BILINEAR OPERATORS WITH HOMOGENEOUS SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES

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1 BILINEAR OPERATORS WITH HOMOGENEOUS SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES JOSHUA BRUMMER AND VIRGINIA NAIBO Abstract. We present a unifying approach to establish mapping properties for bilinear pseudodifferential operators with homogeneous symbols in the settings of function spaces that admit a discrete transform and molecular decompositions in the sense of Frazier and Jawerth. As an application, we obtain related Kato-Ponce inequalities.. Introduction and main results As the main purpose of this note we present a unifying approach towards establishing mapping properties of the form (.) T σ (f, g) Y f X g L + f L g X, where X and Y are function spaces admitting a molecular decomposition and a ϕ-transform in the sense of Frazier-Jawerth as introduced in [0, ], and T σ is a bilinear pseudodifferential operator given by T σ (f, g)(x) := σ(x, ξ, η) f(ξ)ĝ(η)e 2πix (ξ+η) dξ dη x R n, R 2n with a bilinear symbol σ in the class multiindices α, β, γ N n 0, it holds BS m, for some m R, that is, σ is such that for all (.2) σ γ,α,β := sup x γ ξ α η β σ(x, ξ, η) ( ξ + η ) m γ + α+β <. (x,ξ,η) R 3n \{0} When m = 0, the x-independent symbols in BS 0, constitute the well-known class of Coifman-Meyer bilinear multipliers. The bilinear forbidden class BS, 0 is defined as the family of symbols satisfying (.2) with m = 0 and with ξ + η replaced by + ξ + η. Note that if σ belongs to BS, 0 then σ = σ + σ 2 where σ is in BS 0, and σ 2 is a smoothing symbol supported in {(x, ξ, η) : ξ + η }. We refer the reader to the work of Coifman and Meyer in [7] and the references it contains for pioneering work related to such symbols. As we will describe next, these two classes of symbols possess distinct essential features, and, as a noteworthy consequence of our Theorem. below, it will follow that they share various mapping properties of the form (.). Date: September 4, Mathematics Subject Classification. Primary: 47G30, 42B35. Secondary: 46E35. Key words and phrases. Pseudodifferential operators, homogeneous symbols, smooth molecules, Kato- Ponce inequalities. The authors are partially supported by the NSF under grant DMS

2 2 JOSHUA BRUMMER AND VIRGINIA NAIBO Coifman-Meyer bilinear multipliers can be realized as bilinear Calderón-Zygmund operators. As such, they inherit their mapping properties; for instance, Calderón-Zygmund operators are bounded in the settings of Lebesgue spaces, BMO, the Hardy space H (Grafakos- Torres [5]), and in weighted Lebesgue spaces (Lerner et al. [2]). On the other hand, the bilinear forbidden class BS, 0 is known to produce bilinear pseudodifferential operators with a bilinear Calderón Zygmund kernel, but, in general, they are not bilinear Calderón Zygmund operators (Bényi-Torres [4]). In particular, they do not always possess mapping properties of the form L p L p 2 L p with < p, p 2 and /p +/p 2 = /p. Mapping properties for bilinear pseudodifferential operators with symbols in BS, 0 have been studied in Bényi [2] in the setting of Besov spaces, in Bényi-Torres [4] and Bényi-Nahmod-Torres [3] in the scale of Lebesgue-Sobolev spaces, and in Naibo [23] and Koezuka-Tomita [20] in the context of Besov and Triebel-Lizorkin spaces. As our main result, Theorem. below, we prove molecular estimates on T σ, with σ BS m,, when one of its arguments is a fixed function and its other argument is a smooth molecule. Theorem.. Given m R and σ BS m,, there exist σ, σ 2 BS m, with T σ = T σ + T σ 2 and such that if r, 0 < M <, ψ S(R n ), with ψ supported in {ξ R n : 2 < ξ < 2}, and γ N n 0, it holds that and γ T σ (ψ ν,k, g)(x) 2 νn 2 2 ν(m+ γ ) 2 νn r ( + 2 ν x k ) M g L r x R n γ T σ 2(f, ψ ν,k )(x) 2 νn 2 2 ν(m+ γ ) 2 νn r ( + 2 ν x k ) M f L r x R n, for every ν Z, k Z n and f, g S(R n ), and where ψ ν,k (x) = 2 νn 2 ψ(2 ν x k). Here S(R n ) denotes the Schwartz class of smooth rapidly decreasing functions defined on R n ; the notation means C, where C is a constant that may depend on some of the parameters used but not on the functions or variables involved... A sample of applications of Theorem.. In the case r =, Theorem. implies that, up to uniform multiplicative constants, the functions 2 νm T σ (ψ ν,k, g)/ g L and 2 νm T σ 2(f, ψ ν,k )/ f L can be regarded as smooth molecules, as introduced in [0, ] in the settings of Besov and Triebel-Lizorkin spaces. Since smooth molecules also serve as building blocks for a variety of other function spaces, Theorem. will apply to such spaces as well. As a concrete application, we will implement Theorem. in the scales of homogeneous Besov-type and Triebel-Lizorkin-type spaces. These spaces were introduced and studied in Sawano-Yang-Yuan [25] and Yang-Yuan [28, 29] as natural spaces that extend and unify the scales of homogeneous Besov spaces, homogeneous Triebel-Lizorkin spaces, and Q-spaces. The latter were introduced in Essén et al. [9] as a refinement of BMO functions. In addition, as proved in [25], the Besov-type and Triebel-Lizorkin-type spaces also contain or coincide with Besov-Morrey and Triebel-Lizorkin-Morrey spaces. We refer the reader to Section 3 for detailed notation and precise definitions. In the following, S 0 (R n ) denotes the closed subspace of functions in S(R n ) that have vanishing

3 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES 3 moments of all orders; that is, f S 0 (R n ) if and only if f S(R n ) and x α f(x) dx = 0 R n for all α N n 0. For 0 < p, q, set ( ) ( ) (.3) s := n and s min{,} p := n. min{,p} By means of Theorem. and molecular techniques, we obtain the following mapping properties in the scales of homogeneous Besov-type and Triebel-Lizorkin-type spaces. Theorem.2. Let m R and σ BS m,. If 0 < p, q, s p < s < and 0 τ < + s sp, p n it holds that T σ (f, g) Ḃ f Ḃ s+m,τ g L + f L g Ḃs+m,τ f, g S 0 (R n ). If 0 < p <, 0 < q, s < s < and 0 τ < + s s, it holds that p n T σ (f, g) F f F s+m,τ g L + f L g F s+m,τ f, g S 0 (R n ). Theorem.2 can be considered as a bilinear counterpart to Grafakos-Torres [4, Theorems. and.2] (see also Torres [27]), where boundedness properties in homogeneous Besov and Triebel-Lizorkin spaces were addressed for linear pseudodifferential operators with symbols in the class Ṡm,, the linear analog to BS m,. In turn, the (linear) results in [4] were extended to the setting of Besov-type and Triebel-Lizorkin-type spaces in [25, Theorem.5]. We refer the reader to Hart-Torres-Wu [7] where very different techniques are used to obtain estimates in the spirit of those in Theorem.2 in the setting of Sobolev spaces for operators with x-independent symbols and a limited amount of regularity. In Remark 4. we address Theorem.2 in the cases corresponding to s s p and s s and show that analogous estimates are obtained, with a slightly different range for the parameter τ, if a number of cancellation conditions are imposed on the first adjoint of T σ and on the second adjoint of T σ 2, where σ and σ 2 are as in Theorem.. In Remark 4.2 we give a version of Theorem.2 involving the L r norms of f and g instead of their L norms. The next corollary of Theorem.2 follows from the realization of Q-spaces as special cases of Triebel-Lizorkin-type spaces (see Section 3..). Corollary.3. Let s, s + m (0, ) and σ that T σ (f, g) Q s,q f p Q s+m,q p BS m,. If q p and q, it holds g L + f L g Q s+m,q p f, g S 0 (R n )..2. Applications to Kato-Ponce inequalities. As a consequence of Theorem. in the case σ, given a function space X that admits a molecular representation and a ϕ-transform, we obtain the following fractional Leibniz rule or Kato-Ponce inequality: (.4) fg X f X g L + f L g X. Inequalities of the form (.4) were proved by Kato-Ponce [8] in the case where X is the Sobolev space W s,p (R n ), with < p < and 0 < s <, in relation to Cauchy problems for the Euler and Navier-Stokes equations; prior work due to Strichartz [26] treats the range n/p < s <, while the case of s N can be obtained from the Leibniz rule and the Gagliardo- Nirenberg inequality. Later on, Gulisashvili-Kon [6] showed (.4) for the homogeneous space X = Ẇ s,p (R n ), for the same range of parameters, in connection with the study of smoothing properties of Schrödinger semigroups. The estimates (.4) also hold true in the settings of Besov and Triebel-Lizorkin spaces and have applications to partial differential equations (see, for instance, Bahouri-Chemin-Danchin [], Chae [5], Runst-Sickel [24] and

4 4 JOSHUA BRUMMER AND VIRGINIA NAIBO the references they contain). In particular, all such estimates imply that X L (R n ) is an algebra under pointwise multiplication. Closely related versions to (.4) were given by Christ- Weinstein [6] and Kenig-Ponce-Vega [9], in the contexts of Korteweg-de Vries equations, and by Gulisashvili-Kon [6]. Extensions to the cases of indices below one appear in Grafakos- Oh [3] and Muscalu-Schlag [22], and versions in weighted and variable exponent space settings were proved in Cruz-Uribe-Naibo [8]. In particular, in the scales of Besov-type and Triebel-Lizorkin-type spaces, Theorem.2 yields the following new Kato-Ponce inequalities. Corollary.4. If 0 < p, q, s p < s < and 0 τ < + s sp, it holds that p n fg Ḃ f Ḃ g L + f L g Ḃ f, g S 0 (R n ). If 0 < p <, 0 < q, s < s < and 0 τ < + s s, it holds that p n fg F f F If 0 < s <, q p and q, it holds that fg Q s,q p f Q s,q p g L + f L g F f, g S 0 (R n ). g L + f L g Q s,q p f, g S 0 (R n ). The article is organized as follows. In Section 2 we prove Theorem.. Section 3 contains the definitions of Besov-type and Triebel-Lizorkin-type spaces, smooth molecules, and the ϕ-transform. The proof of Theorem.2 and several closing remarks are given in Section Proof of Theorem. Our first step towards the proof of Theorem. will be obtaining a representation of a bilinear pseudodifferential operator with a symbol in BS m, as a superposition of paraproductlike operators. Such representations can be traced back to the pioneering work of Coifman and Meyer; Lemma 2. gives a version of a decomposition suited for our purposes, and its proof follows ideas inspired from [7, pp.54-55]. We then state and prove Lemma 2.2, which procures a formula for the derivatives of the building blocks, appropriately evaluated, given by Lemma 2.. We close this section with the proof of Theorem.. The Fourier transform of a tempered distribution f S (R n ) will be denoted by f; in particular, we use the formula f(ξ) = f(x)e 2πix ξ dx for f S(R n ). R n Let θ be a real-valued infinitely differentiable function supported on ( 2, 2) and such that θ(t) + θ(/t) = for every t > 0. For σ BS m,, m R, define ( ) ( ) σ (x, ξ, η) := σ(x, ξ, η)θ η and σ 2 (x, ξ, η) := σ(x, ξ, η)θ ξ x, ξ, η R n. ξ η Simple computations show that σ, σ 2 σ d γ,α,β BS m, with sup σ γ,ᾱ, ᾱ α, β β β for γ, α, β N n 0 and d =, 2, where the implicit constant depends only on γ, α, β and θ, and we have T σ (f, g) = T σ (f, g) + T σ 2(f, g), f, g S 0 (R n ). Endowing S 0 (R n ) with the topology inherited from S(R n ), a standard argument using integration by parts allows to conclude that T σ is continuous from S 0 (R n ) S(R n ) to S(R n ) and T σ 2 is continuous from S(R n ) S 0 (R n ) to S(R n ). Let Ψ, Φ S(R n ) be such that Ψ and

5 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES 5 Φ are real-valued, supp( Ψ) {ξ : 2 < ξ < 2}, j Z Ψ(2 j ξ) 2 = for every ξ 0, Φ for ξ 4 and Φ 0 for ξ > 0. Lemma 2.. Let σ BS m,. With the notation introduced above and given N > n, there exist sequences of functions {m j(x, u, v)} j Z and {m 2 j(x, u, v)} j Z defined for x, u, v R n such that if γ N n 0, then (2.5) sup x,u,v R n γ xm d j(x, u, v) 2 j(m+ γ ), j Z, d =, 2, and, if f S 0 (R n ), g S(R n ) and x R n, it holds that (2.6) T σ (f, g)(x) = m j(x, u, v) u j f(x) Sj v dudv g(x) ( + u 2 + v 2 ) N and (2.7) T σ 2(g, f)(x) = R 2n j Z R 2n j Z m 2 j(x, u, v) Sj u g(x) v dudv jf(x) ( + u 2 + v 2 ), N where u j f(ξ) = Ψ u (2 j ξ) f(ξ) with Ψ u (x) := Ψ(x + u) and Ŝv j g(ξ) = Φ v (2 j ξ)ĝ(ξ) with Φ v (x) := Φ(x + v). Proof. We will prove (2.6), with the proof of (2.7) following analogously. Since the support of Ψ(2 j ξ) 2 σ (x, ξ, η) is contained in {(x, ξ, η) : η 2 ξ and 2 j < ξ < 2 j+ } {(x, ξ, η) : η 2 j+2 } and Φ(2 j η) for η 2 j+2, we have Ψ(2 j ξ) 2 σ (x, ξ, η) = Φ(2 j η) 2 Ψ(2 j ξ) 2 σ (x, ξ, η) x, ξ, η R n, j Z. From this, the fact that j Z Ψ(2 j ξ) 2 = for ξ 0 and Fubini s theorem, it follows that if f S 0 (R n ) and g S(R n ), then (2.8) T σ (f, g)(x) = σj (x, 2 j ξ, 2 j η) Ψ(2 j ξ) Φ(2 j η) ˆf(ξ)ĝ(η) e 2πix (ξ+η) dξdη, j Z R 2n where σ j (x, ξ, η) := Ψ(ξ) Φ(η)σ (x, 2 j ξ, 2 j η). Given multiindices γ, α, β N n 0, the Leibniz rule implies that γ x α ξ β η σ j can be written as a linear combination of terms of the form (2.9) α β Ψ(ξ) γ Φ(η)( x α 2 ξ β 2 η σ )(x, 2 j ξ, 2 j η)2 j α 2+β 2, α + α 2 = α, β + β 2 = β. Since σ BS m,, the absolute value of each term (2.9) can be bounded by a multiple of α Ψ(ξ) β Φ(η) 2 j α 2 +β 2 ( 2 j ξ + 2 j η ) m+ γ α 2+β 2 2 j(m+ γ ) x, ξ, η R n, where we have used that α β Ψ(ξ) Φ(η) is supported in {(ξ, η) : < ξ + η < 2}, and 2 the implicit constant is independent of j. Define m j(x, u, v) := ( + u 2 + v 2 ) N σ j (x,, )(u, v); by the above we have xm γ j(x, u, v) = ( + u 2 + v 2 ) N xσ γ j (x, ξ, η) ( ξ,η) N e 2πi(u ξ+v η) R ( + 4π 2n 2 u 2 + 4π 2 v 2 ) dξdη N ( ξ,η ) N ( xσ γ 2 < ξ + η <2 j )(x, ξ, η)e 2πi(u ξ+v η) dξdη 2j(m+ γ ).

6 6 JOSHUA BRUMMER AND VIRGINIA NAIBO Finally, using that σj (x, 2 j ξ, 2 j η) = m j(x, u, v)e 2πi(u 2 jξ+v 2 j η) dudv R ( + u 2n 2 + v 2 ) N in (2.8), after interchanging summation and integral signs justified by Fubini s theorem, we get (2.6). For each u, v R n, set σ u,v(x, ξ, η) := j Z m j(x, u, v) Ψ u (2 j ξ) Φ v (2 j η), then T σ u,v (f, g)(x) = j Z m j(x, u, v) u j f(x) S v j g(x). Similarly define σ 2 u,v. In our next lemma we look at derivatives of T σ u,v (ψ ν,k, g) and T σ 2 u,v (f, ψ ν,k ). Lemma 2.2. If γ N n 0, ν Z, k Z n, u, v R n, g S(R n ) and ψ S(R n ) is such that supp( ψ) {ξ R n : < ξ < 2}, then 2 γ T σ u,v (ψ ν,k, g)(x) = 2 νn 2 ν+ j=ν γ +γ 2 +γ 3 =γ C γ,γ 2,γ 3 2 ν γ γ γ x m j(x, u, v) (Φ γ 2 ν j g(2 ν ))(2 ν x + 2 ν j v) Ψ γ 3 ν j (2ν x k + 2 ν j u), where Φ γ 2 ν j, Ψγ 3 ν j S(Rn ) are independent of g and ψ ν,k (x) = 2 νn 2 ψ(2 ν x k). An analogous formula holds for γ T σ 2 u,v (f, ψ ν,k ) with f S(R n ). Proof. In view of the supports of ψ and Ψ, the supports of ψ(2 ν ) and Ψ(2 j ) only intersect if ν j ν +. We then have T σ u,v (ψ ν,k, g)(x) = = = ν+ j=ν ν+ j=ν Denoting ν+ j=ν m j(x, u, v)2 νn 2 m j(x, u, v) Ψu (2 j ξ) Φ v (2 j η) ψ ν,k (ξ)ĝ(η)e 2πix (ξ+η) dξ dη R 2n Ψu (2 j ξ) Φ v (2 j η)e 2πi2 ν k ξ ψ(2 ν ξ)ĝ(η)e 2πix (ξ+η) dξ dη R 2n ( ) ( 2 νn 2 m j (x, u, v) 2 νn ĝ(2 ν η) Φ v (2 ν j η)e 2πi2νx η dη R n Ψu (2 ν j ξ) ψ(ξ)e 2πi(2νx k) ξ dξ R n ( ) ( ) F j (x) := m j(x, u, v) 2 νn ĝ(2 ν η) Φ v (2 ν j η)e 2πi2νx η dη Ψu (2 ν j ξ) ψ(ξ)e 2πi(2νx k) ξ dξ R n R n ).

7 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES 7 and given a multi-index γ N n 0, we have γ F j (x) = C γ,γ 2,γ 3 γ x m j(x, u, v) γ +γ 2 +γ 3 =γ ( ) ( ) 2 νn ĝ(2 ν η)2 ν γ2 η γ 2 Φv (2 ν j η)e 2πi2νx η dη R n = C γ,γ 2,γ 3 2 ν γ γ γ x m j(x, u, v) 2 ν γ3 ξ γ 3 Ψu (2 ν j ξ) ψ(ξ)e 2πi(2νx k) ξ dξ R n γ +γ 2 +γ 3 =γ ( ) ( ) 2 νn ĝ(2 ν η)η γ 2 Φ(2 ν j η)e 2πi(2νx+2ν jv) η dη ξ γ 3 Ψ(2 ν j ξ) ψ(ξ)e 2πi(2νx k+2ν ju) ξ dξ R n R n = C γ,γ 2,γ 3 2 ν γ γ γ x m j(x, u, v)(φ γ 2 ν j g(2 ν ))(2 ν x + 2 ν j v)ψ γ 3 ν j (2ν x k + 2 ν j u), γ +γ 2 +γ 3 =γ where Φ γ 2 ν j (η) := ηγ 2 Φ(2 ν j η) and Ψ γ 3 ν j (ξ) := ξγ 3 Ψ(2 ν j ξ) ψ(ξ). Since xt γ σ u,v (ψ ν,k, g)(x) = ν+ j=ν 2 νn 2 γ F j (x), we get the desired result. Proof of Theorem.. Let σ BS m,, r, 0 < M <, ψ S(R n ) such that ψ is supported in {ξ R n : < ξ < 2} and g 2 S(Rn ). With the notation used above, Lemma 2.2 and (2.5) imply γ T σ u,v (ψ ν,k, g)(x) 2 νn 2 2 ν(m+ γ ) 2 νn 2 2 ν(m+ γ ) 2 νn 2 2 ν(m+ γ ) 2 νn r ν+ j=ν γ +γ 2 +γ 3 =γ ν+ j=ν γ +γ 2 +γ 3 =γ Φ γ 2 ν j g(2 ν ) L Ψ γ 3 ν j (2ν x k + 2 ν j u) Φ γ 2 ν j ( + u ) M ( + 2 ν x k ) M g L r, g(2 ν ) ( + 2 ν j u ) M L r L r ( + 2 ν x k ) M where in the second inequality we have used that Ψ γ 3 ν j S(Rn ). Since dudv T σ (f, g)(x) = T σ u,v (f, g)(x) R ( + u 2n 2 + v 2 ), N by choosing N sufficiently large so that (+ u ) M dudv <, we obtain the desired estimate for γ T σ (ψ ν,k, g)(x). Analogous reasoning leads to the estimate for γ T σ 2(f, ψ ν,k )(x). R 2n (+ u 2 + v 2 ) N 3. Function spaces We recall that S 0 (R n ) denotes the closed subspace of functions in S(R n ) that have vanishing moments of all orders and we endow S 0 (R n ) with the topology inherited from S(R n ). The dual space of S 0 (R n ), S 0(R n ), can be identified with the space of tempered distributions modulo polynomials, S (R n )/P(R n ). Let D be the collection of dyadic cubes in R n. That is, D := {Q ν,k } ν Z,k Z n where Q ν,k := {x R n : k j 2 ν x j < k j +, j =,..., n}.

8 8 JOSHUA BRUMMER AND VIRGINIA NAIBO We denote the edge length of Q ν,k by l(q ν,k ) and set x Q = x ν,k := 2 ν k where Q = Q ν,k. We will consider functions ϕ, ψ S(R n ) such that (3.0) supp( ϕ), supp( ψ) {ξ R n : < ξ < 2}, 2 (3.) ϕ(ξ), ψ(ξ) > c for all ξ such that 3 < ξ < 5 and some c > 0, 5 3 (3.2) ϕ(2 j ξ) ψ(2 j ξ) = for ξ 0. j Z See [2, Lemma 6.9] for a construction of ψ given that ϕ satisfies (3.0) and (3.). If ϕ S(R n ) satisfies (3.0) and (3.), ν Z and k Z n, we recall that ϕ ν,k denotes the L 2 -normalized function ϕ ν,k (x) = 2 νn 2 ϕ(2 ν x k) = 2 νn 2 ϕ(2 ν (x x ν,k )). If ψ S(R n ) verifies (3.0), (3.) and (3.2), then it follows that f = f, ϕ ν,k ψ ν,k, ν Z,k Z n where the series converges for f L 2 (R n ) in the topology of L 2 (R n ), for f S 0 (R n ) in the topology of S(R n ) and for f S (R n ) in S (R n ) modulo polynomials (see [0, ] for details). 3.. Homogeneous Besov-type and Triebel-Lizorkin-type spaces. Let ϕ S(R n ) satisfy conditions (3.0) and (3.), and set ϕ j (x) := 2 jn ϕ(2 j x) for x R n and j Z. Fix s, τ R and 0 < q. For 0 < p, the Besov-type space Ḃ (R n ) is defined as the set of all f S 0(R n ) such that f Ḃ := sup P D P τ j= log 2 (l(p )) [ ] q/p (2 js ϕ j f(x) ) p dx P /q <. For 0 < p <, the Triebel-Lizorkin-type space F (R n ) is defined as the set of all f S 0(R n ) such that p/q /p f F := sup (2 js ϕ P D P τ j f(x) ) q dx <. P j= log 2 (l(p )) These spaces are independent of the choice of ϕ (see [29, Corollary 3.]). As in [29], we will use A (R n ) to denote either Ḃ (R n ) or F (R n ), excluding p = in the latter case Special cases of A (R n ). We refer the reader to [28, Section 3] and [29, Proposition 3.] regarding the following statements. (i) If 0 < p, q, s R and < τ < 0, then A (R n ) equals the equivalence class of all polynomials on R n ; if 0 τ <, they are quasi-banach spaces and contain S 0 (R n ). (ii) If 0 < p, q, s R and τ = 0, then Ḃs,0 (R n ) coincides with the homogeneous Besov space Ḃs (R n ), with equivalent norms. (iii) If 0 < p <, 0 < q, s R and τ = 0, then F s,0 (R n ) coincides with the homogeneous Triebel-Lizorkin space F (R s n ), with equivalent norms. In turn, F p,2(r s n ) coincides with the Sobolev space Ẇ s,p (R n ) for < p < and 0 < s <, with equivalent norms.

9 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES 9 (iv) If 0 < p <, 0 < q and s R, then F s, p (R n ) coincides with the homogeneous Triebel-Lizorkin space F,q(R s n ), with equivalent norms. In particular, F 0, p p,2 (R n ) = BMO(R n ), with equivalent norms. (v) If 0 < p, q < and 0 < s <, then F s, q p q,q (R n ) coincides with the Q-space Q s,q p (R n ), with equivalent norms. Here f Q s,q p (R n ) if and only if f S 0(R n ) with f(x) f(y) measurable on R n R n and { } f Q s,q p (R n ) := sup I /p /q f(x) f(y) q /q dy dx <, I I I x y n+qs where I ranges over all cubes of R n with dyadic edge lengths. In particular, Q s (R n ) := Q s,2 n/s (Rn ) = F s, 2 s n 2,2 (R n ). For 0 < s < if n 2, or for 0 < s if n =, the spaces 2 Q s (R n ) constitute a decreasing family of nontrivial subspaces of BMO, see [9]. (vi) Further special cases of the spaces Ȧ (R n ) involving homogeneous Besov-Morrey and Triebel-Lizorkin-Morrey spaces can be found in [25, Theorem.] Molecules. Based on the pioneering work from [0, ], it was proved in [29, Theorem 3.] that the spaces A (R n ) can be characterized in terms of the so-called ϕ-transform defined by S ϕ (f) = { f, ϕ ν,k } ν,k for f S 0(R n ), where ϕ S(R n ) satisfies (3.0) and (3.). More precisely, if 0 < p, q, s R and 0 τ <, then (3.3) f Ḃ { f, ϕ ν,k } ν,k ḃ and f F { f, ϕ ν,k } ν,k f, where ḃ and f refer to the following spaces of sequences: For 0 < p, the space ḃ (R n ) is defined as the collection of all sequences t = {t Q } Q D C, indexed by the dyadic cubes, such that t ḃ := sup p q/p /q Q s/n /2 t P D P τ Q χ Q (x) dx <. j= log 2 (l(p )) P l(q)=2 j For 0 < p <, the space f (R n ) is defined as the collection of all sequences t = {t Q } Q D C, indexed by the dyadic cubes, such that [ ] p/q /p t f := sup ( Q s/n /2 t P D P τ Q χ Q (x)) q dx <. P Q P As before, we will use ȧ (R n ) to denote either ḃ (R n ) or f (R n ), excluding the case p = in the latter case. Let 0 < p, q, s R, 0 τ < and s := s [s], where [s] denotes the largest integer smaller than or equal to s. Set { sp + n if A J := (R n ) = Ḃ (R n ) s + n if A (R n ) = F (R n ), where s p and s are as in (.3). We say that {m Q } Q D, where m Q : R n C, is a family of smooth synthesis molecules for A (R n ) if there exist δ and M with max{s, (s + nτ) } <

10 0 JOSHUA BRUMMER AND VIRGINIA NAIBO δ and J < M < such that R n m Q (x)x γ dx = 0 if γ max{[j n s], }, m Q (x) γ m Q (x) γ m Q (x) γ m Q (y) Q /2 γ /n δ/n x y δ Q /2 ( + l(q) x x Q ) max{m,m s} x R n, Q /2 γ /n ( + l(q) x x Q ) M x Rn and γ [s + nτ], sup z x y ( + l(q) x z x Q ) x, y M Rn and γ = [s + nτ]. It easily follows that {ϕ ν,k } ν Z,k Z n and {ψ ν,k } ν Z,k Z n are families of smooth synthesis molecules for any A (R n ) with parameters δ = and any M > J. Through analogous ideas on almost-diagonal operators used to prove [, Theorem 3.5] it follows that if 0 < p, q, s R, max{s, (s + nτ) } < δ, J < M <, 0 τ < min{ + M J, + (J s) } if max{[j n s], } 0, 0 τ < min{ + M J, + s+n J } p 2n p n p 2n p n if max{[j n s], } < 0, and {m Q } Q D is a family of synthesis molecules for Ȧ (R n ) with parameters δ and M, then (3.4) Q D t Q m Q Ȧ t ȧ t = {t Q } Q D ȧ, where the implicit constant does not depend on the family of molecules ([29, Theorem 4.2]). 4. Proof of Theorem.2 and closing remarks Proof of Theorem.2. Let ϕ, ψ S(R n ) satisfy (3.0), (3.) and (3.2). Since T σ and T σ 2, as given by Theorem., are continuous from S 0 (R n ) S 0 (R n ) to S(R n ) and h = ν Z,k Z h, ϕ n ν,k ψ ν,k for h S 0 (R n ) with convergence in S 0 (R n ) (see Section 3), we have T σ (f, g) = f, ϕ ν,k T σ (ψ ν,k, g) f, g S 0 (R n ), ν Z,k Z n T σ 2(f, g) = ν Z,k Z n g, ϕ ν,k T σ 2(f, ψ ν,k ) f, g S 0 (R n ), where the convergence is in S(R n ). Theorem. implies that there are constants c and c 2 such that if f, g S 0 (R n ), then { } { } c 2 νm T σ (ψ ν,k, g) c2 2 νm T σ 2(f, ψ ν,k ) and g L f L ν Z,k Z n ν Z,k Z n are families of smooth synthesis molecules for any Ȧ (R n ) if 0 < p, q, s > J n and 0 τ < (with δ = and any M > J; note that the zero moment condition is void since

11 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES J n s < 0). If, in addition, 0 τ < + s+n J, we can apply (3.4) and (3.3) to get p n T σ (f, g) Ȧ {2νm f, ϕ ν,k } ȧ g L = { f, ϕ ν,k } ȧs+m,τ g L f Ȧs+m,τ g L, T σ 2(f, g) Ȧ {2νm g, ϕ ν,k } ȧ f L = { g, ϕ ν,k } ȧs+m,τ f L g Ȧs+m,τ f L, from which the desired estimates follow. Remark 4.. Let m R and σ BS m,. The estimates in Theorem.2 hold true in A for 0 < p, q, s J n and 0 τ < + (J s) if the following cancellation conditions p n are satisfied: T σ (xγ, g) = T σ 2 (f, xγ ) = 0 f, g S 0 (R n ), γ [J n s]. We recall that if T is a bilinear operator continuous from S 0 (R n ) S 0 (R n ) to S(R n ), T and T 2 denote the adjoint operators of T defined from S (R n ) S 0 (R n ) to S 0(R n ) and from S 0 (R n ) S (R n ) to S 0(R n ), respectively, as h, T (f, g) = T (h, g), f = T 2 (f, h), g. The proof of the estimates in this case is the same as above, with the only thing left to check being the zero moment conditions for T σ (ψ ν,k, g) and T σ 2(f, ψ ν,k ) (note that the range assumed for τ comes from the assumptions for the validity of (3.4)). We have, for γ [J n s], x γ T σ (ψ ν,k, g) dx = x γ, T σ (ψ ν,k, g) = T σ (xγ, g), ψ ν,k = 0 g S 0 (R n ), R n and similarly for T σ 2(f, ψ ν,k ). Remark 4.2. Let r and m, σ, p, q, s and τ be as in the hypothesis of Theorem.2 or Remark 4.. By the same reasoning as in the proof of Theorem.2 and Remark 4., we also obtain T σ (f, f g) Ȧ Ȧ s+m+ n g r,τ L r + g Ȧs+m+ n f r,τ L r. Remark 4.3. The implicit constants in the inequalities of Theorem. and Theorem.2 depend linearly on σ K,L for some K, L N, where σ K,L := sup σ γ,α,β. γ K, α+β L From the proofs, it follows that the implicit constants in the inequalities of Theorem. are multiples of σ γ,2n, with N N, N > M + n and where γ and M are as in the statement of the theorem. In turn, this implies that the implicit constants in Theorem.2 can be taken to be multiples of σ [s+nτ]+,2n with N > max{j + n, 2(s + n) J + n}. The latter is also true for the inequalities from Remark 4. with N > J + n + 2( (J s) ). References [] H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 20. [2] Á. Bényi. Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces. J. Math. Anal. Appl., 284():97 03, [3] Á. Bényi, A. R. Nahmod, and R. H. Torres. Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. J. Geom. Anal., 6(3):43 453, 2006.

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13 BILINEAR SYMBOLS, SMOOTH MOLECULES, AND KATO-PONCE INEQUALITIES 3 Joshua Brummer, Department of Mathematics, Kansas State University. 38 Cardwell Hall, 228 N. 7th Street, Manhattan, KS 66506, USA. address: brummerjd@math.ksu.edu Virginia Naibo, Department of Mathematics, Kansas State University. Hall, 228 N. 7th Street, Manhattan, KS 66506, USA. address: vnaibo@math.ksu.edu 38 Cardwell