Conditions for Boundedness into Hardy spaces
Loukas Grafakos, Shohei Nakamura, Hanh Van Nguyen, Yoshihiro Sawano
aa r X i v : . [ m a t h . F A ] F e b CONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES
LOUKAS GRAFAKOS, SHOHEI NAKAMURA, HANH VAN NGUYEN, AND YOSHIHIRO SAWANO
Abstract.
We obtain boundedness from a product of Lebesgue or Hardy spaces into Hardy spacesunder suitable cancellation conditions for a large class of multilinear operators that includes theCoifman-Meyer class, sums of products of linear Calder´on-Zygmund operators and combinationsof these two types.
Contents
1. Introduction 12. Preliminary and related results 42.1. Equivalent definitions of Hardy spaces 42.2. Reductions in the proof of main results 53. The Coifman-Meyer type 63.1. Fundamental estimates for the Coifman-Meyer type 63.2. The proof of Proposition 2.5 for Coifman-Meyer type 114. The product type 124.1. Fundamental estimates for the product type 134.2. The proof of Proposition 2.5 for the product type 185. The mixed type 195.1. Fundamental estimates for the mixed type 205.2. The proof of Proposition 2.5 for the mixed type 246. Examples 25References 261.
Introduction
In this work, we obtain boundedness for multilinear singular operators of various types fromproducts of Lebesgue or Hardy spaces into Hardy spaces, under suitable cancellation conditions.This particular line of investigation was initiated in the work of Coifman, Lions, Meyer and Semmes[1] who showed that certain bilinear operators with vanishing integral map L q × L q ′ into the Hardyspace H for 1 < q < ∞ with q ′ = q/ ( q − H p × H p → H p for theentire range 0 < p , p , p < ∞ and 1 /p = 1 /p + 1 /p , under the necessary cancellation conditions.Additional proofs of these results were provided by Grafakos and Li [10], Hu and Meng [13],and Huang and Liu [14]. All the aforementioned accounts on this topic are based on different The first author would like to thank the Simons Foundation.MSC 42B15, 42B30. approaches and address two classes of operators but [4], [13], and [14] seem to contain flaws intheir proofs; in fact, as of this writing, only the approach in [10] stands, which deals with the caseof finite sums of products of Calder´on-Zygmund operators. In this work we revisit this line ofinvestigation via a new method based on ( p, ∞ )-atomic decompositions. Our approach is powerfulenough to encompass many types of multilinear operators that include all the previously studied(Coifman-Meyer type and finite sums of products of Calder´on-Zygmund operators) as well as mixedtypes. An alternative approach to Hardy space estimates for bilinear operators has appeared inthe recent work of Hart and Lu [12].Recall that the Hardy space H p with 0 < p < ∞ is given as the space of all tempered distributions f for which k f k H p = (cid:13)(cid:13) sup t> | e t ∆ f | (cid:13)(cid:13) L p is finite, where e t ∆ denotes the heat semigroup for 0 < p ≤ ∞ . Note that H p and L p are isomorphicwith norm equivalence when 1 < p ≤ ∞ .In this work we study the boundedness into H p of the following three types of operators: • multilinear singular integral operators of Coifman-Meyer type; • sums of m -fold products of linear Calder´on-Zygmund singular integrals; • multilinear singular integrals of mixed type (i.e., combinations of the previous two types).Let m, n be positive integers. For a bounded function σ on ( R n ) m we consider the multilinearoperator T σ ( f , . . . , f m )( x ) = Z ( R n ) m σ ( ξ , . . . , ξ m ) b f ( ξ ) · · · c f m ( ξ m ) e πix · ( ξ + ··· + ξ m ) dξ · · · dξ m ( x ∈ R n )for f , . . . , f m ∈ S . Here S is the space of Schwartz functions and b f ( ξ ) = R R n f ( x ) e − πix · ξ dx isthe Fourier transform of a given Schwartz function f on R n . The space of tempered distributionsis denoted by S ′ .Certain conditions on σ imply that T σ extends to a bounded linear operator from L p × · · · × L p m to L p as long as 1 < p , . . . , p m ≤ ∞ and 0 < p < ∞ satisfies(1.1) 1 p = 1 p + · · · + 1 p m . Such a condition is the following by Coifman-Meyer (modeled after the classical Mihlin linearmultiplier condition)(1.2) | ∂ α σ ( ξ , . . . , ξ m ) | . ( | ξ | + · · · + | ξ m | ) −| α | , ( ξ , . . . , ξ m ) ∈ ( R n ) m \ { } for α ∈ ( N n ) m satisfying | α | ≤ M for some large M . Such operators are called m -linear Calder´on-Zygmund operators and there is a rich theory for them analogous to the linear one.An m -linear Calder´on-Zygmund operator associated with a Calder´on-Zygmund kernel K on R mn is defined by(1.3) T σ ( f , . . . , f m )( x ) = Z ( R n ) m K ( x − y , . . . , x − y m ) f ( y ) · · · f m ( y m ) dy · · · dy m , where σ is the distributional Fourier transform of K on ( R n ) m that satisfies (1.2). When m = 1,these operators reduce to classical Calder´on-Zygmund singular integral operators.An m -linear operator of product type on R mn is defined by(1.4) T X ρ =1 T σ ρ ( f )( x ) · · · T σ ρm ( f m )( x ) ( x ∈ R n ) , ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 3 where the T σ ρj ’s are linear Calder´on-Zygmund operators associated with the multipliers σ ρj . Interms of kernels these operators can be expressed as T σ ( f , . . . , f m )( x ) = T X ρ =1 m Y j =1 Z R n K ρσ j ( x − y j ) f j ( y j ) dy j , where K ρ , . . . , K ρm are the Calder´on-Zygmund kernels of the operator T ρσ , . . . , T ρσ m , respectivelyfor ρ = 1 , . . . , T .In this work we also consider operators of mixed type , i.e., of the form(1.5) T σ ( f , . . . , f m )( x ) = T X ρ =1 X I ρ ,...,I ρG ( ρ ) G ( ρ ) Y g =1 T σ Iρg ( { f l } l ∈ I ρg )( x ) , where for each ρ = 1 , . . . , T , I ρ , · · · , I ρG ( ρ ) is a partition of { , . . . , m } and each T σ Iρg is an | I ρg | -linear Coifman-Meyer multiplier operator. We write I ρ + · · · + I ρG ( ρ ) = { , . . . , m } to denote suchpartitions.In this work, we study operators of the form (1.3), (1.4), and (1.5). We will be working withindices in the following range 0 < p , . . . , p m ≤ ∞ , < p < ∞ that satisfy (1.1). Throughout this paper we reserve the letter s to denote the following index:(1.6) s = [ n (1 /p − + and we fix N ≫ s a sufficiently large integer, say N = m ( n + 1 + 2 s ).We recall that a ( p, ∞ )-atom is an L ∞ -function a that satisfies | a | ≤ χ Q , where Q is a cube on R n with sides parallel to the axes and Z R n x α a ( x ) dx = 0for all α with | α | ≤ N . By convention, when p = ∞ , a is called a ( ∞ , ∞ )-atom if Q = R n and k a k L ∞ ≤ . No cancellations are required for ( ∞ , ∞ )-atoms.Our main results are as follows: Theorem 1.1.
Let T σ be the operator defined in (1.3) and assume that it satisfies (1.2) . Let < p , . . . , p m ≤ ∞ and < p < ∞ satisfy (1 . . Assume that (1.7) Z R n x α T σ ( a , . . . , a m )( x ) dx = 0 , for all | α | ≤ s and all ( p l , ∞ ) -atoms a l . Then T σ can be extended to a bounded map from H p ×· · · × H p m to H p . Theorem 1.2.
Let T σ be the operator defined in (1.4) , < p , . . . , p m < ∞ , and < p < ∞ satisfies (1 . , where each σ ρj satisfies (1.2) with m = 1 . Assume that (1 . holds for all | α | ≤ s .Then T σ can be extended to a bounded map from H p × · · · × H p m to H p . Theorem 1.3.
Let T σ be the operator defined in (1.5) , < p , . . . , p m ≤ ∞ , and < p < ∞ satisfies (1 . . Suppose that each σ I ρg satisfies (1.2) with m = | I ρg | . Assume that (1 . holds for all | α | ≤ s and that (1.8) sup ρ =1 ,...,T sup I ρ + ··· + I ρG ( ρ ) = { ,...,m } inf l ∈ I tg p l < ∞ . GRAFAKOS, NAKAMURA, NGUYEN, AND SAWANO
Then T σ can be extended to a bounded map from H p × · · · × H p m to H p . Remark 1.4. (1) In Theorem 1.2, we exclude the case p l = ∞ for all l = 1 , . . . , m . In fact, onecan not expect the mapping property of T σ with (1.4) if p l = ∞ for some l = 1 , . . . , m . Similarly,in Theorem 1.3, we need to assume (1.8) instead of the exclusion of the case p l = ∞ for some l = 1 , . . . , m .(2) The convergence of the integral in (1.7) is a consequence of Lemma 3.1 for all x outside theunion of a fixed multiple of the supports of a i , while the function T ( a , . . . , a m ) is integrable for x inside any compact set.A few comments about the notation. For brevity we write d~y = dy · · · dy m and we use thesymbol C to denote a nonessential constant whose value may vary at different occurrences. For( k , . . . , k m ) ∈ Z m , we write ~k = ( k , . . . , k m ). We use the notation A . B to indicate that A ≤ C B for some constant C . We denote the Hardy-Littlewood maximal operator by M :(1.9) M f ( x ) = sup r> r n Z B ( x,r ) | f ( y ) | dy. We say that A ≈ B if both A . B and B . A hold. The cardinality of a finite set J is denoted byeither | J | or ♯J .A cube Q in R n has sides parallel to the axes. We denote by Q ∗ a centered-dilated cube of anycube Q with the length scale factor 3 √ n ; then(1.10) Q ∗ = 3 √ nQ ∗ , Q ∗∗ = 9 nQ. Preliminary and related results
Equivalent definitions of Hardy spaces.
We begin this section by recalling Hardy spaces.Let φ ∈ C ∞ c satisfy(2.1) supp( φ ) ⊂ { x ∈ R n : | x | ≤ } and(2.2) Z R n φ ( y ) dy = 1 . For t >
0, we set φ t ( x ) = t − n φ ( t − x ). The maximal function M φ associated with the smooth bump φ is given by:(2.3) M φ ( f )( x ) = sup t> (cid:12)(cid:12) ( φ t ∗ f )( x ) (cid:12)(cid:12) = sup t> (cid:12)(cid:12)(cid:12) t − n Z R n φ (cid:0) y/t (cid:1) f ( x − y ) dy (cid:12)(cid:12)(cid:12) for f ∈ S ′ ( R n ). For 0 < p < ∞ , the Hardy space H p is characterized as the space of all tempereddistributions f for which M φ ( f ) ∈ L p ; also the H p quasinorm satisfies k f k H p ≈ k M φ ( f ) k L p . Denote by C ∞ c the space of all smooth functions on R n with compact support. The followingdensity property of Hardy spaces will be useful in the proof of the main theorems. Proposition 2.1 ([17, Chapter III, 5.2(b)]) . Let N ≫ s be fixed. Then the following space is densein H p : O N ( R n ) = \ α ∈ N n , | α |≤ N (cid:26) f ∈ C ∞ c : Z R n x α f ( x ) dx = 0 (cid:27) , where C ∞ c is the space of all smooth functions with compact supports in R n . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 5
The definition of the Hardy space is useful as the following theorem implies:
Theorem 2.2 ([15]) . Let < p < ∞ .If f ∈ H p , then there exist a collection of ( p, ∞ ) -atoms { a k } ∞ k =1 and a nonnegative sequence { λ k } ∞ k =1 such that f = ∞ X k =1 λ k a k in S ′ ( R n ) and that we have (cid:13)(cid:13)(cid:13) ∞ X k =1 λ k χ Q k (cid:13)(cid:13)(cid:13) L p . k f k H p . Moreover, if f ∈ C ∞ c and Z R n x α f ( x ) dx = 0 for all α with | α | ≤ [ n (1 /p − + , then we canarrange that λ k = 0 for all but finitely many k . The following lemma, whose proof is just an application of the Fefferman-Stein vector-valuedinequality for maximal function, will be used frequently in the next sections.
Lemma 2.3. If γ > max(1 , p ) , < p < ∞ , λ k ≥ and { Q k } k are sequence of cubes, then (cid:13)(cid:13)(cid:13) X k λ k ( M χ Q k ) γ (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) X k λ k χ Q k (cid:13)(cid:13)(cid:13) L p . In particular (cid:13)(cid:13)(cid:13) X k λ k χ Q ∗∗ k (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) X k λ k χ Q k (cid:13)(cid:13)(cid:13) L p . We will also make use of the following result:
Lemma 2.4.
Let p ∈ (0 , ∞ ) . Assume that q ∈ ( p, ∞ ] ∩ [1 , ∞ ] . Suppose that we are given a sequenceof cubes { Q j } ∞ j =1 and a sequence of non-negative L q -functions { F j } ∞ j =1 . Then (cid:13)(cid:13)(cid:13) ∞ X j =1 χ Q j F j (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) ∞ X j =1 | Q j | Z Q j F j ( y ) q dy ! /q χ Q j (cid:13)(cid:13)(cid:13) L p . Proof.
See [13] for the case of 0 < p ≤ < p < ∞ . (cid:3) Reductions in the proof of main results.
To start the proof of the main results, let p , . . . , p m and p be given as in Theorems 1.1, 1.2 or 1.3 and note that H p l ∩ O N ( R n ) is dense in H p l for 1 ≤ l ≤ m and 0 < p l < ∞ . Recall the integer N ≫ s and fix f l ∈ H p l ∩ O N ( R n ) forwhich 0 < p l < ∞ . By Theorem 2.2, we can decompose f l = P ∞ k l =1 λ l,k l a l,k l , where { λ l,k l } ∞ k l =1 is a non-negative finite sequence and { a l,k l } ∞ k l =1 is a sequence of ( p l , ∞ )-atoms such that a l,k l issupported in a cube Q l,k l satisfying | a l,k l | ≤ χ Q l,kl , Z R n x α a l,k l ( x ) dx = 0 , | α | ≤ N and that(2.4) (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13) L pl . k f l k H pl . GRAFAKOS, NAKAMURA, NGUYEN, AND SAWANO If p l = ∞ and f l ∈ L ∞ , then we can conventionally rewrite f l = λ l,k l a l,k l where λ l,k l = k f l k L ∞ and a l,k l = k f k − L ∞ f is an ( ∞ , ∞ )-atom supported in Q l,k l = R n . In this case the summation in (2.4) isignored since there is only one summand.By the multi-sublinearity of M φ ◦ T σ , we can estimate M φ ◦ T σ ( f , . . . , f m )( x ) ≤ ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m )( x ) . To prove Theorems 1.1, 1.2, and 1.3, it now suffices to establish the following result:
Proposition 2.5.
Let T σ be the operator defined in (1.3) , (1.4) or (1.5) . Let p , . . . , p m and p begiven as in corresponding Theorems 1.1, 1.2 or 1.3. Then we have (2.5) (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) (cid:13)(cid:13)(cid:13) L p . m Y l =1 (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13) L pl . Notice that in view of (2.4) and Proposition 2.5, one obtains the required estimate kT σ ( f , . . . , f m ) k H p = k M φ ◦ T σ ( f , . . . , f m ) k L p . k f k H p · · · k f m k H pm . We may therefore focus on the proof of Proposition 2.5. In the sequel we will prove (2.5). Itsproof will depend on whether T σ is of type (1.3), (1.4) or (1.5). The detail proof for each type isdiscussed in subsequent sections.3. The Coifman-Meyer type
Throughout this section, T σ denotes for the operator defined in (1.3). The main purpose of thissection is to establish (2.5) for T σ .3.1. Fundamental estimates for the Coifman-Meyer type.
We treat the case of Coifman-Meyer multiplier operators whose symbols satisfy (1.2). The study of such operators was initiatedby Coifman and Meyer [2], [3] and was later pursued by Grafakos and Torres [11]; see also [7] foran account. Denoting by K the inverse Fourier transform of σ , in view of (1.2), we have | ∂ βy K ( y , . . . , y m ) | . (cid:0) m X i =1 | y i | (cid:1) − mn −| β | , ( y , . . . , , y m ) = (0 , . . . , β = ( β , . . . , β m ) ∈ N mn = ( N n ) m and | β | ≤ N .Examining carefully the smoothness of the kernel, we obtain the following estimates: Lemma 3.1.
Let a k be ( p k , ∞ ) -atoms supported in Q k for all ≤ k ≤ m . Let Λ be a non-emptysubset of { , . . . , m } . Then we have |T σ ( 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 for all y / ∈ ∪ k ∈ Λ Q ∗ k .Proof. We may suppose that Λ = { , . . . , r } for some 1 ≤ r ≤ m and that ℓ ( Q ) = min { ℓ ( Q k ) : k ∈ Λ } . Let c k be the center of Q k and fix y / ∈ ∪ k ∈ Λ Q ∗ k . Using the cancellation of a we can rewrite T σ ( a , . . . , a m )( y ) = Z R mn K ( y − y , . . . , y − y m ) a ( y ) · · · a m ( y m ) d~y ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 7 = Z R mn (cid:2) K ( y − 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, (3.1)where P N ( y, y , y , . . . , y m ) = X | α |≤ N α ! ∂ α K ( y − c , y − y , . . . , y − y m )( c − y ) α is the Taylor polynomial of degree N of K ( y − · , y − y , . . . , y − y m ) at c and(3.2) K ( y, y , . . . , y m ) = K ( y − 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 can estimate (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 σ ( 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 . (cid:3) Lemma 3.2.
Let a k be ( p k , ∞ ) -atoms supported in Q k for all ≤ k ≤ m . Suppose Q is the cubesuch that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } . Then for fixed ≤ r < ∞ and j ∈ N , we have kT σ ( a , . . . , a m ) χ Q ∗∗ k L r . | Q | r m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn , (3.3) k M ◦ T σ ( a , . . . , a m ) χ Q ∗∗ k L r . | Q | r m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn , (3.4) Furthermore, if Q is a cube such that ℓ ( Q ) ≤ ℓ ( Q ) and j Q ∗∗ ∩ j Q ∗∗ l = ∅ for some l , then (3.5) kT σ ( a , . . . , a m ) χ j Q ∗∗ k L ∞ . m Y l =1 inf z ∈ j Q ∗ M χ j Q ∗∗ l ( z ) n + N +1 mn . In particular, under the above assumption, (cid:16) | j Q ∗∗ | Z j Q ∗∗ |T σ ( a , . . . , a m )( y ) | r dy (cid:17) r . m Y l =1 inf z ∈ j Q ∗ M χ j Q ∗∗ l ( z ) n + N +1 mn . (3.6) GRAFAKOS, NAKAMURA, NGUYEN, AND SAWANO
Proof.
To check (3.3), it is enough to consider 1 < r < ∞ and two following cases. First, if Q ∗∗ ∩ Q ∗∗ k = ∅ for all 2 ≤ k ≤ m , then, by the assumption ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } , Q ∗∗ ⊂ Q ∗∗ k for all 1 ≤ k ≤ m . This impliesinf z ∈ Q ∗ M χ Q ∗∗ k ( z ) ≥ , for all 1 ≤ k ≤ m. Now the boundedness of T σ from L r × L ∞ × · · · × L ∞ to L r yields kT σ ( a , . . . , a m ) χ Q ∗∗ k L r ≤kT σ ( a , . . . , a m ) k L r . k a k L r k a k L ∞ · · · k a m k L ∞ . | Q | r m Y k =1 inf z ∈ Q ∗ M χ Q ∗∗ k ( z ) n + N +1 mn . (3.7)Second, if Q ∗∗ ∩ Q ∗∗ k = ∅ for some k , then the setΛ = { ≤ k ≤ m : Q ∗∗ ∩ Q ∗∗ k = ∅} is a non-empty subset of { , . . . , m } . Fix arbitrarily y ∈ R n . By the cancellation of a , rewrite T σ ( 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 in (3.2). For y ∈ Q we estimate (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 some ξ ∈ Q and for all y l ∈ Q l .Since Q ∗∗ ∩ Q ∗∗ k = ∅ for all k ∈ Λ, | y − ξ | + | y − y k | ≥ | ξ − y k | ≥ C | c − c k | for all y k ∈ Q k and k ∈ Λ. Therefore (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 − , for all y ∈ Q ∗ and y k ∈ Q k for k ∈ Λ. Insert the above inequality into (3.1) to obtain |T σ ( 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 P k ∈ Λ (cid:2) ℓ ( Q ) + | c − c k | + ℓ ( Q k ) (cid:3) n + N +1 . Noting that Q ∗∗ ⊂ Q ∗∗ l for l / ∈ Λ , the last inequality gives(3.8) kT σ ( a , . . . , a m ) k L ∞ . m Y k =1 inf z ∈ Q ∗ M χ Q ∗∗ k ( z ) n + N +1 mn , which yields(3.9) kT σ ( a , . . . , a m ) χ Q ∗∗ k L r . | Q | r m Y k =1 inf z ∈ Q ∗ M χ Q ∗∗ k ( z ) n + N +1 mn . Combining (3.7) and (3.9) and noting that
M χ Q . M χ Q , we obtain (3.3).Similarly, we can prove (3.4)–(3.5). For example, to show (3.4), we again consider the case where Q ∗∗ ∩ Q ∗∗ l = ∅ holds for all l and the case where this fails. In the first case, using the boundednessof M on L r , we arrive at the same situation as above. In the second case, we use the boundednessof M on L ∞ to see k M ◦ T σ ( a , . . . , a m ) χ Q ∗∗ k L r . | Q | r kT σ ( a , . . . , a m ) k L ∞ . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 9
Notice that the right-hand side is already treated in (3.8). (cid:3)
Lemma 3.2 will be used to study the behavior of the operator M φ ◦ T σ inside Q ∗∗ . For the regionoutside of Q ∗∗ , we need the following estimates. Lemma 3.3.
Let a k be ( p k , ∞ ) -atoms supported in Q k for all ≤ k ≤ m . If p k = ∞ then Q k = R n .Suppose that Q is the cube for which ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } . Fix < t < ∞ .(1) If x / ∈ Q ∗∗ and c / ∈ B ( x, n t ) , then (3.10) 1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . m Y l =1 M χ Q l ( x ) n + N +1 mn . (2) If x / ∈ Q ∗∗ and c ∈ B ( x, n t ) , then (3.11) ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn , and (3.12) 1 t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) N − smn . (3) For all x / ∈ Q ∗∗ , we have M φ ◦ T σ ( a , . . . , a m )( x ) . m Y l =1 M χ Q l ( x ) n + N +1 mn + M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ (cid:16) M χ Q l ( z ) N − smn (cid:17) . (3.13) Proof.
Fix x / ∈ Q ∗∗ and denote Λ = { ≤ k ≤ m : x / ∈ Q ∗∗ k } . (1) Suppose c / ∈ B ( x, n t ). For y ∈ B ( x, t ), from (3.1) we rewrite T σ ( a , . . . , a m )( y ) = Z R mn K ( y, y , . . . , y m ) a ( y ) · · · a m ( y m ) d~y, where K is defined in (3.2). Note that for y ∈ B ( x, t ), y ∈ Q and c / ∈ B ( x, n t ), we have t . | x − c | . | y − y | . Since x / ∈ Q ∗∗ k for all k ∈ Λ, | x − c k | . | x − y k | . t + | y − y k | . | y − y | + | y − y k | for all k ∈ Λ and y k ∈ Q k . Consequently, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( y, y , . . . , y m ) m Y l =1 a l ( y l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ℓ ( Q ) N +1 χ Q ( y ) m X l =2 | y − y l | + X k ∈ Λ | x − c k | ! mn + N +1 . (3.14)Integrating (3.14) over ( R n ) m , and using that ℓ ( Q ) ≤ ℓ ( Q l ) for all 2 ≤ l ≤ m , we obtain that |T σ ( a , . . . , a m )( y ) | . ℓ ( Q ) n + N +1 X l ∈ Λ | x − c l | ! n + N +1 . Y l ∈ Λ ℓ ( Q l ) n + N +1 | Λ | | x − c l | n + N +1 | Λ | χ ( Q ∗∗ l ) c ( x ) · Y k / ∈ Λ χ Q ∗∗ k ( x ) . m Y l =1 M χ Q l ( x ) n + N +1 mn . This pointwise estimate proves (3.10). (2)
Assume c ∈ B ( x, n t ). Fix 1 < r < ∞ and estimate the left-hand side of (3.11) by ℓ ( Q ) s +1 t n + s +1 | Q | − r kT σ ( a , . . . , a m ) χ Q ∗∗ k L r . ℓ ( Q ) n + s +1 t n + s +1 m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn , where we used (3.3) in the above inequality. Since x / ∈ Q ∗∗ and c ∈ B ( x, n t ), Q ∗ ⊂ B ( x, n t ) and hence, ℓ ( Q ) /t . M χ Q ( x ). This combined with the last inequality implies(3.11).To verify (3.12), we recall the expression of T σ ( a , . . . , a m )( y ) in (3.1) and the pointwise estimatefor K ( y, y , . . . , y m ) defined in (3.2). Denote J = { ≤ k ≤ m : Q ∗∗ ∩ Q ∗∗ k = ∅} . Using the factsthat | y − y | ∼ | y − c | ≥ ℓ ( Q ) for y / ∈ Q ∗ , y ∈ Q and | y − y | + | y − y l | ≥ | y − y l | & | z − c l | forall z ∈ Q ∗ and l ∈ J , we now estimate |T σ ( a , . . . , a m )( y ) | . Z ( R n ) m ℓ ( Q ) N +1 χ Q ( y ) d~y ℓ ( Q ) + | y − c | + X l ∈ J | z − c l | + m X l =2 | y − y l | ! mn + N +1 for all y ∈ ( Q ∗ ) c and z ∈ Q ∗ . Thus,1 t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . t n + s +1 Z R n × ( R n ) m | y − c | s +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 . (cid:18) ℓ ( Q ) t (cid:19) n + s +1 Y l ∈ J (cid:18) ℓ ( Q l ) | z − c l | (cid:19) N − sm . Note that 1 . inf z ∈ Q ∗ M χ Q ∗∗ l ( z ) if Q ∗∗ ∩ Q ∗∗ l = ∅ ; otherwise, ℓ ( Q l ) / | z − c l | . M χ Q ∗∗ l ( z ) n for all z ∈ Q ∗ . Consequently,1 t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) N − smn , which deduces (3.12). (3) It remains to prove (3.13). Fix x / ∈ Q ∗∗ . To calculate M φ ◦ T σ ( a , . . . , a m )( x ), we need toestimate (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) for each t ∈ (0 , ∞ ). Let consider two cases: c / ∈ B ( x, n t ) and c ∈ B ( x, n t ).In the first case, since φ is supported in the unit ball, (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy. Since c / ∈ B ( x, n t ), (3.10) implies that(3.15) (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . m Y l =1 M χ Q l ( x ) n + N +1 mn . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 11
In the second case, we will exploit the moment condition of T σ ( a , . . . , a m ). Denote(3.16) δ s ( t ; x, y ) = φ t ( x − y ) − X | α |≤ s ∂ α [ φ t ]( x − c ) α ! ( c − y ) α . Since | δ s ( t ; x, y ) | . t − n − s − for all x, y and (1.7), (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z R n δ s ( t ; x, y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . t n + s +1 Z R n | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy = 1 t n + s +1 Z Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy + 1 t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy (3.17) + 1 t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy. Invoking (3.11) and (3.12), we obtain (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ h M χ Q l ( z ) n + N +1 mn + M χ Q l ( z ) N − smn i . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ (cid:16) M χ Q l ( z ) N − smn (cid:17) . (3.18)Combining (3.15) and (3.18) yields the required estimate (3.13). The proof of Lemma 3.3 is nowcompleted. (cid:3) The proof of Proposition 2.5 for Coifman-Meyer type.
We now turn into the proof of(2.5), i.e., estimate(3.19) A = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) (cid:13)(cid:13)(cid:13) L p . For each ~k = ( k , . . . , k m ), we denote by R ~k the cube with smallest length among Q ,k , . . . , Q m,k m .Then we have A . B + G , where(3.20) B = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) χ R ∗∗ ~k (cid:13)(cid:13)(cid:13) L p and(3.21) G = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) χ ( R ∗∗ ~k ) c (cid:13)(cid:13)(cid:13) L p . To estimate B , for some max(1 , p ) < r < ∞ Lemma 2.4 and (3.4) imply B . (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) χ R ∗∗ ~k | χ R ∗∗ ~k | r k M φ ◦ T σ ( a ,k , . . . , a m,k m ) χ R ∗∗ ~k k L r (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17)(cid:16) m Y l =1 inf z ∈ R ∗ ~k M χ Q l,kl ( z ) n + N +1 mn (cid:17) χ R ∗∗ ~k (cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 inf z ∈ R ∗ ~k λ l,k l M χ Q l,kl ( z ) n + N +1 mn (cid:17) χ R ∗∗ ~k (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 inf z ∈ R ∗ ~k λ l,k l M χ Q l,kl ( z ) n + N +1 mn (cid:17) χ R ∗ ~k (cid:13)(cid:13)(cid:13) L p , where we used Lemma 2.3 in the last inequality. Now we can remove the infimum and applyH¨older’s inequality to obtain B . (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 m Y l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13) m Y l =1 ∞ X k l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:13)(cid:13)(cid:13) L p ≤ m Y l =1 (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:13)(cid:13)(cid:13) L pl . m Y l =1 (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q ∗∗ l,kl (cid:13)(cid:13)(cid:13) L pl . m Y l =1 (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13) L pl . (3.22)Once again, Lemma 2.3 was used in the last two inequalities.To deal with G , we use (3.13) and estimate G . G + G , where G = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) m Y l =1 (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:13)(cid:13)(cid:13) L p and G = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17)(cid:16) m Y l =1 inf z ∈ R ∗ ~k M χ Q l,kl ( z ) N − smn (cid:17) ( M χ R ∗ ~k ) n + s +1 n (cid:13)(cid:13)(cid:13) L p . Repeating the argument in estimating for B , noting that ( n + s +1) pn > N ≫ s , we obtain(3.23) G . G + G . m Y l =1 (cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13) L pl . Combining (3.22) and (3.23) deduces (2.5). This completes the proof of Proposition 2.5 for theoperator T σ of type (1.3). Remark 3.4.
The techniques in this paper also work for CZ operators of non-convolution types;this recovers the results in [13]. 4.
The product type
On this whole section, we denote by T σ the operator defined in (1.4) and prove Proposition 2.5for this operator. Now we need to establish some results analogous to Lemmas 3.2 and 3.3. ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 13
Fundamental estimates for the product type.
Let a k be ( p k , ∞ )-atoms supported in Q k for all 1 ≤ k ≤ m . Here and below M ( r ) denotes the power-maximal operator: M ( r ) f ( x ) = M ( | f | r )( x ) r . Suppose Q is the cube such that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } , then we havethe following lemmas. Lemma 4.1.
For all x ∈ Q ∗∗ , we have M φ ◦ T σ ( a , . . . , a m )( x ) χ Q ∗∗ ( x ) . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) . (4.1) Proof.
Fix x ∈ Q ∗∗ . We need to estimate (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy . t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy for each t ∈ (0 , ∞ ). The proof of (4.1) is mainly based on the boundedness of T σ and the smoothnesscondition of each Calder´on-Zygmund kernel in (1.4). Instead of considering the whole sum in (1.4),for notational simplicity, it is convenient to consider one term, i.e.,(4.2) T σ ( f , . . . , f m ) = T σ ( f ) · · · T σ m ( f m )except when cancellation is used, when the entire sum is needed. We consider two cases: t ≤ ℓ ( Q )and t > ℓ ( Q ). Case 1: t ≤ ℓ ( Q ). By H¨older inequality and (3.6), we have1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . m Y l =1 (cid:16) t n Z B ( x,t ) | T σ l a l ( y ) | m dy (cid:17) m . Now, we decompose the above product depending on two sub-cases; B ( t, x ) ∩ Q ∗∗ l = ∅ or not. Then m Y l =1 (cid:16) t n Z B ( x,t ) | T σ l a l ( y ) | m dy (cid:17) m = Y l : B ( t,x ) ∩ Q ∗∗ l = ∅ (cid:16) t n Z B ( x,t ) | T σ l a l ( y ) | m dy (cid:17) m Y l : B ( t,x ) ∩ Q ∗∗ l = ∅ (cid:16) t n Z B ( x,t ) | T σ l a l ( y ) | m dy (cid:17) m . For the first sub-case, we employ (3.6). For the second sub-case, we observe that the assumption t ≤ ℓ ( Q ) ≤ ℓ ( Q l ) imply B ( x, t ) ⊂ Q ∗∗ l . As a result, m Y l =1 (cid:16) t n Z B ( x,t ) | T σ l a l ( y ) | m dy (cid:17) m . Y l : B ( x,t ) ∩ Q ∗∗ l = ∅ χ Q ∗∗ l ( x ) M ( m ) ◦ T σ l ( a l )( x ) Y l : B ( x,t ) ∩ Q ∗∗ l = ∅ M χ Q l ( x ) n + N +1 mn . Thus(4.3) (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) . Case 2: t > ℓ ( Q ). Now we can estimate1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . | Q ∗ | Z R n |T σ ( a , . . . , a m )( y ) | dy = 1 | Q ∗ | Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy + 1 | Q ∗ | Z R n \ Q ∗ |T σ ( a , . . . , a m )( y ) | dy. By the H¨older inequality and (3.6), the similar technique to (4.3) yields1 | Q ∗ | Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . m Y l =1 (cid:16) | Q ∗ | Z Q ∗ | T σ l a l ( y ) | m dy (cid:17) m . m Y l =1 (cid:16) inf z ∈ Q ∗ M χ Q ∗∗ l ( z ) n + N +1 mn + inf z ∈ Q ∗ M ( m ) ◦ T σ l ( a l )( z ) χ Q ∗∗ l ( z ) (cid:17) . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) , (4.4)since x ∈ Q ∗ . For the second term, using the decay of T σ a ( y ) when y / ∈ Q ∗ as in Lemma 3.1, weobtain 1 | Q ∗ | Z R n \ Q ∗ |T σ ( a , . . . , a m )( y ) | dy . | Q ∗ | Z R n \ Q ∗ ℓ ( Q ) n + N +1 | y − c | n + N +1 m Y l =2 | T σ l a l ( y ) | dy. We decompose R n \ Q ∗ into dyadic annuli and estimate1 | Q ∗ | Z R n \ Q ∗ |T σ ( a , . . . , a m )( y ) | dy . ∞ X j =1 j ( − N − | j Q ∗ | Z j Q ∗ χ j Q ∗ ( y ) m Y l =2 | T σ l a l ( y ) | dy . ∞ X j =1 j ( − N − m Y l =2 (cid:16) | j Q ∗ | Z j Q ∗ | T σ l a l ( y ) | m dy (cid:17) m . ∞ X j =1 j ( − N − m Y l =2 (cid:16) inf z ∈ j Q ∗ ( M χ j Q ∗∗ l )( z ) n + N +1 mn + inf z ∈ j Q ∗ M ( m ) ◦ T σ l ( a l )( z ) χ j +1 Q ∗∗ l ( z ) (cid:17) , where we used (3.6) in the last inequality.Since M χ j Q . jn M χ Q , χ j +1 Q ∗∗ l ( x ) ≤ ( M χ j Q ∗∗ l ) n + N +1 mn . j ( n + N +1) m M χ n + N +1 mn Q l . Insert this inequality into the previous estimate to obtain1 | Q ∗ | Z R n \ Q ∗ |T σ ( a , . . . , a m )( y ) | dy . ∞ X j =1 − j ( n + N +1 m − n ) m Y l =1 (cid:16) M χ Q l ( x ) n + N +1 mn + M ( m ) ◦ T σ l ( a l )( x ) M χ Q l ( x ) n + N +1 mn (cid:17) . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) , (4.5)since N ≫ n . Combining (4.3)–(4.5) together completes the proof of (4.1). (cid:3) ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 15
Lemma 4.2.
Assume x / ∈ Q ∗∗ and c / ∈ B ( x, n t ) . Then we have t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) . Proof.
Fix any x / ∈ Q ∗∗ and t > c / ∈ B ( x, n t ). We denote(4.6) J = { ≤ l ≤ m : x / ∈ Q ∗∗ l } , J = { l ∈ J : B ( x, t ) ∩ Q ∗ l = ∅} , J = J \ J . Similar to the previous lemma, it is enough to consider the reduced form (4.2) of T σ . From theH¨older inequality, we have1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . k T σ a χ B ( x,t ) k L ∞ Y l ∈ J k T σ l a l χ B ( x,t ) k L ∞ × Y l ∈ J | B ( x, t ) | Z B ( x,t ) | T σ l a l ( y ) | m dy ! m Y l / ∈ J | B ( x, t ) | Z B ( x,t ) | T σ l a l ( y ) | m dy ! m =: I × II × III × IV . For I, we notice Q ∗ ∩ B ( x, t ) = ∅ since we have x / ∈ Q ∗∗ and c / ∈ B ( x, n t ). So, we have onlyto use the decay estimate for T σ a to getI = k T σ a χ B ( x,t ) k L ∞ . (cid:18) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:19) n + N +1 . For all l ∈ J , since B ( x, t ) ∩ Q ∗ l = ∅ , t & ℓ ( Q l ); and hence, Q ∗ l ⊂ B ( x, n t ). Therefore,(4.7) | B ( x, t ) | Z B ( x,t ) | T σ l a l ( y ) | m dy ! m . (cid:18) | Q l || B ( x, t ) | (cid:19) m . . for all l ∈ J . Now combining the above inequality with the estimates for I yields(4.8) I × III . (cid:18) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:19) n + N +1 m Y l ∈ J (cid:18) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:19) n + N +1 m . As showed about Q ∗ l ⊂ B ( x, n t ) for all l ∈ J . This implies | x − c l | . t . Furthermore, c / ∈ B ( x, n t ) means t . | x − c | which yields | x − c l | . t . | x − c | .Recalling ℓ ( Q ) ≤ ℓ ( Q l ), we see that ℓ ( Q ) | x − c | + ℓ ( Q ) . ℓ ( Q l ) | x − c l | + ℓ ( Q l ) . From (4.8), we obtain I × III . M χ Q ∗∗ ( x ) n + N +1 mn Y l ∈ J M χ Q l ( x ) n + N +1 mn . (4.9)Now, we turn to the estimate for II and IV. For II, we have only to employ the moment conditionof a l to get II = Y l ∈ J k T σ l a l · χ B ( x,t ) k L ∞ . Y l ∈ J M χ Q l ( x ) n + N +1 n . (4.10) For IV, since x ∈ Q ∗∗ l , we can estimateIV . Y l / ∈ J M ( m ) ◦ T σ l ( a l )( x ) χ Q ∗∗ l ( x )(4.11)Putting (4.9)–(4.11) together, we conclude the proof of Lemma 4.2. (cid:3) Lemma 4.3.
Assume x / ∈ Q ∗∗ and c ∈ B ( x, n t ) . Then we have ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . (4.12) Proof.
It is enough to restrict T σ to the form (4.2). By the H¨older inequality we have ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy ≤ ℓ ( Q ) n + s +1 t n + s +1 m Y l =1 (cid:16) | Q ∗ | Z Q ∗ | T σ l a l ( y ) | m dy (cid:17) m . ℓ ( Q ) n + s +1 t n + s +1 m Y l =1 (cid:16) inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 n + inf z ∈ Q ∗ M ( m ) ◦ T σ l ( a l )( z ) χ Q ∗∗ l ( z ) (cid:17) , where the last inequality is deduced from (3.6).Since x / ∈ Q ∗∗ and c ∈ B ( x, n t ), Q ⊂ B ( x, n t ) which implies ℓ ( Q ) /t . M χ Q ( x ) n .As a result, ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . This proves (4.12). (cid:3)
Lemma 4.4.
Assume x / ∈ Q ∗∗ and c ∈ B ( x, n t ) . Then we have t n + s +1 Z R n \ Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . Proof.
Using the decay of T σ a ( y ) when y / ∈ Q ∗ , we obtain1 t n + s +1 Z R n \ Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . t n + s +1 Z R n \ Q ∗ | y − c | s +1 ℓ ( Q ) n + N +1 | y − c | n + N +1 m Y l =2 | T σ l a l ( y ) | dy. ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 17
By dyadic decomposition of R n \ Q ∗ as in the proof of Lemma 4.1, we can estimate1 t n + s +1 Z R n \ Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . ℓ ( Q ) s +1 t n + s +1 ∞ X j =1 j ( s − N − n ) Z j Q ∗ χ j Q ∗ ( y ) m Y l =2 | T σ l a l ( y ) | dy . ℓ ( Q ) n + s +1 t n + s +1 ∞ X j =1 j ( s − N ) m Y l =2 (cid:16) | j Q ∗ | Z j Q ∗ | T σ l a l ( y ) | m dy (cid:17) m . ℓ ( Q ) n + s +1 t n + s +1 ∞ X j =1 j ( s − N ) m Y l =2 (cid:16) inf z ∈ j Q ∗ M χ j Q ∗∗ l ( z ) n + N +1 mn + inf z ∈ j Q ∗ M ( m ) ◦ T σ l ( a l )( z ) χ j +1 Q ∗∗ l ( z ) (cid:17) , where we used (3.6) in the last inequality.We now repeat the argument in establishing (4.5) to obtain1 t n + s +1 Z R n \ Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . ℓ ( Q ) n + s +1 t n + s +1 m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . Moreover, the assumption x / ∈ Q ∗∗ and c ∈ B ( x, n t ) implies ℓ ( Q ) t . M χ Q ( x ) n . Therefore,1 t n + s +1 Z R n \ Q ∗ | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . This proves Lemma 4.4. (cid:3)
Lemma 4.5.
For all x ∈ R n , we have M φ ◦ T σ ( a , . . . , a m )( x ) . m Y l =1 M χ Q l ( x ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( x ) (cid:17) + M χ Q ( x ) n + s +1 n m Y l =1 inf z ∈ Q ∗ M χ Q l ( z ) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l )( z ) (cid:17) . Proof. If x ∈ Q ∗∗ , the desired estimate is a consequence of Lemma 4.1. Fix x / ∈ Q ∗∗ . To estimate M φ ◦ T σ ( a , . . . , a m )( x ), we need to examine (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) for each t ∈ (0 , ∞ ). If c / ∈ B ( x, n t ), then we make use of Lemma 4.2; otherwise, when c ∈ B ( x, n t ) we recall (3.17) and then apply Lemma 4.3 and 4.4 to obtain the requiredestimate in Lemma 4.5. This completes the proof of the lemma. (cid:3) The proof of Proposition 2.5 for the product type.
To process the proof of (2.5), weset A = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) (cid:13)(cid:13)(cid:13) L p . For each ~k = ( k , . . . , k m ), we recall R ~k , the smallest-length cube among Q ,k , . . . , Q m,k m .In view of Lemma 4.5, we have(4.13) A . B := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 m Y l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l,k l ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . In fact, our assumption imposing on s means ( n + s + 1) p/n > M to obtain A . B + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) M χ R ∗ ~k (cid:17) n + s +1 n m Y l =1 λ l,k l inf z ∈ R ∗ ~k (cid:18) M χ Q l,kl (cid:19) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l,k l ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . B. So, our task is to estimate B . Here, we prepare the following lemma. Lemma 4.6.
Let p ∈ (0 , ∞ ) and α > max (1 , p − ) . Assume that q ∈ ( p, ∞ ] ∩ [1 , ∞ ] . Suppose thatwe are given a sequence of cubes { Q k } ∞ k =1 and a sequence of non-negative L q -functions { F k } ∞ k =1 .Then (cid:13)(cid:13)(cid:13) ∞ X k =1 ( M χ Q k ) α F k (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) ∞ X k =1 χ Q k M ( q ) F k (cid:13)(cid:13)(cid:13) L p . Proof.
By Lemma 2.4 and the fact that
M χ Q . χ Q + P ∞ j =1 − jn χ j Q \ j − Q , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 ( M χ Q k ) α F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =0 ∞ X k =1 − αjn χ j Q k F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =0 ∞ X k =1 − αjn χ j Q k (cid:18) | j Q k | Z j Q k F k ( y ) q dy (cid:19) q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . Choose α > β > max(1 , p ) and observe the trivial estimate χ j Q k . (cid:0) jn M χ Q k (cid:1) β . Now, Lemma 2.3 gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 ( M χ Q k ) α F k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =0 ∞ X k =1 χ Q k ( β − α ) jqn | j Q k | Z j Q k F k ( y ) q dy ! q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13) ∞ X k =1 χ Q k M ( q ) F k (cid:13)(cid:13)(cid:13) L p , which yields the desired estimate. (cid:3) Lemma 4.6 can be regard as a substitution of Lemma 2.4.Before applying Lemma 4.6 to B , we observe B ≤ m Y l =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l,k l ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 19
Then applying Fefferman-Stein’s vector-valued inequality and Lemma 4.6, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn (cid:16) M ( m ) ◦ T σ l ( a l,k l ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l M (cid:0) χ Q ∗∗ l,kl (cid:1) n + N +1 mn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l (cid:16) M χ Q l,kl (cid:17) n + N +1 mn M ( m ) ◦ T σ l ( a l,k l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q ∗∗ l,kl M ◦ M ( m ) ◦ T σ l ( a l,k l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . For the second term, we choose q ∈ ( m, ∞ ) and employ Lemma 2.4, and the boundedness of M and T σ l to have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q ∗∗ l,kl M ◦ M ( m ) ◦ T σ l ( a l,k l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q ∗∗ l,kl | Q l,k l | q (cid:13)(cid:13)(cid:13) M ◦ M ( m ) ◦ T σ l ( a l,k l ) (cid:13)(cid:13)(cid:13) qL q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . As a result, A . B . m Y l =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l,kl (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl , which completes the proof of Proposition 2.5.5. The mixed type
In this section, we prove Proposition 2.5 for operators of type (1.5). The main techniques todeal with the operator T σ of mixed type are combinations of two previous types. We now establishsome necessary estimates for T σ . For the mixed type, we need the following lemma which can beshown by a way similar to that in Lemma 3.1. Lemma 5.1.
Let σ be a Coifman-Meyer multiplier, a l be ( p l , ∞ ) -atoms supported on Q l for ≤ l ≤ m . Assume ℓ ( Q ) = min { ℓ ( Q l ) : l = 1 , . . . , m } and write Λ j = { l = 1 , . . . , m : 2 j Q ∗∗ ∩ j Q ∗∗ l = ∅} . The detailed proof is as follows. Fix any y ∈ j +1 Q ∗ \ j Q ∗ . Let us use a notation K ( y, y , . . . , y m ) as in theproof of Lemma 3.1. Then for any y l ∈ Q l , l = 1 , . . . , m , we have | K ( y, y , . . . , y m ) | . ℓ ( Q ) | y − y | + P l ∈ Λ j | y − y l | + P l ≥ | y − y l | ! n + N +1 . ℓ ( Q ) | y − c | + P l ∈ Λ j | y − c l | + P l ≥ | y − y l | ! n + N +1 . In fact, if l ∈ Λ j , 2 j Q ∗∗ ∩ j Q ∗∗ l = ∅ and hence, y ∈ j +1 Q ∗ means | y − y l | ∼ | y − c l | for all y l ∈ Q l for such l . Ofcourse, | y − y | ∼ | y − c | is clear since y / ∈ j Q ∗ . Using this kernel estimate, we may prove the desired estimate. Then for any y ∈ j +1 Q ∗ \ j Q ∗ we have | T σ ( a , . . . , a m )( y ) | . ℓ ( Q ) | y − c | + P l ∈ Λ j | y − c l | ! n + N +1 . Fundamental estimates for the mixed type.
Let a k be ( p k , ∞ )-atoms supported in Q k for all 1 ≤ k ≤ m . Suppose Q is the cube such that ℓ ( Q ) = min { ℓ ( Q k ) : 1 ≤ k ≤ m } . For each1 ≤ g ≤ G , let Q l ( g ) be the smallest cube among { Q l } l ∈ I g and let m g = | I g | be the cardinality of I g . Then we have the following analogues to Lemmas 4.1–4.5. We write m g = ♯I g for each g . Lemma 5.2.
For all x ∈ Q ∗∗ , we have M φ ◦ T σ ( a , . . . , a m )( x ) χ Q ∗∗ ( x ) . G Y g =1 M χ Q l ( g ) ( x ) ( n + N +1) mgnm M ( G ) ◦ T σ Ig ( { a l,k l } l ∈ I g )( x ) + Y l ∈ I g M χ Q l ( x ) n + N +1 mn . (5.1) Proof.
Fix x ∈ Q ∗∗ . We need to estimate (cid:12)(cid:12)(cid:12) Z R n φ t ( x − y ) T σ ( a , . . . , a m )( y ) dy (cid:12)(cid:12)(cid:12) . t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy for each t ∈ (0 , ∞ ). Similar to the previous section, it is enough to consider the following form:(5.2) T σ ( f , . . . , f m ) = G Y g =1 T σ Ig ( { f l } l ∈ I g ) , where { I g } Gg =1 is a partition of { , . . . , m } with 1 ∈ I . By the H¨older inequality, we have(5.3) 1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . G Y g =1 (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G . For each 1 ≤ g ≤ G , we need to examine (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G . We consider two cases as in the proof of Lemma 4.1.
Case 1: t ≤ ℓ ( Q ). We observe that1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy ≤ Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G × Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G . When B ( x, t ) ∩ Q ∗∗ l ( g ) = ∅ , we see that x ∈ Q ∗∗ l ( g ) . This shows Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G (5.4) . Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ χ Q ∗∗ l ( g ) ( x ) M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 21
When B ( x, t ) ∩ Q ∗∗ l ( g ) = ∅ , we may use (3.6) to have(5.5) Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ (cid:16) t n Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy (cid:17) G . Y g : B ( x,t ) ∩ Q ∗∗ l ( g ) = ∅ Y l ∈ I g M χ Q l ( x ) n + N +1 mn . These two estimates (5.4) and (5.5) yield the desired estimate in the Case 1.
Case 2: t > ℓ ( Q ). We split1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . | Q ∗ | Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy + 1 | Q ∗ | Z ( Q ∗ ) c |T σ ( a , . . . , a m )( y ) | dy. For the first term, (5.5) yields1 | Q ∗ | Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . G Y g =1 Y l ∈ I g M χ Q l ( x ) n + N +1 mn + χ Q ∗∗ l ( g ) ( x ) M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . For the second term, by a dyadic decomposition of ( Q ∗ ) c ,1 | Q ∗ | Z ( Q ∗ ) c |T σ ( a , . . . , a m )( y ) | dy = ∞ X j =0 | Q ∗ | Z j +1 Q ∗ \ j Q ∗ | T σ I ( { a l } l ∈ I )( y ) | Y g ≥ | T σ Ig ( { a l } l ∈ I g )( y ) | dy = ∞ X j =0 I j . Now, we fix any j and evaluate each I j . Letting Λ j = { l = 1 , . . . , m : 2 j Q ∗∗ ∩ j Q ∗∗ l = ∅} and usingLemma 5.1, for y ∈ j +1 Q ∗ \ j Q ∗ we obtain | T σ I ( { a l } l ∈ I )( y ) | . ℓ ( Q ) | y − c | + P l ∈ I ∩ Λ j | y − c l | ! n + N +1 . − j ( n + N +1) j ℓ ( Q )2 j ℓ ( Q ) + P l ∈ I ∩ Λ j | c − c l | ! n + N +1 . We estimate this term further. If l ∈ I ∩ Λ j , | c − c l | ∼ | x − c l | since x ∈ Q ∗∗ . On the other hand,if l ∈ I \ Λ j , χ j Q ∗∗ l ( x ) = χ Q ∗∗ ( x ) = 1 since x ∈ Q ∗∗ . So, we have | T σ I ( { a l } l ∈ I )( y ) | . − j ( n + N +1) Y l ∈ I ∩ Λ j (cid:18) j ℓ ( Q l ) | x − c l | (cid:19) n + N +1 m Y l ∈ I \ Λ j χ j Q ∗∗ l ( x ) . − j ( n + N +1) Y l ∈ I \{ } M χ j Q ∗∗ l ( x ) n + N +1 mn . − j ( n + N +1) j n + N +1 m ( m − Y l ∈ I M χ Q l ( x ) n + N +1 mn . This and H¨older’s inequality imply that(5.6) I j . − j ( N +1) j n + N +1 m ( m − Y l ∈ I M χ Q l ( x ) n + N +1 mn Y g ≥ | j Q ∗ | Z j Q ∗ | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G . In the usual way, we claim that Y g ≥ | j Q ∗ | Z j Q ∗ | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G (5.7) . j n + N +1 m ( m − m ) Y g ≥ M χ Q l ( g ) ( x ) ( n + N +1) mgnm M ( G ) ◦ T σ Ig ( { a l,k l } l ∈ I g )( x ) + Y l ∈ I g M χ Q l ( x ) n + N +1 mn . To see this, we again consider two possibilities of g for each j ; 2 j Q ∗∗ ∩ j Q ∗∗ l ( g ) = ∅ or not. In thefirst case, we notice x ∈ Q ∗ ⊂ j Q ∗∗ l ( g ) and that we defined m g = ♯I g , and hence | j Q ∗ | Z j Q ∗ | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G . χ j Q ∗∗ l ( g ) ( x ) M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . j mg ( n + N +1) m M χ Q l ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . In the second case; 2 j Q ∗∗ ∩ j Q ∗∗ l ( g ) = ∅ , we use (3.6) to see | j Q ∗ | Z j Q ∗ | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G . Y l ∈ I g M χ j Q ∗∗ l ( x ) n + N +1 mn . j mg ( n + N +1) m Y l ∈ I g M χ Q l ( x ) n + N +1 mn . These two estimates yields (5.7). Inserting (5.7) to (5.6), we arrive at I j . − j ( n + N +1 m − n ) G Y g =1 M χ Q l ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) + Y l ∈ I g M χ Q l ( x ) n + N +1 mn . Taking N sufficiently large, we can sum the above estimate up over j ∈ N and get desired estimate.This completes the proof of Lemma 5.2 (cid:3) Lemma 5.3.
Assume x / ∈ Q ∗∗ and c / ∈ B ( x, n t ) . Then we have t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . G Y g =1 M χ Q l ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) + Y l ∈ I g M χ Q l ( x ) n + N +1 mn . ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 23
Proof.
Fix x / ∈ Q ∗∗ and t > c / ∈ B ( x, n t ) . Let T σ be the operator of type (1.5).We may consider the reduced form (5.2) of T σ and start from (5.3). We define J = { g = 2 , . . . , m : x / ∈ Q ∗∗ l ( g ) } , J = { g ∈ J : B ( x, t ) ∩ Q ∗ l ( g ) = ∅} , J = J \ J and split the product as follows:1 t n Z B ( x,t ) |T σ ( a , . . . , a m )( y ) | dy . (cid:13)(cid:13)(cid:13) T σ I ( { a l } l ∈ I ) χ B ( x,t ) (cid:13)(cid:13)(cid:13) L ∞ Y l ∈ J (cid:13)(cid:13)(cid:13) T σ Ig ( { a l } l ∈ I g ) χ B ( x,t ) (cid:13)(cid:13)(cid:13) L ∞ × Y g ∈ J | B ( x, t ) | Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G × Y g ∈{ ,...,G }\ J | B ( x, t ) | Z B ( x,t ) | T σ Ig ( { a l } l ∈ I g )( y ) | G dy ! G = I × II × III × IV . To estimate I, we further define the partition of I : I = { l ∈ I : x / ∈ Q ∗∗ l , B ( x, t ) ∩ Q ∗ l = ∅} , I = { l ∈ I : x / ∈ Q ∗∗ l , B ( x, t ) ∩ Q ∗ l = ∅} ,I = I \ ( I ∪ I ) . Since x / ∈ Q ∗∗ and c / ∈ B ( x, n t ), we can see that 1 ∈ I . From Lemma 3.1, we deduce | T σ I ( { a l } l ∈ I )( y ) | . ℓ ( Q ) n + N +1 ( P l ∈ I | y − c l | ) n + N +1 . ℓ ( Q ) n + N +1 ( P l ∈ I | x − c l | ) n + N +1 . (cid:16) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:17) ( m − m ) n + N +1 m Y l ∈ I (cid:16) ℓ ( Q l ) | x − c l | + ℓ ( Q l ) (cid:17) n + N +1 m Y l ∈ I (cid:16) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:17) n + N +1 m for all y ∈ B ( x, t ), where m = | I | is the cardinality of the set I . As in the proof of Lemma 4.2for the product type, if x / ∈ Q ∗∗ l and B ( x, t ) ∩ Q ∗ l = ∅ then | x − c l | . t . | x − c | . This observationimplies ℓ ( Q ) | x − c | + ℓ ( Q ) . ℓ ( Q l ) | x − c l | + ℓ ( Q l )for all l ∈ I . Therefore, we can estimate | T σ I ( { a l } l ∈ I )( y ) | . (cid:16) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:17) ( m − m ) n + N +1 m Y l ∈ I ∪ I M χ Q l ( x ) n + N +1 mn for all y ∈ B ( x, t ). Obviously, 1 . M χ Q l ( x ) for all l ∈ I , and hence we have(5.8) | T σ I ( { a l } l ∈ I )( y ) | . (cid:16) ℓ ( Q ) | x − c | + ℓ ( Q ) (cid:17) ( m − m ) n + N +1 m Y l ∈ I M χ Q l ( x ) n + N +1 mn for all y ∈ B ( x, t ) which gives the estimate for I. For the third term III, we simply haveIII ≤ Y g ∈ J M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . So, we obtainI × III . Y l ∈ I M χ Q l ( x ) n + N +1 mn Y g ∈ J (cid:18) ℓ ( Q ) ℓ ( Q ) + | x − c | (cid:19) mg ( n + N +1) m M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . Y l ∈ I M χ Q l ( x ) n + N +1 mn Y g ∈ J M χ Q l ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) , since g ∈ J implies | x − c l ( g ) | . | x − c | . For the second term II, we use Lemma 5.6 and anargument as for estimate for I to getII = Y l ∈ J (cid:13)(cid:13)(cid:13) T σ Ig ( { a l } l ∈ I g ) χ B ( x,t ) (cid:13)(cid:13)(cid:13) L ∞ . Y g ∈ J Y l ∈ I g M χ Q l ( x ) n + N +1 mn . For the last term IV, we recall g / ∈ J means x ∈ Q ∗∗ l ( g ) and hence,IV . Y g / ∈ J M χ Q l ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( x ) . Combining the estimates for I, II, III and IV, we complete the proof of Lemma 5.3. (cid:3)
Lemma 5.4.
Assume x / ∈ Q ∗∗ and c ∈ B ( x, n t ) . Then we have ℓ ( Q ) s +1 t n + s +1 Z Q ∗ |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n G Y g =1 inf z ∈ Q ∗ M χ Q l ( g ) ( z ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( z ) + Y l ∈ I g M χ Q l ( z ) n + N +1 mn . Lemma 5.5.
Assume x / ∈ Q ∗∗ and c ∈ B ( x, n t ) . Then we have t n + s +1 Z ( Q ∗ ) c | y − c | s +1 |T σ ( a , . . . , a m )( y ) | dy . M χ Q ( x ) n + s +1 n G Y g =1 inf z ∈ Q ∗ M χ Q l ( g ) ( z ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l } l ∈ I g )( z ) + Y l ∈ I g M χ Q l ( z ) n + N +1 mn . The proof of Lemmas 5.4 and 5.5 are very similar to those of Lemma 5.2, so we omit the detailshere.5.2.
The proof of Proposition 2.5 for the mixed type.
Employing the above lemmas, wecomplete the proof of (2.5). For each ~k = ( k , . . . , k m ), recall the smallest-length cube R ~k among Q ,k , . . . , Q m,k m and write Q l ( g ) ,~k ( g ) for the cube of smallest-length among { Q l,k l } l ∈ I g . CombiningLemmas 5.2-5.5, we have the following pointwise estimate M φ ◦ T σ ( a ,k , . . . , a m,k m )( x ) . G Y g =1 b g,~k ( g ) ( x ) + M χ R ∗ ~k ( x ) n + s +1 n G Y g =1 inf z ∈ R ∗ ~k b g,~k ( g ) ( z ) ,b g,~k ( g ) ( x ) = M χ Q ∗ l ( g ) ,~k ( g ) ( x ) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l,k l } l ∈ I g )( x ) + Y l ∈ I g M χ Q l,kl ( x ) n + N +1 mn ONDITIONS FOR BOUNDEDNESS INTO HARDY SPACES 25 for all x ∈ R n . As in the proof for the product type, we let A = (cid:13)(cid:13)(cid:13) ∞ X k ,...,k m =1 (cid:16) m Y l =1 λ l,k l (cid:17) M φ ◦ T σ ( a ,k , . . . , a m,k m ) (cid:13)(cid:13)(cid:13) L p . In view of ( n + s + 1) p/n >
1, using Lemma 2.3 and H¨older’s inequality, we see A . G Y g =1 (cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g (cid:18) Y l ∈ I g λ l,k l (cid:19)(cid:18)(cid:16) M χ Q ∗ l ( g ) ,~k ( g ) (cid:17) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l,k l } l ∈ I g )+ Y l ∈ I g ( M χ Q l,kl ) n + N +1 mn (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L qg . G Y g =1 (cid:16) A g, + A g, (cid:17) , where q g ∈ (0 , ∞ ) is defined by 1 /q g = P l ∈ I g /p l and A g, = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g Y l ∈ I g λ l,k l (cid:16) M χ Q ∗ l ( g ) ,~k ( g ) (cid:17) mg ( n + N +1) mn M ( G ) ◦ T σ Ig ( { a l,k l } l ∈ I g ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qg ,A g, = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g Y l ∈ I g λ l,k l ( M χ Q l,kl ) n + N +1 mn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qg . For A g, , we have only to employ Lemma 2.3 to get the desired estimate. For A g, , take large r and employ Lemma 4.6 to obtain A g, . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g Y l ∈ I g λ l,k l χ Q ∗ l ( g ) ,~k ( g ) M ( r ) ◦ M ( G ) [ T σ Ig ( { a l,k l } l ∈ I g )] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qg . Then it follows from Lemma 2.4 and (3.4) that A g, . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g Y l ∈ I g λ l,k l χ Q ∗ l ( g ) ,~k ( g ) | Q l ( g ) ,~k ( g ) | /q (cid:13)(cid:13)(cid:13) χ Q ∗ l ( g ) ,~k ( g ) M ( r ) ◦ M ( G ) [ T σ Ig ( { a l,k l } l ∈ I g )] (cid:13)(cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qg . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k l ≥ l ∈ I g Y l ∈ I g λ l,k l χ Q ∗ l ( g ) ,~k ( g ) inf z ∈ Q ∗ l ( g ) ,~k ( g ) Y l ∈ I g M χ Q l ( z ) n + N +1 mn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qg ≤ Y l ∈ I g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l ( M χ Q l ) n + N +1 mn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl . Y l ∈ I g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k l =1 λ l,k l χ Q l (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pl , which completes the proof of Lemma 2.5 for for operators of mixed type.6. Examples
We provide some examples of operators of the kinds discussed in this paper: all of the followingare symbols of trilinear operators acting on functions on the real line, thus they are functions on R = R × R × R . The symbol σ ( ξ , ξ , ξ ) = ( ξ + ξ + ξ ) ξ + ξ + ξ is associated with an operator of type (1.3).The symbol σ ( ξ , ξ , ξ ) = ξ (1 + ξ ) ξ + ξ ) + 1(1 + ξ ) ξ (1 + ξ + ξ ) + 1(1 + ξ ) ξ (1 + ξ + ξ ) − ξ (1 + ξ ) ξ ξ (1 + ξ + ξ ) = ( ξ + ξ + ξ )( ξ + ξ + ξ − ξ ξ − ξ ξ − ξ ξ )(1 + ξ ) (1 + ξ + ξ ) provides an example of an operator of type (1.5). Note that each term is given as a product of amultiplier of ξ times a multiplier of ( ξ , ξ ).The symbol σ ( ξ , ξ , ξ ) = ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) − ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) − ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) + ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) + ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) − ξ (1 + ξ ) ξ (1 + ξ ) ξ (1 + ξ ) = − ξ ξ ξ ( ξ − ξ )( ξ − ξ )( ξ − ξ )( ξ + ξ + ξ )(1 + ξ ) (1 + ξ ) (1 + ξ ) yields an example of an operator of type (1.4). The next example: σ ( ξ , ξ , ξ ) = ξ ξ ξ + ξ + ( ξ + ξ ) · − ξ ξ ξ + ξ + ξ shows that the integer G ( ρ ) varies according to ρ . Notice that all four examples satisfy σ ( ξ , ξ , ξ ) = σ ( ξ , ξ , ξ ) = σ ( ξ , ξ , ξ ) = σ ( ξ , ξ , ξ ) = 0when ξ + ξ + ξ = 0. This yields condition (1.7) when s = 0; see [8]. For the case of s ∈ Z + , weconsider σ s +1 , σ s +1 , σ s +1 , for example. References [1] Coifman R. R., Lions P. L., Meyer Y., Semmes S.,
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Department of Mathematics, University of Missouri, Columbia, MO 65211
E-mail address : [email protected] Department of Mathematical Science and Information Science, Tokyo Metropolitan University,1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan
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E-mail address : [email protected] Department of Mathematical Science and Information Science, Tokyo Metropolitan University,1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan
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