A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 2, April 996 A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION CARLOS E. KENIG, GUSTAVO PONCE, AND LUIS VEGA. Introduction In this article we continue our study of the initial value problem (IVP) for the Korteweg-de Vries (KdV) equation with data in the classical Sobolev space H s (R). Thus we consider ( ) u t u + x 3 2 (.) u + x =0, t,x R, 2 u(x, 0) = u 0 (x), where u 0 H s (R). Our principal aim here is to lower the best index s for which one has local well posedness in H s (R), i.e. existence, uniqueness, persistence continuous dependence on the data, for a finite time interval, whose size depends on u 0 H s. Equation in (.) was derived by Korteweg de Vries [2] as a model for long wave propagating in a channel. A large amount of work has been devoted to the existence problem for the IVP (.). For instance, (see [9], [0]), the inverse scattering method applies to this problem,, under appropriate decay assumptions on the data, several existence results have been established, see [5],[6],[4],[28],[33]. Another approach, inherited from hyperbolic problems, relies on the energy estimates,, in particular shows that (.) is locally well posed in H s (R) fors>3/2, (see [2],[3],[2],[29],[30],[3]). Using these results conservation laws, global (in time) well posedness in H s (R), s 2 was established, (see [3],[2],[30]). Also, global in time weak solutions in the energy space H (R) were constructed in [34]. In [3] [22] a local smoothing effect for solutions of (.) was discovered. This, combined with the conservation laws, was used in [3] [22] to construct global in time weak solutions with data in H (R),eveninL 2 (R). In [6], we introduced oscillatory integral techniques, to establish local well posedness of (.) in H s (R), s > 3/4, hence, global (in time) well posedness in H (R), s. (In [6] we showed how to obtain the above mentioned result by Picard iteration in an appropriate function space.) In [4] J. Bourgain introduced new function spaces, adapted to the linear operator t + x, 3 for which there are good bilinear estimates for the nonlinear term x (u 2 /2). Using these spaces, Bourgain was able to establish local well posedness of (.) in H 0 (R) =L 2 (R), hence, by a conservation Received by the editors July 3, 994, in revised form, May, 995. 99 Mathematics Subject Classification. Primary 35Q53; Secondary 35G25, 35D99. Key words phrases. Schrödinger equation, initial value problem, well-posedness. C. E. Kenig G. Ponce were supported by NSF grants. L. Vega was supported by a DGICYT grant. 573 c 996 American Mathematical Society

574 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA law, global well posedness in the same space. The method was first introduced by Bourgain in order to obtain corresponding results in the spatially periodic setting (see (.2)). These were the first results in the periodic setting, for s 3/2. Existence results for weak solutions to (.) corresponding to rougher data are also in the literature. Using the inverse scattering method, T. Kappeler [4] showed that if u 0 is a real measure with appropriate decay at infinity, then (.) has a global in time weak solution (see also [5], [6]). Also, Y. Tsutsumi [35] proved the existence of a global, in time, weak solution for u 0 any positive, finite measure. In [8] we showed that (.) is locally well posed in H s (R), for s> 5/8. Since finite measures are in H s (R), s</2, this establishes (locally in time) the uniqueness for finite measures. Our method of proof in [8] combined the ideas introduced by Bourgain in [4], with some of the oscillatory integral estimates found in [6] [7]. This enabled us to extend the bilinear estimate in [8] for x (u 2 /2) to the Bourgain function spaces associated with the index s, fors> 5/8. In the present paper, we reconsider this bilinear operator in these function spaces, obtaining the best index s ( 3/4) for which it is bounded. This allows us to obtain the corresponding result for (.) in H s (R), for s> 3/4. Since we also show that our estimates for the bilinear operator are sharp, (except for the limiting case s = 3/4 which remains open), our result concerning the local well posedness of the IVP (.) is the optimal one provided by the method below. The proof of the estimates for this bilinear operator is based on elementary techniques. In fact, it is similar to those used by C. Fefferman E. M. Stein [7] to establish the L 4/3 (R 2 ) restriction Fourier transform to the sphere by C. Fefferman [8] to obtain the L 4 (R 2 ) estimate for the Bochner-Riesz operator. These estimates extend to the periodic case. Thus we can give a simplified proof of J. Bourgain [4] result concerning the local well-posedness of the periodic boundary value problem (PBVP) for the KdV equation ( ) u t u + x 3 2 (.2) u + x =0, t R,x T, 2 u(x, 0) = u 0 (x). There is an interesting parallel between the development of the results described above the recent works of S. Klainerman M. Machedon, [9] [20], G. Ponce T. Sideris [27] H. Lindblad [23] on non-linear wave equations. In fact, in [9] Klainerman Machedon consider the IVP t 2wI w I = N I (w, w), t R, x R 3, (.3) w I (x, 0) = f I (x) H s (R 3 ), t w I (x, 0) = g I (x) H s (R 3 ), where w = w I is a vector valued function, the nonlinear terms N I have the form (.4) N I (w, w) = Γ I J,K(w)BJ,K( w I J, w K ), J,K where B I J,K (.5) is any of the null forms 3 Q 0 ( w J, w K )= xi w J xi w K t w J t w K, i= Q α,β ( w J, w K )= xα w J xβ w K xβ w J xα w K,

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 575 with 0 α β 3, x0 = t Γ I J,K (w) is a polynomial in w. Such equations arise in the study of wave maps (for Q 0 ) Yang-Mills systems in a Coulomb gauge form for the general case. The energy method establishes the local well posedness of (.3) if s> + 3/2. For this result, the special form of the non-linear term is irrelevant. However, Klainerman Machedon showed in [9], that for the special non-linearities in (.5), there is a local well posedness for s 2. This was done by studying the bilinear operators Q 0 ( φ, ψ) Q α,β ( φ, ψ) in (.5), where φ, ψ are homogeneous solutions to the linear wave equation, then extended to the inhomogeneous wave equation via Duhamel s formula. Thus, in this context, the idea of establishing bilinear estimates also appeared, simultaneously to Bourgain s work [4]. Shortly after the work [9], Ponce Sideris [27] were able to show, using oscillatory integral estimates (the so called Strichartz estimates [32], see also [26]), that for any quadratic non-linearity N I, one has local well posedness in (.3), for s>2. Moreover, Lindblad [23] has shown that for general quadratic non-linearity N I,this is sharp. Thus, to achieve s = 2, the null forms (.5) are needed. Very recently, Klainerman Machedon [20], motivated by the results in [4], [8] those in the present paper, have established better estimates for Q 0 ( φ, ψ), in function spaces somewhat similar to those studied here, but adapted to the wave operator, have used this to show local well posedness for (.3) for s>3/2, when N I involves only Q 0. The exponent s =3/2issharp. To state our main results we recall the following definitions. Definition. ([4]). For s, b R, X s,b denotes the completion of the Schwartz space S(R 2 ) with respect to the norm (.6) F Xs,b = ( + τ ξ 3 ) 2b ( + ξ ) 2s F (ξ,τ) 2 dξdτ Let Y be the space of all F such that i) F : T R C. ii) F (x, ) S(R) for each x T. iii) x F (x, ) is C. iv) F (0,τ) = 0 for all τ R. For s, b R, Y s,b denotes the completion of Y with respect to the norm (.7) F Ys,b = n 0 ( + τ n 3 ) 2b n 2s F (n, τ) 2 dτ For F X s,b (or F Y s,b ) we define the bilinear operator /2. /2. (.8) B(F, F)(x, t) = 2 x(f 2 (x, t)). Our main results in this work are gathered in the following theorems.

576 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA Theorem.. Given s ( 3/4, 0] there exists b (/2, ) such that (.9) B(F, F) Xs,b c F 2 X s,b. Theorem.2. For s ( /2, 0] b =/2it follows that (.0) B(F, F) Ys,b c F 2 Y s,b. Theorem.3. For any s< 3/4 any b R the estimate (.9) fails. Theorem.4. For any s< /2 any b R the estimate (.0) fails. As a consequence of the estimates (.9)-(.0) we obtain the following results concerning the IVP (.) the PBVP (.2). Theorem.5. Let s ( 3/4, 0]. Then there exists b (/2, ) such that for any u 0 H s (R) there exist T = T ( u 0 H s) > 0 (with T (ρ) as ρ 0) a unique solution u(t) of the IVP (.) in the time interval [ T,T] satisfying (.) u C([ T,T]:H s (R)), (.2) u X s,b L p x,loc (R : L2 t(r)), for p, (.3) x (u 2 ) X s,b (.4) t u X s 3,b. Moreover, for any T (0,T) there exists R = R(T ) > 0 such that the map ũ 0 ũ(t) from {ũ 0 H s (R) : u 0 ũ 0 H s < R} into the class defined by (.8)-(.9) with T instead of T is Lipschitz. In addition, if u 0 H s (R) with s >s, then the above results hold with s instead of s inthesametimeinterval[ T,T]. Theorem.6. Let s ( /2, 0]. Then for any u 0 H s (T) with û 0 (0) = 0 there exist T = T ( u 0 H s) > 0 a unique solution u(t) of the PBVP (.2) in the time interval [ T,T] satisfying (.5) u C([ T,T]:H s (T)) Y s,b, (.6) x (u 2 ) Y s,b. Moreover, for any T (0,T) there exists r = r(t ) > 0 such that the map ũ 0 ũ(t) from {ũ 0 H s (T) : u 0 ũ 0 H s <R}into the class defined by (.2) with T instead of T is Lipschitz. Once the key estimate (.0) has been established, the proof of Theorem.6, as well as its general version, i.e. for data u 0 with arbitrary mean value, follows the arguments in [4], therefore it will be omitted.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 577 Theorem.5 improves the results in [4], s 0, that in [8], s> 5/8. In these works the bounds of the bilinear operator (.8) in the spaces X s,b, were based on several L p L q -estimates, i.e. Strichartz type [32], a sharp version of Kato smoothing effect [3] found in [5], maximal functions those obtained by interpolating these. As it was remarked above our proof is elementary. Indeed, it only uses simple calculus arguments. Theorem.6 was essentially proven in [4]. There the bound for the bilinear operator (.8) was based on the Zygmund result [36] concerning the restriction of Fourier series in L 4 (T 2 ). As in the continuous case our proof of this key estimate is based on simple calculus inequalities. It is interesting to compare the above result for the KdV equation with those obtained, by different arguments, for the modified KdV (mkdv) equation (.7) t v + 3 xv + 3 x(v 3 )=0. It was proven in [7] that s /4 suffices for the local well posedness of the corresponding IVP of the mkdv in H s (R). Similar result, with s>/2, for the PBVP associated to (.4) can be found in [4]. We will show that the method used here does not improve these results. More precisely, let (.8) B(F, F, F)(x, t) = 3 x(f 3 (x, t)) denote the tri-linear operator associated to the mkdv. Theorem.7. For any s</4 any b R the estimate (.9) B(F, F, F) Xs,b c F 3 X s,b fails. Theorem.8. For any s</2 any b R the estimate (.20) B(F, F, F) Ys,b c F 3 Y s,b fails. We observe that in both cases, the IVP the PBVP, the difference between the best Sobolev exponents known for the local well posedness of the KdV the mkdv is one. Heuristically, this is consistent with the Miura transformation [25] which affirms that if v( ) solves the mkdv (.4), then (.2) u = ci x v + v 2 solves the KdV equation. For the generalized KdV equation (.22) t u + 3 x u + k + x(u k+ )=0, x,t R,k Z +,

578 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA one has that if u(x, t) is a solution, then for any λ>0 (.23) u λ (x, t) =λ 2/k u(λx, λ 3 t) also solves (.9). This scaling argument suggests the value s k =(k 4)/2k as a lower bound for the Sobolev exponent which guarantees the local well posedness of the corresponding IVP. For higher nonlinearity, k 4, this lower bound was achieved in [7]. Moreover, it was shown in [] that this is the best possible result. However, this is not always the case; the following example shows that the values of the Sobolev exponent given by the scaling argument the optimal one may be different even for positive values. As before, consider the IVP for a type of nonlinear wave equations t 2 ω ω =( t ω) j, t R,x R 3, j =,2,3,..., (.24) ω(x, 0) = f(x) H s (R 3 ), t ω(x, 0) = g(x) H s (R 3 ). The scaling argument suggests that (.2) is locally well posed for s>s(j) = (5j 7)/(2j 2). When j 3 this result, as well as its sharpness, was established in [27]. For j = 2 it was proven in [27] that s>2 is sufficient for the local existence in [23] that s 2 is necessary (see also [24]). Observe that in this case, j =2, s(2) = 3/2. As was noted before, in [20] it was shown that s>s(2) = 3/2 is actually attained for the null form Q 0 in (.5). This paper is organized as follows. Sections 2-4 are concerned with the nonperiodic case. Theorems..5 will be proven in Sections 2 3 respectively. Section 4 contains the proof of Theorems.3,.7. The periodic case is considered in Sections 5-6. Theorem.2 will be proven in Section 5, Theorems.4,.8 in Section 6. 2. Proof of Theorem. First we will rewrite the estimate (.9) in an equivalent form. Defining (2.) ρ = s [0, 3/4) it follows from (.6) that if F X s,b = X ρ,b,then (2.2) f(ξ,τ) =(+ τ ξ 3 ) b ( + ξ ) ρ F (ξ,τ) L 2 (R 2 ) (2.3) f L 2 ξ L 2 τ = F X s,b = F X ρ,b. Using that (2.4) x (F 2 )(ξ,τ) =cξ( F F)(ξ,τ)

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 579 we can rewrite (.9) in terms of f as B(F, F) Xs,b = ( + τ ξ 3 ) b ( + ξ ) ρ x (F 2 ) L 2 ξ L 2 τ = c ( + ρ ξ 3 ) b ( + ξ ) ρ ξ( F F ) L 2 ξ L 2 τ (2.5) = c ξ ( + τ ξ 3 ) b ( + ξ ) ρ f(ξ,τ )( + ξ ) ρ f(ξ ξ,τ τ )( + ξ ξ ) ρ dξ dτ ( + τ ξ 3 )b ( + τ τ (ξ ξ ) 3 ) b L 2 ξ L 2 τ c F 2 X s,b = c f 2 L. 2 ξ L2 τ Thus, Theorem. can be restated as follows. Theorem 2.. Given ρ = s [0, 3/4) there exists b (/2, ) such that ξ ( + τ ξ 3 ) b ( + ξ ) ρ (2.6) f(ξ,τ )( + ξ ) ρ f(ξ ξ,τ τ )( + ξ ξ ) ρ ( + τ ξ 3 )b ( + τ τ (ξ ξ ) 3 ) b dξ dτ L 2 ξ L 2 τ c f 2 L. 2 ξ L2 τ We will restrict ourselves to the most interesting cases of (2.6), ρ =0,which provides a simplified proof of the L 2 -result in [4], ρ (/2, 3/4). In fact we will prove the following slightly stronger version of Theorem 2.. Theorem 2.2. Given ρ = s (/2, 3/4) there exists b (/2, ) such that for any b (/2,b] with b b min{ρ /2; /4 ρ/3} it follows that ξ ( + τ ξ 3 ) b ( + ξ ) ρ (2.7) f(ξ,τ )( + ξ ) ρ f(ξ ξ,τ τ )( + ξ ξ ) ρ ( + τ ξ 3 )b ( + τ τ (ξ ξ ) 3 dξ dτ ) b L 2 ξ L 2 τ c f 2 L, 2 ξ L2 τ where the constant c depends on ρ, b b b. Moreover (2.7) still holds for ρ =0,b (/2, 3/4] b (/2,b]. We observe that (2.6) follows from (2.7) by taking b = b. The following elementary calculus inequalities will provide the main tool in the proof of Theorem 2.2.

580 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA Lemma 2.3. If l>/2, then there exists c>0such that (2.8) dx ( + x α ) 2l ( + x β ) 2l c ( + α β ) 2l, (2.9) dx ( + x ) 2l α x c ( + α ), /2 (2.0) (2.) dx ( + x α ) 2( l) ( + x β ) c 2l ( + α β ) 2( l) x β dx ( + x ) 2( l) + β)2(l /2) c(. α x ( + α ) /2 Next, using these calculus inequalities, we deduce three lemmas from which the proof of Theorem 2.2 will follow directly. Lemma 2.4. If b (/2, 3/4] b (/2,b], then there exists c>0such that (2.2) ξ ( + τ ξ 3 ) b dτ dξ ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 ) 2b /2 c. Proof. Since b > /2, from (2.8) it follows that (2.3) dτ ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 ) 2b c ( + τ ξ 3 +3ξξ (ξ ξ ) 2b ). To integrate with respect to ξ we change variables (2.4) µ = λ ξ 3 +3ξξ (ξ ξ ), then dµ =3ξ(ξ 2ξ )dξ, (2.5) ξ = 2 { ξ ± } 4τ ξ 3 4µ. 3ξ

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 58 Therefore (2.6) ξ(ξ 2ξ ) = c ξ 4τ ξ 3 4µ dµ (2.7) dξ = c ξ 4τ ξ3 4µ. Now combining these identities with (2.9) we find (2.8) c ξ dξ ( + τ ξ 3 +3ξξ (ξ ξ ) ) 2b Thus the term in (2.0) is bounded by dµ ( + µ ) 2b 4τ ξ 3 4µ c ξ /2 ( + 4τ ξ 3 ) /2. c ξ 3/4 (2.9) ( + τ ξ 3 ) b ( + 4τ ξ 3 ). /4 It is easy to check that the expression in (2.7) is bounded whenever b 3/4. Lemma 2.5. If ρ = s (/2, 3/4), b (/2, 3/4+s/3] b ( 2,b], then there exists c>0such that (2.20) A ξ ( + τ ξ 3 ) b ( + ξ ) ρ ξ (ξ ξ ) 2ρ ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 dτ dξ ) 2b where A = A(ξ,τ) is defined as { A = (ξ,τ ) R 2 ξ, ξ ξ, (2.2) } τ τ (ξ ξ ) 3 τ ξ τ ξ 3 3. /2 <c, Proof. First we observe that in A (2.22) τ ξ 3 +3ξξ (ξ ξ ) = τ ξ + τ τ (ξ ξ ) 3 2 τ ξ 3. Thus, by (2.8) (2.23) dτ ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 ) 2b χ((τ ξ3 +3ξξ (ξ ξ ))/2(τ ξ 3 )) ( + τ ξ 3 +3ξξ (ξ ξ ) ) 2b,

582 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA where χ( ) denotes the characteristic function of the interval [, ]. As in the previous Lemma we use the change of variable (2.24) µ = τ ξ 3 +3ξξ (ξ ξ ). Thus (2.4)-(2.7), (2.23) (2.29) lead to the inequalities A ξ (ξ ξ ) 2ρ ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 ) 2b dξ dτ (2.25) ξ (ξ ξ ) 2ρ χ((τ ξ 3 +3ξξ (ξ ξ ))/2(τ ξ 3 )) c ( + τ ξ 3 dξ +3ξξ (ξ ξ ) ) 2b c µ 2 τ ξ 3 τ ξ 3 µ 2ρ ξ 2ρ+/2 ( + µ ) 2b 4τ ξ 3 4µ dµ Hence the expression c τ ξ 3 2ρ ξ 2ρ+/2 ( + 4τ ξ 3 ). /2 (2.26) φ(ξ,τ) = c ξ 3/4 ρ ( + ξ ) ρ ( + τ ξ 3 ) ρ+b ( + 4τ ξ 3 ) /4 bounds the term in (2.20). From the assumptions on ρ = s b one has that (2.27) φ L (R 2 ), which completes the proof. Lemma 2.6. If ρ = s (/2, 3/4), b (/2, ) b (/2,b] with b b min{ρ /2; /4 ρ/3}, then there exists c > 0such that ( + τ ξ 3 )b (2.28) B ξ 2( ρ) ξξ (ξ ξ ) 2ρ dτdξ ( + ξ ) 2ρ ( + τ ξ 3 ) 2( b) ( + τ τ (ξ ξ ) 3 ) 2b /2 c, where B = B(ξ,τ ) is defined as (2.29) { B = (ξ,τ) R 2 ξ ξ, ξ, τ τ (ξ ξ ) 3 τ ξ 3, τ ξ3 τ ξ 3 }.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 583 Proof. First we observe that in B (2.30) τ ξ 3 +3ξξ (ξ ξ ) = τ ξ 3 (τ τ (ξ ξ ) 3 ) 2 τ ξ 3. By (2.0) it follows that ( + τ ξ (2.3) 3 ) 2( b) ( + τ τ (ξ ξ ) 3 ) 2b c ( + τ ξ 3 +3ξξ (ξ ξ ) ). 2( b) Thus to obtain (2.28) it suffices to bound (2.32) I(D) = ( + τ ξ 3 )b D dτ ξ 2( ρ) ξξ (ξ ξ ) 2ρ ( + ξ ) 2ρ ( + τ ξ 3 +3ξξ (ξ ξ ) ) 2( b) dξ with D = B = B (ξ,τ ), where { B = ξ R ξ ξ, ξ, τ ξ 3 τ ξ 3 (2.33) } τ ξ+3ξξ 3 (ξ ξ ) 2 τ ξ 3. We split B into two subdomains B B 2,where { (2.34) B = ξ B 3 ξξ (ξ ξ ) } 2 τ ξ 3 (2.35) B 2 = { ξ B } 2 τ ξ 3 ξξ 3 (ξ ξ ) 3 τ ξ 3. /2 In B we have that (2.36) 2 τ ξ 3 τ ξ 3 +3ξξ (ξ ξ ) (2.37) ξ 6 τ ξ 3. Hence (2.38) I(B ) c ( + τ ξ 3 ) ρ+b b c( + τ ξ 3 )/2 ρ b +b ξ τ ξ 3 ξ 2( ρ) ( + ξ ) 2ρ dξ /2 which is bounded since, by hypothesis, /2 ρ b + b 0.

584 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA To bound I(B 2 ) we split B 2 into three parts, B 2,, B 2,2 B 2,3. First we have (2.39) B 2, {ξ = B 2 } 4 ξ ξ 00 ξ. In this domain one has (2.40) c( + τ ξ 3 ) ξ 3 ξ 3. Combining (2.35), the hypothesis ρ>/2, a change of variable similar to that in (2.4)-(2.7), µ = τ ξ 3 +3ξξ (ξ ξ ), (2.9) it follows that I(B 2,) c( + τ ξ ) 3 ρ b +( 2s)/3 dξ ( + τ ξ 3 +3ξξ (ξ ξ ) ) 2( b) µ 2 τ ξ 3 /2 (2.4) c( + τ ξ 3 +/3 )ρ/3 b dµ ξ /4 ( + µ ) 2( b) 4τ ξ 3 4µ µ <2 τ ξ 3 /2 c ( + τ ξ 3 )ρ/3 /4+b b ( + 4τ ξ 3, )/4 which is bounded since, by hypothesis, ρ/3 /4+b b 0. Next we consider (2.42) B 2,2 = {ξ B 2 In this domain one has that } ξ ξ /4. (2.43) ξ 2ξ ξ ξ ξ (2.44) c ( + τ ξ 3 ) ξξ (ξ ξ ) ξ 2 ξ c ξ 3. Therefore (2.44), the change of variable µ = λ ξ 3 +3ξξ (ξ ξ ) (2.) give the inequalities I(B 2,2 ) c ( + τ ξ 3 )b /2 ξ 2( 2ρ) ξξ (ξ ξ ) 2ρ (2.45 ) ξ /2 ( + τ ξ 3 )/2 ( + µ ) dµ 2( b) µ 3 τ ξ 3 c( + τ ξ 3 ) θ,

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 585 where (2.46) θ = b +/4+( 2ρ)/3+ρ+b /2=ρ/3 5/2 + b b 0 by using our hypothesis on ρ, b b. Hence I(B 2,2) is bounded. Finally we have { } (2.47) B 2,3 = ξ B 2 00 ξ ξ. In this case one has that (2.48) ξ 2ξ ξ ξ ξ, τ ξ 3 << ξ 3 (2.49) 4τ ξ 3 ξ3. Thus a familiar argument shows that (2.50) I(B 2,3) c( + τ ξ ) 3 ρ b dµ ξ /2 ( + µ ) 2( b) 4τ ξ 3 4µ µ 3 τ ξ 3 /2 c( + τ ξ 3 )ρ 5/6 b +b, which is bounded. This completes the proof of (2.28). Proof of Theorem 2.2. First we consider the case s = 0. Using Cauchy-Schwarz inequality, (2.2) Fubini s theorem it follows that ξ f(ξ,τ ) f(ξ ξ,τ τ )dξ dτ ( + τ ξ 3 ) b ( + τ ξ 3 )b ( + τ τ (ξ ξ ) 3 ) b L 2 ξ L 2 τ ξ ( + τ ξ 3 ) b ( ) /2 dξ dτ (2.5) ( + τ ξ 3 )2b ( + τ τ (ξ ξ ) 3 ) 2b L ξ L τ ( ) /2 f(ξ,τ ) 2 f(ξ ξ,τ τ ) 2 dξ dτ L 2 ξ L2 τ c f 2 L, 2 ξ L2 τ for any b (/2,b]. Taking b = b we obtain the result. Next we turn to the case ρ = s (/2, 3/4). Note that if (2.52) either ξ or ξ ξ,

586 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA we have (2.53) ( + ξ ) ρ ( + ξ ξ ) ρ c( + ξ ) ρ, which reduces the estimate to the case ρ = 0. Thus, we can assume (2.54) ξ ξ ξ. By symmetry we can also assume (2.55) τ τ (ξ ξ ) 3 τ ξ 3. We split the region of integration in (2.7) into two parts (2.56) τ ξ 3 τ ξ3 τ ξ 3 τ ξ 3. For the first part we use Cauchy-Schwarz (2.20) as in (2.5) to obtain the same bound. For the second part we use duality, Cauchy-Schwarz (2.28) to conclude the proof. We observe that Theorem 2.2 is equivalent to the following result. Corollary 2.7. Given s ( 3/4, /2) there exists b (/2, ) such that for any b (/2,b] with b b min{ s /2; /4+s/3} (2.57) B(F, F) Xs,b c F 2 X s,b. Moreover (2.57) still holds for s =0,b (/2, 3/4] b (/2,b]. 3. Proof of Theorem.4 We will denote by {W (t)} the unitary group describing the solution of the linear IVP associated to (.): (3.) { t u + x 3 u =0, t,x R, u(x, 0) = u 0 (x), where (3.2) u(x, t) =W(t)u 0 (x)=s t u 0 (x) with S t ( ) defined by the oscillatory integral (3.3) S t (x) =c We will also use the notation (3.4) f L p xl q = t e ixξ e itξ3 dξ. f(x, t) q dt p/q dx /q,

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 587 (3.5) Ĵ s h(ξ) =(+ ξ ) s ĥ(ξ), Ds x h(ξ)= ξ s ĥ(ξ) (3.6) Λb g(τ) =(+ τ ) b ĝ(τ), Db t g(τ)= τ b ĝ(τ). The relationship between the spaces X s,b in (.6) the group {W (t)} in (3.2) is described by the identity (3.7) F Xs,b = Λ b J s W (t)f L 2 ξ L 2 τ. Hence, if F X s,b then (3.8) ) F = F (x, t) = (( + ξ ) s ( + τ ξ 3 ) b f(ξ,τ) ( ) = ( + ξ ) s fb (ξ,τ) with f L 2 (R 2 ). From the sharp version of the Kato smoothing effect [3] found in [5], i.e. (3.9) D x W (t)v 0 L x L 2 t c v 0 L 2, it follows that for b>/2 (3.0) D θ x f b L 2/( θ) x L 2 t, for θ [0, ]. In particular, this implies that for s (, 0) (3.) F C α loc(r : L 2 t(r)) with α (0, +s). Lemma 3.. If F X s,b with s (, 0) b>/2,then (3.2) F C α loc (R : L2 t (R)) Lp x,loc (R : L2 t (R)) for p, α (0, +s). Let ψ C0 (R) withψ on[,] supp ψ ( 2, 2). We recall the following results proven in [8]. Lemma 3.2. If s 0 b (/2, ], then for δ (0, ) (3.3) ψ(δ t)w (t)v 0 Xs,b cδ ( 2b)/2 v 0 H s, (3.4) ψ(δ t)f Xs,b cδ ( 2b)/2 F Xs,b, (3.5) ψ(δ t) t 0 W (t t )F (t )dt Xs,b cδ ( 2b)/2 F Xs,b

588 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA (3.6) ψ(δ t) t 0 W (t t )F (t )dt H s cδ ( 2b)/2 F Xs,b. Proof. For (3.3)-(3.5) see Lemmas 3.-3.3 in [8]. The proof of (3.6) follows the same argument used in [8] to obtain (3.5). Proof of Theorem.5. First we observe that if u( ) is a solution of the IVP (.), then for any λ>0 (3.7) u λ (x, t) =λ 2 u(λx, λ 3 t) also satisfies the KdV equation with data (3.8) u λ (x, 0) = λ 2 u 0 (λx). Thus for s 0 (3.9) u λ (, 0) H s =0(λ 3/2+s ) as λ 0. In our case s ( 3/4, 0], hence, without loss of generality, we can restrict ourselves to considering the IVP (.) with data u 0 (x) satisfying (3.20) u 0 H s = r<<, i.e., r arbitrary small; see (3.24) below. For u 0 H s (R), s ( 3/4, 0], satisfying (3.20) we define the operator (3.2) Φ u0 (ω) =Φ(ω)=ψ(t)W(t)u 0 ψ(t) 2 We shall prove that Φ( ) defines a contraction on (3.22) B(2cr)= t { } ω X s,b ω Xs,b 2cr. 0 W(t t )ψ 2 (t ) x (ω 2 (t ))dt. First combining (3.3)-(3.5) (.9) (Theorem.) it follows that for ω B 2cr (3.23) Φ(ω) Xs,b cr + c x (ψ(t)ω(,t)) 2 Xs,b cr + c ψω 2 X s,b cr + c ω 2 X s,b cr + c(2cr) 2 2cr for r in (3.20) satisfying (3.24) 4c 2 r.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 589 Similarly using (3.4)-(3.5), (.9) (3.24) it follows that for ω, ω B(2cr) (3.25) Φ(ω) Φ( ω) Xs,b = t 2 ψ(t) W (t t )ψ 2 (t ) x (ω 2 ω 2 )(t )dt 0 Xs,b c ω + ω Xs,b ω ω Xs,b 2c 2 r ω ω Xs,b 2 ω ω X s,b. Thus Φ( ) is a contraction. Therefore there exists a unique u B(2cr) such that (3.26) u(t) =ψ(t) W(t)u 0 2 t 0. W(t t ) x (ψ(t )u(t )) 2 dt Hence, in the time interval [, ], u( ) solves the integral equation associated to the IVP (.). Now we turn to the proof of the persistence property, i.e. (3.27) u C([, ] : H s (R)) the continuous dependence of the solution upon the data in the norm of the space in (3.27). Combining (3.7) Hölder Sobolev inequalities we obtain ψ(ρ )u Xs,0 = J s (ψ(ρ )u) L 2 t L 2 x = W (t)(ψ(ρ )J s u) L 2 t L 2 x = ψ(ρ )W (t)j s u L 2 x L 2 t (3.28) cρ /4 ψ(ρ )W (t)j s u L 2 x L 4 t cρ /4 D /4 t (ψ(ρ )W (t)j s u) L 2 x L 2 t = cρ /4 ψ(ρ )u Xs,/2 cρ /4 ψ(ρ )u Xs,b. Thus, by Hölder s inequality, (3.4) (3.28), for 0 <b b we find that (3.29) ψ(ρ )u Xs,b ψ(ρ )u (b b )/2 X s,0 ψ(ρ )u b /b X s,b cρ (b b )/8 ψ ( ρ )u Xs,b cρ (b b )/8 u Xs,b. Using the integral equation (3.26), (2.57) with some fixed b <b, (3.4), (3.6)

590 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA (3.29), for 0 t <t t t tit follows that (3.30) u(t) u( t) H s W(t t)u( t) u( t) H s t ( +c t W(t t )ψ 2 ) t x (u 2 (t ))dt t t H ( s ) ) W(t t)u( t) u( t) H s +c ( t x ψ 2 u 2 t Xs,b ) 2 W(t t)u( t) u( t) H s +c ( t ψ u t X s,b W(t t)u( t) u( t) H s +c( t) (b b )/4 u 2 X s,b = o() as t 0, which yields the persistence property. The proof of the continuous dependence of the solution upon the data in the L ([0, ] : H s (R))-norm follows a similar argument, therefore it will be omitted. Finally we explain how to extend the uniqueness result in B(2cr) in (3.22) to the whole X s,b. For any δ (0, ) we define { } (3.3) B(2cδ ( 2b)/2 r)= ω X s,b ω Xs,b 2cδ ( 2b)/2 r (3.32) Φ δ,u0 (ω) =Φ δ (ω)=ψ(δ t)w(t)u 0 ψ(δ t) 2 Combining (2.57) with b ( 2,b) such that (3.33) b b =min{ s /2; /4+s/3} with the arguments in (3.23) (3.29) one has t 0 W(t t )ψ 2 (t ) x (ω 2 (t ))dt. (3.34) Φ δ (ω) Xs,b cδ ( 2b)/2 r +4c 3 δ 3( 2b)/2 δ (b b )/4 r 2, (3.35) Φ δ (ω) Φ δ ( ω) Xs,b 2c 3 δ ( 2b) δ (b b )/4 r ω ω Xs,b. By (3.33) it follows that (3.36) 2b +(b b )/4<0, therefore for any δ (0, ) (3.37) Φ δ (B(2cδ ( 2b)/2 r)) B(2cδ ( 2b)/2 r) is a contraction, which completes the proof.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 59 4. Proof of Theorems.3.7 Proof of Theorem.3. Using the notation in (2.)-(2.5), one sees that (.9) is equivalent to ξ ( + τ ξ 3 ) b ( + ξ ) ρ (4.) f(ξ,τ )( + ξ ) ρ f(ξ ξ,τ τ )( + ξ ξ ) ρ dξ dτ ( + τ ξ 3 )b ( + τ τ (ξ ξ ) 3 ) b L 2 ξ L 2 τ c f 2 L. 2 ξ L2 τ First we shall see that if ρ = s >3/4, then b /2. Fix N Z + take (4.2) f(ξ,τ) =χ A (ξ,τ)+χ A (ξ,τ), where χ A ( ) represents the characteristic function of the set A with (4.3) A = (4.4) A = Clearly { (ξ,τ) R } 2 N ξ N +, τ ξ 3 N { } (ξ,τ) R 2 (ξ,τ) A. (4.5) f L 2 ξ L 2 τ cn 4. On the other h, A contains a rectangle with (N,N 3 ) as a vertex, with dimensions 0 2 N 2 N 3/2 longest side pointing in the (, 3N 2 ) direction. Therefore (4.6) (f f)(ξ,τ) c χ R (ξ,τ), N where R is the rectangle centered at the origin of dimensions cn 2 N 3 2 longest side pointing in the (, 3N 2 ) direction. Hence (4.) implies that (4.7) N 2ρ N 2 N 3 2 (b ) N 2 N 4 cn 2, i.e. (4.8) b +4ρ 3 2 as desired.

592 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA To complete the proof we just need to show that if ρ = s >3/4thenb>/2. By polarization duality (4.) is equivalent to the following inequality: ( + ξ ) ρ ( + τ ξ 3 )b (4.9) ξ g(ξ,τ) h(ξ ξ,τ τ )( + ξ ξ ) s ( + ξ ) s ( + τ ξ 3 ) b ( + τ τ (ξ ξ ) 3 ) b dξdτ L 2 ξ L2 τ We take c g L 2 ξ L 2 τ h L 2 ξ L2 τ. (4.0) g(ξ,τ) =χ A (ξ,τ), (A defined in (4.3)) (4.) h(ξ,τ) =χ B (ξ,τ), where (4.2) B = { (ξ,τ) R 2 N+/2 N ξ N+3/4 } N ; τ ξ 3 <. Estimating the first norm in (4.9) in the domain [ (4.3) 4τ ξ 3 ξ N, N ] 4 20 N 0 N from (4.9) we see that (4.4) i.e. if ρ = s >3/4 (4.5) N ρ+/4 N 3b N /2, 3 2 <ρ+3 4 3b. Proof of Theorem.7. We consider the case b =/2, remark that a similar argument to that used in the previous proof shows that this assumption does not imply a loss of generality. First we take F (, ) such that (4.6) F (ξ,τ) =χa (ξ,τ)+χ A (ξ,τ), where A was defined in (4.3). Thus (4.7) ( F F F )(ξ,τ) c (χa χ R )(ξ,τ), N

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 593 where R represents the rectangle centered at the origin of dimension 0 2 N 2 N 3/2 with its longest side pointing in the (, 3N 2 ) direction. Then (4.8) ( F F F )(ξ,τ) c N χ R 0 (ξ,τ), where R 0 is a rectangle like R, of the same dimensions direction, but with (N,N 3 ) as one of its vertices. Hence (4.9) B(F, F, F) Xs,b cn s /4. Since (4.20) F Xs,b cn s /4, (.8) implies that s /4, which completes the proof. 5. Proof of Theorem.2 First we rewrite the inequality (.0). Defining (5.) ρ = s [0, /2) by (.7) it follows that if F Y s,b = Y ρ,b,then (5.2) f(n, τ) =(+ τ n 3 ) b n ρ F(n, τ) L 2 τ (R : l 2 n(t)) (5.3) f L 2 τ l 2 n = F Y ρ,b. Since (5.4) x (F 2 )(n, τ) =cn( F F)(n, τ) we can express (.0) in terms of f as (5.5) B(F, F) Ys,b,0 = ( + τ n 3 ) b n ρ x (F 2 )(n, τ) L 2 τ l 2 n = c ( + τ n 3 ) b n ρ n( F F )(n, τ) L 2 τ l 2 n ( n 2 = ( + τ n 3 ) 2( b) n 2ρ n 0 n (n n ) ρ ( + τ n 3 )/2 n n n 0 f(n,τ ) f(n n,τ τ )dτ ( + τ n 3 )/2 ( + τ τ (n n ) 3 ) /2 = B(f,f)(n, τ) L 2 τ l 2 n c ( n 0 2) /2 ) f(n, τ) 2 dτ = c f 2 L. 2 τ l2 n As in the continuous case we first need some calculus results.

594 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA Lemma 5.. There exists c>0such that for any n 0 any τ R (5.6) n 0 n n log(2 + τ nn (n n ) ) + τ nn (n n ) <c. Proof. Since n 0 it suffices to show that (5.7) n 0 n n ( + µ n (n n ) ) 3/4 L µ l n <c. Let α, β be the roots of the equation (5.8) µ n (n n )=0, i.e. (5.9) µ n (n n )=(n α)(n β), with α = α(n, µ), β = β(n, µ). We shall prove that (5.0) ( + (n α)(n β) ) 3/4 n L α L β <c, which implies (5.7). Notice that for a fixed (α, β) R 2 there exist at most 8 n s such that (5.) n α 2 or n β 2. For the remaining n s one has that (5.2) + n α n β 2 ( + n α )( + n β ), hence 8+ ( + (n n α)(n β) ) 3/4 ( + (n n α 2 α)(n β) ) 3/4 8+ n β 2 /2 /2 ( + n α ) 3/2 ( + n β ) 3/2 <c, n α 2 which yields the lemma. n β 2

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 595 Lemma 5.2. If ρ = s [0, /2], then (5.3) where (5.4) n ( + τ n 3 ) /2 n ρ n (n n ) 2ρ ( + τ n 3 )( + τ τ (n n ) 3 ) dτ n A A { A = A(n, τ) = (n,τ ) Z R n 0,n n, } τ τ (n n ) 3 τ n 3 τ n. /2 <c, Proof. It suffices to prove (5.3) for s =0s= /2. The general case follows by the Three Lines Theorem. As in [4] we shall use the identity (5.5) τ n 3 (τ n 3 ) (τ τ (n n ) 3 )=3nn (n n ), which implies that (5.6) max { τ n 3 ; τ n 3 ; τ τ (n n ) 3 } nn (n n ). Case s =0. Combining (5.6) the definition of A it follows that (5.7) τ n 3 nn (n n ) n 2 /2. Hence we only have to show that (5.8) n A dτ ( + τ n 3 )( + τ τ (n n ) 3 ) <c. Changing variable (5.9) θ = τ n 3 using the notation (5.20) a = τ n 3 +3nn (n n ) we find that (5.2) = dτ ( + τ n 3 )( + τ τ (n n ) 3 ) dθ ( + θ )( + θ a ) c log(2 + a ). + a Inserting (5.2) into (5.8) using Lemma 5. we complete the proof.

596 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA Case ρ = s =/2. In this case we have that in A (5.22) ( + τ n 3 ) /2 n /2 n (n n ) /2. Therefore the proof reduces to the previous case. Lemma 5.3. If ρ = s [0, /2], then (5.23) ( + τ n 3 )/2 n 2( 2ρ) nn (n n ) 2ρ dτ ( + τ n 3 )( + τ τ (n n ) 3 ) n D D /2 <c, where (5.24) D = D(n,τ )= { (n, τ) Z R n n, } τ n 3 τ n 3, τ τ (n n ) 3 τ n 3. Proof. As in the previous case it suffices to consider the cases s =0s= /2. Case s =0. From the definition of D the relations (5.5)-(5.6) it follows that (5.25) τ n 3 n2 /2. Hence the proof reduces to that in Lemma 5.2, case s =0. Case ρ = s =/2. Since (5.26) ( + τ n 3 )/2 n /2 n (n n ) /2 the numerator in the integr in (5.23) cancels the expression outside the parentheses. Hence the proof follows the argument in the previous case. Proof of Theorem.2. Using (5.2)-(5.3), the Cauchy-Schwarz inequality we

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 597 write (5.27) B(F, F) Ys,b = B(f,f) L 2 τ l 2 n ( n 2 = ( + τ n 3 ) n 2ρ n 0 n (n n ) ρ 2 f(n,τ )f(n n,τ τ ) ( + τ n 2 )/2 ( + τ τ (n n ) 3 ) dτ /2 dτ n n n 0 ( ( n 0 n n n 0 ( n 2 ( + τ n 3 ) n 2ρ n (n n ) 2ρ ( + τ n 3 )( + τ τ (n n ) 3 dτ f(n,τ )f(n n,τ τ ) 2 dτ )dτ ) /2 ) ) /2 n n n 0 n ( + τ n 3 ) /2 n ρ ( n 0 n n ( n 0 n n n 0 =Ψ f 2 L 2 τ l2 n. n (n n ) 2ρ /2 ) ( + τ n 3 )( + τ τ (n n ) 3 ) dτ ) /2 f(n,τ ) 2 f(n n,τ τ ) 2 dτ dτ L τ l n To complete the proof we just need to show that Ψ, defined above, is bounded. But this follows by combining Lemmas 5.2 5.3, using duality symmetry arguments. 6. Proof of Theorems.4,.8 We recall the notation introduced in (5.5) (6.) B ρ,b (f,f)(n, τ) = n 0 n n First we shall see that the condition n ( + τ n 3 ) b n ρ n (n n ) ρ f(n,τ )f(n n,τ τ ) ( + τ n 3 )b ( + τ τ (n n ) 3 ) b dτ. (6.2) b = 2 is necessary for the boundedness of B ρ,b (f,f).

598 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA Lemma 6.. If there exists ρ R such that for any f L 2 τ l2 n (6.3) B ρ,b (f,f) L 2 τ l 2 c f 2 n L, 2 τ l2 n then (6.4) b = 2. Proof. Choose (6.5) f(n, τ) =a n χ /2 (τ n 3 ), where {, n =, N, (6.6) a n = 0, elsewhere, {, x <θ, (6.7) χ θ (x) = 0, elsewhere. We evaluate B ρ,b (f,f)(n, τ) atn=n bound it below by (6.8) N ( + τ N 3 ) b for those τ s for which there is a τ interval of size larger than a fixed positive constant such that (6.9) τ 2 τ τ (N ) 3 2. Assuming that (6.0) τ 4 τ τ N 3 ++3N(N ) < 2 we have that (6.9) holds (6.) τ N 3 N 2. Hence, (6.3) implies that (6.2) N c for any N Z+ N 2( b) consequently (6.3) b /2.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 599 To see that b /2 we note that polarization duality show that if (6.3) holds, then a similar inequality must hold for B ρ,b (, ), where (6.4) B ρ,b (f,f)(n,τ )= n 0 n n ( + τ n 3 )b n f(n, τ) n (n n ) ρ f(n n,τ τ ) ( + τ n 3 ) b n ρ ( + τ τ (n n ) 3 ) b dτ, i.e., (6.3) is equivalent to (6.5) B ρ,b (f,f) L 2 τ l 2 c f 2 n L. 2 τ l2 n We choose (6.6) f(n, τ) =b n χ /2 (τ n 3 ), where {, n = N,N, (6.7) b n = 0, elsewhere, we evaluate B ρ,b (f,f)(n,τ )atn = bound it below by (6.8) N ( + τ ) b for those τ s for which there is a τ interval of size larger than a fixed positive constant such that (6.9) τ N 3 2 τ τ (N ) 3 2. Clearly (6.9) holds in an interval of τ s of length larger than 2 in which (6.20) τ N 2. Hence (6.5) implies that (6.2) consequently N c for any N Z+ N 2b (6.22) b 2, which completes the proof. Proof of Theorem.4. By Lemma 6., it suffices to see that (6.3) with b = /2 fails for ρ = s >/2.

600 C. E. KENIG, GUSTAVO PONCE, AND LUIS VEGA We choose (6.23) f(n, τ) =a n χ (τ n 3 ), where {, n = N/2, (6.24) a n = 0, elsewhere. We evaluate B ρ,/2 (f,f)(n, τ) atn=n bound it below by (6.25) N ( + τ N 3 ) /2 N s N 2s for those τ s for which in a unit size interval of τ s we have (6.26) τ (N/2) 3 τ τ (N/2) 3. Assuming that (6.27) τ (N/2) 3 2 τ 2(N/2) 3 it follows that (6.26) is satisfied. Moreover (6.28) τ N 3 /4 2 implies τ N 3 N 3. Hence (6.29) N N 3/2 N ρ c for any N Z +. Therefore (6.30) ρ 2. Proof of Theorem.8. Using the notation in (6.) we define (6.3) n B 3 (f,f,f)(n, τ) = ( + τ n 3 ) b ( + n ) ρ n 2 n n n 2 n 0 f(n,τ )( + n ) ρ ( + τ n 3 )b f(n 2 n,τ 2 τ )( + n 2 n ) ρ ( + τ 2 τ (n 2 n ) 3 ) b f(n n 2,τ τ 2 )( + n n 2 ) ρ ( + τ τ (n n 2 ) 3 ) b dτ dτ 2.

A BILINEAR ESTIMATE WITH APPLICATIONS TO THE KdV EQUATION 60 Thus to obtain (.9) it suffices to show that (6.32) B 3 (f,g,h) L 2 τ l 2 n c f L 2 τ l2 n g L 2 τ l2 n h L 2 τ l2 n fails for any ρ = s >/2. An argument similar to that used in the proof of Lemma allows us to restrict ourselves to the case (6.33) b = 2. We now choose (6.34) f = a n χ /4 (τ n 3 ), g = b n χ /4 (τ n 3 ), h = d n χ /4 (τ n 3 ), where {,n=n, (6.35) a n = 0, elsewhere, {,n= N+, b n = 0, elsewhere, {,n=n, d n = 0,elsewhere. Performing the addition integration in (6.3) in the n,τ variables respectively we find a n b n2 n N 2ρ χ /4 (τ n 3 )χ (τ /4 2 τ (n 2 n ) 3 )dτ (6.36) n = c n a n b n2 n N 2ρ χ /2 (τ 2 n 3 2 +3n 2 n (n 2 n )) = cα n2 N 2ρ χ /2 (τ 2 3N(N )), where (6.37) {,n2 =, α n2 = 0, elsewhere. Now performing the operations in the n 2,τ 2 variables we get N 3ρ d n n2 α n2 (6.38) n 2 χ /2 (τ 2 3N(N ))χ /4 (τ τ 2 (N ) 3 )dτ 2 = cn 3ρ χ(τ N 3 ). By inserting this computation in (6.32) it follows that (6.39) N N 3ρ N ρ c for any N Z+. Hence (6.40) 2ρ + 0, i.e. s = ρ 2.

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