TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4)

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1 PERIODIC L 4 -STRICHARTZ ESTIMATE FOR KDV TADAHIRO OH 1. Introduction In [], Bourgain proved global well-posedness of the periodic KdV in L T): u t + u xxx + uu x 0, x, t) T R. 1.1) The key ingredient in the proof of local well-posedness is the periodic L 4 -Strichartz estimate: where the X s,b -norm is defined by T R) u X 0, 1 T R), 1.) u X s,b T R) n s τ n b ûn, τ) l n L τ. 1.) The result in [] is in fact stated for time-periodic functions: T ) u X 0, 1 T ). 1.4) Such a restriction, however, is not necessary. See Tao [, 4]. Remark 1.1. In [1], Bourgain also proved the periodic L 4 -Strichartz estimate for the Schrödinger equation: T T) u X 0, 8 T R), 1.5) Let us compare 1.5) with the classical result by Zygmund [5]: e t x u0 L 4 T ) u 0 L T). 1.6) By the transference principle Lemma.9 in []), 1.6) yields T ) u X 0,b T ), b > ) From this, we can see that Bourgain s periodic L 4 -Strichartz estimate 1.5) is a significant improvement of 1.6) and 1.7). In the following, we present a heuristic argument to show how the regularity b 1 naturally arises. On the one hand, the X s,b -space given by the norm 1.) is adapted to the linear part of the KdV equation called the Airy equation): u t + u xxx 0. Namely, we can formally view three spatial derivatives as equivalent to one temporal derivative. On the other hand, by the Sobolev inequality, we have T R) u X 1 4, 1 4 T R). 1.8) 1

2 TADAHIRO OH Then, by formally moving the spatial derivative s 1 in 1.8) to the temporal side, 4 we obtain the temporal regularity b 1 in 1.), since Of course, 4 4 this is merely a heuristic argument showing why b 1 is the natural regularity in 1.). The necessity of the regularity b 1 can be shown in Section, following the argument in [1, p.116]. The proof of 1.4) appeared in []. Upon a small modification, it is easy to see that the same proof works for 1.). In [4, Proposition 6.4], Tao presented a short proof of 1.) by estimating a certain [ : Z R]-multiplier. In [], a simple proof of the periodic L 4 -Strichartz estimate 1.5) for the Schrödinger equation is presented after Tzvetkov). In Section, we present a short proof of 1.) based on this argument.. Proof of 1.) We basically follow the argument in [, Proposition.1] with a small but important modification. First, write u as u where u M is the localization of u on M τ n < M. Then, we have M um L u..1) X 0, 1 We also have u L 4 uu L M, M, dyadic u M u M u M L m0.) In the following, we use n 1, τ 1 ) and n, τ ) as the frequencies for u M and u m M, respectively. We also use n, τ) for the frequencies for the product u M u m M, i.e. we have n n 1 + n and τ τ 1 + τ. First, consider the case when n m M. Then, it suffices to show that εm M um L.) for all m 0, where ε > 0 is some constant to be chosen later. Now, we claim that.) follows once we show εm M 1 um L m M u m M L..4) Indeed, by Cauchy-Schwarz inequality with.4), we have RHS of.). LHS of.) εm M M um L M m M) um L In the following, we show.4) when n m M. Without loss of generality, assume that u M L u m M L 1. By Cauchy-Schwarz and Hölder inequalities

3 with Fubini, we have PERIODICL 4 -STRICHARTZ ESTIMATE FOR KDV sup sup û M n 1, τ 1 )û m Mn, τ )dτ 1 ττ 1 +τ um L nn 1 +n nn 1 +n Hence,.4) follows once we show sup nn 1 +n nn 1 +n ττ 1 +τ τ 1 n 1 +OM) τ n +Om M) 1 dτ 1 ττ 1 +τ ττ 1 +τ 1 dτ 1. l n L τ u m M L 1 dτ 1 ε)m M 4.5) Fix n and τ. Under the given constraints, we have τ n 1 + n + O m M). With n n 1 + n, this yields τ n nn 1 n + O m M) n n 1 n ) 1 τ 4 n + O m M) n 1 n ) ) Cn, τ) + O m M), since n m M by assumption. This in particular implies that there are at most O m M 1 ) possible values for n1 and hence for n ). Then,.5) follows from integrating in τ 1 and this observation. Note that we can take ε 1 6. Next, we consider the case when n m M. This is where the argument is different from that in []. As before, our goal is to prove.4) under u M L u m M L 1. Our goal is to show.4): 1 ε)m M.6) By Minkowski inequality followed by Cauchy-Schwarz inequality note that τ 1 n 1 + OM)), we have l û M n 1, τ 1 )û m Mn, τ τ 1 )dτ 1 nn 1 +n nl τ l û M n 1, τ 1 ) û m Mn, ) L τ dτ 1 nn 1 +n n M 1 û M n 1, ) L τ1 û m Mn, ) L τ nn 1 +n l n m 6 M um L u m M L.

4 4 TADAHIRO OH With ε 1, this proves.6), and hence 1.). 6. Necessity of the regularity b 1 In this section, we show that the regularity b 1 in 1.) is indeed necessary. Consider the function ux, t) e inx+τt) dτ. τ N Namely, we have ûn, τ) χ N n)χ N τ), where χ N is the characteristic function of the interval [ N, N]. First, we establish the lower bound on the L 4 -norm of u. u 4 L 4 u L vn, τ) l nl vn, τ) dτ,.1) τ τ N where vn, τ) : û n, τ) χ N Z χ N ) n) χn R χ N ) τ). Note that the first convolution is with respect to the counting measure on Z and the second with respect to the Lebesgue measure on R. Thus, we have { ) χn Z χ N n) N, for n N, ).) χn R χ N τ) N, for τ N. From.1) and.), we obtain Next, we compute the X 0, 1 -norm of u. u X 0, 1 N..) τ N τ n dτ N 1 m N 1 0 N 1 m N m n dα 1 0 N 1 m + α n dα m N m n For fixed n [ N, N], there is one-to-one correspondence between m [ N, N 1] k [ N n, N 1 n ] under m n k. Hence, we have u X 0, 1 N 1 n k k N n N N k0 k The necessity of the regularity b 1 follows from.) and.4).. N..4)

5 PERIODICL 4 -STRICHARTZ ESTIMATE FOR KDV 5 References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal. 199), no., [] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation, Geom. Funct. Anal. 199), no., [] T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 006. xvi+7 pp. [4] T. Tao, Multilinear weighted convolution of L -functions, and applications to nonlinear dispersive equations, Amer. J. Math ), no. 5, [5] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Stud. Math ), Tadahiro Oh, Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ , USA address: hirooh@math.princeton.edu