SOBOLEV SPACE ESTIMATES AND SYMBOLIC CALCULUS FOR BILINEAR PSEUDODIFFERENTIAL OPERATORS

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1 SOBOLEV SPACE ESTIMATES AND SYMBOLIC CALCULUS FOR BILINEAR PSEUDODIFFERENTIAL OPERATORS Abstract. Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so called Coifman- Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too. 1. Introduction 1.1. General approach. This article is part of an ongoing project by the authors devoted to the analysis of the unfolding theory of multilinear pseudodifferential operators with x-dependent symbols. One of our principal goals is to eventually provide a complete understanding of those symbols that naturally arise from the non-smooth multipliers generalizing the bilinear Hilbert transform when they become x-dependent. Little is known for pseudodifferential operators beyond the results available for the so called Coifman-Meyer class (see the references below) and we present some new results in this direction. On the other hand, for bilinear multipliers there are by now numerous results in the literature dealing with boundedness properties on Lebesgue spaces. As we will observe later in this article, such results extend to Sobolev spaces. For the analysis of operators with x-dependent symbols it is important to make available in the multilinear case some of the proved powerful techniques used to study linear ones. This leads us to further develop the symbolic calculus for bilinear operators presented in this article. Understanding properties of the transposes of a bilinear operator is a needed first step in understanding its boundedness properties on Lebesgue spaces. Understanding their composition is useful in the understanding of their behavior in Sobolev spaces. The bilinear Hilbert transform and similar singular multipliers that recently received a lot of attention are of order zero and, in a way, they generalize the notion of product of two functions. As such, their boundedness properties mimic Hölder s relation, i.e. the operators are bounded from L p L q into L r where 1/p + 1/q = 1/r for appropriate values of the exponents. We will look here at pseudodifferential operators which are of order m and that should be thought of as the generalization of products of derivatives of functions or distributions. This is motivated in part by typical estimates needed in partial differential Date: March 30, Mathematics Subject Classification. Primary 47G30. Secondary 42B15, 42C10, 35S99. Key words and phrases. Bilinear pseudodifferential operators, composition, asymptotic expansion, elementary symbols, Littlewood-Paley theory. Second author s research partially supported by the NSF under grant DMS Third author s research partially supported by the NSF under grant DMS

2 2 equations involving nonlinear terms with products of derivatives of function or similar operations. Homogeneity considerations dictate that multilinear operators of order m bigger than zero should satisfy boundedness properties paralleling results about estimates for distributional derivatives, generalizations of Leibniz s rule, embedding type estimates for spaces of smooth functions, etc. Sobolev space norms are then the natural choice to quantify the boundedness properties for operators of order m. In this sense, this article is related to the work by Bényi and Torres in [4], where a first extension to bilinear pseudodifferential operators of the Leibniz s rule for fractional derivatives from the works of Kato-Ponce [24], Christ-Weinstein [9] and Kenig-Ponce-Vega [25] was obtained. Related estimates for mixed derivatives were recently obtained by Muscalu, Pipher, Tao, and Thiele [31] using estimates for singular multipliers with multiparameter dilations. In [4] a calculus for the formal transposes of certain bilinear operator was constructed. Here we shall extend this result to operators with symbols in a much larger class and we will develop also a calculus for the composition of a bilinear operator with a linear one. Symmetric a priori considerations play a crucial role in the technical proofs of L p boundedness of operators of order zero. We will show here that the larger classes of symbols studied behave well with respect to transposition. Likewise, boundedness properties of operators of higher order on Sobolev spaces are obviously connected to the understanding of their compositions with differential operators. Some results about composition of operators were described in the first named author s Ph.D. thesis [1] but have not been published before. They fit exactly into our approach and so they are presented here in a greater generality. The symbolic calculus that we develop opens up the possibility to study multilinear pseudodifferential operators using time-frequency analysis techniques such as those employed for the bilinear Hilbert transform, where computations with the transposes are implicitly exploited. We will attempt such an approach in forthcoming work Notation, background, and description of the main results. Before we state our main results, we need to recall some notation and further motivating facts. For convenience, we take for definition of the Fourier transform of a function f in S(R n ) (the Schwartz space of functions) to be f(ξ) = f(x)e ix ξ dx, R n extended in the usual way to all of S (R n ) (the space of tempered distributions). The inverse Fourier transform is given then by f (ξ) = (2π) n f( ξ). We are interested in bilinear pseudodifferential operators, defined a priori from S(R n ) S(R n ) into S (R n ), of the form (1) T σ (f, g)(x) = σ(x, ξ, η) f(ξ)ĝ(η)e ix (ξ+η) dξdη. R n R n Let BSρ,δ m and BSm ρ,δ; θ denote the classes of symbols σ which satisfy estimates of the form (2) α x β ξ γ η σ(x, ξ, η) C αβγ (1 + ξ + η ) m+δ α ρ( β + γ ), respectively (3) α x β ξ γ η σ(x, ξ, η) C αβγ;θ (1 + η ξ tan θ ) m+δ α ρ( β + γ ),

3 BILINEAR PSEUDODIFFERENTIAL OPERATORS 3 for all (x, ξ, η) R 3n, all multi-indices α, β and γ, and some positive constants C αβγ or C αβγ;θ. If we restrict our attention to the one dimensional case, we see that the latter condition expresses the decay of the derivatives of the symbol in terms of the distance from the frequency pair (ξ, η) to the line Γ θ at angle θ with respect to the axis η = 0 and, up to a constant depending on θ, it is equivalent to the condition, (4) α x β ξ γ η σ(x, ξ, η) C αβγ (1 + dist((ξ, η); Γ θ ) m+δ α ρ( β + γ ). In equation (3), we allow θ ( π/2, π/2], with the convention that θ = π/2 corresponds to the decay in terms of 1 + ξ only. Note also that, in general, the classes BSρ,δ m and BSρ,δ; m π/4 are not comparable. Let us recall a few references to results about multilinear pseudodifferential operators with symbols in the above classes. The study of bilinear operators with symbols in the class BS1,0 0 started in the works of Coifman and Meyer [12], [13], [14]. They used techniques related to Littlewood-Paley theory to prove that this class produces bilinear operators bounded on products of L p spaces for a certain range of p s. More recently the result has been extended to other range of exponents. The extension follows from the Coifman-Meyer result and further use of the Calderón-Zygmund decomposition and multilinear interpolation arguments. This can be also seen, for example, from a result in the work of Grafakos and Torres [21, Corollary 1] and the fact that the class BS1,0 0 is closed by transposition, as proved by Bényi and Torres in [4]. See also the work of Kenig and Stein [26] for the x-independent case. The exotic or forbidden larger class BS1,1 0, called in that way by analogy with the linear counterpart, was shown in [4] to contain unbounded operators on products of Lebesgue spaces and not to be close under transposition. Nevertheless, in [4] the authors have obtained substitute estimates and proved that operators with forbidden symbols are bounded on products of Sobolev spaces with positive smoothness. It is worth noting that the intermediate classes BS1,δ 0 with 0 < δ < 1, give also bounded pseudodifferential operators on products of L p and other spaces; see the work by Bényi [2] and the references therein. All these classes are still related to the general multilinear Calderón-Zygmund theory, since the kernels associated to the operators in these classes, even the unbounded ones in the class BS1,1 0, satisfy the standard size and smoothness conditions of the classical (multilinear) singular integral operators (see [21] for precise statements and references). The kernels and symbols of the operators in BS1,0; 0 θ are far more singular and completely outside the Calderón-Zygmund theory. The investigation of operators related to the class BS1,0; 0 θ was initiated by the work of Lacey and Thiele [28], [29] in the special case of the bilinear Hilbert transform. Actually, the symbol associated to the bilinear Hilbert transform, though x-independent, is even more singular than those in BS1,0; 0 θ. In our notation, and up to a multiplicative constant, the bilinear Hilbert transform is the operator with symbol σ(ξ, η) = sign(ξ η), ξ, η R. We note, however, that treating (non-singular) multipliers that satisfies estimates of the form β ξ γ η m(ξ, η) C βγ (1 + ξ η ) β γ, is technically almost as hard as dealing with the ones with estimates of the form β ξ γ η m(ξ, η) C βγ ξ η β γ. The study of general bilinear multipliers was continued by Gilbert and Nahmod [16], [17], [18], where they proved that, in the one dimensional case, x-independent symbols σ in BS 0 1,0; θ, with θ 0, π/2, π/4, give rise to operators T σ bounded on the same products of

4 4 Lebesgue spaces as the bilinear Hilbert transform. Muscalu, Thiele, and Tao [32] treated the general multilinear setup of this same class 1. When θ = π/4, Γ θ becomes one of the so-called degenerate subspaces (the others are Γ 0 and Γ π/2 ) and the boundedness results for singular multipliers do not apply anymore. It is interesting to note, however, that when the multipliers are still in the Coifman-Meyer class or are tensor products of linear Calderón-Zygmund operators, then the additional cancellation property of the multiplier along the antidiagonal, σ(ξ, ξ) = 0, ξ 0, gives even better estimates such as the boundedness of T σ from L p L p into the Hardy space H 1. See the work of Coifman, Lions, Meyer, and Semmes [11], Coifman and Grafakos [10], Grafakos [19] and Gilbert and Nahmod [15]. Furthermore, results that cover the whole range of H p spaces, p > 0 can be obtained by imposing additional symmetries and cancellation conditions as described by Grafakos and Torres in [20, Theorem 6]. Such symmetries take the form of certain imposed type of estimates on the symbols of an operator and its transposes. In the x-independent case the cancellations referred to amount to the vanishing of the symbols and their derivatives up to a certain order on the three degenerate directions. We note also that if a symbol in BS1,0; 0 π/4 depends only on the sum of the frequency variables, that is σ(x, ξ, η) = σ 0 (x, ξ+η), then σ 0 is a linear symbol in the classical Hörmander s class S1,0 0 and T σ(f, g) = T σ0 (fg), where T σ0 is the linear pseudodifferential operator with symbol σ 0. It is well-known that such T σ0 is bounded on Lebesgue spaces (see e.g., [35, Chapter VI]) and, using Hölder s inequality, one immediately gets boundedness of T σ from L p L q into L r with 1/p + 1/q = 1/r and r > 1. Despite all these partial results for multipliers and particular types of operators, the general x-dependent case presents new fascinating challenges and many of the analogous results for them are still unknown. In this article we present some progress concerning the classes BS1,0 m, BSm 1,1, and BSm 1,0,θ, for order m 0. Specifically we obtain the following results. We start in Section 2 by proving a basic estimate in Sobolev spaces with positive exponent s for operators in BS1,1 m. In the form of generalized Leibniz s rule we show in Theorem 1 that such operators produce a loss from m + s derivatives to s derivatives in the Sobolev scale. With a first rudimentary instance of a symbolic calculus, we improve in Theorem 2 the corresponding estimates for the smaller class BS1,0 m, allowing s = 0. We then try to further understand the rather drastic change in the passage from the class BS1,0 m to the class BS1,0; m π/4, obtaining only results for certain large m (Proposition 1) and providing a counterexamples for m = 0 and making a connection to the bilinear Calderón-Vaillancourt class BS0,0 0. We show in Section 3 how to obtain Sobolev space estimates for general multipliers from L p ones. We present two different formulations, Proposition 2 and Proposition 3, which apply in particular to the bilinear Hilbert Transform and similar multipliers; Corollary 1. We develop in Section 4 the symbolic calculus alluded to before. First we treat the composition of classical linear pseudodifferential operators with bilinear pseudodifferential operators having symbols in the class BS1,0; m π/4, Theorem 3, and we present an application 1 Gilbert and Nahmod as well as Muscalu, Thiele and Tao actually proved a stronger result. Namely they allowed for the x-independent symbols or multiplers to be singular meaning they satisfy (4) but without the 1 in the denominator on the right hand side

5 BILINEAR PSEUDODIFFERENTIAL OPERATORS 5 involving again the class BS0,0 0. Finally, we present the calculus for the transposes of operators with symbols in the classes BS1,0;θ 0, Theorem 4. We note that our main results can be stated in the more general setting of multilinear pseudodifferential operators. For notational convenience and to ease the flow of reading we restrict ourselves to the the bilinear case. The interested reader could easily adapt the statements to the multilinear set up. Acknowledgments. The third author expresses his gratitude to Russell Brown for the interaction and conversations on Sobolev space estimates for product of functions as used in [7], which motivated the results and proofs presented in Section Basic Sobolev space estimates for operators of order m 2.1. The classes BS m 1,0 and BSm 1,1. A bilinear operator T : S S S is linear in every entry and consequently has two formal transposes T 1 and T 2 defined via T (f, g), h = T 1 (h, g), f = T 2 (f, h), g, for all f, g, h in S. The formal transposes of a bilinear pseudodifferential operator T σ with sufficiently smooth symbol are also pseudodifferential operators as in equation (1) and we write σ 1 and σ 2 to denote their corresponding symbols. For operators T σ which are translation invariant, that is σ(x, ξ, η) = σ(ξ, η), the symbols of the transposes are given by σ 1 (ξ, η) = σ( ξ η, η), σ 2 (ξ, η) = σ(ξ, ξ η). For symbols that depend on x the situation is much more complicated but for operators with symbols in BS1,0 m, one can compute the symbols of their transposes via a symbolic calculus and an asymptotic formula. Moreover, recall here the following result (see Remark 2 in [4]). The symbols in BS1,0 m, m 0, produce a class of operators which is closed by transposition. The class BS1,1 m, which contains BSm 1,0, does not however satisfy this property. In the next theorem we prove a positive result about this forbidden class of order m by considering products of Sobolev spaces with positive smoothness. Theorem 1. Every operator T σ with a symbol in the class BS1,1 m, m 0, has a bounded extension from L p m+s Lq m+s into Lr s, provided that 1/p + 1/q = 1/r, 1 < p, q, r <, and s > 0. Moreover, (5) T σ (f, g) L r s C(p, q, r, s, n, m, σ)( f L p m+s g L q + f L p g L q m+s ). The proof of the theorem is adapted after the proof given in [4] for the case m = 0. We will avoid the tedious calculations that were already done there and only point out the differences introduced by considering the general case. We will make use of a decomposition into elementary symbols due to Coifman and Meyer. As in [4], we will need also to invoke a square function estimate for the Sobolev space norm of a sum of functions with appropriate spectra as used by Bourdaud in [6], where the following lemma is traced back to Meyer s work [30].

6 6 Lemma 1. Let {w j } j 0 be a sequence of functions so that supp ŵ j { z c 2 j } for some c > 1. Then, for 1 < r < and s > 0, w j L r s C ( 4 sj w j 2 ) 1/2 L r. j=0 j=0 We note that the restriction 1 < r < in Theorem 1 is imposed by the previous lemma. Nevertheless, by modifying appropriately Leibniz s rule (5), we can allow one of the other two exponents to be ; see the remarks immediately following the proof. Proof. Fix m 0. It is enough to prove boundedness for T σ, where σ is an elementary bilinear symbol of the particular form σ = σ 1 + σ 2 + σ 3, where (6) σ k (x, ξ, η) = m j (x)ϕ k (2 j ξ)χ k (2 j η), k = 1, 2, 3. j=0 Here, supp ϕ 1 { 1 4 ξ 2}, supp χ 1 { η 1 8 }, in ϕ 3, χ 3 the roles of ξ and η are reversed, supp ϕ 2, supp χ 2 { 1 20 ξ 2}, and α m j L C α,m 2 j α 2 jm. It is worth noting that in order to infer results about general symbols, the estimates on the norms of the operators T σk must depend solely on the functions ϕ k, χ k and their derivatives up to a certain order. We will see that this is indeed the case. For f, g in S, let (7) fjk (ξ) = ϕ k (2 j ξ) f(ξ), ĝ jk (η) = χ k (2 j η)ĝ(η). We can then write where T σ (f, g) = T σ1 (f, g) + T σ2 (f, g) + T σ3 (f, g), (8) T σk (f, g)(x) = = m j (x)f jk (x)g jk (x) j=0 2 jm M j,m (2 j x)f jk (x)g jk (x) j=0 The functions M j = M j,m (m is fixed) have the property that α M j L C α,m. Let us now focus on the term T σ1. A standard Littlewood-Paley decomposition of M j allows us to write (9) T σ1 (f, g) = j 0 2 jm N j,j f j1 g j1 + h 1 h 1 2 jm N j,h f j1 g j1, where N j,h L C t 2 (j h)t, h j, for all t > 0 and an appropriate constant C t which depends only on the L norms of the first t derivatives of M j, Nj,j is supported in { ξ 7 2 j }, and N j,h is supported in {3 2 h ξ 7 2 h } for h > j. Since 2jm N j,j f j1 g j1 is j=0

7 BILINEAR PSEUDODIFFERENTIAL OPERATORS 7 supported in a ball of size 2 j and we assume s > 0, we can use Lemma 1 to obtain 2 jm N j,j f j1 g j1 L r s C ( 4 sj 2 jm N j,j f j1 g j1 2 ) 1/2 L r j 0 j 0 (10) C ( 4 (m+s)j N j,j f j1 2 ) 1/2 L p sup g j1 L q j 0 j C ( j 0 4 (m+s)j f j1 2 ) 1/2 L p g L q C f L p m+s g L q. In the estimates above we have used the standard Littlewood-Paley characterization of L p m+s deemed possible by the fact that the function ϕ 1 is supported in an annulus, and also that the maximal function given by sup j g j1 is bounded on L q, q > 1. We note that the constant C in the last estimate in (10) depends only on the L norms of finitely many derivatives of ϕ 1 and χ 3. To estimate the second sum in the decomposition (9), we select t > s + m and get (11) h 1 2 jm N j,h f j1 g j1 L r s C ( h 1 j=0 h 1 h 1 4 sh 2 jm N j,h f j1 g j1 2 ) 1/2 L r j=0 C ( h 1 4 h(m+s t) ( 2 j(t m) 2 jm f j1 g j1 ) 2 ) 1/2 L r h 1 j=0 C ( h 1 4 h(m+s t) ( 2 jt f j1 ) 2 ) 1/2 L p sup g j1 L q j h 1 j=0 C ( j 0 4 j(m+s) f j1 2 ) 1/2 L p g L q C f L p m+s g L q. In the second inequality of the previous estimate we used the fact that N j,h L C t,m 2 (j h)(t m), while in the first to last inequality we used Young s inequality. The last constant in (11) again depends only on the L norms of finitely many derivatives of ϕ 1. From (9), (10) and (11), we obtain (12) T σ1 (f, g) L r s C f L p m+s g L q. In similar fashion, we can check that (13) T σ3 (f, g) L r s C f L p g L q m+s. As for the term T σ2, because both ϕ 2 and χ 2 are supported in annuli, we can get better bounds from either L p m L q L r or L p L q m L r. Indeed, T σ2 (f, g) L r = j 0 m j f j2 g j2 L r (14) ( j 0 m j f j2 2 ) 1/2 ( j 0 g j2 2 ) 1/2 L r C ( j 0 2 jm f j2 2 ) 1/2 L p ( j 0 g j2 2 ) 1/2 L q C f L p m g L q.

8 8 Remark 1. A careful examination of the proof shows that if f, g L r m+s L, 1 < r <, s > 0, m 0, then (15) T σ (f, g) L r s C(r, s, n, m, σ)( f L r m+s g L + g L r m+s f L ). In fact, the Leibniz rule estimate (5) is a little more general and can be replaced with (16) T σ (f, g) L r s C(p 1, p 2, q 1, q 2, r, s, n, m, σ)( f L p 1 m+s g L q 1 + g L p 2 m+s f L q 2 ), where 1/p 1 + 1/q 1 = 1/p 2 + 1/q 2 = 1/r, 1 < p 1, p 2, q 1, q 2, r <. Remark 2. The limitations in the exponents of Lebesgue spaces used comes from Lemma 1, but it may be possible to include other values as well as more general spaces adapted to Littlewood-Paley characterizations using paraproduct techniques such as those in, e.g., the work of Chae [8]. We will not pursue this here. Remark 3. Clearly the same proof provides a more general boundedness result from L p m+s Lq m+t into Lr min(s,t) provided 1/p + 1/q = 1/r, 1 < p, q, r <, and s, t > 0. Remark 4. Theorem 1 implies, in particular, that the smaller class BS1,0 m also yields bounded operators from L p m+s Lq m+s into Lr s for s > 0. Moreover, since this last class is closed under transposition, it is not hard to see using duality and linear interpolation (by freezing the first or second function) that operators with such symbols further map L p m+s Lq m into L r or from L p m L q m+s into Lr, for s > 0. This, however is not the best possible estimate. Recall that bilinear pseudodifferential operators with symbols in BS1,0 0 are bounded from L p L q into L r for 1 < p, q <, while the mentioned argument would only give, say, a weaker boundedness from L p L q s into L r. In the remaining of this section we fix the gap in the previous remark by proving that for the smaller class BS1,0 m Theorem 1 also holds for s = 0 and the Leibniz rule property is preserved. We let J m = (I ) m/2 denote the linear Fourier multiplier operator with symbol < ξ > m, where < ξ >= (1 + ξ 2 ) 1/2. Theorem 2. Let σ be any symbol in BS1,0 m, m 0 and let T σ be its associated operator. Then there exist symbols σ 1 and σ 2 in BS1,0 0 such that, for all f, g S, (17) T σ (f, g) = T σ1 (J m f, g) + T σ2 (f, J m g). In particular, we then have that T σ has a bounded extension from L p m L q m into L r, provided that 1/p + 1/q = 1/r, 1 < p, q <. Moreover, (18) T σ (f, g) L r C(p, q, r, n, m, σ)( f L p m g L q + f L p g L q m ).

9 BILINEAR PSEUDODIFFERENTIAL OPERATORS 9 Proof. Let φ be a C function on R such that 0 φ 1, supp φ [ 2, 2], and φ(r) + φ( 1 r ) = 1 on [0, ) (such a function exists). We have the following sequence of equalities Rn T σ (f, g) = σ(x, ξ, η)(φ( < η >2 < ξ >2 ) + φ( R n < ξ > 2 < η > 2 )) f(ξ)ĝ(η)e ix (ξ+η) dξdη Rn = σ(x, ξ, η)φ( < η >2 R n < ξ > 2 ) < ξ > m Ĵ m f(ξ)ĝ(η)e ix (ξ+η) dξdη (19) Rn + σ(x, ξ, η)φ( < ξ >2 < η > 2 ) < η > m f(ξ) Ĵ m g(η)e ix (ξ+η) dξdη where (20) R n = T σ1 (J m f, g) + T σ2 (f, J m g), σ 1 (x, ξ, η) = σ(x, ξ, η)φ( < η >2 < ξ > 2 ) < ξ > m, σ 2 (x, ξ, η) = σ(x, ξ, η)φ( < ξ >2 < η > 2 ) < η > m. Now, elementary calculations that take into account the support condition on φ show that σ 1 and σ 2 belong to BS1,0 0. We only indicate some of the estimates proving this fact for σ 1. Similar computations hold for σ 2. Since supp φ [ 2, 2], in the estimates for σ 1 we can restrict our attention to frequency variables (ξ, η) satisfying < η > 2 1/2 < ξ >. Taking into account the differential inequalities satisfied by σ BS1,0 m, we obtain α x σ 1 (x, ξ, η) C < ξ > m (< ξ > + < η >) m C3 m/2 ; ξ σ 1 (x, ξ, η) 2 ξ < ξ > m 2 (< ξ > + < η >) m + C < ξ > m (< ξ > + < η >) m < ξ > m (< ξ > + < η >) m < η > 2 < ξ > 4 ξ C3 m/2 (< ξ > + < η >) 1 C3 m/2 (1 + ξ + η ) 1 ; η σ 1 (x, ξ, η) < ξ > m (C(< ξ > + < η >) m 1 + 2(< ξ > + < η >) m η < ξ > 2 ) C3 m/2 (< ξ > + < η >) 1 C3 m/2 (1 + ξ + η ) 1. The estimates for β ξ and γ η can be obtained in similar fashion. The second conclusion follows from the already known boundedness properties on products of Lebesgue spaces of BS 0 1,0. Remark 5. We can again replace (18) with the more general estimate T σ (f, g) L r C(p, q, r, n, m, σ)( f L p 1 m g L q 1 + g L p 2 m f L q 2 ), where 1/p 1 + 1/q 1 = 1/p 2 + 1/q 2 = 1/r, 1 < p 1, p 2 < and 1 < q 1, q 2 < The class BS1,0; m π/4. A natural question to ask is whether the boundedness properties just obtained hold for the class BS1,0; m π/4. We only obtain the following. Proposition 1. There exists a sufficiently large m such that T σ maps L p m L q m into L r, 1/p + 1/q = 1/r, 1 < p, q, r <, for all σ in BS m 1,0; π/4.

10 10 In fact, if one assumes that m k(n, p, q, r), where k(n, p, q, r) is the minimal number of derivatives required for a symbol σ in BS1,0 0 to guarantee the boundedness of T σ from L p L q into L r, then one can easily see that the same arguments used before continue to apply. The reason is that for such a choice of m one does not face in the estimates negative powers of the quantity 1 + ξ + η and a similar reduction to the class BS1,0 0 holds again. One should compare this with the Kato-Ponce commutator estimates [24] that hold for all m 0. Due to the special structure of the operators they considered (derivatives of product of two functions), one can interpolate the estimates that hold for both m large and m = 0 to cover the whole range of m s (see the arguments in [24, p. 905]). Because of the general problem we consider here, we miss the endpoint m = 0. In fact, we observe the following A counterexample for m = 0. Let σ(ξ, η) = m( ξ η, η), where m(ξ, η) = ϕ(ξ)ψ(η) and the functions ϕ, ψ satisfy the following properties: ϕ Cc, ψ C, η γ ψ(η) C γ for all γ, and ψ is not a multiplier for L p for, say large p > 2 (explicit functions ψ with these properties are for example given in [35, p. 322] and the work of Wainger [36]). Note that due to the ξ-compact support of the multiplier m, the symbol σ belongs to the class BS1,0; 0 π/4. Clearly, T m does not map L 2 L p into L r for appropriate 1 < p, r with 1/2 + 1/p = 1/r. By duality and interpolation, it is not hard to see that T σ = Tm 1 cannot map L 2 L 2 into L 1. Unbounded operators on other product of Lebesgue spaces are easy to construct too. Heuristically, and assuming that we are in the one dimensional case, this unboundedness has to do with the fact that in the direction of any of the three degenerate lines, the ξ and η directions decouple ; hence a bilinear multiplier that only decays in one direction is unable to counteract the possibly uncontrollable behavior on the other. Remark 6. If we further assume that a symbol σ is in BS1,0; 0 π/4 or simply in BS0 0,0 satisfies the additional conditions ( 1/2 sup η α σ(x, ξ, η) dξ) 2 dη C α, x (21) ( 1/2 sup ξ dη) α σ(x, ξ, η) 2 dξ C α, x for all multi-indices α, β, γ, then it follows that T σ is bounded from L 2 L 2 into L 1 ; see the article by Bényi and Torres [5, Theorem 1]. These conditions are probably far from minimal. Note that, although they are symmetric in the frequency variables, it is not trivial to decide on the closeness under transposition of the inequalities (21) if σ is x- dependent, since it is hard to grasp how the symbols of the two formal transposes look like. As mentioned in the introduction, experience tells that necessary and sufficient conditions for L p boundedness are closely related to symmetric assumptions on an operator and its transposes. We will come back in the last section to the question of closeness of the classes BS1,0; 0 θ under transposition and show that, as a family, they indeed are. 3. Further Sobolev estimates for general multipliers We include here a few interesting observations about the boundedness of generic bilinear multipliers on products of Sobolev spaces which do not seem to be in the literature. Proposition 2. If the symbol σ is x-independent and T σ is bounded from L p L q into L r, with 1/p + 1/q = 1/r, r > 1, then T σ is also bounded from L p s L q s into L r s for all s 0.

11 BILINEAR PSEUDODIFFERENTIAL OPERATORS 11 Proof. Pick s = N an arbitrarily fixed positive integer. For f, g S we obtain x N T σ (f, g)(x) = i N σ(ξ, η)(ξ + η) N f(ξ)ĝ(η)e ix (ξ+η) dξdη (22) = c αβ T σ ( α f, β g). α + β =N Using the hypothesis and (22) we get N T σ (f, g) L r C N α + β =N α f L p β g L q C N f L p N g L q N. Consequently, we have T σ : L p N Lq N Lr N, as well as (the given boundedness) T σ : L p L q L r. By complex bilinear interpolation we obtain boundedness L p s L q s L r s for all s 0. As mentioned already several times the above estimates for operators of order zero are trivial by Hölder s inequality for the product of two functions. However there are other estimates for product of functions as long as we restrict the range of the index s. A typical case is given by the product of two functions in L 2 1 (Rn ). As observed for example in [7], the product rule for (integer) derivatives combined with the Sobolev embedding easily imply that the product is in L n/(n 1) 1 and not just in L 1 1. More generally, see the book of Runst and Sickel [34], if u is in L p s and v is in L q s, with 1 < p, q <, 1/p + 1/q 1, and 0 s < n min(1/p, 1/q) then uv is in L r s, where 1/r = 1/p + 1/q s/n. Let now a b denote the largest integer strictly smaller than min(a, b). (Note that a b and the integer part of min(a, b) are not necessarily the same.) This type of estimates motivates the following result for bilinear operators. Proposition 3. If the symbol σ is x-independent and T σ is bounded from L p L q into L r, with 1/p + 1/q = 1/r, r > 1, then T σ is also bounded from L p s L q s into L rs s, where 1/r s = 1/p + 1/q s/n and 0 s n/p n/q. Proof. Let N = n/p n/q. We proceed as in the proof of Proposition 2 to obtain the estimate (23) N T σ (f, g) L r N C N α f p L N,α β g q L N,β, α + β =N where for each choice of pairs α, β we require 1/p N,α + 1/q N,β = 1/r N. Pick 1 = 1 p N,α p N α n and 1 = 1 q N,β q N β. n Note that, indeed, 1/p N,α + 1/q N,β = 1/p + 1/q N/n = 1/r N for all α + β = N. Moreover, because of the Sobolev embedding, from (23) we get T σ (f, g) L r N N C N α + β =N α f L p N α β g L q N β C N f L p N g L q N. Use now again complex bilinear interpolation to complete the proof. Remark 7. Similar arguments to the ones used in the proofs above show that the statements of Propositions 2 and 3 remain true for x-dependent symbols in the class BS 0 1,0.

12 12 Corollary 1. The bilinear Hilbert transform is a bounded operator from L p s L q s into L r s for 1/p + 1/q = 1/r < 1 and all s 0 and from L p s L q s into L rs s, when 1/r s = 1/p + 1/q s/n and 0 s n/p n/q. Remark 8. The same applies to the generalizations as bilinear (not necessarily smooth) multipliers in [16], [17]. The bilinear Hilbert transform and those generalizations actually have better L r estimates allowing r > 2/3, but in the corollary we need to restrict ourselves to r > 1 because of the interpolation arguments used in the previous proposition. It is not clear to these authors how to proceed for r 1 and fractional derivatives. 4. Symbolic calculus 4.1. The composition S1,0 k BSm 1,0; π/4. Let L a be a linear pseudodifferential operator given by (24) L a (f)(x) = a(x, ξ) f(ξ)e ix ξ dξ, R n where the symbol is in the class S1,0 m. That is, it satisfies estimates of the form (25) α x β ξ a(x, ξ) C αβ(1 + ξ ) m β for all (x, ξ) R 2n, all multi-indices α and β, and some positive constants C αβ. It is well known that the classes S1,0 m admit a symbolic calculus both for composition and transposition. This calculus shows, for example, that there is an asymptotic formula for the composition of two operators with symbols in such classes which has the product of the symbols as its main term. A similar asymptotic expansion exists for the transposes of such operators. We have already discussed the existence of a symbolic calculus for transposes of the bilinear class BS1,0 m. Concerning the composition of bilinear operators, several definitions are easy to imagine. For example, it was shown in [1] how one such definition allows for a symbolic calculus by reducing matters to the well established linear case. A much more interesting situation arises however when one tries to compose a linear pseudodifferential operator with a bilinear pseudodifferential operator. Theorem 2 above, where compositions of the form T σ (J m, ) or T σ (, J m ) appear, already illustrated this. Another important instance is provided by compositions of the form J k T σ, which are natural in studying T σ on Sobolev spaces. Motivated by the linear counterpart situation and since both classes BS1,0 m and BSm 1,0; π/4 could be viewed as bilinear extension of S1,0 m, one could ask the following natural questions: For a S k 1,0 and σ BSm 1,0, is it true that L at σ = T λ with λ BS m+k 1,0? The equality L a T σ = T λ should be interpreted in its most natural sense: for appropriate functions f and g, L a (T σ (f, g)) = T λ (f, g). The answer, however, to the question is negative. In fact, if we consider both a and σ to be x-independent, then it is easy to see that the composition symbol λ(ξ, η) = a(ξ + η)σ(ξ, η). Even in the simplest case where a S1,0 0 and σ BS1,0 0, the most we can say about λ is that it is in BS0 1,0; π/4. In fact, the failure of λ to belong to the smaller class BS1,0 0 has nothing to do with the x-independence of a. We should then ask:

13 BILINEAR PSEUDODIFFERENTIAL OPERATORS 13 For a S k 1,0 and σ BSm 1,0; π/4, is it true that L at σ = T λ with λ BS m+k 1,0; π/4? We will show below that the answer to this second question is affirmative. The ideas employed are inspired by some of the proofs given in the linear case by Hörmander [23], Kohn and Nirenberg [27], and Stein [35]. There are, nevertheless, some additional technical difficulties present in our analysis because of the bilinear set up. Theorem 3. Let a be a (linear) symbol in S k 1,0 and σ be a (bilinear) symbol in BSm 1,0; π/4. Then there exist a (bilinear) symbol λ in BS k+m 1,0; π/4 so that L at σ = T λ. Moreover, we have the asymptotic expansion (26) λ(x, ξ, η) = (2π) n i α α! α y a(x, y) y=ξ+η x α σ(x, ξ, η), α in the sense that for every N > 0, (27) λ(x, ξ, η) (2π) n α <N i α α! α y a(x, y) y=ξ+η α x σ(x, ξ, η) BS k+m N 1,0; π/4. Proof. We will show first the result for symbols a and σ with compact support. Next we will prove that in fact the estimates involved in the asymptotic formula (27) are independent of the supports of the two symbols. Finally, we will use a limiting argument to prove that the symbolic calculus holds for any two symbols in the linear and bilinear classes respectively. Let us assume then that the symbols a(x, ξ) and σ(x, ξ, η) have compact support. This assumption justifies the computations that follow. We have (28) L a T σ (f, g)(x) = a(x, y) T σ (f, g)(y)e ix y dy = a(x, y) T σ (f, g)(z)e iy z dze ix y dy = a(x, y)σ(z, ξ, η) f(ξ)ĝ(η)e iy z e ix y dye iz (ξ+η) dξdηdzdy = λ(x, ξ, η) f(ξ)ĝ(η)e ix (ξ+η) dξdη, where (29) λ(x, ξ, η) = a(x, y)σ(z, ξ, η)e iy z e ix y e iz (ξ+η) e ix (ξ+η) dydz = a(x, y + ξ + η)σ(z, ξ, η)e iy z e ix (y+ξ+η) e ix (ξ+η) dydz = a(x, y + ξ + η) σ 1 (y, ξ, η)e ix y dy, and σ 1 denotes the Fourier transform with respect to the x-variable of σ(x, ξ, η). Thus, equation (28) shows that T a T σ can be realized as a pseudodifferential operator T λ with symbol λ given by (29). We will show next that λ is indeed in the class BS k+m 1,0; π/4. Let τ be a multi-index. Since y τ σ 1 (y, ξ, η) = σ(z, ξ, η) z τ (e iz y )dz,

14 14 integrating by parts and using the differential inequalities (3) satisfied by σ, we obtain for any integer M > 0 Similarly, for any multi-indices β and γ, σ 1 (y, ξ, η) C M (1 + y ) M (1 + ξ + η ) m. (30) β ξ γ η σ 1 (y, ξ, η) C Mβγ (1 + y ) M (1 + ξ + η ) m β γ, where the constants C Mβγ depend on the x-support of σ. In what follows, for any three multi-indices α, β, γ, the notation triple sum α 1 + α 2 = α β 1 + β 2 = β γ 1 + γ 2 = γ x α β ξ γ η λ(x, ξ, η) = α j,β j,γ j C αj β j γ j α j,β j,γ j α j,β j,γ j will refer to the. Using Leibniz rule and equation (29) we have y α 1 α 2 x β 1 ξ γ 1 η a(x, y + ξ + η) β 2 ξ γ 2 η σ 1 (y, ξ, η)e ix y dy y α 1 C Mαj β j γ j (1 + y + ξ + η ) p 1 (1 + ξ + η ) p 2 (1 + y ) M dy C αβγ (1 + ξ + η ) m+k β γ, where we denoted p 1 = k β 1 γ 1 and p 2 = m β 2 γ 2. We also used the fact that M can be chosen as large as we want, together with a version of Peetre s inequality (31) (1 + y + ξ + η ) s (1 + ξ + η ) s (1 + y ) s, y, ξ, η R n, s R. We note also that the constants C αβγ are still dependent on the spatial support of the bilinear symbol σ. Let us prove now that the bilinear pseudodifferential symbol λ has an asymptotic expansion (27). To this end, we expand a(x, y + ξ + η) in a Taylor series with respect to the y variable around 0. We have (32) a(x, y + ξ + η) = P N (x, y, ξ, η) + R N (x, y, ξ, η) where N is any positive integer and (33) P N (x, y, ξ, η) = α <N α y a(x, y) y=ξ+η y α α!. Recall that equation (29) gave the following representation for the symbol of the composition: λ(x, ξ, η) = a(x, y + ξ + η) σ 1 (y, ξ, η)e ix y dy. Therefore, the Taylor expansion (32) splits λ as a sum of a principal term with a remainder term λ(x, ξ, η) = p N (x, ξ, η) + r N (x, ξ, η).

15 The principal term is given by p N (x, ξ, η) = (34) BILINEAR PSEUDODIFFERENTIAL OPERATORS 15 α <N 1 α! = (2π) n We evaluate the remainder term (35) r N (x, ξ, η) = We start by noticing that α <N α y a(x, y) y=ξ+η y α σ 1 (y, ξ, η)e ix y dy i α α! α y a(x, y) y=ξ+η α x σ(x, ξ, η). R N (x, y, ξ, η) σ 1 (y, ξ, η)e ix y dy. R N (x, y, ξ, η) C y N sup { y α a(x, ty + ξ + η) : α = N} 0 t 1 C N y N Simple calculations show that, for k < N, we have and while for k N we have sup (1 + ty + ξ + η ) k N. 0 t 1 R N (x, y, ξ, η) C N y N (1 + ξ + η ) k N if 2 y ξ + η R N (x, y, ξ, η) C N y N if 2 y ξ + η, R N (x, y, ξ, η) C N (1 + y ) k (1 + ξ + η ) k N. Using these inequalities, as well as (35) and (30), we get r N (x, ξ, η) C M (1 + ξ + η) m R N (x, y, ξ, η) (1 + y ) M dy C N (1 + ξ + η ) k+m N. We made use of the fact that M is a positive integer at our disposal and split the integral as a sum of integrals over the regions 2 y ξ + η and 2 y ξ + η in the case k < N. Note that the constant C N in the estimate of r N depends on the spatial support of the bilinear symbol σ. Analogous computations can be carried out for the derivatives of the remainder, giving estimates of the form α x β ξ γ η r N (x, ξ, η) C Nαβγ (1 + ξ + η ) k+m N β γ. Hence, r N = λ p N BS k+m N 1,0; π/4, which is equivalent to the asymptotic formula (27). We will now show that in fact the constants involved in the asymptotic formula (27) are independent of the spatial support of σ. For the computations that follow we are still assuming that the linear and bilinear symbol have compact support. Let φ be a C function which equals 1 for x 1 and is supported in x 2. We can write σ = σ 1 + σ 2, where σ 1 (x, ξ, η) = φ(x)σ(x, ξ, η) and σ 2 (x, ξ, η) = (1 φ(x))σ(x, ξ, η). Since σ 1 has (fixed) compact x-support, the estimates in the asymptotic formula for the symbol of L a T σ1 hold independently of the supports of the symbols. It is therefore sufficient to show that, if λ 2 is the symbol of L a T σ2, then λ 2 BS k+m N 1,0; π/4 for all positive N and, say, x 1/2. Of course, we also need the constants in our estimates independent of the

16 16 supports of the symbols. We will show that this is indeed the case. Integrating by parts with respect to y and z in the equality λ 2 (x, ξ, η) = a(x, y)σ 2 (z, ξ, η)e i(x z) (y ξ η) dydz we obtain (36) λ 2 (x, ξ, η) = N 1 y a(x, y)(i z ) N 2 σ 2 (z, ξ, η) x z 2N 1 (1 + y ξ η ) N 2. Note that since σ 2 vanishes for z near zero, in (36) we are integrating over z > 1. Hence, we can estimate the integral in (36) by (1 + y ) k 2N 1 (1 + ξ + η ) m (37) C N1,N 2 (1 + x z ) 2N 1 (1 + y ξ η ) 2N 2. The constants C N1,N 2 are obtained from the differential inequalities satisfies by a and σ. Since we have N 1 and N 2 at our disposal, and (1 + ξ + η )(1 + y ξ η ) 1 + y, the last integral (37) can be estimated by C N (1 + ξ + η ) k+m N, where N is an arbitrary positive integer and the constant C N is independent of the supports. Similar calculations show that for arbitrary N we have α x β ξ γ η λ 2 (x, ξ, η) C Nαβγ (1 + ξ + η ) k+m N β γ. We finally eliminate the assumption that the symbols are compactly supported. We will proceed as in [4]. Let a and σ be symbols in S1,0 k and BSm 1,0; π/4 respectively. Let u(x, ξ) and v(x, ξ, η) be two C functions with compact support and such that u(0, 0) = v(0, 0, 0) = 1. Given ɛ > 0, set If we now define λ ɛ by a ɛ (x, ξ) = a(x, ξ)u(ɛx, ɛξ), σ ɛ (x, ξ, η) = σ(x, ξ, η)v(ɛx, ɛξ, ɛη). T λɛ = L aɛ T σɛ, the previous two steps show that λ ɛ belongs to BS k+m 1,0; π/4 and it satisfies an asymptotic expansion (27) in which a, σ, and λ are replaced by a ɛ, σ ɛ, and λ ɛ. Because of the independence of the estimates on the supports of a ɛ and σ ɛ, it is easy to see that λ ɛ converges pointwise to a limit λ which belongs to BS k+m 1,0; π/4, has the asymptotic expansion (27) and T λ = L a T σ. The proof is complete. Remark 9. Given a symbol σ BS1,0;θ m with θ π/4 the argument used in the proof of Theorem 3 fails because Peetre s inequality (31) does not hold in general, if one replaces on the right hand-side ξ + η with η ξ tan θ. Similarly, our argument fails if we replace the class BS1,0; π/4 m with BSm 1,0. As mentioned above, the interest in having a symbolic calculus for composition lies in its potential application to study boundedness on Sobolev spaces. The usual way to go about this is to reduce the study of operators of order m on Sobolev spaces to the study of operators of order 0 on Lebesgue spaces; see, e.g., [35, Chapter VI] for a detailed treatment

17 BILINEAR PSEUDODIFFERENTIAL OPERATORS 17 of the linear case. Unfortunately, as we saw in the counterexample in Section 2, the class BS1,0; 0 π/4 does not yield in general bounded operators on products of Lebesgue spaces. There is, however, a connection of the symbolic calculus with the ( bilinear Calderón- Vaillancourt) class BS0,0 0 which gives boundedness of operators of order m from certain products of Sobolev spaces into some other spaces of functions An application. For a given Banach space of functions X, and some real number k, we will call X (k) the space of functions defined by f X (k) iff (I ) k/2 f X, and the norm f X(k) = (I ) k/2 f X. If X = L p, then X (k) become the well-known Sobolev spaces L p k. For σ BS1,0; m π/4, s, m 0, and a given space X, we can write (38) T σ (f, g) X(s) = (I ) s/2 T σ (f, g) X = T λ (f, g) X, where λ BS s+m 1,0; π/4. In the passage from the second to the third equality we used Theorem 3. Here, the linear symbol is given by a(x, ξ) = (1 + ξ 2 ) s/2 S1,0 s. Therefore, the boundedness of the class BS1,0; m π/4 on the Sobolev-type spaces X (s) reduces to the boundedness of BS s+m 1,0; π/4 on the given space X. Moreover, for any λ BSs 1,0; π/4, we can write (39) T λ (f, g) = T λ((i ) (s+m)/2 f, (I ) (s+m)/2 g), where λ(x, ξ, η) = λ(x, ξ, η)(1 + ξ 2 ) ( s m)/2 (1 + η 2 ) ( s m)/2. It is not hard to see that the best differential inequalities satisfied by this bilinear pseudodifferential symbol are (40) α x β ξ γ η λ(x, ξ, η) C αβγ, that is λ BS 0 0,0. Assume now that the symbol λ yields a bounded operator from a product of two function spaces Y Z into X. If we combine (38) and (39), we get T σ (f, g) X(s) = T λ((i ) (s+m)/2 f, (I ) (s+m)/2 g) X C (I ) (s+m)/2 f Y (I ) (s+m)/2 g Z = C f Y(s+m) g Z(s+m), that is we obtain the boundedness of the class BS1,0; m π/4 from Y (s+m) Z (s+m) into X (s) for all s 0 granted we know that symbols in BS0,0 0 yield bounded operators from Y Z into X. Ideally, if one is interested in boundedness on Sobolev spaces, the choices for X, Y, Z would be L r, L p, L q with 1/p + 1/q = 1/r. Unfortunately, this cannot be achieved, since operators with symbols in BS0,0 0 are also in general unbounded from Lp L q into L r ; see [4, Proposition 1]. As a consequence, we cannot hope for boundedness results on products of classical Sobolev spaces to hold for the classes BS1,0; m π/4. However, it was shown in [3] that BS0,0 0 does yield bilinear pseudodifferential operators bounded from, say, L2 L 2 into the modulation space M 1, (which contains L 1 ; recall that L 2 L 2 L 1 is not true). Thus, we do obtain boundedness of operators with symbols in BS1,0; m π/4 from the Sobolev spaces L 2 m+s L 2 m+s into M 1, (s). More general boundedness results for BSm 1,0; π/4 on products of Sobolev-type modulation spaces M p,q (m+s) follow from their counterpart results

18 18 for BS0,0 0 proved in [3]. The so called modulation spaces are defined through their phasespace distribution (rather than their Littlewood-Paley decomposition) and have found many applications in signal analysis, non-linear approximation, and the formulation of uncertainty principles. Given 1 p, q, and given a window function g S, the modulation space M p,q is the space of all distributions f S for which the following norm is finite: ( ( ) q/p ) 1/q (41) f M p,q = V g f(x, y) p dx dy = V g f L p,q, R n R n with the usual modifications if p and/or q are infinite. The definition is independent of the choice of the window g in the sense of equivalent norms. If 1 p, q <, then M 1,1 is densely embedded into M p,q. In fact, the Schwartz class S is dense in M p,q for 1 p, q <. One can also show that the dual of M p,q is M p,q, where 1 p, q < and p, q are the dual exponents. The following embeddings between Lebesgue or Besov spaces and modulation spaces can be found, for example, in [33]. (a) B s p,q L p M p,p for s > 0, 1 p 2 and 1 q ; (b) B s p,q L p M p,p for s > 0, 2 p and 1 q ; (c) B s p,p M p,p for s > n/p, 1 p. For more details about modulation spaces, the interested reader is referred to [22] and the references therein The family {BS 0 1,0; θ } θ is closed under transposition. Our goal is to prove that as in the x-independent case the classes {BS 0 1,0; θ } θ still possess certain symmetry properties that make them amenable to the time-frequency analysis used to study the bilinear Hilbert transform and similar singular multipliers. We have the following. Theorem 4. Let σ be a symbol in BS1,0;θ 0. Then there exist symbols σ 1 in BS 1,0; θ 1 and σ 2 in BS1,0; 0 θ so that (T 2 σ ) 1 = T σ 1 and (T σ ) 2 = T σ 2. Moreover, we have the asymptotic expansions (42) σ 1 (x, ξ, η) = α and (43) σ 2 (x, ξ, η) = α in the sense that for every N > 0, (44) σ 1 (x, ξ, η) and α <N (45) σ 2 (x, ξ, η) α <N i α α! α x ξ α σ(x, ξ η, η) i α α! α x ξ α σ(x, ξ, ξ η), i α α! α x ξ α σ(x, ξ η, η) BS N 1,0; θ 1 i α α! α x ξ α σ(x, ξ, ξ η) BS N. 1,0; θ 2 The angles θ 1 and θ 2 are computed in the proof. The arguments of proof of the theorem follows closely the ones given by the authors in [4] to prove the invariance of BS 0 1,0 under the two formal transposes. We would only like to recall the principal ideas and point out the differences that appear when considering the new class of symbols BS 0 1,0; θ.

19 BILINEAR PSEUDODIFFERENTIAL OPERATORS 19 Proof. We will show that the symbolic calculus holds for symbols with compact support and that the estimates demonstrating formulas (44), (45) do not depend on the support of the symbol. We can then remove the additional assumption on the symbol by a standard limiting argument. Let us then assume that the symbol σ(x, ξ, η) is with compact support. Elementary computations show that, for h, g S, we have T 1 (h, g)(x) = σ(y, ξ, η)h(y)ĝ(η)e i(y x) ξ e iy η dξdηdy y η ξ (46) = σ(y, ξ η, η)h(y)ĝ(η)e i(y x) ξ e ix η dξdηdy where we denoted y η ξ = T [c] (h, g)(x), (47) c(y, ξ, η) = σ(y, ξ η, η). The operator T [c] is called a compound operator (with compound symbol c). Consider first the case where θ is not any of the degenerate directions 0, π/2, π/4. We will show then that c satisfies the differential inequalities defining the class BS 0 1,0; θ 1, where θ 1 is given by the equation (48) cot θ + cot θ 1 = 1. Now, using the fact that for all ξ and η we have 1+ η+(ξ+η) tan θ C θ (1+ η ξ tan θ 1 ), with C θ = min(1, 1 + tan θ ), we obtain y α β ξ γ η c(y, ξ, η) C αβγ1 γ 2 y α β ξ γ 1 σ(y, ξ η, η) γ 1 + γ 2 = γ γ 1 + γ 2 = γ 2 γ 2 3 C αβγ1 γ 2 ;θ(1 + η + (ξ + η) tan θ ) β γ 1 γ 2 C αβγ;θ (1 + η ξ tan θ 1 ) β γ. Here, 2 and 3 denote the partial derivatives of σ with respect to the frequency variables. We will now show that given the compound operator T [c] defined above, there exists a symbol σ 1 BS 0 1,0; θ 1 such that T 1 = T [c] = T σ 1. We will also show that the asymptotic expansion holds for such symbol σ 1. Since we are assuming that the symbol has compact support, we can write where (49) (2π) n σ 1 (x, ξ, η) = T [c] (h, g)(x) = T σ 1(h, g)(x), z y c(y, z, η)e i(z ξ) (x y) dydz. Select an even, C function φ, identically equal to 1 for y 1, and supported in y 2. We can decompose the symbol σ 1 as (50) (2π) n σ 1 (x, ξ, η) = m(x, ξ, η) + r(x, ξ, η), where r(x, ξ, η) = z (1 φ(x y))c(y, z, η)e i(z ξ) (x y) dydz, y

20 20 m(x, ξ, η) = b2 (x, z, z + ξ, η) dz. z Here, b 2 denotes the Fourier transform with respect to the y-variables of the function The function b verifies b(x, y, ξ, η) = φ(y)c(x + y, ξ, η). (51) α x τ y β ξ γ η b(x, y, ξ, η) C αβγτ;θ (1 + η ξ tan θ 1 ) β γ. With these preliminaries, we can show that σ 1 belongs to BS1,0; 0 θ. Let us first show 1 that the remainder term r is smoothing of infinite order, that is, it is in BS N 1,0; θ, for all N 0. Since we assumed that σ has compact support, we can integrate by parts with respect to y and z to obtain (I y ) N 2 N 1 z ((1 φ(x y))c(y, z, η)) (52) r(x, ξ, η) = (1 + ξ z 2 ) N 2 x y 2N 1 e i(ξ z) (y x) dzdy. y z Our goal is to show that the constants in the estimates satisfied by σ 1 and its asymptotic expansion are independent of the support. Note that 1 φ(x y) vanishes for y near x, so in (52) we are actually integrating over y x > 1. The integral in (52) can then be estimated by (53) C N1,N 2 y z (1 + η z tan θ 1 ) 2N 1 (1 + ξ z ) 2N 2 (1 + x y ) 2N 1 dzdy. The constants C N1,N 2 are obtained from the differential inequalities satisfied by c and N 1, N 2 can be chosen as we wish. Note that (1 + η z tan θ 1 )(1 + ξ z ) C θ 1(1 + η ξ tan θ 1 ), where C θ 1 = 1/2 min(1, cot θ 1 ). This allows us to estimate the last integral in (53) by C N;θ (1 + η ξ tan θ 1 ) N, N an arbitrary positive integer. The constant C N;θ is independent of the support. Similarly, if we take derivatives before integrating by parts, we can also obtain α x β ξ γ η r(x, ξ, η) C N,αβγ;θ (1 + η ξ tan θ 1 ) N, for arbitrary large N. Let us now estimate the main term m in (50). Since b satisfies the estimates (51), we prove as in [4, p. 1168] that for any multi-indices α, β, and γ, α x β ξ γ η b 2 (x, z, z + ξ, η) C Mαβγ;θ (1 + η (z + ξ) tan θ 1 ) β γ (1 + z ) M, where the constants C Mαβγ;θ do not depend on the support. With the same constant C θ 1 as above, we have Therefore (1 + η (z + ξ) tan θ 1 )(1 + z ) C θ 1(1 + η ξ tan θ 1 ). α x β ξ γ η m(x, ξ, η) C M,αβγ;θ (1 + η (z + ξ) tan θ 1 ) β γ (1 + z ) M dz C αβγ;θ (1 + η ξ tan θ 1 ) β γ, with constants C αβγ;θ independent of the support of the symbol.