Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 560976, 5 pages http://dx.doi.org/10.1155/2013/560976

Research Article

On a Class of Bilinear Pseudodifferential Operators

Arpad Benyi1 and Tadahiro Oh2

1 Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA 98225, USA

2 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA

Correspondence should be addressed to Arpad Benyi; arpad.benyi@wwu.edu Received 26 September 2012; Accepted 3 December 2012 Academic Editor: Baoxiang Wang

Copyright © 2013 A. Benyi and T. Oh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hormander class BS0 s, 0 < S < 1. The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory.

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 If xLq ^ Lr boundedness of a class of translation invariant bilinear operators (bilinear multiplier operators) given by

Ta (f,0)W=\ \ a^fiVgWe'^dZdv. (1)

Jr" JR"

We have the following.

Theorem A. If < (1 + + \q\)-m-M for all

e R" and all multi-indices ft, y, then Ta has a bounded extension from Lp x Lq into Lr,for all 1 < p, q < >x> such that 1/p + 1/q = 1/r.

In fact, Coifman and Meyer's approach yields Theorem A only for r > 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 x L1 into L1/2'm 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 variable

x as well. Let us then consider bilinear operators a priori defined from S x S into S' of the form

Ta (f, 9)(x)=\ \a (x, n) f (V g (n) dn.

Jr» JR»

Perhaps unsurprisingly, we impose then similar conditions on the derivatives of the symbol a with the expectation that they would yield indeed bounded operators Ta on appropriate spaces of functions. The estimates that we have in mind define the so-called bilinear Hoormander classes of symbols, denoted by BS™s. We say that a e BS™s if

\daxd^o (x,t,t1)\<(1 + fl + l^p^M-^Mr1) (3)

for all e R" and all multi-indices a, ft, y. Note that we

need smoothness in x as in the linear Hormander 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 a(q) belongs to the class BS0 0, then Ta is bounded from if xLq into Lr for all 1 < p,q < >x> such that 1/p + 1/q = 1/r.

The condition of translation invariance (equivalently, the x-independence of the symbol) is superfluous. Moreover, the previous boundedness result can be shown to hold for the larger class of symbols BSjg 2 BS0o,whereO < S < 1. This is a known fact that is tightly connected to the bilinear Calderon-Zygmund theory developed by Grafakos and Torres in [2]

and the existence of a transposition symbolic calculus proved by Benyi 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 BS0s, 0 < S < 1, are bilinear Calderon-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. If T and its transposes, T*1 and T*2, have symbols in BS0 1, then they can be extended as bounded operators from Lp x Lq into Lr for 1 < p, q < ot and 1/p + 1/q = 1/r.

Third, by [4, Theorem 2.1], we have the following.

Theorem C. Assume that 0 < S < p < 1, S < 1, and a e BS™s. Then, for j = 1,2,T* = Ta,j, where a*' e BS™s.

Finally, since BS0 s c BS0 1, 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, ot).

Theorem 1. If a is a symbol in BS0 s, 0 < S < 1, then Ta has a bounded extension from Lp x Lq into Lr for all 1 < p, q < ot such that 1/p + 1/q= 1/r.

Once we have the boundedness of the class BS0 s on products of Lebesgue spaces, a "reduction method" allows us to deduce also the boundedness of the class BS™s 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 Schrodinger equations. Let ]m = (I - A)m/2 denote the linear Fourier multiplier operator with symbol (¡,)m, where ($) = (1 + |£|2)1/2. By definition, we say that f belongs to the Sobolev space Lpm if Jmf e Lp. We have the following.

Theorem 2. Let a be a symbol in BS™s, 0 < S < 1, m > 0, and let Ta be its associated operator. Then there exist symbols a1 and a2 in BS0 s such that, for all f,g e S,

T (f,g) = Tai (lmf,g) + Tai (fJmg).

In particular, then one has that Ta has a bounded extension from Lpm x Lqm into Lr, provided that 1/p + 1/q = 1/r, 1 < p, q < ot. Moreover,

IT (f. 9)1

Wis\\y\\Lq-

The proofofTheorem 2 followsasimilarpathasthe onein 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 $ be a Cœ-function on R such that 0 < 0 < 1, supp $ c [-2,2], and $(r) + $(1/r)= 1 on [0, ot). Then (4) holds if we let

(x, t,v) = a (x, n) $

o2 (x, ï,,n) = o (x, n) $

(i)2 (n)2

Now, straightforward calculations that take into account the support condition on 0 given that a1 and a2 belong to BS0 s. 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

IIW^IL

\\9\U IIfIILn. (7)

where 1/p1 + 1/q1 = 1/p2 +1/q2 = 1/r, 1 < p1,p2,q1,q2 < ot.

One of the main reasons for the study of the Hormander classes of bilinear pseudodifferential operators is the fact that the conditions imposed on the symbols arise naturally in PDEs. In particular, the bilinear Hormader classes BS™s model the product of two functions and their derivatives.

Example 3. Consider first a bilinear partial differential operator with variable coefficients

DU (f,9)= 11c,r (x)U (8)

lPl<k lrl<e

Note that Dk e = Ta , where the bilinear symbol is given by

au (x,t,n) = (2n)-2nYcßy (x) (tf(in)y.

Assuming that the coefficients c^y have bounded derivatives, it is easy to show that ak e e BS\+e

Example 4. The symbol in the previous example is almost equivalent to a multiplier of the form

t(t,n) = (i

+ Iii2 + I

Indeed, this symbol belongs to BS™0. We can also think of this symbol as the bilinear counterpart of the multiplier (%)m that defines the linear operator ]m.

Example 5. With the notation in Example 4, the multipliers ¡;a_0(%,n) and na_0 (%,r() belong to BS00. In general, the

multipliers ak+e(¡-, n) = % n a_0(i;, n) belong to BS^

Example 6. One of the recurrent techniques in PDE estimates is to truncate a given multiplier at the right scale. Consider now

a & n) = am & r) 1 ca,b (x) <P (2_"0 X {2_hr) , (11)

where f and \ are smooth "cutoff" functions supported in the annulus {\/2 < < 2}, and the coefficients satisfy derivative estimates of the form

ôjaj ma x(a,b)

Elementary calculations show that a e BS™s.

Remark 7. Theorems 1 and 2 lead to the natural question about the boundedness properties of other Hormander classes of bilinear pseudodifferential operators. An interesting situation arises when we consider the bilinear Calderon-Vaillancourt class BS0 0. A result ofBenyi and Torres [9] shows that, in this case, the Lp x Lq ^ Lr boundedness fails. One can impose some additional conditions (besides being in BS0 0) on a symbol to guarantee that the corresponding bilinear pseudodifferential operator is Lp x 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 bookby Grochenig [11] for their definition and basic properties), we can show, for example, that if a e BS0>0 then Ta : L2 x L2 ^ M1'™ (which contains L1). This and other more general boundedness results on modulation spaces for the class BS0 0 were obtained by Benyi 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 BS^n from L2 x L2 into

0,0 m m

M1'™. Interestingly, we can also obtain the Lp x Lq ^ V boundedness of the class BS™0, but we have to require in this case the order m to depend on the Lebesgue exponents; see the work of Benyi 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 BS™s is also given in [13].

In the remainder of the paper we will provide an alternate proof of Theorem 1 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 Calderon-Zygmund theory, which applies to our case. We will borrow some of the ideas from Benyi and Torres [15], which in turn go back to the nice exposition (in the linear case) by Journe [16], by making use of the so-called bilinear elementary symbols.

2. Proof of Theorem 1

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 BS0 s, 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 symbol a in the class BS° s, 0 < S < 1, and an arbitrary large positive integer N. Then, for any f,ge S, Ta(f, g) can be written in the form

Ta (f,g)= 1 dkeTake (f,g) + R(f,g),

where {dke} is an absolutely convergent sequence of numbers,

°ke M £ n) = lKjke (x) Vke {2-)£ 2rln) ' (14)

with each fke a Cm-function supported on the set {1/3 < max (1^1,1^1) <1},

fe Ы\<1 V\ß\,\V\<N,

d"Kjke Ml < 2'

Vjaj > 0,

and R is a bounded operator from Lp x Lq into Lr, for 1/p + 1/q= 1/r, 1 < p,q,r < >x>.

Now, if ake is any of the symbols in (14) and we knew a priori that Tak( are bounded from LpxLq into Lr with operator norms depending only on the implicit constants from (15), the fact that the sequence [dke} is absolutely convergent immediately implies the Lp x Lq ^ Lr boundedness of Ta. Our first step has thus reduced the study of generic symbols in the class BS° s to symbols of the form

a (x, rj) = 1mj (x) f (2 ^, 2 'ц) ,

where \\damj\\L^ < and у is supported in {1/3 <

maxW,W)<1}.

Our second step is to further reduce the simpler looking symbol given in (16) to a sum of bilinear elementary symbols.

Lemma 9. Let a be as in (16). One can further reduce the study to symbols of the form

о = ol+o2 + a3, (17)

where the elementary symbols ak, к = 1,2,3, are defined via

(x, ц) = lmj (x) (pk {2-Ч) Xk {2~'ч) > (18)

with supp ^ с {1/4 < < 2}, supp ^ {M < 1/8}, 9з = Xi'Xs = <Pi, supp(P2, supp^2 С {1/20 < < 2}, and

\\damj\\L^ <2jm.

Therefore, we are now only faced with the question of boundedness for the two operators Ta and Ta , with a^ and o2 defined in Lemma 9; boundedness of Ta3 follows from that of by symmetry. In the following, for f,ge S, we write

f jk (H) = Vk (2-jï)f(ï), 9jk (n) = Xk {2-} n) g (n) ■

In this case, for k = 1,2, we can write

Tak (f 9) (x) = 1mj (x) fjk (x) 9jk (x) ■ (20)

Claim 1. T„ is bounded from Lp x Lq into Lr.

Proof. By (20) and the Cauchy-Schwarz inequality, we can write

K (f,9)^ * T H IM M

< ( IIUl \j=°

1/2 , \ 1/2 ™ 12 \

We used here the fact that the coefficients mj are bounded,

see Lemma 9. Using now Holder's inequality, we have

IK(f,9)\l <

Finally, the conditions on the supports of <p2 and fa allow us to make use of the Littlewood-Paley theory and conclude that

IK(f,9)L <IIfI\Le\\9IIl«■

(23) □

Obtaining the boundedness of the operator TBi is a bit more delicate due to the support condition on fa, specifically having supp fa contained in a disk rather than in an annulus. Nevertheless, we still claim the following.

Claim 2. Tai is bounded from Lp x Lq into Lr.

Proof. Note first that we can still use the Littlewood-Paley theory on the fj1 part of the sum that defines TBi (f, g)(x) = 1*1=0 mj(x)fj1 (x)gjX(x). We have the following inequalities:

'to \in

Z\ffi\2 ) < \\f\\„,

sup \9ji \

At this point, however, we must proceed more cautiously. Observe that

suppfji9ji c supp fji + supp gjlc {2j-3 < \£\ < 2j+3} •

Denoting then hj1 := fj10j1, we now have (f,g)(x) = 1'j=0mjhj1, where \\damj\\L^ < 2jS'"', and hj1 satisfies the support condition (25). Assume for the moment that the following inequality holds:

II œ II / \ 1/2 œI

Tmjhji < (zw2)

j=° Lr \j=° /

Then, the boundedness of the operator Tai can be obtained as follows:

IK (f,9)\Lr =

œ œ 1/2 III

= Tmjhj1 < (i hi )

j=° Lr \j=° IIII

œ 1/2

< sup\9ji\ ( IfI2 1

j>° \j=° Lr

IIfjiI

sup I9ji\

The proof of Claim 2 assumed the estimate (26). Our next claim is that (26) is indeed true.

Claim 3. Assume that \\dam

■j\\L«

2jSM and supp h:

{2j-3 < j^j < 2j+3}. Then, for all r > 1,we have

II œ II / \ 1/2 œI

Tmjhj < (iM2)

j=° Lr \j=° /

In our proof of this claim, we will make use of Journe's lemma [16, p. 69].

Lemma 10. There exists a constant C > 0 such that, for all j > 0, mj = 9j + bj, where |l9j||L~ < C, ||fcj||L~ < C2(s~0^, and supph~9j c{2j/72<lZl<9-2'}.

Proof of Claim 3. First, we consider the case r = 2. With the notation in Lemma 10, it suffices to estimate || l^bjhj||L2

and ||TTO=0 9jhj||^.

The estimate on the "bad part" follows from the triangle and Cauchy-Schwarz inequalities and the control Hbj||L<» <

2{S_1)j;recall that S < 1:

* ZINLKL

L* j=°

* ( T\fc \j=°

j\l<«

1/2 (œ ^\1/2

(2S-2)j ) ( TIIh. f2 )

Let us now look at the "good part". We start by noticing that, given the two summation indices j, k > 0,we have

supp gjhj c

-<\t\<9- 2j 72 |S|

supp 9khk c {

<\$\<9-2k

Thus, for \] - k\ > 11, we have supp g^hj n supp gkhk = 0. In view of this orthogonality, Plancherel's theorem with the estimate < 1 gives

œ 11 œ

Yvjhj <1 1di+11khi+11k <

j=0 L2 = k=0 L2

2 \\h II2

This completes the proof of the case r = 2.In the general case r > 1,weagainseekthecontrolofthe"bad"and"good"parts. The estimate on the "bad" part follows virtually the same as in the case r = 2:

where we used Minkowski's integral inequality in the last step.

For the "good" part, we can think of gkhk as being dyadic blocks in the Littlewood-Paley decomposition of the sum si := Xk=i( mod 11) 0khk. Thus, it will be enough to control uniformly (in the Lr norm) the sums St, 0 < i < 11, in order to obtain the same bound on || gjhjllLr. The control on St however follows from the uniform estimate on the gk's and an immediate application of Littlewood-Paley theory. □

Acknowledgments

Ae . Beenyi's work is partially supported by a Grant from the Simons Foundation (no. 246024). T. Oh acknowledges support from an AMS-Simons Travel Grant.

[3] C. E. Kenig and E. M. Stein, "Multilinear estimates and fractional integration," Mathematical Research Letters, vol. 6, no. 1, pp. 1-15,1999.

[4] A. Benyi, D. Maldonado, V. Naibo, and R. H. Torres, "On the Hörmander classes of bilinear pseudodifferential operators," Integral Equations and Operator Theory, vol. 67, no. 3, pp. 341364, 2010.

[5] F. Bernicot, D. Maldonado, K. Moen, and V Naibo, "Bilinear Sobolev-Poincare inequalities and Leibniz-type rules," Journal of Geometric Analysis. In press, http://arxiv.org/pdf/1104 .3942.pdf.

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[9] Ae . Beenyi and R. H. Torres, "Almost orthogonality and a class of bounded bilinear pseudodifferential operators," Mathematical Research Letters, vol. 11, no. 1, pp. 1-11, 2004.

[10] F. Bernicot and S. Shrivastava, "Boundedness of smooth bilinear square functions and applications to some bilinear pseudodifferential operators," Indiana University Mathematics Journal, vol. 60, no. 1, pp. 233-268, 2011.

[11] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, Mass, USA, 2001.

[12] A. Benyi, K. Gröchenig, C. Heil, and K. Okoudjou, "Modulation spaces and a class of bounded multilinear pseudodifferential operators," Journal of Operator Theory, vol. 54, pp. 301-313,2005.

[13] A. Benyi, F. Bernicot, D. Maldonado, V. Naibo, and R. H. Torres, "On the Hörmander classes of bilinear pseudodifferential operators, II," Indiana University Mathematics Journal. In press, http://arxiv.org/pdf/1112.0486.pdf.

[14] A. Miyachi and N. Tomita, "Calderon-Vaillancourt type theorem for bilinear pseudo-differential operators," Indiana University Mathematics Journal. In press.

[15] A. Benyi and R. H. Torres, "Symbolic calculus and the transposes of bilinear pseudodifferential operators," Communications in Partial Differential Equations, vol. 28, no. 5-6, pp. 11611181, 2003.

[16] J.-L. Journe, Calderon-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderon, vol. 994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.

References

[1] R. R. Coifman and Y. Meyer, Au Delà des Opérateurs PseudoDifférentiels, vol. 57 of Asterisque, 2nd edition, 1978.

[2] L. Grafakos and R. H. Torres, "Multilinear Calderon-Zygmund theory," Advances in Mathematics, vol. 165, no. 1, pp. 124-164, 2002.

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