# On a Class of Bilinear Pseudodifferential Operators

Authors: Árpád Bényi; Tadahiro Oh
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We provide a direct proof for the boundedness of pseudodifferential
operators with symbols in the bilinear Hörmander classBS1,δ0, 0≤δ<1. The
proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory.


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## Body

## 1. Introduction: Main Results and Examples

Coifman and Meyer’s ideas on multilinear operators and their applications in partial differential equations (PDEs) have had a great impact in the future developments and growth witnessed in the topic of multilinear singular integrals. One of their classical results [1, Proposition 2, p. 154] is about the Lp×Lq→Lr boundedness of a class of translation invariant bilinear operators (bilinear multiplier operators) given by
(1)Tσ(f,g)(x)=∫ℝn‍∫ℝn‍σ(ξ,η)f^(ξ)g^(η)eix·(ξ+η)dξdη.
We have the following.Theorem A.
If|∂ξβ∂ηγσ(ξ,η)|≲(1+|ξ|+|η|)-|β|-|γ| for all ξ,η∈ℝn and all multi-indices β,γ, then Tσ has a bounded extension from Lp×Lq into Lr, for all 1<p, q<∞ such that 1/p+1/q=1/r.In fact, Coifman and Meyer’s approach yields Theorem A only forr>1. The optimal extension of their result to the range r>1/2 (as implied in the theorem above) can be obtained using interpolation arguments and an end-point estimate L1×L1 into L1/2,∞ in the works of Grafakos and Torres [2] and Kenig and Stein [3].Bilinear pseudodifferential operators are natural nontranslation invariant generalizations of the translation invariant ones; they allow symbols to depend on the space variablex as well. Let us then consider bilinear operators a priori defined from 𝒮×𝒮 into 𝒮′ of the form
(2)Tσ(f,g)(x)=∫ℝn‍∫ℝn‍σ(x,ξ,η)f^(ξ)g^(η)eix·(ξ+η)dξdη.
Perhaps unsurprisingly, we impose then similar conditions on the derivatives of the symbol σ with the expectation that they would yield indeed bounded operators Tσ on appropriate spaces of functions. The estimates that we have in mind define the so-called bilinear Hörmander classes of symbols, denoted by BSρ,δm. We say that σ∈BSρ,δm if
(3)|∂xα∂ξβ∂ηγσ(x,ξ,η)|≲(1+|ξ|+|η|)m+δ|α|-ρ(|β|+|γ|)
for all x,ξ,η∈ℝn and all multi-indices α,β,γ. Note that we need smoothness in x as in the linear Hörmander classes. As usual, the notation a≲b means that there exists a positive constant K (independent of a,b) such that a≤Kb.With this terminology, we can restate Theorem A as follows.If the
x
-independent symbol
σ
(
ξ
,
η
)
belongs to the class
B
S
1,0
0
, then
T
σ
is bounded from
L
p
×
L
q
into
L
r
for all
1
<
p
,
q
<
∞
such that
1
/
p
+
1
/
q
=
1
/
r
.The condition of translation invariance (equivalently, thex-independence of the symbol) is superfluous. Moreover, the previous boundedness result can be shown to hold for the larger class of symbols BS1,δ0⊇BS1,00, where 0≤δ<1. This is a known fact that is tightly connected to the bilinear Calderón-Zygmund theory developed by Grafakos and Torres in [2] and the existence of a transposition symbolic calculus proved by Bényi et al. [4]. Let us briefly give an outline of how this follows. First, we note that the bilinear kernels associated to bilinear operators with symbols in BS1,δ0, 0≤δ<1, are bilinear Calderón-Zygmund operators in the sense of [2]. Second, we recall that [2, Corollary 1], which is an application of the bilinear T(1) theorem therein, states the following.Theorem B.
IfT and its transposes,   T*1 and  T*2, have symbols in BS1,10, then they can be extended as bounded operators from Lp×Lq into Lr for 1<p, q<∞ and 1/p+1/q=1/r.Third, by [4, Theorem 2.1], we have the following.Theorem C.
Assume that0≤δ≤ρ≤1,   δ<1, and σ∈BSρ,δm. Then, for j=1,2,Tσ*j=Tσ*j, where σ*j∈BSρ,δm.Finally, sinceBS1,δ0⊂BS1,10, we can directly combine Theorems B and C to recover the following optimal extension of the Coifman-Meyer result; note that now the symbol is allowed to depend on x while r is still allowed to be in the optimal interval (1/2,∞).Theorem 1.
Ifσ is a symbol in BS1,δ0, 0≤δ<1, then Tσ has a bounded extension from Lp×Lq into Lr for all 1<p, q<∞ such that 1/p+1/q=1/r.Once we have the boundedness of the classBS1,δ0 on products of Lebesgue spaces, a “reduction method” allows us to deduce also the boundedness of the class BS1,δm on appropriate products of Sobolev spaces. Moreover, our estimates in this case come in the form of Leibniz-type rules; for more on these kinds of properties, see the work of Bernicot et al. [5]. In the particular case when the bilinear operator is just a differential operator, the Leibniz-type rules are referred to as Kato-Ponce’s commutator estimates and are known to play a significant role in the study of the Euler and Navier-Stokes equations, see [6]; see also Kenig et al. [7] for further applications of commutators to nonlinear Schrödinger equations. Let Jm=(I-Δ)m/2 denote the linear Fourier multiplier operator with symbol 〈ξ〉m, where 〈ξ〉=(1+|ξ|2)1/2.  By definition, we say that f belongs to the Sobolev space Lmp if Jmf∈Lp. We have the following.Theorem 2.
Letσ be a symbol in BS1,δm, 0≤δ<1, m≥0, and let Tσ be its associated operator. Then there exist symbols σ1 and σ2 in BS1,δ0 such that, for all f,g∈𝒮,
(4)Tσ(f,g)=Tσ1(Jmf,g)+Tσ2(f,Jmg).
In particular, then one has that Tσ has a bounded extension from Lmp×Lmq into Lr, provided that 1/p+1/q=1/r, 1<p, q<∞. Moreover,
(5)∥Tσ(f,g)∥Lr≲∥f∥Lmp∥g∥Lq+∥f∥Lp∥g∥Lmq.The proof of Theorem2 follows a similar path as the one in the work of Bényi et al. [8, Theorem 2.7]. For the convenience of the reader, we sketch here the main steps in the argument. 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,∞). Then (4) holds if we let
(6)σ1(x,ξ,η)=σ(x,ξ,η)ϕ(〈η〉2〈ξ〉2)〈ξ〉-m,σ2(x,ξ,η)=σ(x,ξ,η)ϕ(〈ξ〉2〈η〉2)〈η〉-m.
Now, straightforward calculations that take into account the support condition on ϕ given that σ1 and σ2 belong to BS1,δ0. The Leibniz-type estimate (5) follows now from Theorem 1 and (4).It is also worthwhile to note that we can replace (5) with a more general Leibniz-type rule of the form
(7)∥Tσ(f,g)∥Lr≲∥f∥Lmp1∥g∥Lq1+∥g∥Lmp2∥f∥Lq2,
where 1/p1+1/q1=1/p2+1/q2=1/r, 1<p1, p2,q1,q2<∞.One of the main reasons for the study of the Hörmander classes of bilinear pseudodifferential operators is the fact that the conditions imposed on the symbols arise naturally in PDEs. In particular, the bilinear Hörmader classesBSρ,δm model the product of two functions and their derivatives.Example 3.
Consider first a bilinear partial differential operator with variable coefficients(8)Dk,ℓ(f,g)=∑|β|≤k‍∑|γ|≤ℓ‍cβγ(x)∂βf∂xβ∂γg∂xγ.
Note that Dk,ℓ=Tσk,ℓ, where the bilinear symbol is given by
(9)σk,ℓ(x,ξ,η)=(2π)-2n∑β,γ‍cβγ(x)(iξ)β(iη)γ.
Assuming that the coefficients cβγ have bounded derivatives, it is easy to show that σk,ℓ∈BS1,0k+ℓ.Example 4.
The symbol in the previous example is almost equivalent to a multiplier of the form(10)σm(ξ,η)=(1+|ξ|2+|η|2)m/2.
Indeed, this symbol belongs to BS1,0m. We can also think of this symbol as the bilinear counterpart of the multiplier 〈ξ〉m that defines the linear operator Jm.Example 5.
With the notation in Example4, the multipliers ξσ-1(ξ,η) and ησ-1(ξ,η) belong to BS1,00. In general, the multipliers σk+ℓ(ξ,η)=ξkηℓσ-1(ξ,η) belong to BS1,0k+ℓ.Example 6.
One of the recurrent techniques in PDE estimates is to truncate a given multiplier at the right scale. Consider now(11)σ(ξ,η)=σm(ξ,η)∑a,b∈ℕ‍ca,b(x)φ(2-aξ)χ(2-bη),
where φ and χ are smooth “cutoff” functions supported in the annulus {1/2≤|ξ|≤2}, and the coefficients satisfy derivative estimates of the form
(12)∥∂xαca,b(x)∥L∞≲2δ|α|max(a,b).
Elementary calculations show that σ∈BS1,δm.Remark 7.
Theorems1 and 2 lead to the natural question about the boundedness properties of other Hörmander classes of bilinear pseudodifferential operators. An interesting situation arises when we consider the bilinear Calderón-Vaillancourt class BS0,00. A result of Bényi and Torres [9] shows that, in this case, the Lp×Lq→Lr boundedness fails. One can impose some additional conditions (besides being in BS0,00) on a symbol to guarantee that the corresponding bilinear pseudodifferential operator is Lp×Lq→Lr bounded; see, for example, [9] and the recent work of Bernicot and Shrivastava [10]. However, there is a nice substitute for the Lebesgue space estimates. If we consider instead modulation spaces Mp,q (see the excellent book by Gröchenig [11] for their definition and basic properties), we can show, for example, that if σ∈BS0,00 then Tσ:L2×L2→M1,∞ (which contains L1). This and other more general boundedness results on modulation spaces for the class BS0,00 were obtained by Bényi et al. [12]. Then, this particular boundedness result with the reduction method employed in Theorem 2 allows us to also obtain the boundedness of the class BS0,0m from Lm2×Lm2 into M1,∞. Interestingly, we can also obtain the Lp×Lq→Lr boundedness of the class BS0,0m, but we have to require in this case the order m to depend on the Lebesgue exponents; see the work of Bényi et al. [13], also Miyachi and Tomita [14] for the optimality of the order m and the extension of the result in [13] below r=1. The most general case of the classes BSρ,δm is also given in [13].In the remainder of the paper we will provide an alternate proof of Theorem1 that does not use sophisticated tools such as the symbolic calculus. The proof is in the original spirit of the work of Coifman and Meyer that made use of the Littlewood-Paley theory. As such, we will only be concerned here with the boundedness into the target space Lr with r>1. Of course, obtaining the full result for r>1/2 is then possible because of the bilinear Calderón-Zygmund theory, which applies to our case. We will borrow some of the ideas from Bényi and Torres [15], which in turn go back to the nice exposition (in the linear case) by Journé [16], by making use of the so-called bilinear elementary symbols.


## 2. Proof of Theorem1

We start with two lemmas that provide the anticipated decomposition of our symbol into bilinear elementary symbols. Since they are the immediate counterparts of [15, Lemma 1 and Lemma 2] to our class BS1,δ0, we will skip their proofs; see also [16, pp. 72–75] and [1, pp. 55–57]. The first reduction is as follows.Lemma 8.
Fix a symbolσ in the class BS1,δ0, 0≤δ<1, and an arbitrary large positive integer N. Then, for any f,g∈𝒮, Tσ(f,g) can be written in the form
(13)Tσ(f,g)=∑k,ℓ∈ℤn‍dkℓTσkℓ(f,g)+R(f,g),
where {dkℓ} is an absolutely convergent sequence of numbers,
(14)σkℓ(x,ξ,η)=∑j=0∞‍κjkℓ(x)ψkℓ(2-jξ,2-jη),
with each ψkℓ a C∞-function supported on the set {1/3≤max(|ξ|,|η|)≤1},
(15)|∂ξβ∂ηγψkℓ(ξ,η)|≲1∀|β|,|γ|≤N,|∂ακjkℓ(x)|≲2jδ|α|∀|α|≥0,
and R is a bounded operator from Lp×Lq into Lr, for 1/p+1/q=1/r, 1<p,q,r<∞.Now, ifσkℓ is any of the symbols in (14) and we knew a priori that Tσkℓ are bounded from Lp×Lq into Lr with operator norms depending only on the implicit constants from (15), the fact that the sequence {dkℓ} is absolutely convergent immediately implies the Lp×Lq→Lr boundedness of Tσ. Our first step has thus reduced the study of generic symbols in the class BS1,δ0 to symbols of the form
(16)σ(x,ξ,η)=∑j=0∞‍mj(x)ψ(2-jξ,2-jη),
where ||∂αmj||L∞≲2jδ|α|, and ψ is supported in {1/3≤max(|ξ|,|η|)≤1}.Our second step is to further reduce the simpler looking symbol given in (16) to a sum of bilinear elementary symbols.Lemma 9.
Letσ be as in (16). One can further reduce the study to symbols of the form
(17)σ=σ1+σ2+σ3,
where the elementary symbols σk, k=1,2,3, are defined via
(18)σk(x,ξ,η)=∑j=0∞‍mj(x)φk(2-jξ)χk(2-jη),
with suppφ1⊆{1/4≤|ξ|≤2}, suppχ1⊆{|η|≤1/8}, φ3=χ1,χ3=φ1, suppφ2, suppχ2⊆{1/20≤|ξ|≤2}, and ||∂αmj||L∞≲2jδ|α|.Therefore, we are now only faced with the question of boundedness for the two operatorsTσ1 and Tσ2, with σ1 and σ2 defined in Lemma 9; boundedness of Tσ3 follows from that of Tσ1 by symmetry. In the following, for f,g∈𝒮, we write
(19)f^jk(ξ)=φk(2-jξ)f^(ξ),g^jk(η)=χk(2-jη)g^(η).
In this case, for k=1,2, we can write
(20)Tσk(f,g)(x)=∑j=0∞‍mj(x)fjk(x)gjk(x).Claim 1.
T
σ
2 is bounded from Lp×Lq into Lr.Proof.
By (20) and the Cauchy-Schwarz inequality, we can write
(21)|Tσ2(f,g)|≤∑j=0∞‍|mj||fj2||gj2|≲(∑j=0∞‍|fj2|2)1/2(∑j=0∞‍|gj2|2)1/2.
We used here the fact that the coefficients mj are bounded, see Lemma 9. Using now Hölder’s inequality, we have
(22)∥Tσ2(f,g)∥Lr≲∥(∑j=0∞‍|fj2|2)1/2∥Lp∥(∑j=0∞‍|gj2|2)1/2∥Lq.
Finally, the conditions on the supports of φ2 and χ2 allow us to make use of the Littlewood-Paley theory and conclude that
(23)∥Tσ2(f,g)∥Lr≲∥f∥Lp∥g∥Lq.Obtaining the boundedness of the operatorTσ1 is a bit more delicate due to the support condition on χ1, specifically having suppχ1 contained in a disk rather than in an annulus. Nevertheless, we still claim the following.Claim 2.
T
σ
1 is bounded from Lp×Lq into Lr.Proof.
Note first that we can still use the Littlewood-Paley theory on thefj1 part of the sum that defines Tσ1(f,g)(x)=∑j=0∞‍mj(x)fj1(x)gj1(x). We have the following inequalities:
(24)∥(∑j=0∞‍|fj2|2)1/2∥Lp≲∥f∥Lp,∥supj≥0|gj1|∥Lq≲∥g∥Lq.
At this point, however, we must proceed more cautiously. Observe that
(25)suppfj1gj1^⊂suppfj1^+suppgj1^⊂{2j-3≤|ξ|≤2j+3}.
Denoting then hj1:=fj1gj1, we now have Tσ1(f,g)(x)=∑j=0∞‍mjhj1, where ∥∂αmj∥L∞≲2jδ|α|, and hj1 satisfies the support condition (25). Assume for the moment that the following inequality holds:
(26)∥∑j=0∞‍mjhj1∥Lr≲∥(∑j=0∞‍|hj1|2)1/2∥Lr.
Then, the boundedness of the operator Tσ1 can be obtained as follows:
(27)∥Tσ1(f,g)∥Lr=∥∑j=0∞‍mjhj1∥Lr≲∥(∑j=0∞‍|hj1|2)1/2∥Lr≲∥supj≥0|gj1|·(∑j=0∞‍|fj1|2)1/2∥Lr≲∥(∑j=0∞‍|fj1|2)1/2∥Lp∥supj≥0|gj1|∥Lq≲∥f∥Lp∥g∥Lq.The proof of Claim2 assumed the estimate (26). Our next claim is that (26) is indeed true.Claim 3.
Assume that∥∂αmj∥L∞≲2jδ|α| and supph^j⊂{2j-3≤|ξ|≤2j+3}. Then, for all r>1, we have
(28)∥∑j=0∞‍mjhj∥Lr≲∥(∑j=0∞‍|hj|2)1/2∥Lr.In our proof of this claim, we will make use of Journé’s lemma [16, p. 69].Lemma 10.
There exists a constantC>0 such that, for all j≥0, mj=gj+bj, where ∥gj∥L∞≤C, ∥bj∥L∞≤C2(δ-1)j, and supphjgj^⊂{2j/72≤|ξ|≤9·2j}.Proof of Claim3.
First, we consider the caser=2. With the notation in Lemma 10, it suffices to estimate ∥∑j=0∞‍bjhj∥L2 and ∥∑j=0∞‍gjhj∥L2.
The estimate on the “bad part” follows from the triangle and Cauchy-Schwarz inequalities and the control∥bj∥L∞≲2(δ-1)j; recall that δ<1:
(29)∥∑j=0∞‍bjhj∥L2≤∑j=0∞‍∥bj∥L∞∥hj∥L2≤(∑j=0∞‍∥bj∥L∞2)1/2(∑j=0∞‍∥hj∥L22)1/2≲(∑j=0∞‍2(2δ-2)j)1/2(∑j=0∞‍∥hj∥L22)1/2≲∥(∑j=0∞‍|hj|2)1/2∥L2.
Let us now look at the “good part”. We start by noticing that, given the two summation indices j,k≥0, we have
(30)suppgjhj^⊂{2j72≤|ξ|≤9·2j},suppgkhk^⊂{2k72≤|ξ|≤9·2k}.
Thus, for |j-k|≥11, we have suppgjhj^∩suppgkhk^=∅. In view of this orthogonality, Plancherel’s theorem with the estimate ∥gj∥L∞≲1 gives
(31)∥∑j=0∞‍gjhj∥L2≤∑i=011‍∥∑k=0∞‍gi+11khi+11k∥L2≲(∑j=0∞‍∥gjhj∥L22)1/2≤(∑j=0∞‍∥gj∥L∞2∥hj∥L22)1/2≲∥(∑j=0∞‍|hj|2)1/2∥L2.
This completes the proof of the case r=2. In the general case r>1, we again seek the control of the “bad” and “good” parts. The estimate on the “bad” part follows virtually the same as in the case r=2:
(32)∥∑j=0∞‍bjhj∥Lr≤∥(∑j=0∞‍|bj|2)1/2(∑j=0∞‍|hj|2)1/2∥Lr≤∥(∑j=0∞‍|bj|2)1/2∥L∞∥(∑j=0∞‍|hj|2)1/2∥Lr≲∥(∑j=0∞‍|hj|2)1/2∥Lr,
where we used Minkowski’s integral inequality in the last step.
For the “good” part, we can think ofgkhk as being dyadic blocks in the Littlewood-Paley decomposition of the sum Si:=∑k≡i(mod11)‍gkhk. Thus, it will be enough to control uniformly (in the Lr norm) the sums Si, 0≤i≤11, in order to obtain the same bound on ∥∑j≥0‍gjhj∥Lr. The control on Si however follows from the uniform estimate on the gk’s and an immediate application of Littlewood-Paley theory.



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