ) d~y = Z R mn K ( y, y , y , . . . , y m ) a ( y ) · · · a m ( y m ) d~y, (3.6)where P N ( y, y , y , . . . , y m ) = X | α |≤ N α ! ∂ α K ( y, c , y , . . . , y m )( y − c ) α is the Taylor polynomial of degree N of K ( y, · , y , . . . , y m ) at c and(3.7) K ( y, y , . . . , y m ) = K ( y, y , . . . , y m ) − P N ( y, y , y , . . . , y m ) . By the smoothness condition of the kernel and the fact that | y − y k | ≈ | y − c k | for all k ∈ Λ and y k ∈ Q k we have that (cid:12)(cid:12) K ( y, y , . . . , y m ) − P N ( y, c , y , . . . , y m ) (cid:12)(cid:12) . | y − c | N +1 (cid:16) X k ∈ Λ | y − c k | + m X j =2 | y − y j | (cid:17) − mn − N − . Thus, | T ( a , . . . , a m )( y ) | . Z R mn | y − c | N +1 | a ( y ) | · · · | a m ( y m ) | (cid:16) P k ∈ Λ | y − c k | + P mj =2 | y − y j | (cid:17) mn + N +1 d~y . Z R ( m − n ℓ ( Q ) n + N +1 (cid:16) P k ∈ Λ | y − c k | + P mj =2 | y j | (cid:17) mn + N +1 dy · · · dy m . ℓ ( Q ) n + N +1 (cid:16) P k ∈ Λ | y − c k | (cid:17) n + N +1 , which implies (3.4).To prove (3.5), fix y ∈ ( Q ∗ ∩ . . . ∩ Q ∗ m ) c ; then there exists a non-empty subset Λ of { , . . . , m } such that y / ∈ Q ∗ k for all k ∈ Λ and y ∈ Q ∗ l for l / ∈ Λ . Then by (3.4) we have that | T ( a , . . . , a m )( y ) | . min { ℓ ( Q k ) : k ∈ Λ } n + N +1 (cid:0) P k ∈ Λ | y − c k | (cid:1) n + N +1 . Y k ∈ Λ (cid:16) ℓ ( Q k ) ℓ ( Q k ) + | y − c k | (cid:17) n + N +1 mn . m Y k =1 (cid:16) ℓ ( Q k ) ℓ ( Q k ) + | y − c k | (cid:17) n + N +1 mn . Inequality (3.5) follows from the definition of the maximal operator. (cid:3) Lemma 3.7. Given w ∈ A q , ≤ q < ∞ , for ≤ k ≤ m let a k be an ( N, ∞ ) atom supported in Q k and let c k be the center of Q k . Suppose Q is the cube such that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } .Then (3.8) k T ( a , . . . , a m ) χ Q ∗ k L q ( w ) . w ( Q ) q m Y l =1 inf z ∈ Q M ( χ Q l )( z ) n + N +1 mn . Proof. Since the A p classes are nested, we may assume without loss of generality that q > 1. Toprove (3.8) we consider two cases: Q ∗ ∩ Q ∗ k = ∅ for all 2 ≤ k ≤ m or this intersection is empty forat least one value of k . In the first case, since ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } we have Q ∗ ⊂ Q ∗ k for all 1 ≤ k ≤ m . This implies inf z ∈ Q M ( χ Q k )( z ) & , for all 1 ≤ k ≤ m , and so Lemma 3.5 yields(3.9) k T ( a , . . . , a m ) χ Q ∗ k L q ( w ) ≤ k T ( a , . . . , a m ) k L q ( w ) . k a k L q ( w ) k a k L ∞ · · · k a m k L ∞ . w ( Q ) q m Y k =1 inf z ∈ Q M ( χ Q k )( z ) n + N +1 mn . In the second case, since Q ∗ ∩ Q ∗ k = ∅ for some k , the setΛ = { ≤ k ≤ m : Q ∗ ∩ Q ∗ k = ∅} is non-empty. Fix any point y ∈ R n . Then arguing as in the previous proof we have that(3.10) T ( a , . . . , a m )( y ) = Z R mn K ( y, y , y , . . . , y m ) a ( y ) · · · a m ( y m ) d~y, where K ( y, y , . . . , y m ) is defined by (3.7). For y ∈ Q we have that for some ξ ∈ Q and for all y l ∈ Q l , 1 ≤ l ≤ m ,(3.11) (cid:12)(cid:12) K ( y, y , . . . , y m ) (cid:12)(cid:12) ≤ Cℓ ( Q ) N +1 (cid:16) | y − ξ | + m X j =2 | y − y j | (cid:17) − mn − N − . For all k ∈ Λ, since Q ∗ ∩ Q ∗ k = ∅ , | y − ξ | + | y − y k | ≥ | ξ − y k | & | c − c k | . Therefore, for all y ∈ Q ∗ and y k ∈ Q k , k ∈ Λ, (cid:12)(cid:12) K ( y, y , . . . , y m ) (cid:12)(cid:12) . ℓ ( Q ) N +1 (cid:16) X k ∈ Λ | c − c k | + m X j =2 | y − y j | (cid:17) − mn − N − , If we combine this inequality with (3.10), we get | T ( a , . . . , a m )( y ) | . ℓ ( Q ) n + N +1 (cid:0) P k ∈ Λ | c − c k | (cid:1) n + N +1 . ℓ ( Q ) n + N +1 (cid:0) P k ∈ Λ [ ℓ ( Q ) + | c − c k | + ℓ ( Q k )] (cid:1) n + N +1 . ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 15 Since Q ∗ ⊂ Q ∗ l for all l / ∈ Λ , the last inequality gives us(3.12) k T ( a , . . . , a m ) k L ∞ . m Y k =1 inf z ∈ Q M ( χ Q k )( z ) n + N +1 mn ;since w ∈ A q is doubling, this implies that(3.13) k T ( a , . . . , a m ) χ Q ∗ k L q ( w ) . w ( Q ) q m Y k =1 inf z ∈ Q M ( χ Q k )( z ) n + N +1 mn . This completes the proof. (cid:3) Proof of Theorem 1.1 For 1 ≤ k ≤ m , let w k ∈ A ∞ and fix arbitrary functions f k ∈ H p k ( w k ) ∩ O N ( R n ). By Theo-rem 2.6, we have the finite atomic decompositions(4.1) f k = N X j k =1 λ k,j k a k,j k , where λ k,j k ≥ a k,j k are ( N, ∞ )-atoms that satisfysupp( a k,j k ) ⊂ Q k,j k , | a k,j k | ≤ χ Q k,jk , Z Q k,jk x α a k,j k ( x ) dx = 0for all | α | ≤ N , and(4.2) (cid:13)(cid:13)(cid:13) X j k λ j k χ Q jk (cid:13)(cid:13)(cid:13) L pk ( w k ) ≤ C k f k k H pk ( w k ) . Set w = Q mk =1 w ppk k . Again by Theorem 2.6, it will suffice to prove that(4.3) k T ( f , . . . , f m ) k L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . Since T is m -linear, we have that for a.e. x ∈ R n ,(4.4) T ( f , . . . , f m )( x ) = X j · · · X j m λ ,j . . . λ m,j m T ( a ,j , . . . , a m,j m )( x ) . Given a cube Q, let Q ∗ = 2 √ nQ . For each m -tuple, ( j , . . . , j m ), define R j ,...,j m to be the smallestcube among Q ∗ ,j , . . . , Q ∗ m,j m . To estimate k T ( f , . . . , f m ) k L p ( w ) we will split T ( f , . . . , f m ) intotwo parts: | T ( f , . . . , f m )( x ) | ≤ G ( x ) + G ( x ) , where G ( x ) = X j · · · X j m λ ,j . . . λ m,j m | T ( a ,j , · · · , a m,j m ) | χ R j ,...,jm ( x )and G ( x ) = X j · · · X j m λ ,j · · · λ m,j m | T ( a ,j , . . . , a m,j m ) | χ ( R j ,...,jm ) c ( x ) . We first estimate k G k L p ( w ) . By (3.5) we have that | T ( a ,j , . . . , a m,j m )( x ) | χ ( R j ,...,jm ) c ( x ) . m Y k =1 M ( χ Q k,jk )( x ) n + N +1 mn ; thus G . X j · · · X j m m Y k =1 λ k,j k M ( χ Q k,jk ) n + N +1 mn = m Y k =1 h X j k λ k,j k M ( χ Q k,jk ) n + N +1 mn i . By condition (1.3), H¨older’s inequality and the weighted Fefferman-Stein vector-valued inequality(see Remark 2.3), we get(4.5) k G k L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . We now estimate the norm of G . Since w ∈ A ∞ by Lemma 2.1, we can choose q > max(1 , p )such that w ∈ A q . Then by Lemma 3.5 we have that (cid:16) w ( R j ,...,j m ) Z R j ,...,jm | T ( a ,j , · · · , a m,j m ) | q ( x ) w ( x ) dx (cid:17) q . m Y k =1 inf z ∈ R j ,...,jm M ( χ Q k,jk )( z ) n + N +1 mn . If we combine this inequality, Lemma 3.4, H¨older’s inequality and the Fefferman-Stein vector-valuedinequality imply that (again see Remark 2.3), we get the following estimate: k G k L p ( w ) . (cid:13)(cid:13)(cid:13) X j ...,j m m Y k =1 λ k,j k (cid:16) w ( R j ,...,j m ) Z R j ,...,jm | T ( a ,j , · · · , a m,j m ) | q ( x ) w ( x ) dx (cid:17) q χ R j ,...,jm (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13) X j ...,j m (cid:16) m Y k =1 λ k,j k (cid:17) · (cid:16) m Y k =1 inf z ∈ R j ,...,jm M ( χ Q k,jk )( z ) n + N +1 mn (cid:17) χ R j ,...,jm (cid:13)(cid:13)(cid:13) L p ( w ) . (cid:13)(cid:13)(cid:13) m Y k =1 (cid:16) X j k λ k,j k M ( χ Q k,jk ) n + N +1 mn (cid:17)(cid:13)(cid:13)(cid:13) L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k M ( χ Q k,jk ) n + N +1 mn (cid:13)(cid:13)(cid:13) L pk ( w k ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . If we combine the estimates for G and G , we get the desired inequality.5. Proof of Theorem 1.2 The proof of Theorem 1.2 is very similar to the proof of Theorem 1.1. Instead of estimating thenorm of T , we will estimate the norm of M φ ◦ T , where M φ is the non-tangential maximal operator M φ f ( x ) = sup In this section, we fix the smooth approximate identity φ supported in the unit ball. Thefollowing lemma was first proved in [22]; it is the essential part in the proof of Theorem 1.2 andso we repeat the proof here for the convenience of the reader. Hereafter, given a cube Q , let Q ∗∗ = 4 nQ . Lemma 5.1. For ≤ k ≤ m , let a k be ( N, ∞ ) atoms with supp( a k ) ⊂ Q k . Suppose that Q issuch that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } . Then for all x / ∈ Q ∗∗ , we have (5.1) M φ T ( a , . . . , a m )( x ) . m Y l =1 M ( χ Q l )( x ) n + N +1 mn + M ( χ Q )( x ) n + sw +1 n m Y l =1 inf z ∈ Q M ( χ Q l )( z ) N − swmn , where T is the operator in Theorem 1.1.Proof. Fix x ∈ ( Q ∗∗ ) c , 0 < t < ∞ and y ∈ R n such that | y − x | < t . To prove (5.1) it will sufficeto show that(5.2) | φ t ∗ T ( a , . . . , a m )( y ) | . m Y l =1 M ( χ Q l )( x ) n + N +1 mn + M ( χ Q )( x ) n + sw +1 n m Y l =1 inf z ∈ Q M ( χ Q l )( z ) N − swmn , where the implicit constant does not depend on x, y and t . We will consider two cases. Case 1: t > n | x − c | . We will exploit the cancellation in (1.2) to show that(5.3) | φ t ∗ T ( a , . . . , a m )( y ) | . M ( χ Q )( x ) n + sw +1 n m Y l =1 inf z ∈ Q M ( χ Q l )( z ) N − swmn . By (1.2) we have φ t ∗ T ( a , . . . , a m )( y ) = Z φ t ( y − z ) T ( a , . . . , a m )( z ) dz = Z (cid:16) φ t ( y − z ) − X | α |≤ s w ∂ α [ φ t ]( y − c ) α ! ( c − z ) α (cid:17) T ( a , . . . , a m )( z ) dz. Note that by Taylor’s theorem, (cid:12)(cid:12)(cid:12) φ t ( y − z ) − X | α |≤ s w ∂ α [ φ t ]( y − c ) α ! ( c − z ) α (cid:12)(cid:12)(cid:12) . | z − c | s w +1 t n + s w +1 for all y, z ∈ R n and all t ∈ (0 , ∞ ). Since t & | x − c | and x / ∈ Q ∗∗ , we have | φ t ∗ T ( a , . . . , a m )( y ) | . Z | z − c | s w +1 t n + s w +1 | T ( a , . . . , a m )( z ) | dz . (cid:16) ℓ ( Q ) | x − c | (cid:17) n + s w +1 ℓ ( Q ) n + s w +1 Z | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz . M ( χ Q )( x ) n + sw +1 n ℓ ( Q ) n + s w +1 Z | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz. Hence, to prove (5.3) it remains to show that(5.4) 1 ℓ ( Q ) n + s w +1 Z | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz . m Y l =1 inf z ∈ Q M ( χ Q l )( z ) N − swmn . If we split the integral on the left-hand side of (5.4) over Q ∗ and ( Q ∗ ) c , we can estimate as follows: Z | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz . Z Q ∗ | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz + Z ( Q ∗ ) c | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz . ℓ ( Q ) s w +1 Z Q ∗ | T ( a , . . . , a m )( z ) | dz + Z ( Q ∗ ) c | z − c | s w +1 | T ( a , . . . , a m )( z ) | dz. By (3.8), we can estimate the first integral in the last inequality by(5.5) ℓ ( Q ) s w +1 Z Q ∗ | T ( a , . . . , a m )( z ) | dz . ℓ ( Q ) n + s w +1 m Y l =1 inf z ∈ Q M ( χ Q l )( z ) n + N +1 mn . To estimate second integral, we need to exploit carefully the smoothness of the kernel. Recall therepresentation of T ( a , . . . , a m )( z ) in (3.10). Denote J = { ≤ l ≤ m : Q ∗∗ ∩ Q ∗∗ l = ∅} . For z / ∈ Q ∗ , ξ ∈ Q , we have | z − ξ | ≈ | z − c | ≥ ℓ ( Q ). Also for l ∈ J and z l ∈ Q ∗ l , | z − ξ | + | z − z l | ≥ | ξ − z l | & | c − c l | . We now estimate K ( z, z , . . . , z m ) in (3.11) to get | T ( a , . . . , a m )( z ) | . Z ( R n ) m ℓ ( Q ) N +1 χ Q ( z ) dz · · · dz m ℓ ( Q ) + | z − c | + X l ∈ J | c − c l | + m X l =2 | z − z l | ! mn + N +1 for all z ∈ ( Q ∗ ) c . Thus, Z ( Q ∗ ) c | y − c | s w +1 | T ( a , . . . , a m )( y ) | dy . Z R n × ( R n ) m | y − c | s w +1 ℓ ( Q ) N +1 χ Q ( y ) d~ydy (cid:18) ℓ ( Q ) + | y − c | + X l ∈ J | c − c l | + m X l =2 | y − y l | (cid:19) mn + N +1 . ℓ ( Q ) n + s w +1 Y l ∈ J (cid:16) ℓ ( Q l ) ℓ ( Q ) + | c − c l | (cid:17) N − swm . Note that 1 . inf z ∈ Q M ( χ Q l )( z ) if Q ∗∗ ∩ Q ∗∗ l = ∅ and for all l ∈ J , ℓ ( Q l ) ℓ ( Q ) + | c − c l | . inf z ∈ Q M ( χ Q l )( z ) n . Therefore,(5.6) Z ( Q ∗ ) c | y − c | s w +1 | T ( a , . . . , a m )( y ) | dy . ℓ ( Q ) n + s w +1 m Y l =1 inf z ∈ Q M ( χ Q l )( z ) N − swmn . Now we combine (5.5) and (5.6) we get (5.4), which completes the proof of Case 1. ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 19 Case 2: t ≤ n | x − c | . In this case, we will show that(5.7) | φ t ∗ T ( a , . . . , a m )( y ) | . m Y l =1 M ( χ Q l )( x ) n + N +1 mn . Since supp( φ ) ⊂ B (0 , 1) and | y − x | < t ,(5.8) | φ t ∗ T ( a , . . . , a m )( y ) | ≤ Z B ( y,t ) t − n (cid:12)(cid:12) φ (cid:0) t − ( y − z ) (cid:1) T ( a , . . . , a m )( z ) (cid:12)(cid:12) dz . sup z ∈ B ( y,t ) | T ( a , . . . , a m )( z ) | . sup z ∈ B ( x, t ) | T ( a , . . . , a m )( z ) | . Let Λ = { ≤ l ≤ m : x / ∈ Q ∗∗ k } . For z ∈ B ( x, t ), ξ ∈ Q , we have | x − c | ≤ | x − z | + | z − c | ≤ t + | ξ − c | + | z − ξ | ≤ n | x − c | + 12 | x − c | + | z − ξ | ;hence, t . | x − c | . | z − ξ | . For l ∈ Λ and z l ∈ Q l , since x / ∈ Q ∗∗ l , | x − c l | ≤ | x − z l | ≤ | x − z | + 2 | z − z l | ≤ t + 2 | z − z k | . | z − ξ | + | z − z k | . Recall the formula for T ( a , . . . , a m )( z ) in (3.10); we estimate K ( z, z , . . . , z m ) in (3.11) to get | T ( a , . . . , a m )( z ) | . Z ( R n ) m ℓ ( Q ) N +1 χ Q ( z ) dz · · · dz m (cid:16) P ml =2 | z − z l | + P k ∈ Λ | x − c k | (cid:17) mn + N +1 for all z ∈ B ( x, t ). From this we get that(5.9) sup z ∈ B ( x, t ) | T ( a , . . . , a m )( z ) | . Y l ∈ Λ ℓ ( Q l ) n + N +1 | Λ | χ ( Q ∗∗ l ) c ( x ) | x − c l | n + N +1 | Λ | · Y k / ∈ Λ χ Q ∗∗ k ( x ) . m Y l =1 M ( χ Q l )( x ) n + N +1 mn . Combining (5.8) and (5.9) gives (5.7). This completes Case 2 and so completes the proof. (cid:3) The next lemma is an immediate consequence of Lemma 3.7 and the fact that M φ is boundedon L q ( w ) if w ∈ A q (since it is controlled pointwise by the Hardy-Littlewood maximal operator;cf. [17]). Lemma 5.2. Given w ∈ A q , ≤ q < ∞ , for ≤ k ≤ m let a k be an ( N, ∞ ) atom supported in Q k . Suppose Q is the cube such that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } . Then k M φ T ( a , . . . , a m ) χ Q ∗∗ k L q ( w ) . w ( Q ) q m Y l =1 inf z ∈ Q M ( χ Q l )( z ) n + N +1 mn . (5.10) Proof of Theorem 1.2. Fix w k ∈ A ∞ , 1 ≤ k ≤ m , and define w = Q mk =1 w ppk k . Fix f k ∈ H p k ( w k ) ∩O N ( R n ), 1 ≤ k ≤ m . We will show that(5.11) k M φ T ( f , . . . , f m ) k L p ( w ) . k f k H p ( w ) · · · k f m k H pm ( w m ) . Form the atomic decompositions of the functions f k as in the proof of Theorem 1.1 to get (4.1)and (4.2). Then to prove (5.11), it is enough show that(5.12) k M φ T ( f , . . . , f m ) k L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . Since M φ ◦ T is multi-sublinear, we can write M φ T ( f , . . . , f m )( x ) ≤ G ( x ) + G ( x ) , where G ( x ) = X j · · · X j m λ ,j . . . λ m,j m M φ T ( a ,j , · · · , a m,j m ) χ R j ,...,jm ( x )and G ( x ) = X j · · · X j m λ ,j · · · λ m,j m M φ T ( a ,j , . . . , a m,j m ) χ ( R j ,...,jm ) c ( x ) . Here R j ,...,j m is the smallest cube among Q ∗∗ ,j , . . . , Q ∗∗ m,j m .A similar argument as in the proof of Theorem 1.1 with Lemma 5.2 in place of Lemma 3.7 gives(5.13) k G k L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . We now estimate the norm of G . By Lemma 5.1 we get that G ( x ) . G ( x ) + G ( x ) , where G ( x ) = X j · · · X j m λ ,j · · · λ m,j m m Y k =1 M ( χ Q k,jk )( x ) n + N +1 mn and G ( x ) = X j · · · X j m λ ,j · · · λ m,j m M ( χ R j ,...,jm )( x ) n + sw +1 n m Y l =1 inf z ∈ R j ,...,jm M ( χ Q l,jl )( z ) N − swmn . The function G can be estimated by essentially the same argument used for G to get(5.14) k G k L p ( w ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . To estimate G , since ( n + s w +1) pn > 1, we use (1.4) and the Fefferman-Stein vector-valued in-equality (cf. Remark 2.3) to get k G k L p ( w ) . (cid:13)(cid:13)(cid:13) X j · · · X j m λ ,j · · · λ m,j m χ R j ,...,jm m Y l =1 inf z ∈ R j ,...,jm M ( χ Q l,jl )( z ) N − swmn (cid:13)(cid:13)(cid:13) L p ( w ) ≤ (cid:13)(cid:13)(cid:13) X j · · · X j m λ ,j · · · λ m,j m m Y k =1 M ( χ Q k,jk ) N − swmn (cid:13)(cid:13)(cid:13) L p ( w ) ≤ m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k M ( χ Q k,jk ) N − swmn (cid:13)(cid:13)(cid:13) L pk ( w k ) . m Y k =1 (cid:13)(cid:13)(cid:13) X j k λ k,j k χ Q k,jk (cid:13)(cid:13)(cid:13) L pk ( w k ) . (5.15)If we combine (5.13), (5.14) and (5.15), we get (5.12) and this completes the proof. (cid:3) ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 21 Variable Hardy spaces: proof of Theorems 1.7 and 1.8 In this section we prove Theorems 1.7 and 1.8. In fact, we will prove two more general resultsthat include these theorems as special cases. To do so, we first recall some basic facts about thevariable Lebsesgue spaces. For complete information we refer the reader to [7].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 ) . Given p ( · ) ∈ P ( R n ) define L p ( · ) = L p ( · ) ( R n ) to be the set of all measurable functions f such thatfor 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 “norm” k f k L p ( · ) = inf { λ > ρ ( f /λ ) ≤ } . If p − ≥ L p ( · ) is a Banach space; if p ( · ) = p a constant, then L p ( · ) = L p with equality of norms.If the maximal operator is bounded on L p ( · ) we write that p ( · ) ∈ B . A necessary condition forthis to be the case is that p − > 1. A sufficient condition is that 1 < p − ≤ p + < ∞ and p ( · ) is log-H¨older continuous: i.e., (1.6) and (1.7) hold. However, this continuity condition is not necessary:see [7] for a detailed discussion of this problem.Given p ( · ) ∈ P ( R n ), the variable Hardy space H p ( · ) is defined to be the set of all distributions f such that M N f ∈ L p ( · ) . Again, we here assume N > Theorem 6.1. Let F = (cid:8) ( f , . . . , f m , F ) (cid:9) be a family of ( m +1) -tuples of non-negative, measurablefunctions on R n . Suppose that there exist indices < p , . . . , p m , p < ∞ satisfying p = p + · · · + p m such that for all weights w k ∈ A , ≤ k ≤ m , and w = Q mk =1 w ppk k , (6.1) k F k L p ( w ) . k f k L p ( w ) · · · k f m k L pm ( w m ) for all ( f , . . . , f m , F ) such that F ∈ L p ( w ) , and where the implicit constant depends only on n , p k and [ w k ] A , ≤ k ≤ m . Let q ( · ) , . . . , q m ( · ) , q ( · ) ∈ P be such that q ( · ) = 1 q ( · ) + · · · + 1 q m ( · ) ,p k < ( q k ) − , ≤ k ≤ m , and q k ( · ) /p k ∈ B . Then (6.2) k F k L q ( · ) . k f k L q · ) · · · k f m k L qm ( · ) provided k F k L q ( · ) < ∞ . The implicit constant only depends on n and q k ( · ) , ≤ k ≤ m . Remark 6.2. In [11], the hypothesis on the exponents q k ( · ) was stated as ( q k ( · ) /p k ) ′ ∈ B , wherethis exponent is the conjugate exponent, defined pointwise by p ( x ) + p ′ ( x ) = 1. It was stated in this way for technical reasons related to the proof. However, these two hypotheses are equivalent:see [7, Corollary 4.64].The one technical obstacle in applying Theorem 6.1 is constructing the family F to satisfy thehypotheses that the left-hand sides of (6.1) and (6.2) are finite and that the resulting family islarge enough that the desired result can be proved via a density argument. In our case we will usethe atomic decomposition in the weighted and variable Hardy spaces. As we noted in Section 2,given w ∈ A ∞ and 0 < p < ∞ , every f ∈ H p ( w ) can be written as the sum(6.3) f = X k λ k a k , where λ k ≥ a k are ( N, ∞ ) atoms, provided N ≥ s w . Moreover, this series convergesboth in the sense of distributions and in H p ( w ). (See [33, Chapter VIII].) The same is true in thevariable Hardy spaces. More precisely: suppose p ( · ) ∈ P is such that there exists 0 < p < p − with p ( · ) /p ∈ B . Then given N > n ( p − − f ∈ H p ( · ) , there exists a sequence of ( N, ∞ )atoms a k and constants λ k such that (6.3) holds, and the series converges both in the sense ofdistributions and in H p ( · ) . (See [13, Theorem 6.3]; here we have slightly modified the definitionof atoms, but the change is immediate.) It follows immediately from these two results that finitesums of ( N, ∞ ) atoms, for N sufficiently large, are dense in H p ( w ) and H p ( · ) . Remark 6.3. In applying the density of finite sums of atoms, we are not making use of the finiteatomic decomposition norm (as in Theorem 2.6 for weighted spaces or in the corresponding resultfor variable Hardy spaces in [13]). We will only use that these sums are dense with respect to thegiven Hardy space norm. Theorem 6.4. Let q ( · ) , . . . , q m ( · ) , q ( · ) ∈ P be such that q ( · ) = q ( · ) + · · · + q m ( · ) and < ( q k ) − ≤ ( q k ) + < ∞ , ≤ k ≤ m . Suppose further that there exist < p , . . . , p m , p < ∞ with p = p + · · · + p m , < p k < ( q k ) − , and q k ( · ) /p k ∈ B . If T is an m -CZO as in Theorem 1.1 satisfying (1.1) forall | α | ≤ N , where N ≥ max (cid:26) (cid:22) mn (cid:16) p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + ( m − n, then T : H q ( · ) × · · · × H q ( · ) → L q ( · ) . Remark 6.5. Theorem 1.7 follows at once since if q k ( · ) is log-H¨older continuous, then q k ( · ) /p k ∈ B . Proof. Fix an integer K such that K > max (cid:26) (cid:22) n (cid:16) p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) . Define the family F = (cid:8) ( f , . . . , f m , F ) (cid:9) , where for each 1 ≤ k ≤ m , f k = L X j =1 λ j a j is a finite linear combination of ( K , ∞ ) atoms, and F = max (cid:8) | T ( f , . . . , f m ) | , R (cid:9) χ B (0 ,R ) , where 0 < R < ∞ . ULTILINEAR CALDER ´ON-ZYGMUND OPERATORS ON WEIGHTED AND VARIABLE HARDY SPACES 23 Now fix any collection of weights w , . . . , w m ∈ A . Then for 1 ≤ k ≤ m , r w k = 1, so K > s w k .Therefore, given any ( m + 1)-tuple ( f , . . . , f m , F ) ∈ F , f k ∈ H p k ( w k ), and by Theorem 1.1, k F k L p ( w ) ≤ k T ( f , . . . , f m ) k L p ( w ) . k f k H p ( w ) · · · k f m k H pm ( w m ) < ∞ . Moreover, we have that f k ∈ H q k ( · ) and k F k q ( · ) ≤ R k χ B (0 ,R ) k q ( · ) < ∞ . Hence, by Theorem 6.1 we have that k F k q ( · ) . k f k H q · ) · · · k f m k H qm ( · ) < ∞ . By Fatou’s lemma in the scale of variable Lebesgue spaces [7, Theorem 2.61], we get k T ( f , . . . , f m ) k q ( · ) . k f k H q · ) · · · k f m k H qm ( · ) < ∞ . Since finite sums of ( K , ∞ ) atoms are dense in H q k ( · ) , 1 ≤ k ≤ m , a standard density argumentshows that this inequality holds for all f k ∈ H q k ( · ) , 1 ≤ k ≤ m . This completes the proof. (cid:3) The proof of the following result is identical to the proof of Theorem 6.4, except that in thedefinition of the family F we replace T by M N T (for N sufficiently large) and use Theorem 1.2instead of Theorem 1.1. Theorem 1.8 again follows as an immediate corollary. Theorem 6.6. Given q ( · ) , . . . , q m ( · ) , q ( · ) and p , . . . , p m , p as in Theorem 6.4, let T be an m -CZOas in Theorem 1.1 satisfying (1.1) for all | α | ≤ N , where N ≥ (cid:22) mn (cid:16) p − (cid:17)(cid:23) + + max (cid:26) (cid:22) mn (cid:16) p k − (cid:17)(cid:23) + , ≤ k ≤ m (cid:27) + mn. 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Yan, Multilinear version of reversed H¨older inequality and its applications to multilinear Calder´on-Zygmund operators , J. Math. Soc. Japan (2012), no. 4, 1053–1069. 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 E-mail address ::