Hardy Space Estimates for Littlewood-Paley-Stein Square Functions and Calderón-Zygmund Operators
aa r X i v : . [ m a t h . C A ] A p r HARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUAREFUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS
JAROD HART AND GUOZHEN LUA
BSTRACT . In this work, we give new sufficient conditions for Littlewood-Paley-Stein square func-tion and necessary and sufficient conditions for a Calder´on-Zygmund operator to be bounded onHardy spaces H p with indices smaller than 1. New Carleson measure type conditions are definedfor Littlewood-Paley-Stein operators, and the authors show that they are sufficient for the associatedsquare function to be bounded from H p into L p . New polynomial growth BMO conditions are alsointroduced for Calder´on-Zygmund operators. These results are applied to prove that Bony paraprod-ucts can be constructed such that they are bounded on Hardy spaces with exponents ranging all theway down to zero.
1. I
NTRODUCTION
The purpose of this work is to prove new Hardy space H p ( R n ) bounds for Littlewood-Paley-Stein square functions and Calder´on-Zygmund integral operators where the index p is allowedto be small. Part of the novelty of the work here is that it draws an explicit connection betweenCalder´on-Zygmund operators and Littlewood-Paley-Stein square functions.It is well known by now that one way to define the real Hardy spaces H p for 0 < p < ¥ is byusing certain convolution-type Littlewood-Paley-Stein square functions. This has been exploredby many mathematicians; some of the fundamental developments of this idea can be found in thework of Stein [20, 21] and Fefferman and Stein [10]. In particular, Fefferman and Stein provedthat one can define H p = H p ( R n ) using square functions of the form S Q f ( x ) = (cid:229) k ∈ Z | Q k f ( x ) | ! , associated to integral operators Q k f = y k ∗ f for an appropriate choice of Schwartz function y ∈ S , where y k ( x ) = kn y ( k x ) . There are also results in the direction of determining the mostgeneral classes of such convolution operators that can be used to define Hardy spaces, or moregenerally Triebel-Lizorkin spaces; see for example the work of Bui, Paluszy´nski, and Taibelson [4,5]. Generalized classes of non-convolution type Littlewood-Paley-Stein square function operatorswere studied, for example, in [8, 9, 19]. Although all of the bounds in these articles are relegated toLebesgue spaces with index p ∈ ( , ¥ ) , which for this range of indices coincide with Hardy spaces. Date : June 14, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Square Function, Littlewood-Paley-Stein, Bilinear, Calder´on-Zygmund Operators.Hart was partially supported by an AMS-Simons Travel Grant. Lu was supported by NSF grant
In the current work, we consider a general class of non-convolution type Littlewood-Paley-Steinsquare function operators acting on Hardy spaces with indices smaller than 1.Before we state our Hardy space estimates for Littlewood-Paley-Stein square functions, wedefine our classes of Littlewood-Paley-Stein square function operators. Given kernel functions l k : R n → C for k ∈ Z , define L k f ( x ) = Z R n l k ( x , y ) f ( y ) dy for appropriate functions f : R n → C . Define the square function associated to { L k } by S L f ( x ) = (cid:229) k ∈ Z | L k f ( x ) | ! . We say that a collection of operators L k for k ∈ Z is a collection of Littlewood-Paley-Stein oper-ators with decay N and smoothness L + d , written { L k } ∈ LPSO ( N , L + d ) , for N >
0, an integer L ≥ < d ≤
1, if there exists a constant C such that | l k ( x , y ) | ≤ C F Nk ( x − y ) (1.1) | D a l k ( x , y ) | ≤ C | a | k F Nk ( x − y ) for all | a | = a + · · · + a n ≤ L (1.2) | D a l k ( x , y ) − D a l k ( x , y ′ ) | ≤ C | y − y ′ | d k ( L + d ) (cid:0) F Nk ( x − y ) + F Nk ( x − y ′ ) (cid:1) for all | a | = L .(1.3)Here we use the notation F Nk ( x ) = kn ( + k | x | ) − N for N > x ∈ R n , and k ∈ Z . We also usethe notation D a F ( x , y ) = ¶ a x F ( x , y ) and D a F ( x , y ) = ¶ a y F ( x , y ) for F : R n → C and a ∈ N n . It caneasily be shown that LPSO ( N , L + d ) ⊂ LPSO ( N ′ , L + d ′ ) for all 0 < d ′ ≤ d ≤ < N ′ ≤ N .Our goal in studying square functions of the form S L is to prove boundedness properties from H p into L p . Note that it is not reasonable to expect S L to be bounded from H p into H p when0 < p ≤ S L f ≥
0. It is also not hard to see that the condition { L k } ∈ LPSO ( N , L + d ) alone,for any N > L ≥
0, and 0 < d ≤
1, is not sufficient to guarantee that S L to be bounded from H p into L p for any 0 < p < ¥ . In fact, this is not true even in the convolution setting. This can be seenby taking l k ( x , y ) = j k ( x − y ) for some j ∈ S with non-zero integral, where j k ( x ) = kn j ( k x ) .The square function S L associated to this convolution operator is not bounded from H p into L p for any 0 < p < ¥ . Hence some additional conditions are required for L k in order to assure H p to L p bounds. For 1 < p < ¥ , this problem was solved in terms of Carleson measure conditionson L k ( x ) ; see for example [6, 17, 7, 19]. We give sufficient conditions for such bounds when theindex p is allowed to range smaller than 1. The additional cancellation conditions we impose on L k involve generalized moments for non-concolution operators L k . Define the moment function [[ L k ]] b ( x ) by the following. Given { L k } ∈ LPSO ( N , L + d ) and a ∈ N n with | a | < N − n [[ L k ]] a ( x ) = k | a | Z R n l k ( x , y )( x − y ) a dy for k ∈ Z and x ∈ R n . It is worth noting that [[ L k ]] ( x ) = L k ( x ) , which is a quantity that is closelyrelated to L bounds for S L , see for example [8, 9, 19]. We use these moment functions to providesufficient conditions of H p to L p bounds for S L in the following theorem. ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS3
Theorem 1.1.
Let { L k } ∈ LPSO ( N , L + d ) , where N = n + L + d for some integer L ≥ and < d ≤ . If dµ a ( x , t ) = (cid:229) k ∈ Z | [[ L k ]] a ( x ) | d t = − k dx (1.4) is a Carleson measure for all a ∈ N n with | a | ≤ L, then S L can be extended to a bounded operatorfrom H p into L p for all nn + L + d < p ≤ . Here we say that a non-negative measure dµ ( x , t ) on R n + + = R n × ( , ¥ ) is a Carleson measureif there exists C > dµ ( Q × ( , ℓ ( Q ))) ≤ C | Q | for all cubes Q ⊂ R n , where ℓ ( Q ) denotesthe sidelength of Q . We only prove a sufficient condition here for boundedness of S L from H p into L p , but it is reasonable to expect that the Carleson measure conditions in (1.4) are also necessary.We hope to resolve this issue entirely with a full necessary and sufficient condition in future work.We also provide a quick corollary of Theorem 1.1 to the type of operators studied in [8, 9, 19],among others. Corollary 1.2.
Let { L k } ∈ LPSO ( n + d , d ) and < d ≤ . If S L is bounded on L , then S L extendsto a bounded operator from H p into L p for all nn + d < p ≤ . Corollary 1.2 easily follows from Theorem 1.1 and the following observation. If S L is boundedon L , then dµ ( x , t ) , as defined in (1.4) for a =
0, is a Carleson measure; see [6, 17] for proof ofthis observation.Another purpose of this work is to prove a characterization of Hardy space bounds for Calder´on-Zygmund operators. Some of the earliest development of singular integral operators on Hardyspaces is due to Stein and Weiss [22], Stein [21], and Feffermand and Stein [10]. It was proved byFefferman and Stein [10] that if T is a convolution-type singular integral operator that is boundedon L , then T is bounded on H p for p < p < ¥ where 0 ≤ p < T . This situation is considerably more complicated in the non-convolution setting, whichcan be observed in the T T L p bounds for non-convolution Calder´on-Zygmund operators when 1 < p < ¥ , which coincideswith the Hardy space bounds for this range of indices. In [23, 13, 11], the authors give sufficient T H p for 0 < p ≤
1. Theconditions in [23, 13, 11] are too strong though, in the sense that they are not necessary for Hardyspace bounds. The fact that the conditions in [23, 13, 11] are not necessary can be seen by thefull necessary and sufficient conditions provided in [1] when p < p ≤
1, where p = nn + g and g is a regularity parameter for the kernel of T . This can also be seen by considering the Bonyparaproduct, which we prove (in Theorem 1.5) is bounded on H p for p < p ≤ p can betaken arbitrarily close to zero. One of the main purposes of this article is to prove at full necessaryand sufficient T H p bounds from [10, 1, 23, 11, 13].We say that a continuous linear operator T from S into S ′ is a Calder´on-Zygmund oper-ator with smoothness M + g , for any integer M ≥ < g ≤
1, if T has function kernel K : R n \{ ( x , x ) : x ∈ R n } → C such that h T f , g i = Z R n K ( x , y ) f ( y ) g ( x ) dy dx JAROD HART AND GUOZHEN LU whenever f , g ∈ C ¥ = C ¥ ( R n ) have disjoint support, and there is a constant C > K satisfies | D a D b K ( x , y ) | ≤ C | x − y | n + | a | + | b | for all | a | , | b | ≤ M , | D a D b K ( x , y ) − D a D b K ( x ′ , y ) | ≤ C | x − x ′ | g | x − y | n + M + | b | + g for | b | ≤ | a | = M , | x − x ′ | < | x − y | / , | D a D b K ( x , y ) − D a D b K ( x , y ′ ) | ≤ C | y − y ′ | g | x − y | n + | a | + M + g for | a | ≤ | b | = M , | y − y ′ | < | x − y | / . We will also define moment distributions for an operator T ∈ CZO ( M + g ) , but we require somenotation first. For an integer M ≥
0, define the collections of smooth functions of polynomialgrowth O M = O M ( R n ) and of smooth compactly supported function with vanishing moments D M = D M ( R n ) by O M = (cid:26) f ∈ C ¥ ( R n ) : sup x ∈ R n | f ( x ) | · ( + | x | ) − M < ¥ (cid:27) and D M = (cid:26) f ∈ C ¥ ( R n ) : Z R n f ( x ) x a dx = | a | ≤ M (cid:27) . Let h ∈ C ¥ ( R n ) be supported in B ( , ) , h ( x ) = x ∈ B ( , ) , and 0 ≤ h ≤
1. Define for R > h R ( x ) = h ( x / R ) . We reserve this notation for h and h R throughout. In [23, 13, 11], theauthors define T f for f ∈ O M where T is a linear singular integral operator. We give an equivalentdefinition to the ones in [23, 13, 11]. Let T be a CZO ( M + g ) and f ∈ O M for some integer M ≥ < g ≤
1. For y ∈ C ¥ ( R n ) , choose R ≥ ( y ) ⊂ B ( , R / ) , and define h T f , y i = lim R → ¥ h T ( h R f ) , y i − (cid:229) | b |≤ M Z R n D b K ( , y ) b ! x b ( h R ( y ) − h R ( y )) f ( y ) y ( x ) dy dx . This limit exists based on the kernel representation and kernel properties for T ∈ CZO ( M + g ) andis independent of the choice of h , see [23, 13, 11] for proof of this fact. The choice of R hereis not of consequence as long as R is large enough so that supp ( y ) ⊂ B ( , R / ) ; we choose itminimal to make this definition precise. The definition of h T f , y i depends on y here through thesupport properties of y ∈ C ¥ , but for y ∈ D M , it follows that h T f , y i = lim R → ¥ h T ( h R f ) , y i sincethe integral term above vanishes for such y . Now we define the moment distribution [[ T ]] a ∈ D ′ M for T ∈ CZO ( M + g ) and a ∈ N n with | a | ≤ M by h [[ T ]] a , y i = lim R → ¥ Z R n K ( u , y ) y ( u ) h R ( y )( u − y ) a dy du for y ∈ D | a | , where K ∈ S ′ ( R n ) is the distribution kernel of T . We abuse notation here in thatthe integral in this definition is not necessarily a measure theoretic integral; rather, it is the dualpairing between elements of S ( R n ) and S ′ ( R n ) . Throughout this work, we will use K to denotedistributional kernels and K to denote function kernels for Calder´on-Zygmund operators. Whenwe write K in an integral over R n , the integral is understood to be a the pairing of K ∈ S ′ ( R n ) with an element of S ( R n ) . It is not hard to show that this definition is well-defined by techniquesfrom [23, 13, 11]. This distributional moment associated to T generalizes the notion of T ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS5 in [8] in the sense that h [[ T ]] , y i = h T , y i for all y ∈ D and hence [[ T ]] = T
1. We will also usea generalized notion of
BMO here to extend the cancellation conditions T , T ∗ ∈ BMO , whichwere used in the T M ≥ F ∈ D ′ M / P , that is D ′ M modulopolynomials. We say that F ∈ BMO M if (cid:229) k ∈ Z Mk | Q k F ( x ) | dx d t = − k is a Carleson measure for any y ∈ D M , where Q k f = y k ∗ f and y k ( x ) = kn y ( k x ) . This definitionagrees with the classical definition of BMO . That is, for F ∈ BMO , (cid:229) k ∈ Z | Q k F ( x ) | dx d t = − k is a Carleson measure, and hence F ∈ BMO by the
BMO characterization in terms of Carlesonmeasures in [6, 17]. A similar polynomial growth
BMO M was defined by Youssfi [24]. We use thispolynomial growth BMO M to quantify our cancellation conditions for operators T ∈ CZO ( M + g ) in the following result. Theorem 1.3.
Let T ∈ CZO ( M + g ) be bounded on L and define L = ⌊ M / ⌋ and d = ( M − L + g ) / . If T ∗ ( x a ) = in D ′ M for all | a | ≤ L and [[ T ]] a ∈ BMO | a | for all | a | ≤ L, then T extends to abounded operator on H p for nn + L + d < p ≤ . Recall here that the operator T ∗ is defined from S into S ′ via h T ∗ f , g i = h T g , f i , and thedefinition of T ∗ is extended to an operator from O M to D ′ M by the methods discussed above. Notealso that this is not a full necessary and sufficient theorem for Hardy space bounds as describedabove. This theorem will be used to prove the boundedness of certain paraproduct operators, whichin turn allow us to prove the full necessary and sufficient theorem, which is stated in Theorem 1.6at the end of this section.The choice of L and d here are such that L ≥ < d ≤
1, and 2 ( L + d ) = M + g . Itis also not hard to see that T ∗ ( x a ) = | a | ≤ L if and only if [[ T ∗ ]] a = | a | ≤ L . Weprove Theorem 1.6 by decomposing an operator T ∈ CZO ( M + g ) into a collection of operators { L k } ∈ LPSO ( n + L + d , L + d ′ ) for 0 < d ′ < d and applying Theorem 1.1. This decompositionof T into a collection of Littlewood-Paley-Stein operators is stated precisely in the next theorem. Theorem 1.4.
Let T ∈ CZO ( M + g ) for some integer M ≥ and < g ≤ be bounded on L , andfix y ∈ D M . Also let L = ⌊ M / ⌋ and d = ( M − L + g ) / . If T ∗ ( x a ) = in D ′ M for all | a | ≤ L, then { L k } ∈ LPSO ( n + L + d , L + d ′ ) for all < d ′ < d , where L k = Q k T and Q k f ( x ) = y k ∗ f ( x ) .Furthermore, for nn + L + d < p ≤ , T extends to a bounded operator on H p if and only if S L extendsto a bounded operator from H p into L p . Throughout, we write L p = L p ( R n ) and H p = H p ( R n ) for 0 < p < ¥ . We will also applyTheorem 1.6 to Bony paraproducts operator, which were originally defined in [3] and famouslyapplied in the T y ∈ D L + for some L ≥ j ∈ C ¥ . Define Q k f = y k ∗ f and P k f = j k ∗ f . For b ∈ BMO , define P b f ( x ) = (cid:229) j ∈ Z Q j (cid:0) Q j b · P j f (cid:1) ( x ) . (1.5) JAROD HART AND GUOZHEN LU
It easily follows that P b ∈ CZO ( M + g ) for all M ≥ < g ≤
1. It is well known that P ∗ b ( ) = y and j appropriately, it also follows that P b ( ) = b in BMO as well. We arenot interested in an exact identification of P b ( ) in this work, so we don’t worry about the extraconditions that should be imposed on y and j to assure that P b ( ) = b . Theorem 1.5.
Let P b be as in (1.5) for b ∈ BMO, y ∈ D L + , and j ∈ C ¥ . Then P b is bounded onH p for all nn + L + < p ≤ . By Theorem 1.5 it is possible to construct P b so that it is bounded on H p for p > y ∈ D L + for L sufficiently large. It should be noted that some Hardy spaceestimates for a variant of the Bony paraproduct in (1.5) were proved in [15]. Although we use adifferent construction of the paproduct, so we will prove Theorem 1.5 here as well. Finally, westate the first necessary and sufficient boundedness theorem for Calder´on-Zygmund operators onHardy spaces. Theorem 1.6.
Let T ∈ CZO ( M + g ) be bounded on L and define L = ⌊ M / ⌋ and d = ( M − L + g ) / . Then T ∗ ( x a ) = in D ′ M for all | a | ≤ L if and only if T extends to a bounded operator on H p for nn + L + d < p ≤ . Note that Theorem 1.3 is made obsolete by Theorem 1.6. We state Theorem 1.3 separately sincewe will use it to prove the stronger Theorem 1.6. More precisely, we will prove Theorem 1.3, applyTheorem 1.3 to prove H p bounds for Bony paraproducts in Theorem 1.5, and finally we will proveTheorem 1.6 with the help of Theorem 1.5 and a result from [23, 11, 13]. In this way, Theorems1.3, 1.5, and 1.6 are proved in that order, with each depending on the previous results.The rest of the article is organized as follows. In Section 2, we establish some notation andpreliminary results. Section 3 is dedicated to Littlewood-Paley-Stein square functions and provingTheorem 1.1. In section 4, we prove the singular integral operator results in Theorems 1.3 and 1.4.In section 5, we apply Theorem 1.6 to the Bony paraproducts to prove Theorem 1.5. In the lastsection, we use Theorem 1.5 and a result from [23, 11, 13] to prove Theorem 1.6.2. P RELIMINARIES
We use the notation A . B to mean that A ≤ CB for some constant C . The constant C is allowedto depend on the ambient dimension, smoothness and decay parameters of our operators, indicesof function spaces etc.; in context, the dependence of the constants is clear. Recall that we define F Nk ( x ) = kn ( + k | x | ) − N . It is easy to verify that F Nk ( x ) ≤ F e Nk ( x ) for e N ≤ N , and it is well knownthat F Nj ∗ F Nk ( x ) . F N min ( j , k ) ( x ) . We will use these inequalities many times throughout this work without specifically referring tothem.We will use the following Frazier and Jawerth type discrete Calder´on reproducing formula [12](see also [16] for a multiparameter formulation of this reproducing formula): there exist f j , ˜ f j ∈ S for j ∈ Z with infinite vanishing moment such that f ( x ) = (cid:229) j ∈ Z (cid:229) ℓ ( Q )= − ( j + N ) | Q | f j ( x − c Q ) ˜ f j ∗ f ( c Q ) in L (2.1) ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS7 for f ∈ L . The summation in Q here is over all dyadic cubes with side length ℓ ( Q ) = − ( j + N ) ,where N is some large constant, and c Q denotes the center of cube Q . Throughout this paper, wereserve the notation f j and ˜ f j for the operators constructed in this discrete Calder´on decomposition.We will also use a more traditional formulation of Calder´on’s reproducing formula: fix j ∈ C ¥ ( B ( , )) with integral 1 such that (cid:229) k ∈ Z Q k f = f in L (2.2)for f ∈ L , where y ( x ) = n j ( x ) − j ( x ) , y k ( x ) = kn y ( k x ) , and Q k f = y k ∗ f . Furthermore, wecan assume that y has an arbitrarily large, but fixed, number of vanishing moments. Again we willreserve the notation y k and Q k for convolution operators with convolution kernels in D M for some M ≥
0. For this work, the most important difference between the functions y and f is that y iscompactly supported, while f is necessarily not compactly supported. We will use formula (2.1)to decompose square functions and formula (2.2) to decompose Calder´on-Zygmund operators.There are many equivalent definitions of the real Hardy spaces H p = H p ( R n ) for 0 < p < ¥ .We use the following one. Define the non-tangential maximal function N j f ( x ) = sup t > sup | x − y |≤ t (cid:12)(cid:12)(cid:12)(cid:12) Z R n t − n j ( t − ( y − u )) ∗ f ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) , where j ∈ S with non-zero integral. It was proved by Fefferman and Stein in [10] that one candefine || f || H p = || N j f || L p to obtain the classical real Hardy spaces H p for 0 < p < ¥ . It was alsoproved in [10] that for any j ∈ S and f ∈ H p for 0 < p < ¥ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup k ∈ Z | j k ∗ f | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p . We will use a number of equivalent semi-norms for H p . Let y ∈ D M for some integer M > n ( / p − ) , and let y k and Q k be as above, satisfying (2.2). For f ∈ S ′ / P (tempered distributionsmodulo polynomials), f ∈ H p if and only if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | Q k f | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p < ¥ , and this quantity is comparable to || f || H p . The space H p can also be characterized by the operators f j and ˜ f j from the discrete Littlewood-Paley-Stein decomposition in (2.1). This characterizationis given by the following, which can be found in [16, 18]. Given 0 < p < ¥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) j ∈ Z (cid:229) ℓ ( Q )= − ( j + N ) | ˜ f j ∗ f ( c Q ) | c Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p ≈ || f || H p , where c E ( x ) = x ∈ E and c E ( x ) = x / ∈ E for a subset E ⊂ R n . The summation againis indexed by all dyadic cubes Q with side length ℓ ( Q ) = − ( j + N ) For a continuous function
JAROD HART AND GUOZHEN LU f : R n → C and 0 < r < ¥ , define M rj f ( x ) = M (cid:229) ℓ ( Q )= − ( j + N ) f ( c Q ) c Q r ( x ) r , (2.3)where M is the Hardy-Littlewood maximal operator. The following estimate was also proved in[16]. Proposition 2.1.
For any n > , nn + n < r < p ≤ , and f ∈ H p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) j ∈ Z (cid:0) M rj ( ˜ f j ∗ f ) (cid:1) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p , where M rj is defined as in (2.3) . The next result is a rehash of an estimate proved in [16]; their estimate was in the multiparametersetting, whereas the one here is the single parameter version.
Proposition 2.2.
Let f : R n → C a non-negative continuous function, n > , and nn + n < r ≤ .Then (cid:229) ℓ ( Q )= − ( j + N ) | Q | F n + n min ( j , k ) ( x − c Q ) f ( c Q ) . max ( , j − k ) n M rj f ( x ) for all x ∈ R n , where M rj is defined in (2.3) and the summation indexed by ℓ ( Q ) = − ( j + N ) is thesum over all dyadic cubes with side length − ( j + N ) and c Q denotes the center of cube Q.Proof. Define A = { Q dyadic : ℓ ( Q ) = − ( j + N ) and | x − c Q | ≤ − ( j + N ) } A ℓ = { Q dyadic : ℓ ( Q ) = − ( j + N ) and 2 ℓ − − ( j + N ) < | x − c Q | ≤ ℓ − ( j + N ) } for ℓ ≥
1. Now for each Q ∈ A F n + n min ( j , k ) ( x − c Q ) = min ( j , k ) n ( + min ( j , k ) | x − c Q | ) n + n ≤ min ( j , k ) n ≤ jn , and for each Q ∈ A ℓ when ℓ ≥ F n + n min ( j , k ) ( x − c Q ) = min ( j , k ) n ( + min ( j , k ) | x − c Q | ) n + n ≤ min ( j , k ) n ( + min ( j , k ) ( ℓ − − ( j + N ) )) n + n ≤ min ( j , k ) n − ( n + n ) min ( j , k ) − ( n + n ) ℓ + n + n +( n + n )( j + N ) . max ( , j − k ) n − ( n + n ) ℓ jn ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS9
Since S ℓ A ℓ makes up the collection of all dyadic cubes with side length 2 − ( j + N ) , it follows that (cid:229) ℓ ( Q )= − ( j + N ) | Q | F n + n min ( j , k ) ( x − c Q ) f ( c Q ) = ¥ (cid:229) ℓ = (cid:229) Q ∈ A ℓ − ( j + N ) n F n + n min ( j , k ) ( x − c Q ) f ( c Q ) . (cid:229) Q ∈ A f ( c Q ) + max ( , j − k ) n ¥ (cid:229) ℓ = − ℓ ( n + n ) (cid:229) Q ∈ A ℓ f ( c Q ) ≤ max ( , j − k ) n ¥ (cid:229) ℓ = − ℓ ( n + n ) (cid:229) Q ∈ A ℓ f ( c Q ) r ! r . For Q ∈ A ℓ and y ∈ Q it follows that | x − y | ≤ | x − c Q | + | y − c Q | ≤ − ( j + N ) + ℓ − ( j − N ) ≤ ℓ + − ( j + N ) , Hence S Q ∈ A ℓ Q ⊂ B ( x , ℓ + − ( j + N ) ) . We also have that | A ℓ | ≥ n ( ℓ − ) ; so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ Q ∈ A ℓ Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ − ( j + N ) n n ( ℓ − ) = − n ( ℓ − ( j + N )) n ≥ | B ( , ) | − − n | B ( , ℓ − ( j + N ) | . Now we estimate the sum in Q above: (cid:229) Q ∈ A ℓ f ( c Q ) r ≤ | S Q ∈ A ℓ Q | Z S Q ∈ A ℓ Q c S Q ∈ A ℓ Q ( y ) (cid:229) Q ∈ A ℓ f ( c Q ) r dy ≤ | S Q ∈ A ℓ Q | Z S Q ∈ A ℓ Q ( ℓ + ) n (cid:229) Q ∈ A ℓ f ( c Q ) r c Q ( y ) dy . ℓ n | B ( x , ℓ + − ( j + N ) ) | Z B ( x , ℓ + − ( j + N ) ) (cid:229) Q ∈ A ℓ f ( c Q ) r c Q ( y ) dy = ℓ n | B ( x , ℓ + − ( j + N ) ) | Z B ( x , ℓ + − ( j + N ) ) (cid:229) Q ∈ A ℓ f ( c Q ) c Q ( y ) ! r dy . ℓ n M " (cid:229) Q ∈ A ℓ f ( c Q ) c Q ! r ( x ) . Then we have that (cid:229) ℓ ( Q )= − ( j + N ) | Q | F n + n min ( j , k ) ( x − c Q ) f ( c Q ) . max ( , j − k ) n ¥ (cid:229) ℓ = − ℓ ( n + n − n / r ) ( M " (cid:229) Q ∈ A ℓ f ( c Q ) c Q ! r ( x ) ) r . max ( , j − k ) n M (cid:229) ℓ ( Q )= − ( j + N ) f ( c Q ) c Q r ( x ) r . (cid:3) We will also need some Carleson measure estimates for the result in Theorem 1.1. The nextproof is a well known argument that can be found in [6, 17].
Proposition 2.3.
Suppose dµ ( x , t ) = (cid:229) k ∈ Z µ k ( x ) d t = − k dx (2.4) is a Carleson measure, where µ k is a non-negative, locally integrable function for all k ∈ Z . Alsolet j ∈ S , and define P k f = j k ∗ f , where j k ( x ) = kn j ( k x ) for k ∈ Z . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | P k f | p µ k ! p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p for all < p < ¥ and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | P k f | µ k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p for all < p ≤ . Proof.
Let f ∈ H p , and we begin the proof of the the first estimate above by looking at Z R n (cid:229) k ∈ Z | P k f ( x ) | p µ k ( x ) dx = p Z ¥ dµ (cid:18)(cid:26) ( x , t ) : (cid:12)(cid:12)(cid:12)(cid:12) Z R n t − n j ( t − ( x − y )) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) > l (cid:27)(cid:19) l p d ll . Define E l = { x : | N j f ( x ) | > l } , and it follows that (cid:26) ( x , t ) : (cid:12)(cid:12)(cid:12)(cid:12) Z R n t − n j ( t − ( x − y )) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) > l (cid:27) ⊂ b E l , where b E = { ( x , t ) : B ( x , t ) ⊂ E } . Therefore Z R n (cid:229) k ∈ Z | P k f ( x ) | p µ k ( x ) dx ≤ p Z ¥ dµ ( b E l ) l p d ll . p Z ¥ | E l | l p d ll = || N j f || pL p = || f || pH p . Here we use that dµ ( b E ) . | E | for any open set E ⊂ R n , which is a well known estimate for Carlesonmeasures. In the case p =
2, the second estimate coincides with the first and hence there is no moreto prove. When 0 < p <
2, we set r = p > r is r ′ = − p . Now ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS11 applying the first estimate above, we finish the proof. Z R n (cid:229) k ∈ Z | P k f ( x ) | µ k ( x ) ! p dx ≤ Z R n sup k | P k f ( x ) | ( − p ) p / (cid:229) k ∈ Z | P k f ( x ) | p µ k ( x ) ! p dx ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( N j f ) ( − p ) p / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L r ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | P k f ( x ) | p µ k ( x ) ! p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L r = (cid:12)(cid:12)(cid:12)(cid:12) N j f (cid:12)(cid:12)(cid:12)(cid:12) p ( − p ) L p Z R n (cid:229) k ∈ Z | P k f ( x ) | p µ k ( x ) dx ! p . || f || p ( − p ) / H p || f || p / H p = || f || pH p . (cid:3)
3. H
ARDY S PACE E STIMATES FOR S QUARE F UNCTIONS
In this section we prove Theorem 1.1. To do this, we first prove a reduced version of the theorem.
Lemma 3.1.
Assume { L k } ∈ LPSO ( n + L + d , L + d ) for some integer L ≥ and < d ≤ . If L k ( y a ) = for all k ∈ Z and | a | ≤ L, then || S L f || L p . || f || H p for all f ∈ H p ∩ L and nn + L + d < p ≤ . We call this a reduced version of Theorem 1.1 because we have strengthened the assumptionsof from the Carleson measure estimates for (1.4) to the vanishing moment type assumption above; L k ( y a ) = | a | ≤ L . Proof.
Fix n ∈ ( n / p − n , L + d ) , which is possible since our assumption on p implies that np − n < L + d . Also fix r ∈ ( , ) such that nn + n < r < p . Let f ∈ H p ∩ L , and we decompose L k f ( x ) = (cid:229) j ∈ Z (cid:229) Q | Q | ˜ f j ∗ f ( c Q ) L k y c Q j ( x ) = (cid:229) j ∈ Z (cid:229) Q | Q | ˜ f j ∗ f ( c Q ) Z R n l k ( x , y ) y c Q j ( y ) dy . The summation in Q is over all dyadic cubes with side lengths ℓ ( Q ) = − ( j + N ) . Then we have thefollowing almost orthogonality estimates (cid:12)(cid:12)(cid:12)(cid:12) Z R n l k ( x , y ) f c Q j ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n l k ( x , y ) f c Q j ( y ) − (cid:229) | a |≤ L D a f c Q j ( x ) a ! ( y − x ) a ! dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z R n F n + L + d k ( x − y )( j | x − y | ) L + d (cid:16) F n + L + d j ( y − c Q ) + F n + L + d j ( x − c Q ) (cid:17) dy . ( L + d )( j − k ) Z R n F n + L + d k ( x − y ) (cid:16) F n + L + d j ( y − c Q ) + F n + L + d j ( x − c Q ) (cid:17) dy . ( L + d )( j − k ) F n + L + d min ( j , k ) ( x − c Q ) . Also, using the vanishing moment properties of f j , we have the following estimate, (cid:12)(cid:12)(cid:12)(cid:12) Z R n l k ( x , y ) f c Q j ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n l k ( x , y ) − (cid:229) | a |≤ L D a l k ( x , c Q ) a ! ( x − y ) a ! f c Q j ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z R n F n + L + d k ( x − y )( k | y − c Q | ) L + d F n + L + d j ( y − c Q ) dy + Z R n F n + L + d k ( x − c Q )( k | y − c Q | ) L + d F n + L + d j ( y − c Q ) dy . ( L + d )( k − j ) Z R n F n + L + d k ( x − y ) F n + L + d j ( y − c Q ) dy + ( L + d )( k − j ) Z R n F n + L + d k ( x − c Q ) F n + L + d j ( y − c Q ) dy . ( L + d )( k − j ) F n + L + d min ( j , k ) ( x − c Q ) . Therefore (cid:12)(cid:12)(cid:12)(cid:12) Z R n l k ( x , y ) f c Q j ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) . − ( L + d ) | j − k | F n + n min ( j , k ) ( x − c Q ) . Applying Proposition 2.2 yields | L k f ( x ) | . (cid:229) j ∈ Z (cid:229) Q | Q | ˜ f j ∗ f ( c Q ) − ( L + d ) | j − k | F n + n min ( j , k ) ( x − c Q ) . (cid:229) j ∈ Z − ( L + d ) | j − k | n max ( , k − j ) M rj ( ˜ f j ∗ f )( x ) ≤ (cid:229) j ∈ Z − e | j − k | M rj ( ˜ f j ∗ f )( x ) , where e = L + d − n >
0; recall that these parameter are chosen such that n < L + d . ApplyingProposition 2.1 to M rj ( ˜ f j ∗ f ) (recall that r was chosen such that nn + n < r < p ) yields the appropriateestimate below, || S L f || L p . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z " (cid:229) j ∈ Z − e | j − k | M rj ( ˜ f j ∗ f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) j , k ∈ Z − e | j − k | (cid:2) M rj ( ˜ f j ∗ f ) (cid:3) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p . This completes the proof of Lemma 3.1. (cid:3)
Next we construct paraproducts to decompose L k . Fix an approximation to identity operator P k f = j k ∗ f , where j k ( x ) = kn j ( k x ) and j ∈ S with integral 1. Define for a , b ∈ N n M a , b = ( − ) | b |−| a | b ! ( b − a ) ! Z R n j ( y ) y b − a dy a ≤ b a b . ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS13
Here we say a ≤ b for a = ( a , ..., a n ) , b = ( b , ..., b n ) ∈ N n if a i ≤ b i for all i = , ..., n . It is clearthat | M a , b | < ¥ for all a , b ∈ N n since j ∈ S . Also note that when | a | = | b | M a , b = (cid:26) b ! a = b a = b and | a | = | b | . (3.1)We consider the operators P k D a defined on S ′ , where D a is taken to get the distributional deriv-ative acting on S ′ . Hence P k D a f ( x ) is well defined for f ∈ S ′ since P k D a f ( x ) = (cid:10) j xk , D a f (cid:11) =( − ) | a | (cid:10) D a ( j xk ) , f (cid:11) and D a ( j xk ) ∈ S . In fact, this gives a kernel representation for P k D a ; esti-mates for this kernel are addressed in the proof of Proposition 3.2. We also have [[ P k D a ]] b ( x ) = | b | k Z R n j k ( x − y ) ¶ a y (( x − y ) b ) dy = k | a | M a , b . For k ∈ Z , define L ( ) k f ( x ) = L k f ( x ) − [[ L k ]] ( x ) · P k f ( x ) , and(3.2) L ( m ) k f ( x ) = L ( m − ) k f ( x ) − (cid:229) | a | = m ( − ) | a | [[ L ( m − ) k ]] a ( x ) a ! · − k | a | P k D a f ( x ) . (3.3)for 1 ≤ m ≤ L . Proposition 3.2.
Let { L k } ∈ LPSO ( N , L + d ) , where N = n + L + d for some integer L ≥ and < d ≤ , and assume that dµ a ( x , t ) = (cid:229) k ∈ Z | [[ L k ]] a ( x ) | d t = − k dx (3.4) is a Carleson measure for all a ∈ N n such that | a | ≤ L. Also let L ( m ) k be as in as in (3.2) and (3.3) for ≤ m ≤ L. Then L ( m ) k ∈ LPSO ( N , L + d ) for the same N, L, and d , and satisfy the following: (1) [[ L ( m ) k ]] a = for all a ∈ N n with | a | ≤ m ≤ L. (2) dµ m ( x , t ) is a Carleson measure for all ≤ m ≤ L, where dµ m is defineddµ m ( x , t ) = (cid:229) k ∈ Z (cid:229) | a |≤ L | [[ L ( m ) k ]] a ( x ) | d t = − k dx . Proof.
Since { L k } ∈ LPSO ( n + L + d , L + d ) , we know that | [[ L k ]] a ( x ) | . | a | ≤ L .Then to verify that { L ( m ) k } ∈ LPSO ( n + L + d , L + d ) for 0 ≤ m ≤ L , it is sufficient to show that { − k | a | P k D a } ∈ LPSO ( n + L + d , L + d ) for all a ∈ N n . For f ∈ S ′ , we have the followingintegral representation for 2 − k | a | P k D a f , which was alluded to above,2 − k | a | P k D a f ( x ) = ( − ) | a | − k | a | h D a ( j xk ) , f i = ( − ) | a | ( D a j ) k ∗ f ( x ) . Since j ∈ S , it easily follows that D a j ∈ S for all a ∈ N n and that { − k | a | P k D a } ∈ LPSO ( n + L + d , L + d ) . Now we prove (1) by induction: the m = [[ L ( ) k ]] = L k − [[ L k ]] · P k = [[ L k ]] − [[ L k ]] = . Now assume that (1) holds for m −
1, that is, assume [[ L ( m − ) k ]] a = | a | ≤ m −
1. Then for | b | ≤ m − [[ L ( m ) k ]] b = [[ L ( m − ) k ]] b − (cid:229) | a | = m [[ L ( m − ) k ]] a a ! ( − ) | a | M a , b = . The first term here vanished by the inductive hypothesis. The second term is zero since | b | < m = | a | and hence M a , b =
0. For | b | = m , [[ L ( m ) k ]] b = [[ L ( m − ) k ]] b − (cid:229) | a | = m [[ L ( m − ) k ]] a a ! ( − ) | a | M a , b = [[ L ( m − ) k ]] b − [[ L ( m − ) k ]] b = , where the sum collapses using (3.1). By induction, this verifies (1) for all m ≤ L . Given the Car-leson measure assumption for dµ a ( x , t ) in (3.4), one can easily prove (2) if the following statementholds: for all 0 ≤ m ≤ L (cid:229) | a |≤ L | [[ L ( m ) k ]] a ( x ) | ≤ ( + C ) m + (cid:229) | a |≤ L | [[ L k ]] a ( x ) | , where C = (cid:229) | a | , | b |≤ L | M a , b | . (3.5)We verify (3.5) by induction. For m =
0, let | b | ≤ L , and it follows that [[ L ( ) k ]] b = [[ L k ]] b − [[ L k ]] · [[ P k ]] b = [[ L k ]] b − [[ L k ]] · M , b Then (cid:229) | b |≤ L | [[ L ( ) k ]] b | ≤ (cid:229) | b |≤ L | [[ L k ]] b | + (cid:229) | b |≤ L | [[ L k ]] || M , b | ≤ ( + C ) (cid:229) | b |≤ L | [[ L k ]] b | . Now assume that (3.5) holds for m −
1, and consider (cid:229) | b |≤ L | [[ L ( m ) k ]] b | ≤ (cid:229) | b |≤ L | [[ L ( m − ) k ]] b | + (cid:229) | b |≤ L (cid:229) | a | = m | [[ L ( m − ) k ]] a || M a , b |≤ + (cid:229) | a |≤ m , | b |≤ L | M a , b | ! (cid:229) | b |≤ L | [[ L ( m − ) k ]] b |≤ ( + C ) (cid:229) | b |≤ L | [[ L ( m − ) k ]] b | ≤ ( + C ) m + (cid:229) | b |≤ L | [[ L k ]] b | . We use the inductive hypothesis in the last inequality here to bound the [[ L ( m − ) ]] b . Then byinduction, the estimate in (3.5) holds for all 0 ≤ m ≤ L , and completes the proof. (cid:3) Now we use Lemma 3.1 and the paraproduct operators L ( m ) k along with Propositions 2.3 and 3.2to prove Theorem 1.1. ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS15
Proof of Theorem 1.1.
By density, it is sufficient to prove that || S L f || L p . || f || H p for f ∈ H p ∩ L .We bound L k in the following way using the definitions of L ( m ) k in (3.2) and (3.3); | L k ( x ) f | ≤ | L k ( x ) · P k f ( x ) | + | L ( ) k f ( x ) |≤ | L k ( x ) · P k f ( x ) | + | L ( ) k f ( x ) | + (cid:229) | a | = | [[ L ( ) k ]] a ( x ) | − k | a | | P k D a f ( x ) |≤ | L k ( x ) · P k f ( x ) | + | L ( L ) k f ( x ) | + L (cid:229) m = (cid:229) | a | = m | [[ L ( m − ) k ]] a ( x ) | − k | a | | P k D a f ( x ) | . By Propositions 2.3 and 3.2, it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | L k P k f | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p + L (cid:229) m = (cid:229) | a | = m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | [[ L ( m − ) k ]] a −| a | k P k D a f | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p . Also by Lemma 3.1, it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k ∈ Z | L ( L ) k f | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . || f || H p . Therefore S L can be extended to a bounded operator from H p into L p . (cid:3)
4. H
ARDY S PACE B OUNDS FOR S INGULAR I NTEGRAL O PERATORS
In this section, we prove Theorem 1.3. This is a reduced version of Theorem 1.6 in the sensethat we have strengthened the assumptions on T , and hence obtain only a sufficient condition, notnecessary. We will apply Theorem 1.1 to prove Theorem 1.3. In order to do so, we prove thedecomposition result in Theorem 1.4. Proof of Theorem 1.4.
Let y ∈ D M . It is not hard to check that T ∗ y xk ( y ) is the kernel of Q k T ,where y xk ( y ) = y k ( y − x ) . Also let L = ⌊ M / ⌋ and d = ( M − L + g ) /
2. We first verify (1.1)-(1.3)for | x − y | > − k . Assume that | x − y | > − k . Then for | a | ≤ L | ¶ a y T ∗ y x ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¶ a y Z R n K ( u , y ) − (cid:229) | b |≤ M D b K ( x , y ) b ! ( u − x ) b ! y k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n D a K ( u , y ) − (cid:229) | b |≤ M D b D a K ( x , y ) b ! ( u − x ) b ! y k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z R n | x − u | M + g | x − y | n + | a | + M + g | y k ( u − x ) | du . − k ( M + g ) ( − k + | x − y | ) n + | a | + M + g Z R n | y k ( u − x ) | du . | a | k F n + M + | a | + g k ( x − y ) ≤ | a | k F n + L + d k ( x − y ) . If M ≥
1, then this estimate holds for all | a | ≤ L +
1. In this case, the above estimate implies that(1.3) also holds for N = n + L + d and any 0 < d ≤
1. So it remains to verify (1.3) for M = L = d = g /
2. If | y − y ′ | ≥ − k , then property (1.3) easily follows from theestimate just proved with a =
0. Otherwise we assume that | y − y ′ | < − k , and it follows that | x − y ′ | ≥ | x − y | − | y − y ′ | > | x − y | / ≥ − k . Then | T ∗ y x ( y ) − T ∗ y x ( y ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z R n (cid:0) K ( u , y ) − K ( u , y ′ ) (cid:1) y k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z R n (cid:0)(cid:0) K ( u , y ) − K ( u , y ′ ) (cid:1) − ( K ( x , y ) − K ( x , y ′ )) (cid:1) y k ( u − x ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n (cid:229) | b | = | D b K ( x , y ) − D b K ( x , y ′ ) | | u − x | | y k ( u − x ) | du for some x = cx + ( − c ) u with 0 < c < . Z R n | y − y ′ | g | x − u || x − y | n + + g | y k ( u − x ) | du . | y − y ′ | g − k ( − k + | x − y | ) n + + g = d k | y − y ′ | d F n + L + d k ( x − y ) . Recall this is the situation where M = L = d = g /
2, and | y − y ′ | ≤ − k , and hence in the lastline n + g = n + L + d and 2 g k | y − y ′ | g ≤ d k | y − y ′ | d . This completes the proof of (1.1)-(1.3) for | x − y | > − k .When | x − y | ≤ − k , we decompose Q k T further. Let j ∈ C ¥ with integral 1 such that e y ( x ) = n j ( x ) − j ( x ) and e y ∈ D M . Then T ∗ y xk ( y ) = lim N → ¥ P N T ∗ y xk ( y ) = ¥ (cid:229) ℓ = k e Q ℓ T ∗ y xk ( y ) + P k T ∗ y xk ( y ) . (4.1)This equality holds pointwise almost everywhere since T is a continuous operator from L to L and y xk ∈ D M . Note that e y , y ∈ D M , and it is only this property that will be used throughout therest of this proof. So we abuse notation to make this proof a bit easier to read. For the remainderof the proof, we will simply write e y ℓ = y ℓ and e Q ℓ = Q ℓ with the understanding that these two canactually be allowed to be different elements of D M . Let a ∈ N n with | a | ≤ L . Using the hypothesis T ∗ ( x µ ) = | µ | ≤ L we write (cid:12)(cid:12) D a (cid:10) T y y ℓ , y xk (cid:11)(cid:12)(cid:12) ≤ | A ℓ, k ( x , y ) | + | B ℓ, k ( x , y ) | , where A ℓ, k ( x , y ) = ℓ | a | Z | u − y |≤ − ℓ T (( D a y ) y ℓ )( u ) y xk ( u ) − (cid:229) | a |≤ L D a y xk ( y ) a ! ( u − y ) a ! du , B ℓ, k ( x , y ) = ℓ | a | Z | u − y | > − ℓ T (( D a y ) y ℓ )( u ) y xk ( u ) − (cid:229) | a |≤ L D a y xk ( y ) a ! ( u − y ) a ! du . ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS17
The A ℓ, k term is bounded as follows, | A ℓ, k ( x , y ) | ≤ ℓ | a | || T (( D a y ) y ℓ ) · c B ( y , − ℓ ) || L × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y xk ( u ) − (cid:229) | a |≤ L D a y xk ( y ) a ! ( · − y ) a ! · c B ( y , − ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ¥ . ℓ | a | − ℓ n / || T (( D a y ) y ℓ ) || L ( L + d )( k − ℓ ) kn . ℓ | a | − ℓ n / || ( D a y ) y ℓ || L ( L + d )( k − ℓ ) kn . k | a | d ( k − ℓ ) F n + L + d k ( x − y ) . Let 0 < d ′ < d ′′ < d . The B ℓ, k term is bounded using the kernel representation of T | B ℓ, k ( x , y ) | ≤ ℓ | a | Z | u − y | > − ℓ Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( u , v ) − (cid:229) | b |≤ L D b K ( u , y ) b ! ( v − y ) b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ( D a y ) y ℓ ( u ) | dv × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y xk ( u ) − (cid:229) | µ |≤ L D µ ( y xk )( y ) µ ! ( u − y ) µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) du . ℓ | a | ¥ (cid:229) m = Z m − ℓ < | u − y |≤ m + − ℓ Z R n | v − y | L + d | u − y | n + L + d | ( D a y ) y ℓ ( v ) | dv kn ( k | u − y | ) L + d ′′ du . ℓ | a | ¥ (cid:229) m = Z m − ℓ < | u − y |≤ m + − ℓ Z R n − ( L + d ) ℓ ( n + L + d )( m − ℓ ) | ( D a y ) y ℓ ( v ) | dv kn ( k m − ℓ ) L + d ′′ du . ℓ | a | ¥ (cid:229) m = ( m − ℓ ) n − ( L + d ) ℓ − ( n + L + d )( m − ℓ ) kn ( L + d ′′ )( k + m − ℓ ) . k | a | ( L −| a | + d ′′ )( k − ℓ ) kn ¥ (cid:229) m = ( d ′′ − d ) m . k | a | d ′′ ( k − ℓ ) F n + L + d k ( x − y ) . It is not crucial here that we took d ′ < d ′′ < d , but this estimate will be used again later where ourchoice of d ′ < d ′′ will be important. It follows that the kernel T ∗ y xk ( y ) of Q k T satisfies (cid:12)(cid:12) ¶ a y T ∗ y xk ( y ) (cid:12)(cid:12) = ℓ | a | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) ℓ> k (cid:10) T (( D a y ) y ℓ ) , y xk (cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k | a | (cid:229) ℓ> k d ′′ ( k − ℓ ) F n + L + d k ( x − y ) . k | a | F n + L + d k ( x − y ) . This verifies that T ∗ y xk ( y ) satisfies (1.1) for | x − y | ≤ − k . We also verify the d -H¨older regularityestimate (1.2) for T ∗ y xk ( y ) with d ′ in place of d : let a ∈ N n with | a | = L . It trivially follows from the above estimate that (cid:229) ℓ ≥ k : 2 − ℓ < | y − y ′ | (cid:12)(cid:12)(cid:12)D D a T ( y y ℓ − y y ′ ℓ ) , y xk E(cid:12)(cid:12)(cid:12) . (cid:229) ℓ ≥ k : 2 − ℓ < | y − y ′ | d ′′ ( k − ℓ ) ( ℓ | y − y ′ | ) d ′ k | a | (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) . k | a | (cid:229) ℓ ≥ k − ℓ < | y − y ′ | ( d ′′ − d ′ )( k − ℓ ) ( k | y − y ′ | ) d ′ (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) . k ( | a | + d ′ ) | y − y ′ | d ′ (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) . On the other hand, for the situation where | y − y ′ | ≤ − ℓ , we consider (cid:229) ℓ ≥ k : 2 − ℓ ≥| y − y ′ | (cid:12)(cid:12)(cid:12)D D a T ( y y ℓ − y y ′ ℓ ) , y xk E(cid:12)(cid:12)(cid:12) ≤ | A ℓ, k ( x , y , y ′ ) | + | B ℓ, k ( x , y , y ′ ) | , where A ℓ, k ( x , y , y ′ ) = ℓ | a | Z | u − y |≤ − ℓ T (( D a y ) y ℓ − ( D a y ) y ′ ℓ )( u ) × y xk ( u ) − (cid:229) | a |≤ L D a y xk ( y ) a ! ( u − y ) a ! du , and B ℓ, k ( x , y , y ′ ) = ℓ | a | Z | u − y | > − ℓ T (( D a y ) y ℓ − ( D a y ) y ′ ℓ )( u ) × y xk ( u ) − (cid:229) | a |≤ L D a y xk ( y ) a ! ( u − y ) a ! du . The A ℓ, k term is bounded as follows, | A ℓ, k ( x , y , y ′ ) | ≤ ℓ | a | || T (( D a y ) y ℓ − ( D a y ) y ′ ℓ ) · c B ( y , − ℓ ) || L × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y xk ( u ) − (cid:229) | µ |≤ L D µ y xk ( y ) µ ! ( u − y ) µ ! · c B ( y , − ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ¥ . ℓ | a | − ℓ n / || T (( D a y ) y ℓ − ( D a y ) y ′ ℓ ) || L ( L + d )( k − ℓ ) kn . ℓ | a | ( ℓ | y − y ′ | ) d ′ ( L + d )( k − ℓ ) kn ≤ k | a | ( d − d ′ )( k − ℓ ) ( k | y − y ′ | ) d ′ (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) . ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS19
Recall the selection of d ′′ such that 0 < d ′ < d ′′ < d . The B ℓ, k term is bounded using the kernelrepresentation of T | B ℓ, k ( x , y , y ′ ) | = ℓ | a | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z | u − y | > − ℓ Z R n K ( u , v ) − (cid:229) | n |≤ L D n K ( u , y ) n ! ( v − y ) n ! × (( D a y ) y ℓ ( v ) − ( D a y ) y ′ ℓ ( v )) y xk ( u ) − (cid:229) | µ |≤ L D µ y xk ( y ) µ ! ( u − y ) µ ! du dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ℓ | a | Z | u − y | > − ℓ Z R n | v − y | L + d | u − y | n + L + d × | ( D a y ) y ℓ ( v ) − ( D a y ) y ′ ℓ ( v ) | dv kn ( k | u − y | ) L + d ′′ du . ℓ | a | ¥ (cid:229) m = Z m − ℓ < | u − y |≤ m + − ℓ Z R n − ( L + d ) ℓ ( n + L + d )( m − ℓ ) ( ℓ | y − y ′ | ) d ′ × (cid:0) F n + ℓ ( y − v ) + F n + ℓ ( y ′ − v ) (cid:1) dv kn ( k | u − y | ) L + d ′′ du . ℓ | a | ¥ (cid:229) m = n ( m − ℓ ) − ( L + d ) ℓ ( n + L + d )( ℓ − m ) ( ℓ | y − y ′ | ) d ′ kn ( L + d ′′ )( k + m − ℓ ) . k | a | ( ℓ − k ) | a | d ′ ( ℓ − k ) ( k | y − y ′ | ) d ′ kn ( L + d ′′ )( k − ℓ ) ¥ (cid:229) m = ( d ′′ − d ) m . k | a | ( k | y − y ′ | ) d ′ ( d ′′ − d ′ )( k − ℓ ) (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) . It follows that ¥ (cid:229) ℓ = k | A ℓ, k ( x , y , y ′ ) | + | B ℓ, k ( x , y , y ′ ) | . k | a | ( k | y − y ′ | ) d ′ (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) ¥ (cid:229) ℓ = k ( d ′′ − d ′ )( k − ℓ ) . k | a | ( k | y − y ′ | ) d ′ (cid:16) F n + L + d k ( x − y ) + F n + L + d k ( x − y ′ ) (cid:17) We now check that P k T ∗ y xk ( y ) , the second term from (4.1), also satisfies the appropriate size andregularity estimates. For all a ∈ N n | ¶ a y P k T ∗ y xk ( y ) | = | a | k | (cid:10) T ( D a j ) yk , y xk (cid:11) | ≤ | a | k || T || , kn . | a | k F n + L + d k ( x − y ) . Here || T || , is the L operator norm of T . Therefore T ∗ y xk ( y ) satisfies size and regularity properties(1.1) and (1.2) with d ′ in place of d , and hence { Q k T } ∈ LPSO ( n + L + d , L + d ′ ) for all d ′ ∈ ( , d ) .It is trivial now to note that for nn + L + d < p ≤ T is bounded on H p if and only if S L is boundedfrom H p into L p since || T f || H p ≈ || S L f || L p by the Littlewood-Paley-Stein characterization of H p in [10]. (cid:3) Lemma 4.1.
Let T ∈ CZO ( M + g ) be bounded on L and satisfy T ∗ ( x a ) = for all | a | ≤ L = ⌊ M / ⌋ . For y ∈ D M , definedµ y ( x , t ) = (cid:229) | a |≤ L (cid:229) k ∈ Z | [[ Q k T ]] a ( x ) | d t = − k dx , where Q k f = y k ∗ f and y k ( x ) = kn y ( k x ) . If [[ T ]] a ∈ BMO | a | for all | a | ≤ L, then dµ y is aCarleson measure for any y ∈ D M + L .Proof. Assume that [[ T ]] a ∈ BMO | a | for all | a | ≤ L . Let y ∈ D M + L , and it follows that { Q k T } ∈ LPSO ( L , d ′ ) for all d ′ < d , where Q k f is defined as above and L = ⌊ M / ⌋ and d = ( M − L + g ) / Q b k f = y b k ∗ f , where y b ( x ) = ( − ) | b | y ( x ) x b . It follows that y b ∈ D M + L −| b | .Now let a ∈ N n such that | a | ≤ L . Note that for b ≤ a , it follows that y b ∈ D M , and hence { Q b k T } ∈ LPSO ( n + L + d , L + d ′ ) for all 0 < d ′ < d as well. Then it follows that [[ Q k T ]] a ( x ) = | a | k Z R n T ∗ y xk ( y )( x − y ) a dy = lim R → ¥ | a | k Z R n K ( u , y ) y xk ( u ) h R ( y )( x − y ) a du dy = lim R → ¥ (cid:229) b ≤ a c a , b | a | k Z R n K ( u , y ) y xk ( u )( x − u ) b ( u − y ) a − b du dy = lim R → ¥ (cid:229) b ≤ a c a , b ( | a |−| b | ) k Z R n K ( u , y )( y b k ) x ( u ) h R ( y )( u − y ) a − b du dy = (cid:229) b ≤ a c a , b ( | a |−| b | ) k D [[ T ]] a − b , ( y b k ) x E . Let Q ⊂ R n be a cube with side length ℓ ( Q ) . It follows that (cid:229) − k ≤ ℓ ( Q ) Z Q | [[ Q k T ]] a ( x ) | dx ≤ (cid:229) − k ≤ ℓ ( Q ) Z Q (cid:229) b ≤ a c a , b ( | a |−| b | ) k (cid:12)(cid:12)(cid:12)D [[ T ]] a − b , ( y b k ) x E(cid:12)(cid:12)(cid:12)! dx . (cid:229) b ≤ a (cid:229) − k ≤ ℓ ( Q ) Z Q ( | a |−| b | ) k (cid:12)(cid:12)(cid:12)D [[ T ]] a − b , ( y b k ) x E(cid:12)(cid:12)(cid:12) dx . | Q | . The last inequality holds since [[ T ]] a − b ∈ BMO | a |−| b | and y b k ∈ D M ⊂ D | a |−| b | for all b ≤ a . (cid:3) Motivated by the proof of Lemma 4.1, we pause for a moment to introduce an alternative testingcondition to [[ T ]] a ∈ BMO | a | in Theorem 1.6. The following proposition introduces a perturbationof the definition of [[ T ]] a with necessary and sufficient conditions for [[ T ]] a ∈ BMO | a | for | a | ≤ L . Proposition 4.2.
Let T ∈ CZO ( M + g ) with T ∗ ( y a ) = for | a | ≤ L. Then [[ T ]] a ∈ BMO | a | for all | a | ≤ L if and only if dµ y ( x , t ) = (cid:229) | a |≤ L (cid:229) k ∈ Z k | a | | h T G x a , y xk i | d t = − k dxis a Carleson measure for all y ∈ D M + L , where Q k f = y k ∗ f and G x a ( u ) = ( u − x ) a . ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS21
The quantity h T G x a , y i is very closely related to h [[ T ]] a , y i . One can obtain the distribution T G x a by replacing ( u − y ) a with ( x − y ) a in the definition of [[ T ]] a . This gives an alternative testingcondition for [[ T ]] a ∈ BMO | a | that could be convenient in some situations. Proof.
Similar to the proof of Lemma 4.1, it follows that2 | a | k h T G x a , y xk i = lim R → ¥ | a | k Z R n K ( u , y ) y xk ( u ) h R ( y )( x − y ) a du dy = (cid:229) b ≤ a c a , b ( | a |−| b | ) k D [[ T ]] a − b , ( y b k ) x E . Here c a , b are binomial coefficients and are bounded uniformly for | a | , | b | ≤ L depending on L .Likewise we have that2 | a | k h [[ T ]] a , y xk i = (cid:229) b ≤ a c a , b ( | a |−| b | ) k D T G x a − b , ( y b k ) x E . Lemma 4.2 easily follows. (cid:3)
Finally we prove Theorem 1.3.
Proof of Theorem 1.3.
By density, it is sufficient to prove the appropriate estimates for f ∈ H p ∩ L .Let y ∈ D M + L such that Calder´on’s reproducing formula (2.2) holds for Q k f = y k ∗ f , where L = ⌊ M / ⌋ . By Theorem 1.4, it follows that { L k } = { Q k T } ∈ LPSO ( n + L + d , L + d ′ ) for all0 < d ′ < d = ( M − L + g ) /
2. So fix a d ′ ∈ ( , d ) close enough to d so that nn + L + d < nn + L + d ′ < p .By Lemma 4.1, it follows that dµ ( x , t ) = (cid:229) k ∈ Z (cid:229) | a |≤ L | [[ Q k T ]] a ( x ) | dx d t = − k is a Carleson measure. By Theorems 1.1 and 1.4, it also follows that S L can be extended to abounded operator from H p into L p , and hence T can be extended to a bounded operator on H p . (cid:3)
5. A N A PPLICATION TO B ONY T YPE P ARAPRODUCTS
In this section, we apply Theorem 1.6 to show that the Bony paraproduct operators from [3] arebounded on H p , which was stated in Theorem 1.5. Let y ∈ D L + for some L ≥ j ∈ C ¥ .Define Q k f = y k ∗ f and P k f = j k ∗ f . For b ∈ BMO , recall the definition of P b in (1.5) P b f ( x ) = (cid:229) j ∈ Z Q j (cid:0) Q j b · P j f (cid:1) ( x ) . It follows that P b ∈ CZO ( M + g ) for all M ≥ < g ≤
1. We will focus on the properties T ∗ ( x a ) = [[ T ]] a ∈ BMO | a | for | a | ≤ L . Once we prove these two things, we obtain Theorem1.5 by applying Theorem 1.6. We first give the definition of the Fourier transform that we will useand prove a lemma that will be used to prove the Hardy space bounds for P b . For f ∈ L ( R n ) and x ∈ R n , define b f ( x ) = F [ f ]( x ) = Z R n f ( x ) e ix · x dx . Lemma 5.1.
Let y ∈ D M + for some integer M, and − M ≤ s ≤ M. Define V ( x ) and V k ( x ) by b V ( x ) = | x | s · b y ( x ) and V k ( x ) = kn V ( k x ) . Also defineT V f ( x ) = (cid:229) k ∈ Z V k ∗ f ( x ) . Then T V is bounded on H and on BMO.Proof. We verify this lemma by showing that the convolution kernel of T V has uniformly boundedFourier transform. The kernel of T V is K ( x ) = (cid:229) k ∈ Z V k ( x ) . Then | b K ( x ) | ≤ (cid:229) k ∈ Z | b V ( − k x ) | = (cid:229) k ∈ Z ( − k | x | ) s | b y ( − k x ) | . (cid:229) k ∈ Z ( − k | x | ) s min ( − k | x | , k | x | − ) M + . (cid:229) k ∈ Z min ( − k | x | , k | x | − ) . . Note that since y ∈ D M + , it follows that | b y ( x ) | ≤ min ( | x | , | x | − ) M + . It follows that T V is boundedon H and on BMO ; see [10]. (cid:3)
Proof of Theorem 1.5.
As remarked above, it is clear that P b ∈ CZO ( M + g ) for all M ≥ < g ≤
1. So it is enough to show that T ∗ ( x a ) = [[ T ]] a ∈ BMO | a | for | a | ≤ L . For f ∈ D L ,we check the first condition. D P ∗ b ( x a ) , f E = lim R → ¥ (cid:229) j ∈ Z (cid:10) Q j (cid:0) Q j b · P j f (cid:1) , h R · x a (cid:11) = lim R → ¥ (cid:229) j ∈ Z Z R n Q j b ( u ) P j f ( u ) Q j ( h R · x a )( u ) du = (cid:229) j ∈ Z Z R n Q j b ( u ) P j f ( u ) Q j ( x a )( u ) du = ARDY SPACE ESTIMATES FOR LITTLEWOOD-PALEY-STEIN SQUARE FUNCTIONS AND CALDER ´ON-ZYGMUND OPERATORS23 since Q j ( x a ) = | a | ≤ L . We also verify the BMO | a | conditions. Let | a | ≤ L , and compute (cid:10) [[ P b ]] a , y xk (cid:11) = lim R → ¥ (cid:229) j ∈ Z Z R n y j ( u − v ) Q j b ( v ) (cid:18) Z R n j j ( v − y )( u − y ) a h R ( y ) dy (cid:19) y xk ( u ) dv du = (cid:229) µ ≤ a c a , µ lim R → ¥ (cid:229) j ∈ Z Z R n y j ( u − v ) Q j b ( v ) × (cid:18) Z R n j j ( v − y )( u − v ) ( µ ) ( v − y ) a − µ h R ( y ) dy (cid:19) y xk ( u ) dv du = (cid:229) µ ≤ a c a , µ C a − µ (cid:229) j ∈ Z −| a | j Z R n y ( µ ) j ( u − v ) Q j b ( v ) y xk ( u ) dv du = (cid:229) µ ≤ a c a , µ C a − µ (cid:229) j ∈ Z −| a | j Q k Q ( µ ) j Q j b ( x ) , where y ( µ ) ( x ) = x µ y ( x ) , y ( µ ) j ( x ) = jn y ( µ ) ( j x ) , and Q ( µ ) j f ( x ) = y ( µ ) j ∗ f ( x ) . Now we consider2 | a | ( k − j ) F h Q k Q ( µ ) j Q j f i ( x ) = | a | ( k − j ) b y ( − k x ) d y ( µ ) ( − j x ) b y ( − j x ) b f ( x )= (cid:16) ( − k | x | ) −| a | b y ( − k x ) (cid:17) (cid:16) ( − j | x | ) | a | d y ( µ ) ( − j x ) b y ( − j x ) (cid:17) b f ( x )= F (cid:2) W k ∗ V j ∗ f (cid:3) ( x ) , where W and V are defined by b W ( x ) = | x | −| a | b y ( x ) , d V ( µ ) ( x ) = | x | | a | d y ( µ ) ( x ) b y ( x ) , W k ( x ) = kn W ( k x ) ,and V ( µ ) j ( x ) = jn V ( µ ) ( j x ) . Here c a , µ are binomial coefficients, and C µ = R R n j ( x ) x µ dx . ByLemma 5.1, it follows that T V ( µ ) f ( x ) = (cid:229) j ∈ Z V ( µ ) j ∗ f ( x ) defines an operator that is bounded on BMO . Then (cid:229) j ∈ Z | a | ( k − j ) Q k Q ( µ ) j Q j b ( x ) = (cid:229) j ∈ Z W k ∗ V ( µ ) j ∗ b ( x ) = W k ∗ ( T V ( µ ) b )( x ) , and we have the following Z Q (cid:229) − k ≤ ℓ ( Q ) | a | k | (cid:10) [[ P b ]] a , y xk (cid:11) | = Z Q (cid:229) − k ≤ ℓ ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) µ ≤ a c a , µ C a − µ (cid:229) j ∈ Z | a | ( k − j ) Q k Q ( µ ) j Q j b ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:229) µ ≤ a | c a , µ C a − µ | Z Q (cid:229) − k ≤ ℓ ( Q ) (cid:12)(cid:12) W k ∗ ( T V ( µ ) b )( x ) (cid:12)(cid:12) . Note that | b W ( x ) | . min ( | x | , | x | − ) as well, and since T V ( µ ) b ∈ BMO with || T V ( µ ) b || . || b || BMO , italso follows that1 | Q | Z Q (cid:229) − k ≤ ℓ ( Q ) | a | k | (cid:10) [[ P b ]] a , y xk (cid:11) | . (cid:229) µ ≤ a | c a , µ C a − µ | Z Q (cid:229) − k ≤ ℓ ( Q ) (cid:12)(cid:12) W k ∗ ( T V ( µ ) b )( x ) (cid:12)(cid:12) . || T V ( a ) b || BMO . || b || BMO . Therefore [[ P b ]] a ∈ BMO | a | for | a | ≤ L , and by Theorem 1.6 it follows that P b is bounded on H p for all nn + L + d < p ≤
1, where L = ⌊ M / ⌋ and d = ( M − L + ) / (cid:3)
6. P
ROOF OF T HEOREM
Theorem 6.1 ([13]) . Let T ∈ CZO ( M + g ) be bounded on L and define L = ⌊ M / ⌋ and d =( M − L + g ) / . If T ∗ ( x a ) = in D ′ M for all | a | ≤ L and T = in D , then T is bounded on H p for all nn + L + d < p ≤ . In the notation of [13], this theorem is stated with q =
2, 0 < p ≤ J = n / p , L = ⌊ J − n ⌋ = ⌊ n / p − n ⌋ , a =
0, and H p = ˙ F , p . Proof of Theorem 1.6.
Let T ∈ CZO ( M + g ) be bounded on L and define L = ⌊ M / ⌋ and d =( M − L + g ) /
2. Assume that T ∗ ( x a ) = D ′ M for all | a | ≤ L . Then T ∈ BMO , and by Theorem1.5 there exists P ∈ CZO ( M + ) such that P ( ) = T ( ) , P ∗ ( y a ) = | a | ≤ M , and P is boundedon H p for all nn + L + < p ≤
1. Then T = S + P , where S = T − P . Noting that S ∗ ( y a ) = | a | ≤ L and S =
0, by Theorem 6 it follows that S is bounded on H p for all nn + L + d . Therefore T isbounded on H p for all nn + L + d < p ≤ T is bounded on H p for all nn + L + d < p ≤
1. For y ∈ D L , it follows that T y ∈ H p ∩ L for all nn + L + d < p ≤
1. It is not hard to show that Z R n T y ( x ) x a dx is an absolutely convergent integral for any | a | < sup { n / p − n : nn + L + d < p ≤ } = L + d . ByTheorem 7 in [14], it follows that Z R n T y ( x ) x a dx = a ∈ N n with | a | < L + d . Since d >
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