ON THE BOUNDEDNESS OF BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES

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1 ON THE BOUNDEDNESS OF BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES DIEGO MALDONADO AND VIRGINIA NAIBO Abstract. We rove maing roerties of the form T : Ḃ α,q L 2 Ḃα 2,q 2 3 and T : Ḃ α,q Ḃα 2,q 2 2 L 3, for certain related indices, 2, 3, q, q 2, α, α 2 R, where T is a bilinear Hörmander-Mihlin multilier or a molecular araroduct. Alications to bilinear Littlewood-Paley theory are discussed.. Introduction Beginning from the classical works of R. Coifman and Y. Meyer [7], [8], [9] on bilinear seudo-differential oerators and J.-M. Bony [6] and H. Triebel [30], [3] on bilinear araroducts through the recent rogress in the develoment of the bilinear Calderón- Zygmund theory [8], [9], [20], [2], [22], the bilinear Hilbert transform [24], [25], and molecular araroducts [3], [4], [29], bilinear oerators continue to be obect of intense study. Of articular interest are the recent bilinear estimates in the scales of Besov and Triebel-Lizorkin saces of the form T : Ḃ α,q Ḃα 2,q 2 2 Ḃα 3,q 3 3 and T : F α,q F α 2,q 2 2 F α 3,q 3 3, for related indices, 2, 3, q, q 2, q 3, α, α 2, α 3, and families of bilinear oerators T including bilinear multiliers, bilinear Calderón-Zygmund oerators, molecular araroducts, and bilinear seudo-differential oerators established in, for instance, [], [2], [3], [4], [5], [3], [4], [6], [7], [8], [9], [20], [2], [22], [27], [29], [33], [34]. The urose of this article is to address Besov-Lebesgue boundedness roerties of the form (.) T : Ḃ α,q L 2 Ḃα 2,q 2 3 and T : Ḃ α,q Ḃα 2,q 2 2 L 3, (as well as its corresonding non-homogeneous versions) that comlement the existing results in the literature on the subect. Our key tool is a lemma (Lemma 2. below), which, desite its simlicity, rovides an insightful viewoint into the nature of the bilinear estimates of the form (.). From this ersective, in Sections 3 and 4 we rove Besov-Lebesgue estimates (.) for bilinear multiliers of Hörmander-Mihlin tye and for bilinear molecular araroducts, resectively, without resorting to the usual tools of molecular decomositions of Besov saces or reduced bilinear symbols. In Section 5 we introduce a vector-valued interretation of the resent ideas, along with its alications to bilinear Littlewood-Paley theory. Date: October 8, Mathematics Subect Classification. 42B25, 42B20, 47G30. Key words and hrases. Bilinear multiliers, araroducts, almost-orthogonality. First author artially suorted by NSF grant DMS

2 2 DIEGO MALDONADO AND VIRGINIA NAIBO 2. The Basic Lemma We first fix some notation that will be used throughout the aer. The class of Schwartz functions in R n will be denoted by S(R n ) and we set S 0 (R n ) := {f S(R n ) : γ ˆf(0) = 0, for all γ}. We write ψ Ψ if ψ S(R n ), su( ˆψ) {ξ : /2 ξ 2} and ˆψ in {ξ : 3/5 ξ 5/3}. For α R, 0 <, q, and f S(R n ) we define ( (2.) f Ḃα,q := ν Z 2 ναq ν (f) q L ) /q, where ν (f) = ψ ν f and ψ ν (x) = 2 νn ψ(2 ν x), with the usual interretation when q =. The homogeneous Besov saces Ḃα,q is the set of temered distributions f, modulo olynomials, such that f Ḃα,q is finite. The definition is indeendent of the choice of ψ and the dual of Ḃ α,q is Ḃ α,q, where and q denote the conugate airs of and q, resectively. The scale of homogeneous Triebel-Lizorkin saces F α,q is defined similarly, for q <, with the sum and the integral in (2.) taken in reverse order, see [2] and [32] for more details. Finally, C will denote a constant that may deend only on the arameters involved, and that may change from line to line. Our starting oint is the following lemma, which is essentially based on a bilinear Schurtye inequality and Calderón s reroducing formula. Lemma 2.. Let T : S(R n ) S(R n ) S(R n ) be a continuous bilinear oerator and denote by T 2 its second adoint. Let, q, r, s, with / + /q = /r, and suose that there exist l > 0 and C > 0 such that for all, k Z and h =, 2, 3, (2.2) su x h R n ( ) T 2 ψ () ( x ), ψ (2) k (x 2 ) (x 3 ) 3 dx m C 2 l k, m= m h for some ψ (), ψ (2) Ψ, such that Calderón s formula holds true for ψ (2), i.e, (2.3) h = Z (2) (2) (h), h S 0 (R n ). Then, for all α R with α < l there exists a constant C > 0, deending on C, n, and α, such that T (f, g) Ḃα,s C r f Ḃα,s g L q, f, g S(R n ). Proof. Let K(x, y, z) denote the Schwartz distributional kernel of T. We exress this by formally writing T (f, g)(x) = K(x, y, z) f(y) g(z) dydz. In this sense, the second adoint oerator T 2 has kernel K 2 (x, y, z) = K(z, y, x), see Section 2 in [20], for more details. For k Z and ψ () Ψ, define the bilinear oerators Θ k as Θ k (f, g) := () k T (f, g) = ψ() k T (f, g), that is, ( Θ k (f, g)(x) = ψ () k (x u) K(u, y, z) du ) f(y) g(z) dydz.

3 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 3 Consequently, the bilinear oerator (f, g) Θ k ( K T k (x, y, z) := ψ () k (2) ) f, g has kernel (x w)k(w, u, z)ψ(2) (u y) du dw = T 2 ( ψ (2) ( y), ψ () k (x ) ) (z). Set f = f, f 2 = g, =, 2 = q and 3 = r. For f 3 L r (R n ), Hölder s inequality and inequality (2.2) yield the following bilinear Schur-tye inequalities Thus, (2.4) Θ k ( (2) f, g)(x 3 )f 3 (x 3 ) dx 3 R n 3 = 3 m= C 2 l k m= K T k (x, x 2, x 3 ) K T k (x, x 2, x 3 ) /m f m (x m ) dx m ( K T k (x, x 2, x 3 ) f m (x m ) m dx dx 2 dx 3 ) /m 3 f m L m. m= Θ k ( (2) f, g) C2 l k f L r L g L q. 3 f m (x m ) dx m Next, let {G k } be a sequence of functions such that 2 αks G k s. By using L r Hölder s inequality, (2.3), (2.4), and choosing λ such that 0 < λ < l α, we obtain Θ k (f, g)(x) G k (x) dx = ( ) Θ k (2) (2) f, g (x)g k (x) dx Z /s 2 λ k s 2 kαs G k s ( ) 2 λ k s 2 kαs Θk (2) L r (R n ) (2) s f, g Z Z C /s 2 (λ l) k s 2 kαs (2) f s L (R n ) g s L q (R n ) Z = C /s 2 (k )αs 2 (λ l) k s 2 αs (2) f s g L L (R n ) q (R n ) Z C ( ) /s 2 (λ l+ α ) k s 2 αs (2) f s g L L (R n ) q (R n ) Z C f Ḃα,s g L q (R n ), and the lemma follows by duality. m= L r (R n ) /s

4 4 DIEGO MALDONADO AND VIRGINIA NAIBO Remark. By duality, if the condition (2.2) in Lemma 2. holds with T or T instead of T 2, then the resulting bounds are of the form or resectively. T (f, g) L r C f Ḃα,s T (f, g) Ḃα,s r C f L g Ḃ α,s q g Ḃα,s q, f, g S(R n ),, f, g S(R n ), 3. Boundedness of bilinear Hörmander-Mihlin multiliers In this section we consider bilinear multiliers of the form T σ (f, g)(x) = σ(ξ, η) ˆf(ξ)ĝ(η)e ix (ξ+η) dξdη, where σ(ξ, η) is an infinitely differentiable function defined on R n R n \ {(0, 0)} verifying the Hörmander-Mihlin condition, namely, (3.) γ ξ β η σ(ξ, η) C γ,β ( ξ + η ) γ β, for all (ξ, η) R n R n \ {(0, 0)} and all multiindices γ and β. Here γ = γ + + γ n if γ = (γ,, γ n ), and similarly for β. In [8], L. Grafakos and R. Torres used the molecular decomosition of homogeneous Besov saces to study maing roerties of T σ in the diagonal Besov cases of the form T σ : Ḃ α, Ḃα 2,q q Ḃα +α 2,r r, α, α 2 > 0, <, q, r <, / + /q = /r, under the following cancelation conditions on σ(ξ, η) (3.2) ρ ξ σ(0, η) = 0, for all η 0, (3.3) ρ ξ σ(η, η) = 0, for all η 0, (3.4) ησ(ξ, ρ 0) = 0, for all ξ 0, for suitably many multiindices ρ, see [8, Theorem 3]. Under the same cancelation hyotheses, maing roerties of T σ : F α,q F α 2,q 2 2 F α 3,q 3 3 for the scale of homogeneous Triebel-Lizorkin saces F α,q have been addressed by Á. Bényi in [, Proosition 3] and L. Grafakos and R. Torres in [9, Theorem 7]. For s R, let [s] denote the largest integer smaller that s. We will use Lemma 2. and only two of the cancelation hyotheses above to rove Theorem 3.. Consider, q, r, + q = r, and α Rn. Let σ(ξ, η) be an infinitely differentiable function satisfying (3.) for all γ, β n + and the cancelation conditions (3.2) and (3.3) for all multiindices ρ satisfying ρ [ α ] + n +. Then (3.5) T σ (f, g) Ḃα,s r C f Ḃα,s g L q. Proof. Let ψ Ψ such that Calderón s formula holds true for ψ. For k, Z define (3.6) K k (x, y, z) := Tσ 2 (ψ ( y), ψ k (x )) (z) = σ(ξ, η) ˆψ(2 ξ) ˆψ(2 k (ξ + η))e iξ(x y) e iη(x z) dξdη. By Lemma 2. it will be enough to rove that K k satisfy the conditions (2.2) for some l > α. We consider three different cases given by k 3, k 3, and 2 k 2.

5 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 5 Case k 3. Let l = [ α ] +. We will rove that there exists C α such that (3.7) K k (x, y, z) C α 2 l k 2 kn ( + 2 k x z ) n+ 2 n ( + 2 x y ) n+, for any k, Z, k 2. Then conditions (2.2) follow for this range of and k. We make the following change of variables (ξ, η) (2 ξ, 2 k η). Then K k (x, y, z) = 2 kn 2 n σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 k ξ + η) e i2 ξ(x y) e i2k η(x z) dξdη. Without loss of generality assume that x y x y and x z x z. Choose multiindices γ and β so that γ 0 and γ m = 0 for m = 2,, n, β 0 and β m = 0, m = 2,, n, and γ = β = 0, or γ = β = n +, or γ = n + and β = 0, or γ = 0 and β = n +, according to whether 2 x y and 2 k x z are smaller or larger than ξ(x y) k η(x z). Noticing that e i2ξ(x y) = γ ξ ei2 and e i2kξ(x z) = β η e i2 and integrating by (i2 (x y )) γ (i2 k (x z )) β arts when γ 0 or β 0 we get where F (x, y, z) = γ ξ β η K k (x, y, z) = C γ ξ β η 2 n 2 kn (2 (x y )) γ (2 k F (x, y, z) (x z )) β ( σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 ) k ξ + η) e i2ξ(x y) e i2kη(x z) dξ dη. Then (3.7) will follow if we show that F (x, y, z) C 2 l k. Using Leibniz rule we obtain ( σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 ) k ξ + η) = µ γ,ν β c ν,µ 2 µ 2 k ν ( µ ξ ν η σ)(2 ξ, 2 k η) γ µ ξ η β ν ( ˆψ(ξ) ˆψ(2 k ξ + η)). Using (3.2), the mean value theorem reeatedly, and condition (3.) we obtain 2 µ 2 k ν ( µ ξ ν η σ)(2 ξ, 2 k η) C ( µ+ρ ξ η ν σ)(τ, 2 k η) (2 ξ ) ρ 2 µ 2 k ν ξ ρ C 2 ( k)( ρ + µ ) η µ + ν + ρ, where τ R n is in the segment oining 0 R n and 2 ξ and the multiindex ρ is chosen aroriately and such that ρ = l. We now have ( ) F (x, y, z) 2 k l c ν,µ 2 ( k) µ γ µ ξ η β ν ˆψ(ξ) ˆψ(2 k ξ + η) dξ dη. 2 ξ 2 4 η 9 4 µ γ ν β This yields (3.7) since the last integral is uniformly bounded as k < 0. Case k 3. In this case we make the change of variables (ξ, η) (ξ + η, η) in (3.6) obtaining K k (x, y, z) = σ(ξ + η, η) ˆψ(2 (ξ + η)) ˆψ(2 k ξ) e iη(z y) e iξ(x y) dξdη.

6 6 DIEGO MALDONADO AND VIRGINIA NAIBO Define σ(ξ, η) = σ(ξ + η, η) and note that σ satisfies conditions (3.) and (3.2) since σ satisfies (3.) and (3.3). Now, the change of variables (ξ, η) (2 k ξ, 2 η) gives K k (x, y, z) = 2 n 2 kn σ(2 k ξ, 2 η) ˆψ(2 k ξ + η) ˆψ(ξ) e i2η(z y) e i2kξ(x y) dξdη. We are now in the exact same situation as in the revious case. Therefore, K k (x, y, z) C 2 l k 2 kn ( + 2 k x y ) n+ 2 n ( + 2 z y ) n+, for l = [ α ] +. Case k 2. As in the first case ( k 3), we obtain where F (x, y, z) = K k (x, y, z) = C γ ξ β η 2 n 2 kn (2 (x y )) γ (2 k F (x, y, z) (x z )) β ( σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 ) k ξ + η) e i2ξ(x y) e i2kη(x z) dξ dη. and we assume, without loss of generality, that x y x y and x z x z. We take γ = β = 0, or γ = β = n +, or γ = n + and β = 0, or γ = 0 and β = n +, according to whether 2 x y and 2 k x z are smaller or larger than. Since k 2 the desired result will follow if we rove that F (x, y, z) is bounded as a function of x, y, z,, k, k 2, for any values of α and β. From Leibniz rule, γ ξ β η = ( σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 k ξ + η) µ γ,ν β c ν,µ 2 µ 2 k ν ( µ ξ ν η σ)(2 ξ, 2 k η) γ µ ξ η β ν and using the condition (3.), ( γ ξ β η σ(2 ξ, 2 k η) ˆψ(ξ) ˆψ(2 k ξ + η)) µ γ,ν β µ γ,ν β c ν,µ 2 µ 2 k ν (2 ξ + 2 k η ) µ ν γ µ ξ η β ν ) ( ˆψ(ξ) ˆψ(2 k ξ + η)), ( ) c ν,µ 2 (k ) ν ξ µ ν γ µ ξ η β ν ˆψ(ξ) ˆψ(2 k ξ + η). ( ) ˆψ(ξ) ˆψ(2 k ξ + η) Then F (x, y, z) is bounded since k < 2 and su(ψ) {ξ : 2 ξ 2}. Corollary 3.2. Let T σ be as in Theorem 3., <, q, r <, with / + /q = /r, and α > 0, then T σ (f, g) B α,s C f r B α,s g L q. = f Ḃα,s + f L, the result follows from (3.5) and Proof. Since α > 0 imlies f B α,s the fact that bilinear Hörmander-Mihlin multiliers obey the inequality T σ (f, g) L r C f L g L q, as roved by R. Coifman and Y. Meyer in [8], K. Yabuta in [35], and later extended to other indices by L. Grafakos and R. Torres in [20].

7 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 7 Theorem 3. does not require that the symbol σ satisfies condition (3.4). If σ satisfies conditions (3.), (3.2), (3.3) and (3.4), then it easily follows that the symbol of Tσ 2, which is given by σ(ξ, (ξ + η)), satisfies conditions (3.), (3.2) and (3.3). By duality we then have the following Corollary 3.3. Consider, q, r, + q = r, and α Rn. Let σ(ξ, η) be an infinitely differentiable function satisfying (3.) for all γ, β n + and the cancelation conditions (3.2), (3.3) and (3.4), for all multiindices ρ satisfying ρ [ α ] + n +. Then (3.8) T σ (f, g) L r C f Ḃα,s g Ḃ α,s q Estimates similar to (3.8), but for non-homogeneous Besov saces, have been considered by Á. Bényi in [2] when the bilinear multilier σ(ξ, η) is relaced by a bilinear symbol in the forbidden class BS, 0 (however, this class and the Hörmander-Mihlin class are not comarable). 4. Molecular araroducts For ν Z and k Z n, let P νk be the dyadic cube (4.) P νk = {(x,..., x n ) R n : k i 2 ν x i < k i +, i =,..., n}. The lower left-corner of P = P νk is denoted by x P = x νk = 2 ν k, its size by P = 2 νn, and its characteristic function by χ Pνk. The collection of all dyadic cubes will be denoted by D, i.e. D = {P νk : ν Z, k Z n }. Following [2],. 48, a smooth molecule of regularity M and decay N > n associated to P is a function φ P = φ Pνk = φ νk : R n C that satisfies (4.2) γ φ νk (x) C γ,n 2 νn/2 2 γ ν ( + 2 ν x 2 ν, for all γ M and some N > n. k ) N A family of smooth molecules {φ P } P D = {φ νk } ν Z, n that satisfies the additional conditions (4.3) φ νk (x)x γ dx = 0, for all γ L, ν Z, k Z n, where L will be secified in articular uses, will be called a family of smooth molecules with cancelation. Let {φ Q }, {φ2 Q }, {φ3 Q } be three families of smooth molecules, the molecular araroduct (or model araroduct, [29],. 23) associated to these families is defined by (4.4) T (f, g) = Q D Q /2 f, φ Q g, φ 2 Q φ 3 Q, f, g S(R n ).. T has a bilinear kernel given by (4.5) K(x, y, z) = Q D Q 2 φ Q (y)φ 2 Q(z)φ 3 Q(x). A molecular araroduct has the advantage of involving molecules adated to dyadic cubes, a more flexible construction than the usual dilations and translations of two fixed rofiles ψ and φ defining the Bony araroduct (4.6) Π(f, g) = Z(ψ f) (φ g),

8 8 DIEGO MALDONADO AND VIRGINIA NAIBO where ψ, φ S(R n ) and ψ Ψ and su( ˆφ) {ξ R n : ξ /4}. As oosed to the functions φ and ψ in (4.6), which are L -normalized, the smooth molecules in (4.4) are L 2 - normalized. Nevertheless, the concet of the molecular araroduct (4.4) includes (modulo smoothing oerators) the one of Bony araroduct. Indeed, given ψ and φ as in (4.4), we reason as follows: consider φ S(R n ) with su( φ ) {ξ R n : /6 ξ 9/2} and φ in {ξ R n : /4 ξ 9/4} to obtain and, consequently, φ (ξ + η) ψ (ξ) φ (η) = ψ (ξ) φ (η), Z, ξ, η R n, (4.7) φ ((ψ f) (φ g)) = (ψ f) (φ g), Z. Setting φ 2 := ψ and φ 3 := φ, and using (4.7), we can write Π(f, g)(x) = ν Z φ ν ((φ 2 ν f) (φ 3 ν g))(x) ( = =: ν Z φ ν(x w)φ 2 ν(w y)φ 3 ν(w z) dw K Π (x, y, z)f(y)g(z) dydz, whose bilinear kernel can be exanded as K Π (x, y, z) = ν Z n = Q νk 2 ν Z n = Q νk 2 φ Q (x)φ 2 Q(y)φ 3 Q(z) + E(x, y, z). ν Z n ) f(y)g(z) dydz Q νk 2 3νn 2 2 νn 2 φ (2 ν (x w))2 νn 2 φ 2 (2 ν (w y))2 νn 2 φ 3 (2 ν (w z)) dw 2 νn 2 φ (2 ν (x w))2 νn 2 φ 2 (2 ν (w y))2 νn 2 φ 3 (2 ν (w z)) dw Q νk Q νk Here φ Q (x) = 2νn/2 φ (2 ν x k), for Q = Q νk and =, 2, 3, are smooth molecules and the error term E(x, y, z), deending on the differences 2 νn 2 φ (2 ν (x w)) 2 νn 2 φ (2 ν (x x νk )), =, 2, 3, is the kernel of a smoothing oerator. Due to the size condition (4.2), E(x, y, z) is usually disregarded, since during the estimates the averages over Q νk above can be relaced by the values of the integrand at x νk. A detailed study of the maing roerties of the form T : X Y Z for molecular araroducts T, where X, Y, and Z are related functional saces including Besov, Triebel- Lizorkin, Hardy, Sobolev, and Lebesgue saces, (but not estimate (4.3) in Theorem 4.4 below), can be found in [5]. For the case of Dini continuous molecules, see [27]. End-oint results of the form T : F α,q Y F α,q, for certain Triebel-Lizorkin saces Y are roven in [33] and [34]. Our Besov-Lebesgue estimates for molecular araroducts will be based on three known almost-orthogonality estimates, which we included here for the reader s convenience. Namely, Proosition 4.. (Frazier-Jawerth, Aendix B in []) Suose that ϕ ν and ϕ µ are functions defined on R n such that for some x ν, x µ in R n, some N > n + L + with L a

9 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 9 non-negative integer, and some N 2 > n the following conditions hold: (4.8) ϕ ν (x) (4.9) and (4.0) γ xϕ µ (x) 2 νn/2 ( + 2 ν x x ν ) max(n,n 2 ), R n ϕ ν (x)x γ dx = 0 for all γ L, 2 µ γ 2 µn/2 ( + 2 µ x x µ ) N 2 for all γ L +. Then, for ν µ there exists a constant C = C(N, N 2, L) > 0 such that the following estimate is valid (4.) ϕ ν (x)ϕ µ (x) dx C 2 (ν µ)(l++n/2) R n ( + 2 µ x ν x µ ) N. 2 Lemma 4.2. (see, for instance, [5,.A-36]). Let a, b R n, µ, ν R, and P, Q > n. Then Rn (4.2) 2 µn 2 νn ( + 2 µ x a ) P ( + 2 ν x b ) Q dx C 2 min(µ,ν)n P,Q,n ( + 2 min(µ,ν) a b ) min(p,q). Finally, for three real numbers, a, a 2, a 3, we denote by med(a, a 2, a 3 ) one of the a s that satisfies min(a, a 2, a 3 ) a max(a, a 2, a 3 ). Proosition 4.3. (Proosition 3.6 in [5]) For every N > n + there is a constant C, deending only on N and n, such that for any w = (γ, ν, µ, λ) Z 4 and any x, y, z R n l Z n 2 γn2νn/22µn/22λn/2 [( + 2 ν x 2 γ l )( + 2 µ y 2 γ l )( + 2 λ z 2 γ l )] 5N C 2 max(µ,ν,λ)n/2 2 med(µ,ν,λ)n/2 2 min(µ,ν,λ)n/2 (( + 2 min(ν,µ) x y )( + 2 min(µ,λ) y z )( + 2 min(ν,λ) x z )) N. We are now in osition to state our Besov-Lebesgue estimates for molecular araroducts. Theorem 4.4. Let {φ Q }, {φ2 Q }, {φ3 Q } be three families of molecules and let T be its associated molecular araroduct (4.4). Given α R, suose that {φ Q } and {φ3 Q } satisfy (4.3) with some L > 2[ α ] and (4.2) with M = L + and N > 5n + 5, and {φ 2 Q } satisfies (4.2) with M = 0 and N > 5n + 5. Then, for any, q, r, s with / + /q = /r, there exists a constant C = C(α, n,, q, r, s) such that (4.3) T (f, g) Ḃα,s r C f Ḃα,s g L q, f, g S(R n ). Proof. We first notice that the second transose of T is given by T 2 (f, g)(x) = Q D Q 2 f, φ Q g, φ 3 Q φ 2 Q(x) and that, for ψ Ψ verifying (2.3), the functions 2 n 2 ψ ( x 2 ) and 2 nk 2 ψ k (x ) satisfy (4.8), (4.9), and (4.0) for all N, N 2, and L. Without loss of generality, we can consider

10 0 DIEGO MALDONADO AND VIRGINIA NAIBO only the k. Since the case k > will follow similarly, as identical conditions are required to the families {φ Q } and {φ3 Q }. Proosition 4. yields T 2 (ψ ( x 2 ), ψ k (x ))(x 3 ) Q 2 ψ ( x 2 ), φ Q ψ k (x ), φ 3 Q φ 2 Q(x 3 ) Q=Q νl ν Z,l Z n Q D 2 n 2 (ν++k) 2 ν (L++ n 2 ) 2 k ν (L++ n 2 ) 2 n 2 ν [( + 2 min(,ν) x 2 2 ν l )( + 2 min(k,ν) x 2 ν l )( + 2 ν x 3 2 ν l )] N 2. By using w = (ν, min(k, ν), min(, ν), ν) in Proosition 4.3, inequality (4.2), and the fact that min(k, ν) min(, ν) ν, it follows that T 2 (ψ ( x 2 ), ψ k (x ))(x 3 ) C ν Z C ν Z 2 n 2 (+k+2ν) 2 ν (L++ n 2 ) 2 k ν (L++ n 2 ) [( + 2 min(k,ν) x 2 x )( + 2 min(k,ν) x x 3 )( + 2 min(,ν) x 3 x 2 )] N 2/5 2 n 2 (+k+2ν) 2 ν (L++ n 2 ) 2 k ν (L++ n 2 ) [( + 2 min(k,ν) x 2 x )( + 2 min(,ν) x 3 x 2 )] N 2/5. Finally, let I(, k) denote the integral of T 2 (ψ ( x 2 ), ψ k (x ))(x 3 ) with resect to any two of the variables x, x 2, x 3 to obtain I(, k) C ν Z 2 n 2 (+k+2ν) 2 ν (L++ n 2 ) 2 k ν (L++ n 2 ) 2 n min(k,ν) n min(,ν) 2 = C ν Z 2 ν (L+) 2 k ν (L+) 2 τ(k,,ν), where the ower τ(k,, ν) is given by τ(k,, ν) = n 2 (k + + 2ν) n 2 ν n k ν n min(k, ν) n min(, ν), 2 and, in fact, a brief comutation shows that τ(k,, ν) 0 for all ν,, k Z. Consequently, I(, k) C ν Z 2 ν (L+) 2 k ν (L+) C2 2 k (L+), and the theorem follows from Lemma 2.. Remark 2. For α > 0 and <, q, r < with / + /q = /r, the non-homogeneous version of (4.3) follows as in Corollary 3.2, since molecular araroducts involving two families of smooth molecules with cancelation then verify see, for instance, [5], [3], [4], [29]. T (f, g) L r C f L g L q, f, g S(R n ), 5. Bilinear Littlewood-Paley Theory Given a function ψ in the Schwartz class S(R n ) such that R ψ(x)dx = 0, an immediate n alication of the Fourier transform gives the bound (5.) 0 Ψ t (f) 2 L 2 dt t C f 2 L 2,

11 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES where Ψ t (f)(x) = R ψ n t (x y)f(y)dy and ψ t (x) = t n ψ(x/t). In [28], S. Semmes identified sufficient conditions on a family of functions θ t (x, y), t > 0, x, y R n (more general than ψ t (x y)) so that the non-convolution oerator Θ t (f)(x) = θ t (x, y)f(y) dy R n verifies the square function estimate in L 2 (R n ) (5.2) 0 Θ t (f) 2 L 2 dt t C f 2 L 2. In the discrete case, when a family θ k (x, y), k Z, is considered, inequality (5.2) then becomes ( ) /2 (5.3) Θ k (f) 2 L 2 C f L 2. The alluded sufficient conditions have to do with decay, regularity, and cancelation roerties of the kernels θ t (x, y) (or θ k (x, y)). In the following we will assume that {θ k } is a family of comlex-valued functions defined on R n R n R n satisfying the following conditions: There are L, M, N N and constants c α, and A, such that for all k Z, A 2 2nk (5.4) θ k (x, y, z) ( + 2 k x z ) N ( + 2 k x y ) N, (5.5) y α θ k (x, y, z) c α 2 2nk 2 k α, x, y, z R n, α M +, (5.6) θ k (x, y, z) y α dy = 0, x, z R n, α L. R n Notice that, as oosed to the condition (4.2), condition (5.5) above does not involve any decay in the variables x, y, or z. Theorem 5.. Let α R and suose that the kernels {θ k } of the bilinear oerators (5.7) Θ k (f, g)(x) = θ k (x, y, z)f(y)g(z) dy dz, verify (5.4), (5.5), and (5.6) with constants L, M, and N such that 2 α < min(m +, L + ), M + n + < N, and L + n + < N, 2n < N. Then, there is a constant C deending only on L, M, A, N, n, s, and α, such that for all, q, r, s with / + /q = /r, (5.8) ( 2 αks Θ k (f, g) s L r ) /s C f Ḃα,s g L q, f, g S(R n ). The essential stes in the roof of Theorem 5. can be taken to also rove Corollary 5.2. Let Θ k be defined as in (5.7) such that the kernels θ k satisfy (5.4) for some N > 2n, (5.6) with L = 0, and the following Hölder regularity condition in the y-variable (5.9) θ k (x, y, z) θ k (x, y, z) c γ 2 2nk (2 k y y ) γ,

12 2 DIEGO MALDONADO AND VIRGINIA NAIBO for some γ (0, ] and all x, y, y, z R n and k Z. Then, there is a constant C deending only on s, γ, and n, such that for all, q, r, s with / + /q = /r, ( (5.0) Θ k (f, g) s L r ) /s C f Ḃ0,s g L q, f, g S(R n ). In articular, the case s = = q = 2 yields ( ) /2 (5.) Θ k (f, g) 2 L C f L 2 g L 2, f, g S(R n ), which is the natural bilinear version of (5.3). Remark 3. Notice that cancelation in the y-variable only is assumed in Theorem 5. and Corollary 5.2. Corollary 5.2 has been roved in the context of saces of homogeneous tye in [26]. Lemma 5.3. Let l > 0 and {θ k } satisfy (5.5), (5.4), and (5.6) with constants L, M, and N satisfying l min(m +, L +, N n), M + n + < N, and L + n + < N. Then, for all x, u, z R n,, k Z, and ψ Ψ, θ k (x, y, z) ψ (y u) dy C 2 l 2 k 2 kn R n ( + 2 k x z ) N 2 where C is a constant deending on L, M, A, N, n, and ψ. 2 min(,k)n ( + 2 min(,k) x u ) N 2 Proof of Lemma 5.3. It is enough to rove the following two inequalities. For all x, z, u R n,, k Z we have (5.2) θ k (x, y, z) ψ (y u) dy C 2 kn 2 min(,k)n R n ( + 2 k x z ) N ( + 2 min(,k) x u ) N. and (5.3) θ k (x, y, z) ψ (y u) dy C 2 l k 2 kn 2 min(,k)n R n Proof of (5.2): Using condition (5.4), the roerties of ψ, and inequality (4.2), we estimate θ k (x, y, z) ψ (y u) dy A C ψ 2 nk Rn 2 nk 2 n R n ( + 2 k x z ) N ( + 2 k x y ) N ( + 2 y u ) N dy A C ψ C N,n Proof of (5.3): Case > k. Using the fact that 2 kn ( + 2 k x z ) N 2 min(,k)n ( + 2 min(,k) x u ) N. R n ψ (y u) (y u) α dy = 0, α N 0,

13 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 3 we obtain θ k (x, y, z) ψ (y u) dy = R n = R n θ k (x, y, z) R n α =M+ α M α! α y θ k (x, u, z)(y u) α ψ (y u) dy α! α y θ k (x, ξ, z)(y u) α ψ (y u) dy where ξ is in the segment oining y and u. By (5.5) and the roerties of ψ we get θ k (x, y, z) ψ (y u) dy C M,ψ 2 2nk 2 k(m+) y u M+ 2 n R n R n ( + 2 y u ) N dy = = I + II. y u 2 k y u >2 k + Estimate for I. We have y u > 2 k > 2. Then (+2 y u ) N 2 N y u N I C M,ψ 2 2nk 2 k(m+) 2 (n N) y u >2 k y u M+ N dy Recalling that M + n + < N, k < and l N n, we then have I C M,N,n,ψ 2 2nk 2 k (N n) C L,N,n 2 nk 2 min(,k)n 2 k l. Estimate for II. We have II = C M,ψ 2 2nk 2 k(m+) = y u 2 + y u 2 k y u M+ 2 < y u 2 k = II + II 2. For II we use that and we get +2 y u 2 n ( + 2 y u ) N dy II C M,ψ,n 2 2nk 2 k (M+) C M,ψ,n 2 kn 2 min(,k)n 2 k l and therefore where in the last inequality we have used that l M +. For II 2 we use that, and after integrating in olar coordinates (+2 y u ) N (2 y u ) N and recalling that M + n N + < 0, k <, and l M +, we get II 2 C M,ψ 2 2nk 2 k(m+) 2 (n N) y u M+ N 2 < y u 2 k C M,N,n,ψ 2 2nk 2 k (M+) C M,N,n,ψ 2 nk 2 min(,k)n 2 k l. Case k. Using the cancelation roerty (5.6) for θ k, θ k (x, y, z) ψ (y u) dy = θ k (x, y, z) ψ (y u) R n R n = θ k (x, y, z) R n α =L+ α L α! α ψ (x u)(y x) α dy α! α ψ (ξ u)(y x) α dy

14 4 DIEGO MALDONADO AND VIRGINIA NAIBO where ξ is in the segment oining x and y. By condition (5.4) we then get θ k (x, y, z) ψ (y u) dy A C ψ,l 2 2nk 2 (n+l+) L+ y x R n R n ( + 2 k x y ) N dy = We now roceed as before obtaining x y >2 + x y 2. I A C ψ,l,n 2 kn 2 n 2 k (N n) A C ψ,l,n 2 kn 2 n 2 k l, where we have used that L + n + N and l N n, and II A C ψ,l,n 2 kn 2 n 2 k (L+) A C ψ,l,n 2 kn 2 n 2 k l, where we have used that L + n + N and l L +. Proof of Theorem 5.. As in the roof of Lemma 2., let K k (x, x 2, x 3 ) be the bilinear kernel of the oerator (f, g) Θ k (ψ f, g). That is, (5.4) K k (x, x 2, x 3 ) = θ k (x, y, x 3 )ψ (y x 2 ) dy. R n By Lemma 5.3, for all, k Z and h =, 2, 3, we have, for 2 α < l min(m +, L + ), (5.5) su x h R n K k (x, x 2, x 3 ) This yields, as in the roof of Lemma 2., 3 dx m C 2 l k /2. m= m h Θ k ( f, g) L r C2 l k /2 f L g L q, and, since l/2 > α, the duality argument in the roof of Lemma 2. comletes the roof. Our inequalities (5.8) and (5.0) come as an addition to the related known results on bilinear Littlewood-Paley theory. Namely, the inequality ( ) /2 (5.6) S k (f, g) 2 C γ l f L g L q, L r obtained by G. Diestel in [0] for <, q, r <, /r = / + /q, where the rough araroduct oerator S k is defined by S k (f, g) = γ k ˆf(ξ)χ [a k,a k ](ξ)ĝ(η)χ [ b k,b k ](η)e 2πi(ξ+η)x dξdη, R R for a, b (0, ). And also with the square-function inequality ( ) /2 (5.7) S k (f, g) 2 C f L g L q, n L 2

15 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 5 for all 2, q, with / + /q = /2, roved in the context of Gabor analysis by M. Lacey in [23], with S k (f, g)(x) = f(x + y)g(x y)f k (y) dy, R n k Z n, where the smooth function F has Fourier transform suorted on the unit cube of R n and, for k Z n, F k (ξ) := ˆF (ξ k). For bilinear oerators Θ k of the form (5.7), Theorem 5. immediately imlies Corollary 5.4. Given α R, let Θ k be as in Theorem 5.. Then, for the constant C as in Theorem 5., we have ( ) /2 2 2αk Θ k (f, g) 2 C f Ḃα,2 g L q, f, g S(R n ), L 2 for 2, q with / + /q = /2 and ( ) /2 2 2αk Θ k (f, g) 2 C f Ḃα, g L q, f, g S(R n ), L r for, q, r with / + /q = /r. Remark 4. We oint out that the techniques used in this section rovide new results even in the linear case. Indeed, by considering a family θ k (x, y), k Z, that satisfies A 2 nk (5.8) θ k (x, y) ( + 2 k x y ) N, (5.9) y α θ k (x, y) c α 2 nk 2 k α, x, y R n, α M +, (5.20) θ k (x, y) y α dy = 0, x R n, α L, R n (for suitable L, M, and N), a bound of the form ( ) /s (5.2) 2 αks Θ k (f) s L C f Ḃα,s, f S(R n ), follows. Thus extending Semmes s inequality (5.2) to the scale of homogeneous Besov saces Ḃ α,s with α R and, s. References [] Á. Bényi, Bilinear singular integral oerators, smooth atoms and molecules, J. Fourier Anal. Al. 9 no. 3 (2003), [2] Á. Bényi, Bilinear seudodifferential oerators on Lischitz and Besov saces, J. Math. Anal. Al. 284 (2003), [3] Á. Bényi and R. H. Torres, Symbolic calculus and the transoses of bilinear seudodifferential oerators, Comm. P.D.E. 28 (2003), 6-8. [4] Á. Bényi and R. H. Torres, Almost orthogonality and a class of bounded bilinear seudodifferential oerators, Math. Res. Letters. (2004), 2. [5] Á. Bényi, D. Maldonado, A. Nahmod, and R. Torres, Bilinear araroducts revisited, Math. Nachr., to aear. [6] J.-M. Bony, Calcul symbolique et roagation des singularités our les équations aux dérivées artielles non-linéaires, Annales Scientifiques de l École Normale Suérieure Sér. 4, 4 no. 2 (98),

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17 BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES 7 Kansas State University. Deartment of Mathematics. 38 Cardwell Hall. Manhattan, KS address: dmaldona@math.ksu.edu Kansas State University. Deartment of Mathematics. 38 Cardwell Hall. Manhattan, KS address: vnaibo@math.ksu.edu