HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS ...

HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND ´ CALDERON-ZYGMUND OPERATORS JAROD HART AND GUOZHEN LU A BSTRACT. In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calder´on-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p1 × H p2 into L p for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calder´on-Zygmund operator to be bounded from H p1 × H p2 into H p for indices smaller than 1. Subtle difficulties arise in the bilinear nature of these problems that are related to frequency properties of products of functions. Moreover, three types of bilinear paraproducts are defined and shown to be bounded from H p1 × H p2 into H p for indices smaller than 1. The first is a bilinear Bony type paraproduct that was defined in [33]. The second is a paraproduct that resembles the product of two Hardy space functions. The third class of paraproducts are operators given by sums of molecules, which were introduced in [2].

1. I NTRODUCTION There is a rich theory of Hardy spaces in harmonic analysis. Some of the early groundbreaking work in the area came from Stein, Weiss, Coifman, and C. Fefferman, among many others, see for example [49, 48, 18, 10]. In more recent years, Hardy space theory has been studied in product (aka multiparameter) settings and to a lesser extent in multlinear settings. The challenges in these areas are formidable. There are many instances where results from the classical theory, which one may initially expect to hold in the product and multilinear setting, fail. This phenomenon has been observed many times in the multiparameter setting, for example in the absence of a weak (1, 1) estimate for the strong maximal operator (see for example [37]) and the difference between various definitions of Hardy and BMO spaces in the product setting (see for example [6, 7, 39, 19, 40, 31]). This type of difficulty presents in multilinear analysis as well; of interest in this work are the failure of some boundedness properties for mutlilinear operators on products of Hardy spaces. The study of multilinear Calder´on-Zygmund operators was initiated by Coifman and Meyer [12, 13, 14]. A fruitful theory has grown around these operators, see for example [9, 26, 27, 23, 2, 43, 29, 36, 34, 41]. A multilinear and multi-parameter version of the Coifman-Meyer Fourier multiplier theorem was established in [44, 45] using time-frequency analysis (see also [8] using the Littlewood-Paley analysis), and a pseudo-differential analogue was carried out in [15]. There has also been some work done for the operators in the context of distributions spaces (TriebelLizorkin and Besov spaces), see for example [28, 1, 43, 2]. For appropriate indices, some of these distribution spaces coincide with Hardy spaces, which are the focus of this work. One of the main results we prove in this article is a bilinear T 1 type theorem that extends the bilinear T 1 theorem Date: June 9, 2015. 2010 Mathematics Subject Classification. 42B20, 42B25, 42B30. 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 #DMS1301595.

1

2

JAROD HART AND GUOZHEN LU

from [9, 26, 27, 33] to a Hardy space setting. This work also provides bilinear versions of the linear results in [18, 50, 22, 20, 35]; in particular there is a close parallel with the Hardy space results in [35]. There has also been a considerable development of bilinear Littlewood-Paley-Stein theory in recent years, see for example [42, 43, 32, 29, 25, 5, 24]. All of these articles deal with Littlewood-Paley-Stein operator mapping properties from L p1 × L p2 into L p for p1 , p2 > 1. Here we extend this to H p1 × H p2 to L p boundedness properties for indices 0 < p1 , p2 ≤ 1. To highlight a challenge that arises in bilinear Hardy space theory that is not present in the linear theory, consider the following problem, to which we give a solution in this work. Given a bilinear operator T acting on a product of Hardy spaces H p1 × H p2 , what conditions are sufficient for T to map H p1 × H p2 into H p ? At the root of this problem is a simple, but menacing, fact. Given a nonzero real-valued function f ∈ H 1 (with sufficient decay so that f ∈ L1/2 ), it follows that f · f = f 2 ∈ / H 1/2 . Note that it is that fact that f 2 ≥ 0 has non-zero integral that bars it from membership in H 1/2 (any integrable function in H 1/2 must have mean zero). This is a failure of a bilinear analogue of classical theory in the following sense. The operator P( f1 , f2 )(x) = f1 (x) f2 (x) is analogous to the identity operator I f (x) = f (x) in the linear theory. Clearly the identity operator I is bounded on any reasonable function space, including H p for all 0 < p ≤ 1, and the bilinear operator P enjoys many similar properties to I. For example P is bounded from L p1 × L p2 into L p for all 0 < p1 , p2 , p < ∞ satisfying p11 + p12 = 1p (this is just H¨older’s inequality), and by a result in [23], P is also bounded from H p1 × H p2 into L p for any 0 < p1 , p2 , p ≤ 1 such that p11 + p12 = 1p . Although, the product operator P fails to satisfy many Hardy space bounds that one might initially expect, and this presents a much more difficult problem. Namely, P is not bounded from H p1 ×H p2 into H p , whenever 0 < p1 , p2 , p ≤ 1 and p11 + p12 = 1p . The product structure of bilinear operators severely complicates oscillatory properties of functions. For this reason, addressing Hardy space H p1 × H p2 to H p bounds for bilinear operators is connected to the deep mathematical problem of understanding the oscillatory behavior of products of functions. One result in this work (Theorem 2.5) is a paraproduct operator Π( f1 , f2 ) that resembles the product P( f1 , f2 ) = f1 · f2 in some senses, but satisfies Π( f1 , f2 ) ∈ H 1/2 for all f1 , f2 ∈ H 1 , along with other properties. The example in the last paragraph gives some initial insight into what we can expect as far as results for the bilinear singular integral operators we work with in this article. Consider again the pointwise product operator P( f1 , f2 )(x) = f1 (x) f2 (x). This is arguably the simplest bilinear operator that one can consider, but it is not bounded from H p1 × H p2 into H p for 0 < p ≤ 1. So the conditions we impose on our bilinear operators below must not be verified by P. The appropriate bilinear operators to have the Hardy space boundedness properties must “improve” the functions f1 and f2 in the following sense. Formally, f1 ∈ H p1 and f2 ∈ H p2 must have some regularity and vanishing moment properties up to order (comparable to) 1/p1 and 1/p2 respectively. Since we wish to obtain T ( f1 , f2 ) ∈ H p , T ( f1 , f2 ) must satisfy better regularity and vanishing moment properties than either f1 or f2 (roughly speaking, up to order 1/p = 1/p1 + 1/p2 ). This precludes the typical paradigm for interpreting the regularity properties for inputs and outputs of a bilinear Calder´on-Zygmund operators as set by the product operator P. That is, one typically considers T ( f1 , f2 ) to satisfy the regularity properties that f1 (x) · f2 (x) would satisfy; which means that a general bilinear Calder´on-Zygmund operator cannot be smoothing. We look at a smaller class of bilinear Calder´on-Zygmund operators that are smoothing in some sense; hence from this point of view we can again rule out the product operator P as a representative example. An example of a

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 3

smoothing bilinear Calder´on-Zygmund operator was given in [2] by a certain paraproduct, in which p p p the authors prove boundedness properties on homogeneous Sobolev spaces, from L˙ s 1 × L˙ t 2 to L˙ s+t . We consider these smoothing operators since the output has regularity s + t, whereas the inputs, p p f1 ∈ L˙ s 1 and f2 ∈ L˙ t 2 , have lesser regularity, s and t respectively. A general Calder´on-Zygmund operator that is bounded L2 ×L2 to L1 does not satisfy this type of smoothing boundedness property. Later we apply our main Calder´on-Zygmund operator result (Theorem 2.2) to the paraproducts from [2] to prove that they are also bounded from H p1 × H p2 into H p under appropriate conditions, see Theorem 2.6. There has been work done on bilinear operators that are related to the Hardy space mapping problem we are considering that also hint at what conditions may be needed for bilinear operators to be bounded in H p for 0 < p ≤ 1. In [23], Grafakos and Kalton give conditions for a bilinear Calder´on-Zygmund operator to be bounded from H p1 × H p2 into L p (we restrict to the bilinear setting for this discussion). In fact, they only require that T is bounded from L2 × L2 into L1 and additional kernel regularity for this conclusion. There is no further cancellation required for T beyond what is necessary for Calder´on-Zygmund operators to be bounded on Lebesgue spaces with indices larger than 1, see for example [9, 26, 27, 33]. Note that the product operator still falls into the class of operators to which the results in [23] apply; as previously mentioned P is bounded from H p1 × H p2 into L p for indices smaller than 1. This highlights the drastic difference between the work in [23] and this article. Another work that is closely related to this article is [36]. In that work, the authors prove some H p1 × H p2 to H p estimates when p ≤ 1. Their result, which is for p close to 1, relies heavily on atomic decompositions of Hardy spaces and the boundedness results from [23]. We give sufficient conditions for a bilinear operator T to be decomposed in terms of Littelwood-Paley-Stein theory, and prove estimates for Hardy spaces with indices p ranging all the way down to zero without using atomic decompositions. This is inspired by the works in the multi-parameter Hardy space theory in [30, 31] where a discrete Littlewood-Paley theory is carried out to prove boundedness of singular integral operators on multi-parameter Hardy spaces. 2. M AIN R ESULTS We take a moment to describe our general approach to the results in this article. Our primary goal is to prove H p1 × H p2 to H p estimates for bilinear Calder´on-Zygmund operators. The way we approach this is to decompose T into smooth truncation operators Θk = Qk T for k ∈ Z, and reduce the H p estimates for T to estimates in for Θk using the Littlewood-Paley characterization of H p from [18]. That is, we choose Qk to be Littlewood-Paley-Stein type operators, sufficient for the following semi-norm equivalence, ! 1 ! 1 2 2 2 2 = ||SΘ ( f1 , f2 )||L p , ||T ( f1 , f2 )||H p ≈ ∑ |Qk T ( f1 , f2 )| = |Θ ( f , f )| 1 2 k ∑ k∈Z p k∈Z p L

L

where SΘ is the square function associated to the collection Θk = Qk T . These operators Θk for k ∈ Z define what we call a collection of bilinear Littlewood-Paley-Stein operators. In this way we reduce our H p1 × H p2 to H p estimates for T to H p1 × H p2 to L p estimates for SΘ . The latter estimates for SΘ are the square function type estimates that we will prove, thereby yielding the Calder´on-Zygmund operator estimates as well. We also obtain paraproduct boundedness properties by applying our Calder´on-Zygmund operator estimates.

4

JAROD HART AND GUOZHEN LU

2.1. Bilinear Littlewood-Paley-Stein Theory. Given kernel functions θk : R3n → C for k ∈ Z, define Z

Θk ( f1 , f2 )(x) =

R2n

θk (x, y1 , y2 ) f1 (y1 ) f2 (y2 )dy1 dy2

for appropriate functions f1 , f2 : Rn → C. Define the square function associated to {Θk } !1 2

2

∑ |Θk ( f1, f2)(x)|

SΘ ( f1 , f2 )(x) =

.

k∈Z

We say that a collection of operators Θk for k ∈ Z is a collection of bilinear Littlewood-Paley-Stein operators with decay and smoothness (N, L), written {Θk } ∈ BLPSO(N, L), for an integer L ≥ 0 and N > 0 if there exists a constant C such that (2.1)

|Dα1 D2 θk (x, y1 , y2 )| ≤ C2(|α|+|β|)k ΦN k (x − y1 , x − y2 ) for all |α|, |β| ≤ L. β

2kn (1 + 2k |x| + 2k |y|)−N for N > 0, x, y ∈ Rn , and k ∈ Here we use the notation ΦN k (x, y) = 2 α Z. We also use the notation D0 F(x, y1 , y2 ) = ∂αx F(x, y1 , y2 ), Dα1 F(x, y1 , y2 ) = ∂αy1 F(x, y1 , y2 ), and Dα2 F(x, y1 , y2 ) = ∂αy2 F(x, y1 , y2 ) for F : R3n → C and α ∈ Nn0 . Given {Θk } ∈ BLPSO(N, L) and α, β ∈ Nn0 with |α| + |β| < N − 2n, define the (α, β) order moment function associated to {Θk } by

[[Θk ]]α,β (x) = 2

k(|α|+|β|)

Z R2n

θk (x, y1 , y2 )(x − y1 )α (x − y1 )β dy1 dy2

for k ∈ Z and x ∈ Rn . It is worth noting that [[Θk ]]0 (x) = Θk (1, 1)(x), which is an object that is closely related to boundedness properties of SΘ , see for example [42, 43, 32, 25, 29, 24]. Our main square function boundedness result is the following theorem. Theorem 2.1. Let L ≥ 1 be an integer and N = 2n + L(2n + L + 5)/2. If {Θk } ∈ BLPSO(N, L) and (2.2)

dµ(x,t) =

∑

∑

|[[Θk ]]α,β (x)|2 δt=2−k dx

k∈Z |α|,|β|≤L−1

is a Carleson measure, then SΘ can be extended to a bounded operator from H p1 × H p2 into L p for n n < p ≤ 1 and n+L < p1 , p2 ≤ 1 satisfying p11 + p12 = 1p . all 2n+L Here δt=2−k is the delta point mass measure on (0, ∞) concentrated at 2−k . That is, for a set E ⊂ (0, ∞), the measure δt=2−k is defined by δt=2−k (E) = 1 if 2−k ∈ E and δt=2−k (E) = 0 if 2−k ∈ / E. n+1 n Also, a non-negative measure dµ(x,t) on R+ = R × (0, ∞) is a Carleson measure if there is a C > 0 such that dµ(Q × (0, `(Q))) ≤ C|Q| for all cubes Q ⊂ Rn , where `(Q) is the side length of Q and |Q| is the Lebesgue measure of Q.

2.2. Bilinear Calder´on-Zygmund Theory. Let S = S (Rn ) be the Schwartz class of smooth, rapidly decreasing functions endowed with the standard Schwartz semi-norm topology, and let S 0 = S 0 (Rn ) be its continuous dual, the class of tempered distributions. We say that a continuous bilinear operator T from S × S into S 0 is a bilinear Calder´on-Zygmund operator with smoothness M, written T ∈ BCZO(M), if T has function kernel K : R3n \{(x, x, x) : x ∈ Rn } → C such

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 5

that hT ( f1 , f2 ), gi =

Z R3n

K(x, y1 , y2 ) f1 (y1 ) f2 (y2 )g(x)dy1 dy2 dx

/ and there is a whenever f1 , f2 , g ∈ C0∞ have disjoint support, supp( f1 ) ∩ supp( f2 ) ∩ supp(g) = 0, constant C > 0 such that the kernel function K satisfies C β µ |Dα0 D1 D2 K(x, y1 , y2 )| ≤ for all |α|, |β|, |µ| ≤ M (|x − y1 | + |x − y2 |)2n+|α|+|β|+|µ| for x, y1 , y2 ∈ Rn such that |x − y1 | + |x − y2 | > 0. Define the transposes of T by the dual relations  

hT ( f1 , f2 ), gi = T ∗1 (g, f2 ), f1 = T ∗2 ( f1 , g), f2 for f1 , f2 , g ∈ S , which also satisfy T ∗ j ∈ BCZO(M) for j = 1, 2 whenever T ∈ BCZO(M). We also define moment distributions for an operator T ∈ BCZO(M), but we require some notation first. For an integer M ≥ 0, define the collections of functions   n ∞ n −M OM = OM (R ) = f ∈ C (R ) : sup | f (x)| · (1 + |x|) < ∞ and x∈Rn   Z n ∞ n α DM = DM (R ) = f ∈ C0 (R ) : f (x)x dx = 0 for all |α| ≤ M . Rn

Define the topology of DM by the sequential characterization, for fk , f ∈ DM for k ∈ N, fk → f in DM if there exists a compact set K such that supp( fk ), supp( f ) ⊂ K for all k ∈ N and lim ||Dα fk − Dα f ||L∞ = 0

k→∞

0 is defined to be all linear functionals W : D → C such that for all α ∈ Nn0 . Then DM M

fk → f in DM implies hW, fk i → hW, f i . Let η ∈ C0∞ (Rn ) be supported in B(0, 2) and η(x) = 1 for x ∈ B(0, 1). Define for R > 0, ηR (x) = η(x/R). We reserve the notation ηR for functions constructed in this way. In [1], B´enyi defined T ( f1 , f2 ) for f1 ∈ OM1 and f2 ∈ OM2 where T is a bilinear singular integral operator. We give an equivalent definition. Let T be a BCZO(M + 1) and f1 ∈ OM1 and f2 ∈ OM2 for some integers M1 , M2 ≥ 0 such that M1 + M2 ≤ M. For ψ ∈ DM , define hT ( f1 , f2 ), ψi = lim hT (ηR f1 , ηR f2 ), ψi . R→∞

Also, for

f1 ∈ OM , f2 ∈ C0∞ ,

and ψ ∈ DM , define hT ( f1 , f2 ), ψi = lim hT (ηR f1 , f2 ), ψi . R→∞

These limits exist based on the kernel representation and kernel properties for T ∈ BCZO(M + 1), 0 see [1] for proof of this fact. Now we define the moment distribution [[T ]]α,β ∈ D|α|+|β| for T ∈ BCZO(M + 1) and α, β ∈ Nn0 with |α| + |β| ≤ M by Z

 [[T ]]α,β , ψ = lim K (x, y1 , y2 )(x − y1 )α ηR (y1 )(x − y2 )β ηR (y2 )ψ(x)dy1 dy2 dx R→∞ R3n K ∈ S 0 (R3n )

for ψ ∈ DM . Here is the distribution kernel of T , and the integral above is interpreted as a dual pairing between K ∈ S 0 (R3n ) and elements of S (R3n ). This distributional moment associated to T generalizes the notion of T (1, 1) as used in [9, 26, 27, 33] in the sense that

6

JAROD HART AND GUOZHEN LU

h[[T ]]0,0 , ψi = hT (1, 1), ψi for all ψ ∈ D0 . We will also use a generalized notion of BMO to extend the cancellation conditions T (1, 1), T ∗1 (1, 1), T ∗2 (1, 1) ∈ BMO, which were used in the bilinear T 1 0 /P , distributions D 0 modulo theorems from [9, 26, 27, 33]. Let M ≥ 0 be an integer and F ∈ DM M polynomials. We say that F ∈ BMOM if the non-negative measure on Rn+1 , +

∑ 22Mk |Qk F(x)|2dx δt=2−k k∈Z

is a Carleson measure. This definition agrees with the classical definition of BMO by the characterization of BMO in terms of Carleson measures in [4, 38]. It should be noted that we defined this polynomial growth BMOM space in [35], and a similar polynomial growth BMOM was defined by Youssfi [51]. We use this polynomial growth BMOM to quantify cancellation conditions for operators T ∈ BCZO(M) in the following result. Theorem 2.2. Assume that T ∈ BCZO(M), where M = L(2n + L + 5)/2 for some integer L ≥ 1, and that T is bounded from L2 × L2 into L1 . If (2.3)

0 T ∗1 (xα , ψ) = T ∗2 (ψ, xα ) = 0 in D|α| for all |α| ≤ 2L + n and ψ ∈ D2L+n ,

(2.4)

[[T ]]α,β ∈ BMO|α|+|β| for all |α|, |β| ≤ L − 1,

then T can be extended to bounded operator from H p1 × H p2 into H p for all n 1 1 1 n+L < p1 , p2 ≤ 1 satisfying p1 + p2 = p .

n 2n+L

< p ≤ 1 and

We prove Theorem 2.2 by decomposing an operator T ∈ BCZO(M) into a collection of operators {Θk } ∈ BLPSO(2n + M, L) in Theorem 2.1, where L depends on the regularity parameter M. This decomposition of T into a collection of bilinear Littlewood-Paley-Stein operators is stated precisely in the next theorem. Theorem 2.3. Let L ≥ 1 be an integer, N > 2n, and T ∈ BCZO(M) for some integer M ≥ max(N − 2n, 2L + 1). Also assume that T is bounded from L2 × L2 into L1 . If T satisfies (2.3), then {Θk } = {Qk T } ∈ BLPSO(N, L) for any ψ ∈ DM , where Qk f = ψk ∗ f and ψk (x) = 2kn ψ(2k x). 2.3. Applications to Paraproducts. In this article, we use Theorem 2.2 to prove that three types of paraproducts are bounded on Hardy spaces. The first is a bilinear Bony type paraproduct, which is a bilinear version of Bony’s paraproduct in [3] and was originally introduced in [33]. It is constructed as follows. Let ψ ∈ DM and ϕ ∈ C0∞ , and define ψk (x) = 2kn ψ(2k x), ϕk (x) = 2kn ϕ(2k x), Qk f = ψk ∗ f , and Pk f = ϕk ∗ f . For b ∈ BMO, define the bilinear Bony paraproduct
(2.5) Πb ( f1 , f2 )(x) = ∑ Q j Q j b · Pj f1 · Pj f2 (x). j∈Z

If ψ and ϕ are chosen appropriately, then this paraproduct satisfies Πb (1, 1) = b, see [33]. We are not concerned with this particular property here, so we allow for a more general selection of ψ and ϕ. We will apply Theorem 2.2 to Πb to prove the following theorem. Theorem 2.4. Let L be a non-negative integer, b ∈ BMO, ψ ∈ D3L+n , and ϕ ∈ C0∞ , then Πb , as n n defined in (2.5), is bounded from H p1 × H p2 into H p for all 2n+L < p ≤ 1 and n+L < p1 , p2 ≤ 1 1 1 1 satisfying p = p1 + p2 .

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 7

The motivation for our second paraproduct operator is to find a replacement for the product f1 (x) f2 (x) that is in H p for 0 < p ≤ 1 for appropriate f1 and f2 . Once again, we construct this operator a bit more generally. We will use the following definition for the Fourier transform; for f ∈ L1 , define

F [ f ](ξ) = fb(ξ) =

Z Rn

f (x)e−ixξ dx.

b is supported away from the origin, and let ϕ ∈ S . Let Let ψ ∈ S such that its Fourier transform ψ ψk , ϕk , Qk , and Pk be as above. Define Π( f1 , f2 )(x) =

(2.6)

∑ Qk (Pk f1 · Pk f2) (x). k∈Z

We prove the following Hardy space bounds for Π. Theorem 2.5. Let Π( f1 , f2 ) be as in (2.6). Then Π is bounded from H p1 × H p2 into H p for all 0 < p1 , p2 , p ≤ 1 satisfying 1p = p11 + p12 . In order to construct the paraproduct Π( f1 , f2 ) to resemble the product f1 · f2 , we choose ψ and b(ξ) = 1 for |ξ| ≤ 1 and supp(ϕ b) ⊂ B(0, 2). Define ϕ in the following way. Let ϕ ∈ S such that ϕ ψ(x) = 2−2n ϕ(2−2 x)−2−3n ϕ(2−3 x). With this choice of ψ and ϕ, it follows that Qk = Pk Qk . Under these conditions, it also follows that Π(1, f2 )(x) =

∑ Qk (Pk (1) · Pk f2) (x) = ∑ Qk Pk f2(x) = ∑ Qk f2(x) = f2(x) and k∈Z

Π( f1 , 1)(x) =

k∈Z

∑ Qk (Pk f1 · Pk (1)) (x) = ∑ Qk Pk f1(x) = ∑ Qk f1(x) = f1(x) k∈Z

Hp

k∈Z

k∈Z

k∈Z

H p ∩ L2 .

in for all f1 , f2 ∈ This gives precise meaning to how Π( f1 , f2 )(x) “resembles” the product f1 (x) · f2 (x). We should remark here that the constructions in Theorems 2.4 and 2.5 use slightly different techniques, but are interchangeable in some senses. In Theorem 2.4, we choose the convolution n kernels ψk ∈ D3L+n , and conclude bounds for Hardy spaces with indices bounded below by 2n+L n , where L can be taken arbitrarily large. One can construct the bilinear Bony paraproduct and n+L b is supported away from the origin, and the conclusion is strengthened Πb with ψ ∈ S such that ψ to all 0 < p1 , p2 , p ≤ 1 similar to Theorem 2.5. Similarly, the paraproduct Π can be constructed with functions ψk ∈ DM , and obtain the same conclusion as in Theorem 2.4 with the appropriate lower bounds for 0 < p1 , p2 , p ≤ 1. Finally, we consider a class of paraproducts defined in [2]. For a dyadic cube Q, a function φQ : Rn → C is an (M, N)-smooth molecule associated to Q if there exists a constant C = CM,N independent of Q such that `(Q)−n/2 `(Q)−|α| |D φQ (x)| ≤ C (1 + `(Q)−1 |x − xQ |)N α

for all |α| ≤ M, where xQ denotes the lower-left corner of Q. A family of (M, N)-smooth molecules ψQ indexed by dyadic cubes Q is an (M, N, L)-smooth family of molecules with cancellation if ψQ is an (M, N)-smooth molecule associated to Q and Z

(2.7)

Rn

ψQ (x)xα dx = 0

8

JAROD HART AND GUOZHEN LU

for all |α| ≤ L. Let φ1Q , φ2Q , φ3Q be three families of molecules indexed by dyadic cubes Q, and define the paraproduct T , as in [2], by



 T ( f1 , f2 )(x) = ∑ f1 , φ1Q f2 , φ2Q φ3Q (x). Q

The sum here indexed by Q is over all dyadic cubes. In [2], the authors prove the boundedness of these operators on many different function spaces, for example from L p1 × L p2 → L p , H p1 × H p2 → L p , L∞ × L∞ → BMO, as well as a number of estimates for weak L p spaces and weighted spaces. Here we extend their results to boundedness properties from products of Hardy spaces into Hardy spaces. Theorem 2.6. Let L ≥ 0 be an integer, M = L(2n + L + 5)/2, and N > 10n + 15M + 5. Assume that φ1Q , φ2Q , φ3Q are three families of (M, N)-smooth molecules. Furthermore assume that φ3Q is a collection of (M, N, 2L + n)-smooth molecules with cancellation, and either φ1Q or φ2Q is a family of (M, N, L − 1)-smooth molecules with cancellation. Then T can be extended to a bounded operator n n < p ≤ 1 and n+L such that p11 + p12 = 1p . from H p1 × H p2 into H p for all 2n+L The selection of M and N in this result are not optimal. Here we use the selection of parameters that are sufficient to show that T ∈ BCZO(M) based on the work in [2], in which the authors note the parameters are not chosen to be optimal. This article is organized in the following way. In Section 3, we establish some notation, and state some preliminary results. Section 4 is dedicated to proving the bilinear square function estimates in Theorem 2.1. Section 5 is used to prove our bilinear Calder´on-Zygmund operator estimates in Theorems 2.2 and 2.3. Finally in Section 6, we apply Theorem 2.2 to the paraproduct operators in Theorems 2.4, 2.5, and 2.6. 3. P RELIMINARIES We use the notation A . B to mean that A ≤ CB for some constant C. The constant C is allowed to depend on the ambient dimension, smoothness and decay parameters of our operators, indices of function spaces etc.; in context, the dependence of the constants is clear. We will use the following Frazier and Jawerth type discrete Calder´on reproducing formula [21] (see also [30] for a multiparameter formulation of this reproducing formula): there exist φ j , φ˜ j ∈ S for j ∈ Z with infinite vanishing moment such that (3.1) f (x) = ∑ ∑ |Q| φ j (x − cQ)φ˜ j ∗ f (cQ) in L2 j∈Z `(Q)=2−( j+N0 )

for f ∈ L2 . The summation in Q here is over all dyadic cubes with side length `(Q) = 2−( j+N0 ) for some N0 > n, and cQ denotes the center of cube Q. We will also use a more traditional formulation of Calder´on’s reproducing formula: fix ϕ ∈ C0∞ (B(0, 1)) such that

∑ Qk f = f in L2

(3.2)

k∈Z

∈ L2 ,

ψ(x) = 2n ϕ(2x) − ϕ(x),

for f where ψk (x) = 2kn ψ(2k x), and Qk f = ψk ∗ f . Furthermore, we can assume that ψ has an arbitrarily large, but fixed, number of vanishing moments.

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 9

There are many equivalent definitions of the real Hardy spaces H p = H p (Rn ) for 0 < p < ∞. We use the following one. Define the non-tangential maximal function Z ϕ −n −1 N f (x) = sup sup t ϕ(t (y − u)) ∗ f (u)du , Rn t>0 |x−y|≤t

where ϕ ∈ S with non-zero integral. It was proved by Fefferman and Stein in [18] that one can define || f ||H p = ||N ϕ f ||L p to obtain the classical real Hardy spaces H p for 0 < p < ∞. It was also proved in [18] that for any ϕ ∈ S and f ∈ H p for 0 < p < ∞, sup |ϕk ∗ f | . || f ||H p . k∈Z

Lp

We will use a number of equivalent quasi-norms for H p . Let ψ ∈ DM for some integer M > n(1/p− 1), and let ψk and Qk be as above. For f ∈ S 0 /P (tempered distributions modulo polynomials), f ∈ H p if and only if ! 1 2 ∑ |Qk f |2 < ∞, p k∈Z L space H p

and this quantity is comparable to || f || . The can also be characterized by the operators φ j and φ˜ j from the discrete Littlewood-Paley-Stein decomposition in (3.1). This characterization is given by the following, which can be found in [30]. Given 0 < p < ∞   1 2  2 ˜  | φ ∗ f (c )| χ ∑ ≈ || f ||H p , j Q Q ∑ j∈Z −( j+N ) 0 `(Q)=2 p Hp

L

where χE (x) = 1 for x ∈ E and χE (x) = 0 for x ∈ / E for a subset E ⊂ Rn . For a continuous function 1 (Rn ) and 0 < r < ∞, define f ∈ Lloc   r   1r   r     M j f (x) = M (3.3) (x) , ∑ f (cQ)χQ   −( j+N ) `(Q)=2

0

where M is the Hardy-Littlewood maximal operator. The following estimate was also proved by Han and Lu in [30]. Proposition 3.1. For 0 < r < p ≤ 1 and f ∈ H p ! 1 2
∑ M jr (φ˜ j ∗ f ) 2 j∈Z

. || f ||H p ,

Lp

where

M jr

is defined as in (3.3).

The next result is a reformulation of an estimate proved by Han and Lu in [30]; this version of the result was proved in [35].

10

JAROD HART AND GUOZHEN LU

Proposition 3.2. Let f : Rn → C a non-negative continuous function, ν > 0, and Then

n n+ν

< r ≤ 1.

max(0, j−k)(N−n) |Q| Φn+ν M jr f (x) min( j,k) (x − cQ ) f (cQ ) . 2

∑

`(Q)=2−( j+N0 )

for all x ∈ Rn , where M jr is defined in (3.3) and the summation indexed by `(Q) = 2−( j+N0 ) is the sum over all dyadic cubes with side length 2−( j+N0 ) and cQ denotes the center of cube Q. The next result is proved using some well-known techniques for Carleson measure. The proof can be found in [35]. Proposition 3.3. Suppose dµ(x,t) =

(3.4)

∑ µk (x)δt=2−k dx k∈Z

is a Carleson measure, where µk is a non negative, locally integrable function for all k ∈ Z. Also let ϕ ∈ S , and define Pk f = ϕk ∗ f , where ϕk (x) = 2kn ϕ(2k x) for k ∈ Z. Then ! 1 p ∑ |Pk f | p µk . || f ||H p for all 0 < p < ∞ k∈Z p L

and

! 1 2 2 ∑ |Pk f | µk k∈Z

. || f ||H p

for all 0 < p ≤ 2.

Lp

We will also need Hardy space estimates for linear Littlewood-Paley-Stein square function operators that was proved in [35]. First we set some notation for linear Littlewood-Paley-Stein operators. Given kernel functions λk : R2n → C for k ∈ Z, define Z

Λk f (x) =

Rn

λk (x, y) f (y)dy

for appropriate functions f : Rn → C. Define the square function associated to {Λk } !1 2

SΛ f (x) =

2

∑ |Λk f (x)|

.

k∈Z

We say that a collection of operators Λk for k ∈ Z is a collection of Littlewood-Paley-Stein operators with decay and smoothness (N, L), written {Λk } ∈ LPSO(N, L), for an integer L ≥ 0 and N > 0 if there exists a constant C such that (3.5)

|Dα1 λk (x, y)| ≤ C2|α|k ΦN k (x − y) for all |α| ≤ L.

kn k −N for N > 0, x ∈ Rn , and k ∈ Z. We also Here we use the notation ΦN k (x) = 2 (1 + 2 |x|) α α α write D0 F(x, y) = ∂x F(x, y) and D1 F(x, y) = ∂y F(x, y) for F : R2n → C and α ∈ Nn0 . This is a slightly different definition for LPSO(N, L) than what was used in [35]. In that article, we defined LPSO(N, L + δ) for integers L ≥ 0 and 0 < δ ≤ 1 in terms of δ-H¨older conditions on λk (x, y) in the y variable. Here we require {Λk } ∈ LPSO(N, L) to have kernel that is L times differentiable in y instead of L − 1 times differentiable with Lipschitz L − 1 order derivatives as in [35]. It is not

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 11

hard to see that the definition we use for LPSO(N, L) here is contained in the class defined in [35]; hence all results from [35] for LPSO(N, L) are still applicable in the current work. Given {Λk } ∈ LPSO(N, L) and α ∈ Nn0 with |α| < N − n, define [[Λk ]]α (x) = 2k|α|

Z Rn

λk (x, y)(x − y)α dy

for k ∈ Z and x ∈ Rn . The next theorem was also proved in [35]. Theorem 3.4. Let {Λk } ∈ LPSO(n + 2L, L) for some integer L ≥ 1. If dµα (x,t) =

(3.6)

∑ |[[Λk ]]α(x)|2δt=2−k dx k∈Z

is a Carleson measure for all n n+L < p ≤ 1.

α ∈ Nn0

with |α| ≤ L − 1, then SΛ is bounded from H p into L p for all

Note that we use Λk to denote linear operators and Θk to denote bilinear operators. We will keep this convention throughout the paper. 4. H ARDY S PACE E STIMATES FOR B ILINEAR S QUARE F UNCTIONS To start this section, we prove a reduced version of Theorem 2.1, where we strengthen the cancellation assumptions on Θk from the ones in (2.2) to the conditions in the next lemma. Once we establish Lemma 4.1, we extend to the general situation in Theorem 2.1 by using a paraproduct type decomposition. Lemma 4.1. Let {Θk } ∈ BLPSO(N, L), where N = 2n + 3L for some integer L ≥ 1. If Z Rn

θk (x, y1 , y2 )yα1 dy1 =

Z Rn

θk (x, y1 , y2 )yα2 dy2 = 0

for all k ∈ Z and |α| ≤ L − 1, then SΘ can be extended to a bounded operator from H p1 × H p2 into n n L p for all 2n+L < p ≤ 1 and n+L < p1 , p2 ≤ 1 such that 1p = p11 + p12 . Proof. Let p, p1 , p2 be as above. Choose ν such that np − 2n < ν < L, which is possible since n n 1 1 1 n 2n+L < p. Since ν > p − 2n and p1 + p2 = p , there exists s ∈ (0, 1) such that sν > p1 − n and n n (1 − s)ν > pn2 − n. Finally fix r1 , r2 > 0 such that n+sν < r1 < p1 and n+(1−s)ν < r2 < p2 . Note that this also implies that n + sν ≤ n/p1 < n + L and n + (1 − s)ν ≤ n/p2 < n + L, which will be used n+(1−s)ν n+sν later to conclude that Φn+L (x) and Φn+L (x). k (x) ≤ Φk k (x) ≤ Φk By density, it is sufficient to prove the appropriate estimate for fi ∈ H pi ∩ L2 for i = 1, 2. We decompose Θk ( f1 , f2 ) using (3.1) for each f1 and f2 , Θk ( f1 , f2 )(x) =

∑ ∑

j1 , j2 ∈Z Q1 ,Q2

=

∑ ∑

j1 , j2 ∈Z Q1 ,Q2

cQ

cQ

|Q1 | |Q2 |φ˜ j1 ∗ f1 (cQ1 )φ˜ j2 ∗ f2 (cQ2 )Θk (φ j1 1 , φ j2 2 )(x)

|Q1 | |Q2 |φ˜ j1 ∗ f1 (cQ1 )φ˜ j2 ∗ f2 (cQ2 )

Z R2n

cQ

cQ

θk (x, y1 , y2 )φ j1 1 (y1 )φ j2 2 (y2 )dy1 dy2 .

The summation in Q1 and Q2 are over all dyadic cubes with side lengths `(Q1 ) = 2−( j1 +N0 ) and `(Q2 ) = 2−( j2 +N0 ) respectively. Now we establish an almost orthogonally estimate for the integral term in the previous equation. Using the vanishing moment properties of θk and the regularity of

12

JAROD HART AND GUOZHEN LU

φ j1 , we obtain the following. Z cQ2 cQ1 (y )dy dy (y )φ θ (x, y , y )φ 2 1 2 1 1 2 k j2 j1 R2n ! α φcQ1 (x) Z D c c j1 Q Q = θk (x, y1 , y2 ) φ j1 1 (y1 ) − ∑ (y1 − x)α φ j2 2 (y2 )dy1 dy2 R2n α! |α|≤L−1 Z

.

j1 L ΦN k (x − y1 , x − y2 )(2 |x − y1 |)   n+L n+L × Φ j1 (y1 − cQ1 ) + Φ j1 (x − cQ1 ) Φn+L j2 (y2 − cQ2 )dy1 dy2

R2n

.2

L( j1 −k)

Z R2n

Φ2n+2L (x − y1 , x − y2 ) k

  n+L n+L × Φ j1 (y1 − cQ1 ) + Φ j1 (x − cQ1 ) Φn+L j2 (y2 − cQ2 )dy1 dy2 n+(1−s)ν

. 2L( j1 −k) Φn+sν min( j1 ,k) (x − cQ1 )Φmin( j2 ,k) (x − y2 ). A similar estimate holds for 2L( j2 −k) in place of 2L( j1 −k) . Using the vanishing moment properties of φ j1 , we have the following estimate, Z cQ1 cQ2 θ (x, y , y )φ (y )φ (y )dy dy 1 2 1 2 1 2 k j1 j2 R2n ! Z α θ (x, c , y ) D c c Q1 2 Q Q 1 k θk (x, y1 , y2 ) − ∑ = (y1 − cQ1 )α φ j1 1 (y1 )φ j2 2 (y2 )dy1 dy2 R2n α! |α|≤L−1 Z

.

R2n

k L N+L N ΦN k (x − y1 , x − y2 )(2 |y1 − cQ1 |) Φ j1 (y1 − cQ1 )Φ j2 (y2 − cQ2 )dy1 dy2

× .2

L(k− j1 )

Z R2n

Z R2n

k L N+L N ΦN k (x − cQ1 , x − y2 )(2 |y1 − cQ1 |) Φ j1 (y1 − cQ1 )Φ j2 (y2 − cQ2 )dy1 dy2 N N ΦN k (x − y1 , x − y2 )Φ j1 (y1 − cQ1 )Φ j2 (y2 − cQ2 )dy1 dy2 n+(1−s)ν

. 2L(k− j1 ) Φn+sν min( j1 ,k) (x − cQ1 )Φmin( j2 ,k) (x − cQ2 ). Once again, this estimate holds with 2L(k− j2 ) in place of 2L(k− j1 ) . Therefore Z cQ1 cQ2 θ (x, y , y )φ (y )φ (y )dy dy 1 2 1 2 1 2 k j1 j2 R2n n+(1−s)ν

. 2−L max(| j1 −k|,| j2 −k|) Φn+sν min( j1 ,k) (x − cQ1 )Φmin( j2 ,k) (x − cQ2 ).

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 13

Applying Proposition 3.2 yields |Θk ( f1 , f2 )(x)| .

∑ ∑

|Q1 | |Q2 |φ˜ j1 ∗ f1 (cQ1 )φ˜ j2 ∗ f2 (cQ2 )

j1 , j2 ∈Z Q1 ,Q2 n+(1−s)ν

× 2−L max(| j1 −k|,| j2 −k|) Φn+sν min( j1 ,k) (x − cQ1 )Φmin( j2 ,k) (x − cQ2 ) .

∑

2−L max(| j1 −k|,| j2 −k|) 2sν max(0,k− j1 ) 2(1−s)ν max(0,k− j2 ) M jr11 (φ˜ j1 ∗ f1 )(x)M jr22 (φ˜ j2 ∗ f2 )(x)

∑

2−ε max(| j1 −k|,| j2 −k|) M jr11 (φ˜ j1 ∗ f1 )(x)M jr22 (φ˜ j2 ∗ f2 )(x),

j1 , j2 ∈Z

.

j1 , j2 ∈Z

where ε = L − ν > 0. Applying Proposition 3.1 for both M jr11 (φ˜ j1 ∗ f1 ) and M jr22 (φ˜ j2 ∗ f2 ) yields the appropriate estimate below,  #2  12 " ε ε r2 ˜ r1 ˜ | j −k| − | j −k| − 1 2   2 2 2 M j1 (φ j1 ∗ f1 )M j2 (φ j2 ∗ f2 ) 2 ||SΘ ( f1 , f2 )||L p . ∑ ∑ k∈Z j , j ∈Z 1 2 p L ! 1 2 h i2 r1 ˜ r2 ˜ − 2ε | j1 −k| − 2ε | j2 −k| 2 M ( φ ∗ f ) M ( φ ∗ f ) . 2 j1 j2 1 2 ∑ j1 j2 p j1 , j2 ,k∈Z L ! 1 ! 1 h i2 2 h i2 2 ∑ M r2 (φ˜ j ∗ f2 ) . ∑ M jr11 (φ˜ j1 ∗ f1 ) 2 j2 j1 ∈Z p j2 ∈Z p L

1

L

2

. || f1 ||H p1 || f2 ||H p2 . This completes the proof of Lemma 4.1.



Next we construct paraproducts to decompose Θk . These are the same operators that were constructed in [35], although we modify the decomposition to fit the bilinear framework. Fix an approximation to identity operator Pk f = ϕk ∗ f , where ϕk (x) = 2kn ϕ(2k x) and ϕ ∈ S with integral 1. Define for α, β ∈ Nn0  Z β!  |α|  ϕ(y)yβ−α dy α ≤ β  (−1) n (β − α)! R Mα,β = .    0 α 6≤ β Here we say α ≤ β for α = (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ Nn0 if αi ≤ βi for all i = 1, ..., n. It is clear that |Mα,β | < ∞ for all α, β ∈ Nn0 since ϕ ∈ S . It is not hard to verify that when |α| = |β|,  (−1)|β| β! α = β (4.1) Mα,β = . 0 α 6= β and |α| = |β| Consider the operators Pk Dα f defined for f ∈ S 0 , where Dα is the distributional derivative on S 0.



Hence Pk Dα f (x) is well defined for f ∈ S 0 since Pk Dα f (x) = ϕxk , Dα f = (−1)|α| Dα (ϕxk ), f and Dα (ϕxk ) ∈ S . In fact, this gives a kernel representation for Pk Dα ; estimates for this kernel are

14

JAROD HART AND GUOZHEN LU

addressed in the proof of Proposition 4.3. Also, α

[[Pk D ]]β (x) = 2

|β|k

Z Rn

ϕk (x − y)∂αy ((x − y)β )dy = 2k|α| Mα,β .

For {Θk } ∈ BLPSO(N, L) and k ∈ Z, define (0)

Θk ( f1 , f2 )(x) = Θk ( f1 , f2 )(x) − [[Θk (·, f2 )]]0 (x) · Pk f1 (x) − [[Θk ( f1 , ·)]]0 (x) · Pk f2 (x) + [[Θk ]]0,0 (x) · Pk f1 (x)Pk f2 (x), and

(4.2)

(m−1)

(m)

(m−1)

Θk ( f1 , f2 )(x) = Θk

( f1 , f2 )(x) −

∑

(−1)|α|

[[Θk

|α|=m

−

(4.3)

(m−1) ( f1 , ·)]]β (x) |β| [[Θk

(−1)

∑

β!

|β|=m

+

∑

· 2−k|β| Pk Dβ f2 (x)

(m−1) ]]α,β (x) |α|+|β| [[Θk

(−1)

α!β!

|α|=|β|=m

(·, f2 )]]α (x) −k|α| ·2 Pk Dα f1 (x) α!

· 2−k(|α|+|β|) Pk Dα f1 (x)Pk Dβ f2 (x)

for 1 ≤ m ≤ L.

Lemma 4.2. Let {Θk } ∈ BLPSO(N, L) for some integer L ≥ 1 and N > 2n. Also let 0 ≤ M < N − n be an integer. Then Z

(4.4)

Rn

θk (x, y1 , y2 )yα1 dy1 = 0

for all x, y2 ∈ Rn and |α| ≤ M if and only if [[Θk (·, f )]]α (x) = 0 for all x ∈ Rn , |α| ≤ M, and f ∈ C0∞ . Likewise for integrals in y2 and [[Θk ( f , ·)]]α .

Here we define [[Θk (·, f )]]α by applying the definition of [[ · ]]α for linear operator to the linear operator g 7→ Θk (g, f ) with f fixed. A similar notation is used for [[Θk ( f , ·)]]α .

Proof. Note that the condition 0 ≤ M < N − n implies that [[Θk (·, f )]]α is well defined for |α| ≤ M. Assume that (4.4) holds. Then for any |α| ≤ M [[Θk (·, f )]]α = 2

|α|k

= 2|α|k

Z

θk (x, y1 , y2 )(x − y1 )α f (y2 )dy1 dy2  Z Z |ν| ν θk (x, y1 , y2 )y1 dy1 xµ f (y2 )dy2 = 0. ∑ (−1) cµ,ν

R2n

µ+ν=α

Rn

Rn

Here cµ,ν are binomial coefficients. Now assume that [[Θk (·, f )]]α (x) = 0 for all x ∈ Rn , |α| ≤ M, and f ∈ C0∞ . Let ϕ ∈ C0∞ be radial with integral 1 and define ϕk (x) = 2kn ϕ(2k x). The radial condition

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 15

here is not strictly necessary, but it simplifies notation. Then for |α| ≤ M, we have Z R

θk (x, y1 , y2 )yα1 dy1 n

Z

= =

∑

n µ+ν=α R

θk (x, y1 , y2 )(−1)|α|+|µ| xµ (x − y1 )ν dy1 θk (x, y1 , u)(x − y1 )ν ϕN (u − y2 )dy1 du

∑

(−1)

∑

(−1)|α|+|µ| xµ lim [[Θk (·, ϕN1 )]]ν (x) = 0.

x lim

N→∞

µ+ν=α

=

Z

|α|+|µ| µ

R2n

y

N→∞

µ+ν=α

We use that θk (x, y1 , y2 ) is a bounded L1 (Rn ) continuous function in x for y1 6= y2 to use the approximation to identity property ϕN ∗ θk (x, ·, y2 )(y2 ) → θk (x, y1 , y2 ) as N → ∞ pointwise for y1 6= y2 and in L p (Rn ) for 1 < p < ∞. This completes the proof.  Proposition 4.3. Let {Θk } ∈ BLPSO(N, L) for some integer L ≥ 1 and N ≥ 2n + L(2n + L − 1)/2, and assume that dµ(x,t) =

(4.5)

∑

|[[Θk ]]α,β (x)|2 δt=2−k dx

∑

k∈Z |α|,|β|≤L−1 (m)

(m)

is a Carleson measure. Also let Θk be as in as in (4.2) and (4.3) for 0 ≤ m ≤ L − 1. Then Θk ∈ em , L) where N em = N − L(2n + L − 1)/2, and they satisfy the following for 0 ≤ m ≤ L − 1: BLPSO(N (1) For all α ∈ Nn0 with |α| ≤ m ≤ L − 1, we have Z R

(m) θ (x, y1 , y2 )yα1 dy1 n k

Z

=

(m)

Rn

θk (x, y1 , y2 )yα2 dy2 = 0.

(2) dµ(x,t) is a Carleson measure, where dµ is defined by L−1

dµ(x,t) =

∑∑

∑

(m)

|[[Θk ]]α,β (x)|2 δt=2−k dx.

k∈Z m=0 |α|,|β|≤L−1

(m) em , L) by induction. First we check that {Θ(0) } ∈ Proof. We will show that {Θk } ∈ BLPSO(N k e0 , L) = BLPSO(N − n, L). Using the definition in (4.2), it is sufficient to show that BLPSO(N Θk ( f1 , f2 ), [[Θk (·, f2 )]]0 · Pk f1 , [[Θk ( f1 , ·)]]0 · Pk f2 , and [[Θk ]]0,0 · Pk f1 · Pk f2 each define operators e0 , L). The first and last terms trivially satisfy these properties; note that {Θk } ∈ of type BLPSO(N e0 , L) by assumption and [[Θk ]]0,0 ·Pk f1 ·Pk f2 is in any BPLSO(N, L) class BLPSO(N, L) ⊂ BLPSO(N since [[Θk ]]0,0 . 1 and the convolution kernel of Pk is in Schwartz class S . We consider the second term [[Θk (·, f2 )]]0 · Pk f1 , whose kernel is

ϕk (x − y1 )

Z Rn

θk (x, u1 , y2 )du.

16

JAROD HART AND GUOZHEN LU

It is not hard to verify that this kernel satisfies the appropriate estimates to be a member of e0 , L). The kernel condition (2.1) is verified by the following. For |α|, |β| ≤ L BLPSO(N   Z Z α β α β D D ϕk (x − y1 ) θk (x, u1 , y2 )du1 = ∂y1 ϕk (x − y1 ) ∂y2 θk (x, u1 , y2 )du1 1 2 Rn Rn |α|k

.2

ΦN−n (x − y1 ) k

Z Rn

2|β|k ΦN k (x − u1 , x − y2 )du1

. 2(|α|+|β|)k ΦkN−n (x − y1 )ΦkN−n (x − y2 ) 0 ≤ 2(|α|+|β|)k ΦN k (x − y1 , x − y2 ).

e

Here we use that 2kn 2kn 22kn ≤ = ΦRk (x, y). (1 + 2k |x|)R (1 + 2k |y|)R (1 + 2k |x| + 2k |y|)R e0 , L) as well. Now we proBy symmetry [[Θk ( f1 , ·)]]0 · Pk f2 defines an operator of type BLPSO(N (m−1) em−1 , L). Similar to the reduction for the m = ceed by induction. Assume that {Θ } ∈ BLPSO(N ΦRk (x)ΦRk (y) =

k

(m−1)

(·, f2 )]]α · 2−k|α| Pk Dα f1 for |α| = m and em , L). The kernel of [[Θ(m−1) (·, f2 )]]α · the symmetric term define a collections of type BLPSO(N k 2−k|α| Pk Dα f1 is 0 case, this can be easily reduced to showing that [[Θk

|α|

Z

α

(−1) (D ϕ)k (x − y1 )

(m−1)

Rn

θk

(x, u, y2 )2k|α| (x − u)α du.

(m−1)

The kernel condition (2.1) for [[Θk (·, f2 )]]α · 2−k|α| Pk Dα f1 is verified as follows using the inductive hypothesis. For |µ|, |ν| ≤ L and |α| = m   Z µ ν (m−1) |α| α k|α| α D D (−1) (D ϕ)k (x − y1 ) θ (x, u, y )2 (u − y ) du 2 1 2 k 1 Rn . 2|µ|k |(Dα+µ ϕ)k (x − y1 )| .2 ≤2

|µ|k

em ΦN k (x − y1 )

(|µ|+|ν|)k

Z Rn

Z

(m−1)

Rn

|Dν2 θk

(x, u, y2 )2k|α| (u − y1 )α |du

e N

2|ν|k Φk m−1 (x − u, x − y2 )2k|α| (u − y1 )α |du

em ΦN k (x − y1 )

Z

e N

Φk m−1

−|α|

(x − u, x − y2 )du

Rn em−1 −n−m e N m ≤ 2(|µ|+|ν|)k ΦN (x − y2 ) k (x − y1 )Φk em ≤ 2(|µ|+|ν|)k ΦN k (x − y1 , x − y2 ).

em , defined recursively by N e0 = N − n and N em = N em−1 − n − m This gives the appropriate formula N e e e for m ≥ 1. This yields the formula Nm = N − (m + 1)(2n + m)/2. Since Nm ≤ Nm−1 for all 1 ≤ m ≤ eL−1 = N − L(2n + L − 1)/2 as in the statement of Proposition L − 1, this gives the requirement N 4.3. Since {Θk } ∈ BLPSO(N, L) for some N ≥ 2n + 2L, we know that [[Θk ]]α,β (x) exists for all (m)

|α|, |β| ≤ L − 1. We prove (1) by induction and using the reduction in Lemma 4.2 for each Θk :

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 17

the m = 0 case for (1) is not hard to verify; for all f1 ∈ C0∞ (0)

[[Θk ( f1 , ·)]]0 (x) = [[Θk ( f1 , ·)]]0 (x) − [[Θk ( f1 , ·)]]0 (x) − [[Θk ]]0,0 (x)Pk f1 (x) + [[Θk ]]0,0 (x)Pk f1 (x) = 0, (0)

and likewise [[Θk (·, f2 )]]0 = 0 for all f2 ∈ C0∞ . Now assume that (1) holds for m − 1. Then for |µ| ≤ m − 1 and f1 ∈ C0∞ (m) [[Θk ( f1 , ·)]]µ

(m−1) = [[Θk ( f1 , ·)]]µ −

(m−1) ]]α,µ |α| [[Θk

(−1)

∑

α!

|α|=m

−

(m−1) ( f1 , ·)]]β |β| [[Θk

(−1)

∑

+

∑

· 2−k|β| [[Pk Dβ ]]µ

β!

|β|=m

(m−1) ]]α,β |α|+|β| [[Θk

(−1)

α!β!

|α|=|β|=m

· 2−k|α| Pk Dα f1

· 2−k(|α|+|β|) Pk Dα f1 [[Pk Dβ ]]µ = 0

by the inductive hypothesis and the fact that [[Pk Dα ]]µ = [[Pk Dβ ]]µ = 0 for |µ| ≤ m − 1 < m = |α| = |β|. For |µ| = m, (m) [[Θk ( f1 , ·)]]µ

(m−1) = [[Θk ( f1 , ·)]]µ −

(m−1) ]]α,µ |α| [[Θk

(−1)

∑

|α|=m

−

∑

(m−1) ( f1 , ·)]]β |β| [[Θk

(−1)

+

∑

· 2−k|β| [[Pk Dβ f2 ]]µ

β!

|β|=m

(m−1) ]]α,β |α|+|β| [[Θk

(−1)

α!β!

|α|=|β|=m (m−1) = [[Θk ( f1 , ·)]]µ −

∑

· 2−k(|α|+|β|) Pk Dα f1 [[Pk Dβ f2 ]]µ

(m−1) ]]α,µ |α| [[Θk

(−1)

∑

· 2−k|α| Pk Dα f1

α!

|α|=m (m−1) − [[Θk ( f1 , ·)]]µ +

· 2−k|α| Pk Dα f1

α!

(m−1) ]]α,µ |α| [[Θk

(−1)

α!

|α|=m

· 2−k|α| Pk Dα f1 = 0,

where the summations in β collapse using (4.1) and that Mβ,µ = (−1)|β| when µ = β and Mβ,µ = 0 when |β| = |µ| but β 6= µ. By symmetry the same holds for [[Θk (·, f2 )]]µ and hence by induction and Lemma 4.2 this verifies (1) for all m ≤ L − 1. Given the Carleson measure assumption for dµ(x,t) in (4.5), one can easily prove (2) if the following statement holds: for all 0 ≤ m ≤ L − 1 (4.6)

∑

|α|,|β|≤L−1

(m)

2(m+1)

|[[Θk ]]α,β (x)| ≤ C0

∑

|[[Θk ]]α,β (x)|,

|α|,|β|≤L−1

where C0 =

∑

(1 + |Mα,β |).

|α|,|β|≤L−1

18

JAROD HART AND GUOZHEN LU

We verify (4.6) by induction. For m = 0, let |α|, |β| ≤ L − 1, and it follows that (0)

[[Θk ]]α,β = [[Θk ]]α,β − [[Θk ]]0,β [[Pk ]]α − [[Θk ]]α,0 [[Pk ]]β + [[Θk ]]0,0 [[Pk ]]α [[Pk ]]β Then

∑

(0)

|[[Θk ]]α,β | ≤

|[[Θk ]]α,β | + |[[Θk ]]0,β M0,α |

∑

|α|,|β|≤L−1

|α|,|β|≤L−1

+

|[[Θk ]]α,0 M0,β | + |[[Θk ]]0,0 M0,α M0,β |

∑

|α|,|β|≤L−1

! ≤

!

(1 + |M0,α | + |M0,β | + |M0,α M0,β |)

∑

|µ|,|ν|≤L−1

! ≤

|[[Θk ]]α,β |

∑

|α|,|β|≤L−1

≤ C02

!

(1 + |M0,α |)(1 + |M0,β |)

∑

|[[Θk ]]α,β |

∑

|α|,|β|≤L−1

|µ|,|ν|≤L−1

|[[Θk ]]α,β |.

∑

|µ|,|ν|≤L−1

Now assume that (4.6) holds for m − 1, and consider

∑

(m)

|[[Θk ]]α,β | ≤

|α|,|β|≤L−1

+

(m−1)

|[[Θk

∑

]]α,β | +

|α|,|β|≤L−1

∑

(m−1)

|[[Θk

∑

∑

∑

(m−1)

|[[Θk

]]α,0 Mν,β | +

|α|,|β|≤L−1 |ν|=m

∑

(m−1)

|[[Θk

∑

∑

]]0,0 Mµ,α Mν,β |

|α|,|β|≤L−1 |µ|=|ν|=m

" ≤

]]0,β Mµ,α |

|α|,|β|≤L−1 |µ|=m

#!

1+

|α|,|β|≤L−1

∑

|Mµ,α | +

|µ|=m

∑

|Mν,β | +

|ν|=m

|Mµ,α Mν,β |

∑

|µ|=|ν|=m

! ×

∑

(m−1)

|[[Θk

]]α,β |

|α|,|β|≤L−1

! ≤

∑

(1 + |Mµ,α |)(1 + |Mν,β |)

∑

|α|,|β|≤L−1 |µ|,|ν|≤m

≤ C02

∑

!

(m−1)

|[[Θk

∑

(m−1) |[[Θk ]]α,β |

|α|,|β|≤L−1 2(m+1)

]]α,β | ≤ C0

|α|,|β|≤L−1

∑

|[[Θk ]]α,β |

|α|,|β|≤L−1

We use the inductive hypothesis in the last inequality here to bound the [[Θ(m−1) ]]α,β . Then by induction, the estimate in (4.6) holds for all 0 ≤ m ≤ L − 1, and this completes the proof.  Define the linear operators (4.7)

Λ1,0,α f (x) = [[Θk ( f , ·)]]α (x), k

(4.8)

Λ1,m,α f (x) = [[Θk k

for 1 ≤ m ≤ L − 1.

(m−1)

( f , ·)]]α (x),

Λk2,0,α f (x) = [[Θk (·, f )]]α (x), and

(m−1)

Λ2,m,α f (x) = [[Θk k

(·, f )]]α (x)

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 19

Lemma 4.4. Let L ≥ 1 be an integer, and assume that {Θk } ∈ BLPSO(N, L), where N = 2n + (m) L(2n + L + 3)/2. Define Θk and Λi,m,α for i = 1, 2, 0 ≤ m ≤ L − 1, and |α| ≤ L − 1 as in (4.2), k i,m,α (4.3), (4.7), and (4.8). Then {Λk } ∈ LPSO(n + 2L, L) for each i = 1, 2, 0 ≤ m ≤ L − 1, and |α| ≤ L − 1. Furthermore, if Θk satisfies the Carleson condition in (4.5), then 2 L−1

(4.9)

dν(x,t) =

∑∑∑

|[[Λi,m,α ]]β (x)|2 δt=2−k dx k

∑

k∈Z i=1 m=0 |α|,|β|≤L−1 n < p ≤ 1, i = 1, 2, m = is a Carleson measure and SΛi,m,α is bounded from H p into L p for all n+L 0, 1, ..., L − 1, and |α| ≤ L, where SΛi,m,α is the square function operator associated to {Λi,m,α } !1 2

SΛi,m,α f (x) =

∑

|Λki,m,α f (x)|2

.

k∈Z

Proof. We first look at Λ1,0,α f = [[Θk ( f , ·)]]α for i = 1, 2 and |α| ≤ L − 1. We wish to show that k 1,0,α {Λk } ∈ LPSO(n + 2L, L). The kernel of Λ1,0,α is bounded in the following way. For |β| ≤ L k β |D1 λ1,0,α (x, y)| ≤ k

Z

.2

|D1 θk (x, y, y2 )2|α|k (x − y2 )α |dy2 β

Rn Z |β|k

. 2|β|k

k |α| ΦN k (x − y, x − y2 )(2 |x − y2 |) dy2

n ZR

N−|α|

Φk

Rn

(x − y, x − y2 )dy2

N−n−(L−1)

. 2|β|k Φk

(x − y). (x − y) ≤ 2|β|k Φn+2L k

Here we use that our hypotheses on N imply that N ≥ 2n + 3L − 1. Then {Λ1,0,α } ∈ LPSO(n + k 2,0,α 2L, L) for all |α| ≤ L − 1, and by symmetry it follows that {Λk } ∈ LPSO(n + 2L, L) for |α| ≤ L − 1 as well. If L = 1, this completes the estimates. Now we continue to estimate these operators for 1 ≤ m ≤ L − 1, where L ≥ 2. Fix 0 ≤ m ≤ L − 1 and |α| ≤ L − 1. The kernel λ1,m,α f (x) is given by k (x, y) = λ1,m,α k

Z

(m−1)

(x, y, y2 )(x − y2 )α dy2 . Rn (m−1) em−1 , L) where N em−1 = N − m(2n + m − that {Θk } ∈ BLPSO(N {Λ1,m,α } ∈ LPSO(n + 2L, L). We check the kernel conditions for k θk

By Proposition 4.3, we know 1)/2, and we now show that λ1,m,α hold. For 1 ≤ m ≤ L − 1, |α| ≤ L − 1, and |β| ≤ L k β (x, y)| ≤ |D1 λ1,m,α k

Z

≤2

Rn

β (m−1)

|D1 θk

|β|k

Z ZR

(x, y, y2 )2|α|k (x − y2 )α |dy2

e N

n

Φk m−1 (x − y, x − y2 )2|α|k |(x − y2 )α |dy2

−|α| (x − y, x − y2 )dy2 Rn e N −n−(L−1) ≤ 2|β|k Φk m−1 (x − y) ≤ 2|β|k Φn+2L (x − y). k

≤ 2|β|k

e N

Φk m−1

20

JAROD HART AND GUOZHEN LU

The last inequality is given by our assumption that N ≥ 2n + L(2n + L + 3)/2 since for 0 ≤ m ≤ L − 1, we have em−1 − n − (L − 1) ≥ N eL−2 − n − (L − 1) = N eL−1 = N − L(2n + L − 1)/2 N ≥ 2n + L(2n + L + 3)/2 − L(2n + L − 1)/2 = 2n + 2L ≥ n + 2L. Also, by (2) in Proposition 4.3 it follows that dν(x,t) defined in (4.9) is a Carleson measure; this is because [[Λ1,0,α ]]β = [[Θk ]]β,α , k (m−1)

[[Λ1,m,α ]]β = [[Θk k

[[Λk2,0,α ]]β = [[Θk ]]α,β ,

]]β,α ,

(m−1)

[[Λ2,m,α ]]β = [[Θk k

and

By Theorem 2.1 from [35], it follows that SΛi,m,α is bounded from 1.

(m)

Now we use Lemma 4.1 and the paraproduct operators Θk to prove Theorem 2.1.

Proof of Theorem 2.1. Let

n 2n+L

< p ≤ 1 and

n n+L

for 0 < m ≤ L − 1.

]]α,β Hp

into L p for all

0, we use ψ ∈ Dn to conclude n+1 that |ψk ∗ ϕ(x)| . 2−nk Φn+1 k (x) + Φ0 (x) . Hence for k > 0, we have  
nk/2 −nk n+1 2 + 1 . 2−nk/2 . . 2 ||ψk ∗ ϕ||L2 . 2−nk ||Φn+1 || + ||Φ || 2 2 L L 0 k Then the above sum is absolutely convergent, and the proof of the lemma is complete.



Proof of Theorem 2.3. Assume that T and ψ are as in the statement of Theorem 2.3. Without loss of generality we assume that supp(ψ) ⊂ B(0, 1). To compute the kernel for Θk f = Qk T , we consider the following decomposition: Let ϕ ∈ C0∞ (B(0, 1)) with integral 1 such that Z Rn

ϕ(x)xα dx = 0

e k (x) = ϕk+1 (x) − ϕk (x), and it follows that ψ e ∈ DM . Then for 0 < |α| ≤ M. Define ψ E R−1 D x e Θk ( f1 , f2 )(x) = lim ∑ T (Q`1 f1 , PR f2 ), ψk + hT (Pk f1 , PR f2 ), ψxk i R→∞

= lim

R→∞

`1 =k

R−1 R−1 D

∑ ∑

`1 =k `2 =k

E R−1 D E e` f1 , Q e` f2 ), ψx + ∑ T (Q e` f1 , Pk f2 ), ψx T (Q k k 1 2 1 `1 =k

+

R−1 D

∑

E e` f2 ), ψx + hT (Pk f1 , Pk f2 ), ψx i T (Pk f1 , Q k k 2

`2 =k

Z

(5.1)

= lim

R→∞ R2n

R−1 R−1

R−1

∑ ∑ I`1,`2,k (x, y1, y2) + ∑ II`1,k (x, y1, y2)

`1 =k `2 =k

`1 =k

!

R−1

+

∑ III`2,k (x, y1, y2) + IVk (x, y1, y2)

f1 (y1 ) f2 (y2 )dy1 dy2

`2 =k

where D E e y`1 , ψ e y`2 ), ψxk , I`1 ,`2 ,k (x, y1 , y2 ) = T (ψ 1 2 D E y1 y2 x e III`1 ,k (x, y1 , y2 ) = T (ψ`1 , ϕk ), ψk ,

and

D E y e y`2 ), ψxk , II`2 ,k (x, y1 , y2 ) = T (ϕk1 , ψ 2 

y y IVk (x, y1 , y2 ) = T (ϕk1 , ϕk2 ), ψxk .

24

JAROD HART AND GUOZHEN LU

To show that Θk satisfies (2.1), we first assume that |x − y1 | + |x − y2 | > 22−k . Without loss of generality, we assume that `1 ≥ `2 . For |α|, |β| ≤ L we have D E α β β y2 x α e y1 |α|`1 +|β|`2 e ), ψ , (D ψ ) T ((D ψ ) D D I (x, y , y ) = 2 1 2 `1 ,`2 ,k 1 2 1 k 2 `2 `1 ! µ Z D K(u, y , v ) 1 2 1 = 2|α|`1 +|β|`2 K(u, v1 , v2 ) − ∑ (v1 − y1 )µ R3n µ! |µ|≤M−1 α e y1 β e y2 x × (D ψ)`1 (v1 )(D ψ)`2 (v2 )ψk (u)du dv1 dv2 |α|`1 +|β|`2

.2

|v1 − y1 |M e )y`1 (v1 )(Dβ ψ e )y`2 (v2 )ψxk (u)|du dv1 dv2 |(Dα ψ 2n+M 1 2 R3n (|u − v1 | + |u − v2 |)

Z

−`1 (M−|α|−|β|)

.2

2(2n+M)k (1 + 2k |x − y1 | + 2k |x − y2 |)2n+M

. 2(k−max(`1 ,`2 ))(M−|α|−|β|) 2(|α|+|β|)k Φk2n+M (x − y1 , x − y2 ). Then ∞

∞

∑ ∑

`1 =k `2 =k

α β D1 D2 I`1 ,`2 ,k (x, y1 , y2 ) . 2(|α|+|β|)k Φ2n+M (x − y1 , x − y2 ) k

∞

∞

∑ ∑ 2(k−`1)

M−|α|−|β| 2

2(k−`1 )

M−|α|−|β| 2

`1 =k `2 =k (|α|+|β|)k

.2

ΦN k (x − y1 , x − y2 ).

The second term satisfies a similar estimate Z α β y y e )`1 (v1 )(Dβ ϕ)k2 (v2 )ψxk (u)du dv1 dv2 K(u, v1 , v2 )(Dα ψ D1 D2 II`1 ,k (x, y1 , y2 ) = 2|α|`1 2|β|k 1 R3n ! µ Z D K(u, y , v ) 1 2 1 (v1 − y1 )µ = 2|α|`1 2|β|k K(u, v1 , v2 ) − ∑ R3n µ! |µ|≤M−1 β y2 x α e y1 × (D ψ)`1 (v1 )(D ϕ)k (v2 )ψk (u)du dv1 dv2 .2

|α|`1 |β|k

2

|v1 − y1 |M e )y`1 (v1 )(Dβ ϕ)yk2 (v2 )ψxk (u)|du dv1 dv2 |(Dα ψ 2n+M 1 R3n (|u − v1 | + |u − v2 |)

Z

. 2(|α|+|β|)k 2(k−`1 )(M−|α|) Φ2n+M (x − y1 , x − y2 ). k Again summing over `1 ≥ k, we obtain ∞ α β D D II (x, y , y ) (x − y1 , x − y2 ) ∑ 1 2 `1,k 1 2 . 2(|α|+|β|)k Φ2n+M k `1 =k

. 2(|α|+|β|)k ΦN k (x − y1 , x − y2 ).

∞

∑ 2(k−`1)(M−|α|)

`1 =k

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 25

The third term III`2 ,k (x, y1 , y2 ) is also bounded by a constant times 2(|α|+|β|)k ΦN k (x − y1 , x − y2 ) by symmetry. We estimate the last term. D E x β y2 α y1 α β (|α|+|β|)k |D1 D2 IVk (x, y1 , y2 )| = 2 T ((D ϕ)k , (D ϕ)k ), ψk ! µ Z D K(x, v , v ) 1 2 0 = 2(|α|+|β|)k K(u, v1 , v2 ) − ∑ (u − x)µ R3n µ! |µ|≤M−1 α y1 β y2 x × (D ϕ)k (v1 )(D ϕ)k (v2 )ψk (u)du dv1 dv2 (|α|+|β|)k

.2

. 2(|α|+|β|)k

|u − x|M y y |(Dα ϕ)k1 (v1 )(Dβ ϕ)k2 (v2 )ψxk (u)|du dv1 dv2 2n+M R3n (|x − v1 | + |x − v2 |)

Z

2−Mk = 2(|α|+|β|)k ΦN k (x − y1 , x − y2 ). −k 2n+M (2 + |x − y1 | + |x − y2 |)

So we that that θk (x, y1 , y2 ) as given in (5.1) satisfies (2.1) for |x − y1 | + |x − y2 | ≥ 22−k . Now we assume that |x − y1 | + |x − y2 | ≤ 22−k . In this situation 22kn . ΦN k (x − y1 , x − y2 ), so it is sufficient N 2kn to prove (2.1) with 2 in place of Φk (x − y1 , x − y2 ). We again use the decomposition in (5.1). To , we assume without loss of generality that `1 ≥ `2 . Since we have assumed that D bound I`1 ,`2 ,k E y1 y2 e` , ψ e ` ), xµ = 0 for |µ| ≤ 2L + n, we can write T (ψ 1 2 D E α β e y`1 , ψ e y`2 ), ψxk ≤ |A`1 ,`2 ,k (x, y1 , y2 )| + |B`1 ,`2 ,k (x, y1 , y2 )|, where D1 D2 T (ψ 1 2 A`1 ,`2 ,k (x, y1 , y2 ) = 2

`1 |α|+`2 |β|

Z

y

|u−y1 |≤21−`1

y

e )`1 , (Dβ ψ e )`2 )(u) T ((Dα ψ 1 2

! α ψx (y ) D 1 k (u − y1 )α du, × ψxk (u) − ∑ α! |α|≤2L+n B`1 ,`2 ,k (x, y1 , y2 ) = 2

`1 |α|+`2 |β|

Z

y

|u−y1 |>21−`1

y

e )`1 , (Dβ ψ e )`2 )(u) T ((Dα ψ 1 2

! α ψx (y ) D 1 k (u − y1 )α du. × ψxk (u) − ∑ α! |α|≤2L+n The A`1 ,`2 ,k term is bounded as follows, y

y

e )`1 , (Dβ ψ e )`2 ) · χB(y ,21−`1 ) ||L1 |A`1 ,`2 ,k (x, y1 , y2 )| ≤ 2`1 |α|+`2 |β| ||T ((Dα ψ 1 2 1 ! Dα ψxk (y1 ) x × ψk − ∑ (· − y1 )α · χB(y1 ,21−`1 ) α! |α|≤2L+n

L∞

e )y`1 , (Dβ ψ e )y`2 )||L2 2(n+2L+1)(k−`1 ) 2kn .2 2 ||T ((Dα ψ 1 2 `1 |α|+`2 |β| −`1 n/2 α e y1 β e y2 ∞ (n+2L+1)(k−`1 ) kn .2 2 ||(D ψ)`1 ||L2 ||(D ψ)`2 ||L 2 2 `1 |α|+`2 |β| (n+2L+1)(k−`1 ) kn `2 n k(|α|+|β|) k−max(`1 ,`2 ) `1 |α|+`2 |β| −`1 n/2

.2

2

2 2

.2

2

.

26

JAROD HART AND GUOZHEN LU

Let 0 < δ < 1. The B`1 ,`2 ,k term is bounded using the kernel representation of T , which is viable e y`1 ). Since ψ e `1 ∈ D2L , we have since |u − y1 | > 21−`1 implies u ∈ / supp(ψ 1 Z Z e )y`2 (v2 )dv1 v2 e )y`1 (v1 )(Dβ ψ K(u, v1 , v2 )(Dα ψ |B`1 ,`2 ,k (x, y1 , y2 )| = 2`1 |α|+`2 |β| 2 1 1−` 2n 1 |u−y1 |>2 R ! Dα ψxk (y1 ) × ψxk (u) − ∑ (u − y1 )α du α! |α|≤2L+n ! Z Z K(u, y , v ) 1 2 (v1 − y1 )µ = 2`1 |α|+`2 |β| K(u, v1 , v2 ) − ∑ |u−y1 |>21−`1 R2n µ! |µ|≤2L y

y

e )`1 (v1 )(Dβ ψ e )`2 (v2 )dv1 v2 × (Dα ψ 1 2 ! α ψx (y ) D 1 k × ψxk (u) − ∑ (u − y1 )α du α! |α|≤2L+n ∞ Z

`1 |α|+`2 |β|

.2

|v1 − y1 |2L+1 2n+2L+1 R2n (|u − y1 | + |u − v2 |)

Z

∑

m−` m+1−`1 m=1 2 1 k

`1 =k+1 `1 =k+1

for all |α|, |β| ≤ L. The estimate for II`1 ,`2 ,k is obtained by essentially the same E argument. That is, D e y`1 , ψ e y`2 ), xµ = 0 for |µ| ≤ 2L + n, we estimate II`2 ,k in the following way using the hypothesis T (ψ 1 2 D E α β y e y`2 ), ψxk ≤ |A`2 ,k (x, y1 , y2 )| + |B`2 ,k (x, y1 , y2 )|, D1 D2 T (ϕk1 , ψ 2 where k|α|+`2 |β|

A`2 ,k (x, y1 , y2 ) = 2

Z |u−y1 |≤21−`2

y

y

e )`2 )(u) T ((Dα ϕ)k1 , (Dβ ψ 2 ! α ψx (y ) D 1 k × ψxk (u) − ∑ (u − y1 )α du α! |α|≤2L+n

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 27

and B`2 ,k (x, y1 , y2 ) = 2k|α|+`2 |β|

Z

y

y

e )`2 )(u) T ((Dα ϕ)k1 , (Dβ ψ 2

|u−y1 |>21−`2

×

ψxk (u) −

By the argument above, we obtain the estimate R−1 α β D D II (x, y , y ) ∑ 1 2 `2,k 1 2 . 2k(|α|+|β|)22kn `2 =k+1

! Dα ψxk (y1 ) α (u − y1 ) du. ∑ α! |α|≤2L+n

∑ 2δ(k−`2) . 2k(|α|+|β|)22kn.

`2 >k

A similar argument can be applied to III`1 ,k as well. The estimate for IVk trivially follows from the L2 × L2 to L1 boundedness of T , D E

 β y y1 y2 y α β x k(|α|+|β|) |D1 D2 T (ϕk , ϕk ), ψk | = 2 T ((Dα1 ϕ)k1 , (D2 ϕ)k2 ), ψxk . 2k(|α|+|β|) 22nk . Therefore all of the sums on the right hand side of (5.1) converge absolutely, and hence ∞

θk (x, y1 , y2 ) =

∞

∞

∞

∑ ∑ I`1,`2,k (x, y1, y2) + ∑ II`2,k (x, y1, y2) + ∑ III`1,k (x, y1, y2) + IVk (x, y1, y2)

`1 =k `2 =k

`2 =k

`1 =k

satisfies (2.1). Hence {Θk } = {Qk T } ∈ BLPSO(N, L).



The proof of Theorem 2.2 follows immediately from Theorems 2.1 and 2.3.

6. A PPLICATIONS TO PARAPRODUCT 6.1. Bilinear Bony Paraproducts. Let b ∈ BMO, and recall the bilinear Bony paraproduct
Πb ( f1 , f2 )(x) = ∑ Q j Q j b · Pj f1 · Pj f2 (x) j∈Z

Now we prove Theorem 2.4

Proof. It is easily shown that Πb ∈ BCZO(M) for all M ∈ N. So it is sufficient to show conditions (2.3) and (2.4) from Theorem 2.2. We first show (2.3). For f1 ∈ D2L+n , f2 ∈ D|α| , and |α| ≤ 2L + n Z

α

hΠb ( f1 , f2 ), x i = lim

∑

R→∞

n j∈Z R

Z

=

∑ j∈Z

Rn

Q j b(u)Pj f1 (u)Pj f2 (u)Q j (ηR xα )(u)du

Q j b(u)Pj f1 (u)Pj f2 (u)Q j (xα )(u)du = 0,

28

JAROD HART AND GUOZHEN LU

and likewise hΠb ( f2 , f1 ), xα i = 0. The BMO|α|+|β| conditions in (2.4) are more difficult to verify, but hold nonetheless. For |α|, |β| ≤ L − 1, we first compute 

2k(|α|+|β|) Qk [[Πb ]]α,β (x) = 2k(|α|+|β|) [[Πb ]]α,β , ψxk = lim 2

k(|α|+|β|)

R→∞

Z

∑

4n j∈Z R

ψ j (u − v)Q j b(v)ϕ j (v − y1 )ϕ j (v − y2 )ηR (y1 )ηR (y2 ) × (u − y1 )α (u − y2 )β ψxk (u)du dv dy1 dy2

= lim

∑ ∑ cα,µcβ,ν2k(|α|+|β|) ∑

R→∞ µ≤α

Z

4n j∈Z R

ν≤β

ψ j (u − v)Q j b(v)ϕ j (v − y1 )ϕ j (v − y2 )ηR (y1 )ηR (y2 )

× (u − v)µ+ν (v − y1 )α−µ (v − y2 )β−ν ψxk (u)du dv dy1 dy2 =

∑ ∑ cα,µcβ,νCα−µCβ−ν2

µ≤α ν≤β

=

Z

∑

2n j∈Z R

(µ+ν)

ψj

(µ+ν)

∑ ∑ cα,µcβ,νCα−µCβ−ν2(k− j)(|α|+|β|) ∑ Qk Q j

µ≤α ν≤β

=

(k− j)(|α|+|β|)

(u − v)Q j b(v)ψxk (u)du dv

Q j b(x)

j∈Z

∑ ∑ cα,µcβ,νCα−µCβ−νWk ∗ (TV (µ+ν) b)(x),

µ≤α ν≤β

R (µ+ν) (ξ) = where cα,µ are binomial coefficients, Cµ = Rn ϕ(x)xµ dx, and V , W , and TV are defined via V\ \ (µ+ν) (ξ)ψ b (ξ) = |ξ|−(|α|+|β|) ψ b (ξ), W b (ξ), V j(µ+ν) (x) = 2 jnV (µ+ν) (2 j x), W j (x) = 2 jnW (2 j x), |ξ||α|+|β| ψ and

TV (µ+ν) f (x) =

(µ+ν)

∑ Vj

∗ f (x).

j∈Z

This is verified by the following computation. h i (µ+ν) \ (µ+ν) (2− j ξ)ψ b (2−k ξ)ψ b (2− j ξ) fb(ξ) 2(k− j)(|α|+|β|) F Qk Q j Q j f (ξ) = 2(k− j)(|α|+|β|) ψ \ (µ+ν) (2− j ξ)ψ b (2−k ξ)ψ b (2− j ξ) fb(ξ) = (2−k |ξ|)−(|α|+|β|) (2− j |ξ|)|α|+|β| ψ (µ+ν) (2− j ξ) fb(ξ) b (2−k ξ)V\ =W h i (µ+ν) = F Wk ∗V j ∗ f (ξ).

It was shown in [35] that TV (µ+ν) is bounded on BMO. Then TV (µ+ν) b ∈ BMO with ||TV (µ+ν) b||BMO . ||b||BMO , and

∑ 22k(|α|+|β|)|Qk [[Πb]]α,β(x)|2δt=2−k dx k∈Z

2 = ∑ ∑ ∑ cα,µ cβ,νCα−µCβ−νWk ∗ (TV (µ+ν) b)(x) δt=2−k dx k∈Z µ≤α ν≤β is a Carleson measure since TV (µ+ν) b ∈ BMO and Wk (x) = 2knW (2k x) has integral zero.



´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 29

6.2. Product Function Paraproducts. Now we prove Theorem 2.5. First recall the definition of Π( f1 , f2 ). Π( f1 , f2 ) =

∑ Πk ( f1, f2),

where Πk ( f1 , f2 ) = Qk (Pk f1 · Pk f2 ) .

k∈Z

Proof of Theorem 2.5. It is easy to verify that Π ∈ BCZO(M) for all M ≥ 1. Then to show that Π is bounded as described in Theorem 2.5, it is sufficient to show (2.3) and (2.4) from Theorem 2.2. The kernel of Πk is given by Z

πk (x, y1 , y2 ) =

Rn

ψk (x − u)ϕk (u − y1 )ϕk (u − y2 )du.

We first prove (2.3); for α ∈ Nn0 , f1 ∈ D|α| , and f2 ∈ D2L+n , we have α

hΠ( f1 , f2 ), x i = lim

R→∞

Z

= lim

R→∞

∑

2n k∈Z R

Z

∑

4n k∈Z R

ψk (x − u)ϕk (u − y1 )ϕk (u − y2 ) f1 (y1 ) f2 (y2 )ηR (x)xα du dx dy1 dy2

ϕk (u − y1 )ϕk (u − y2 ) f1 (y1 ) f2 (y2 )

Z Rn

 α

ψk (x − u)ηR (x)x dx du dy1 dy2 = 0

To prove (2.4), let α, β ∈ Nn0 and f ∈ D|α|+|β| . It follows that Z

 [[Π]]α,β , f = lim ∑ ψk (x − u)ϕk (u − y1 )ϕk (u − y2 )ηR (y1 )ηR (y2 ) R→∞

4n k∈Z R

× (x − y1 )α (x − y2 )β f (x)du dx dy1 dy2 Z

= lim

R→∞

∑ ∑

cµ1 ,ν1 cµ2 ,ν2

∑

R4n

k∈Z µ1 +ν1 =α µ2 +ν2 =β

ψk (x − u)ϕk (u − y1 )ϕk (u − y2 )ηR (y1 )ηR (y2 )

× (x − u)µ1 (u − y1 )ν1 (x − u)µ2 (u − y2 )ν2 f (x)du dx dy1 dy2 Z

= lim

R→∞

∑ ∑

∑

k∈Z µ1 +ν1 =α µ2 +ν2 =β

× × =

cµ1 ,ν1 cµ2 ,ν2

∑ ∑

∑

Z Rn

R2n

ψk (x − u)(x − u)µ1 +µ2 f (x)  ν1

ϕk (u − y1 )(u − y1 ) ηR (y1 )dy1

Z

 ν2

ϕk (u − y2 )(u − y2 ) ηR (y2 )dy2 du dx  Z Z µ1 +µ2 du f (x)dx = 0. cµ1 ,ν1 cµ2 ,ν2 Cν1 Cν2 ψk (u)u Rn

Rn

k∈Z µ1 +ν1 =α µ2 +ν2 =β

Rn

Here we define Cν for ν ∈ Nn0 Z

Cν =

Rn

ϕ(x)xν dx,

and cν,µ are binomial coefficients. Therefore Π is bounded from H p1 × H p2 into H p for all n p ≤ 1 and n+L < p1 , p2 ≤ 1, as in the statement of Theorem 2.5.

n 2n+L

< 

30

JAROD HART AND GUOZHEN LU

6.3. Molecular Parparoducts. Finally, we address the paraproducts in Theorem 2.6. Recall the paraproduct operators



 T ( f1 , f2 )(x) = ∑ |Q|−1/2 f1 , φ1Q f2 , φ2Q φ3Q (x), Q

where

φ1Q , φ2Q , φ3Q

are (M, N)-smooth molecules indexed by dyadic cubes.

Proof of Theorem 2.6. It is easy to see that T ∈ BCZO(M) and is bounded from L2 × L2 into L1 , which was proved in [2]. So we must verify conditions (2.3) and (2.4) of Theorem 2.2. As was shown in [2], the kernel of T is K(x, y1 , y2 ) = ∑ |Q|−1/2 φ1Q (y1 )φ2Q (y2 )φ3Q (x). Q

For f1 ∈ D|α| and f2 ∈ D2L+n with |α| ≤ L, we have





 hT ( f1 , f2 ), xα i = lim ∑ |Q|−1/2 f1 , φ1Q f2 , φ2Q φ3Q , xα ηR R→∞

Q

−1/2

= ∑ |Q|



f1 , φ1Q



Q

f2 , φ2Q

Z Rn

φ3Q (x)xα dx = 0.

It is not hard to show that this series is absolutely summable independent of R when f1 ∈ D|α| and f2 ∈ C0∞ ; hence we can bring the limit in R inside the sum. A similar computation shows that hT ( f1 , f2 ), xα i = 0 for f1 ∈ D2L+n and f2 ∈ D|α| . To show (2.4), we first verify that we can use an alternate formula to compute [[T ]]α,β . For α, β ∈ Nn0 with |α| + |β| ≤ L − 1 and ψ ∈ D|α|+|β| , fix R0 large enough so that supp(ψ) ⊂ B(0, R0 /4). Then for S, R > R0 , we have Z R3n

K (x, y1 , y2 )(x − y1 )α (x − y2 )β ηR (y1 )ηS (y2 )ψ(x)dy1 dy2 dx Z

=

R3n

K(x, y1 , y2 )(x − y1 )α (x − y2 )β (ηR (y1 )ηS (y2 ) − ηR0 (y1 )ηR0 (y1 ))ψ(x)dy1 dy2 dx

Z

+

R3n

K (x, y1 , y2 )(x − y1 )α (x − y2 )β ηR0 (y1 )ηR0 (y1 )ψ(x)dy1 dy2 dx,

where the integrand of the first integral is bounded by an L1 (R3n ) function independent of both R and S. Then the limit as S, R → ∞ exists (as a two dimensional limit), and Z

 lim lim K (x, y1 , y2 )(x − y1 )α (x − y2 )β ηR (y1 )ηS (y2 )ψ(x)dy1 dy2 dx = [[T ]]α,β , ψ . R→∞ S→∞ R3n

Therefore

 [[T ]]α,β , ψ = lim lim ∑ |Q|−1/2 R→∞ S→∞

= lim

R→∞

Q

−1/2

∑ |Q| Q

Z R2n

Z R3n

φ1Q (y1 )(x − y1 )α ηS (y1 )φ2Q (y2 )(x − y2 )β ηR (y2 )φ3Q (x)ψ(x)dy1 dy2 dx

Z R

φ1Q (y1 )(x − y1 )α dy1 n



Then we apply Theorem 2.2 to complete the proof.

φ2Q (y2 )(x − y2 )β ηR (y2 )φ3Q (x)ψ(x)dy2 dx = 0. 

´ HARDY SPACE ESTIMATES FOR BILINEAR SQUARE FUNCTIONS AND CALDERON-ZYGMUND OPERATORS 31

R EFERENCES ´ B´enyi, Bilinear singular integral operators, smooth atoms and molecules, J. Fourier Anal. Appl. 9, (2003), no. 1. A. 3, 301-319. ´ B´enyi, D. Maldonado, A. Nahmod, and R. H. Torres, Bilinear paraproducts revisited, Math. Nachr. 283, (2010), 2. A. no. 9, 1257-1276. 3. J. M. Bony, Calcul symbolique et propagation des singulararit´es pour les e´ qurations aux d´eriv´ees partielles non ´ lin´eaires, Ann. Sci. Ecole Norm. Sup., 14, (1981), no. 2, 109-246. 4. L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80, (1958), 921-930. 5. L. Chaffee, J. Hart, and L. Oliviera, Weighted multilinear square function bounds, Michigan Math. J. 63, (2014), no. 2, 371-400. 6. S.-Y. Chang and R. Fefferman, The Calder´on-Zygmund decomposition on product domains, Amer. J. Math., 104, (1982), no. 3, 455-468. 7. S.-Y. Chang and R. Fefferman, Some recent developments in Fourier analysis and H p -theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12, (1985), no. 1, 1-43 8. J. Chen and G. Lu, H˝ormander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal. 101 (2014), 98-112. 9. M. Christ and J. L. Journ´e, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159, (1987), no. 1-2, 51-80. 10. R. R. Coifman, A real variable characterization of H p , Studia math., 51, (1974), 269-274. 11. R. R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators I, Rev. Mat. Iberoamericana 8 (1992), no. 1, 45-67. 12. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Am. Math. Soc., 212, (1975), 315-331. 13. R. R. Coifman and Y. Meyer, Commutateurs d’int´egrales singuli`ers et op´erateurs multilin eaires, Ann. Inst. Fourier, Grenoble, 28, (1978), 177-202. 14. R. R. Coifman and Y. Meyer, Au del`a des op´erateurs pseudo-diff´erentiels, Ast´erisque 57, 1978. 15. W. Dai and G. Lu, L p estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. Fr., arXiv:1308.4062. 16. G. David and J. L. Journ´e, A boundedness criterion for generalized Calder´on-Zygmund operators, Ann. of Math. (2), 120, (1984), no. 2, 371-397. 17. G. David, J. L. Journ´e, and S. Semmes, Op´erateurs de Calder´on-Zygmund, fonctions para-accr´etives et interpolation, Rev. Mat. Iberoamericana 1, (1985), no. 4, 1-56. 18. C. Fefferman, E. Stein, H p spaces of several variables, Acta Math., 129, (1972), no. 3-4, 137-193. 19. S. Ferguson, M. Lacey, A characterization of product BMO by commutators. Acta Math. 189 (2002), no. 2, 143-160. 20. M. Frazier, Y.S. Han, B. Jawerth, and G. Weiss, The T 1 theorem for Triebel-Lizorkin spaces, Lecture Notes in Math., 1384, Springer, Berlin, 1989. 21. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34, (1985), no. 4, 777-799. α,q 22. M. Frazier, R. H. Torres, and G. Weiss, The boundedness of Calder´on-Zygmund operators on the spaces F˙p , Rev. Mat. Iveramericana 4, (1988), no. 1, 41-72. 23. L. Grafakos and N. Kalton, Multilinear Calder´on-Zygmund oprators on Hardy spaces, Collect. Math. 52, (2001), no. 2, 169-179. 24. L. Grafakos, L. Liu, D. Maldonado, and D. Yang, Multilinear analysis on metric spaces, Dissertationes Math. (Rpzprawy Mat.) 497, (2014), 121. 25. L. Grafakos and L. Oliveira, Carleson measures associated with families of multilinear operators, Studia Math. 211, (2012), no. 1, 71-94. 26. L. Grafakos and R. H. Torres, Multilinear Calder´on-Zygmund theory, Adv. Math., 165, (2002), 1, 124-164. 27. L. Grafakos and R. H. Torres, On multilinear singular integrals of Calder´on-Zygmund type, Publ. Mat., Vol. Exra, (2002), 57-91. 28. L. Grafakos and R. H. Torres, Discrete decompositions for bilinear operators and almost diagonal conditions, Trans. Amer. Math. Soc. 354, (2002), no. 3, 1153-1176

32

JAROD HART AND GUOZHEN LU

29. A. Grau de la Herr´an, J. Hart, and L. Oliveira, Multilinear local Tb theorem for square functions, Ann. Acad. Sci. Fenn. Math. 38, (2013), no. 2, 697-720. 30. Y. Han and G. Lu, Some recent works on multiparamter Hardy space theory and discrete Littlewood-Paley-Stein analysis, Trends in partial differential equations, 99-191, Adv. Lect. Math. (ALM), 10, Int. Press, Somerville, MA, 2010. 31. Y. Han, G. Lu, and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE 7, (2014), no. 7, 1465-1534. 32. J. Hart, Bilinear square functions and vector-valued Calder´on-Zygmund operators, J. Fourier Analysis Appl. 18, (2012), no.6, 1291-1313. 33. J. Hart, A new proof of the bilinear T(1) theorem, Proc. Amer. Math. Soc. 142, (2014), no. 9, 3169-3181. 34. J. Hart, A bilinear T b theorem for singular integral operators, J. Funct. Anal. 268, (2015), no. 12, 3680-3733. 35. J. Hart and G. Lu, Hardy space estimates for Littlewood-Paley square functions and singular integral operators, to appear in J. Fourier Anal. Appl. 36. G. Hu and Y. Meng, Multilinear Calder´on-Zygmund operator on products of Hardy spaces, Acta Math. Sin. (Engl. Ser.) 28, (2012), no. 2, 281-294. 37. B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals, Funda. Math., 25, (1935), 217-234. 38. P. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Lecture Notes in Math., 1384, Springer, Berlin, 1989. 39. J.-L. Journ´e, Calder´on-Zygmund operators on product spaces, Rev. Mat. Iberoamericana, 1, (1985), no. 3, 55-91. 40. G. Lu and Z. Ruan, Duality theory of weighted Hardy spaces with arbitrary number of parameters, Forum Math. 26 (2014), no. 5, 1429-1457. 41. G. Lu and P. Zhang, Multilinear Calder´on-Zygmund operators with kernels of Dini’s type and applications, Nonlinear Anal. 107 (2014), 92-117. 42. D. Maldonado, Multilinear Singular Integrals and Quadratic Estimates, Doctoral Dissertation, University of Kansas, 2005. 43. D. Maldonado and V. Naibo, On the boundedness of bilinear operators on products of Besov and Lebesgue spaces, J. Math. Anal. Appl. 352, (2009), no. 2, 591-603. 44. C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math. 193 (2004), no. 2, 269-296. 45. C. Muscalu, J. Pipher, T. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam. 22 (2006), no. 3, 963-976. 46. S. Semmes, Square function estimates and the T(b) theorem, Proc. Amer. Math. Soc., 110, no. 3, (1990), 721-726. 47. E. Stein, On the functions of Littlewood-Paley-Stein; Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 88, (1958), 430-466. 48. E. Stein, On the theory of harmonic functions of several variables. II. Behavior near the boundary, Acta Math. 106, (1961), 137-174. 49. E. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of H p -spaces, Acta Math. 103, (1960), 25-62. 50. R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 90, (1991), 442. 51. A. Youssfi, Bilinear operators and the Jacobian-determinant on Besov spaces, Indiana Univ. Math. J. 45, (1996), 381-396. D EPARTMENT OF M ATHEMATICS , WAYNE S TATE U NIVERSITY , D ETROIT, MI 48202 E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS , WAYNE S TATE U NIVERSITY , D ETROIT, MI 48202 E-mail address: [email protected]