MAXIMIZERS FOR THE STRICHARTZ AND THE SOBOLEV-STRICHARTZ INEQUALITIES FOR THE SCHRÖDINGER EQUATION

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1 Electronic Journal of Differential Euations, Vol. 9(9), No. 3, pp ISSN: UR: or ftp ede.math.tstate.edu (login: ftp) MAXIMIZERS FOR THE STRICHARTZ AND THE SOBOEV-STRICHARTZ INEQUAITIES FOR THE SCHRÖDINGER EQUATION SHUANGIN SHAO Abstract. In this paper, we first show that there eists a maimizer for the non-endpoint Strichartz ineualities for the Schrödinger euation in all dimensions based on the recent linear profile decomposition result. We then present a new proof of the linear profile decomposition for the Schröindger euation with initial data in the homogeneous Sobolev space; as a conseuence, there eists a maimizer for the Sobolev-Strichartz ineuality. 1. Introduction We consider the free Schrödinger euation i t u + u =, (1.1) with initial data u(, ) = u () where u : R R d C is a comple-valued function and d 1. We denote the solution u by using the Schrödinger evolution operator e it : u(t, ) := e it u () := û (ξ)dξ, (1.) R d e i ξ it ξ where û is the spatial Fourier transform of u defined via û (ξ) := e i ξ u ()d, (1.3) R d where ξ (abbr. ξ) denotes the Euclidean inner product of and ξ in the spatial space R d. Formally the solutions to this euation have a conserved mass R d u(t, ) d. (1.4) A family of well-known ineualities, the Strichartz ineualities, is associated with (1.1), which is very useful in nonlinear dispersive euations. It asserts that, for any u (R d ), there eists a constant C d,,r > such that e it u t r (R Rd ) C d,,r u (R d ) (1.5) Mathematics Subect Classification. 35Q55. Key words and phrases. Maimizers; Profile decomposition; Schrödinger euation; Strichartz ineuality. c 9 Teas State University - San Marcos. Submitted October 13, 8. Published January, 9. 1

2 S. SHAO EJDE-9/3 holds if and only if (, r, d) is Schrödinger admissible; i.e., + d r = d, (, r, d) (,, ),, r. (1.6) d For (, r) = (, d ) when d 3 or (, r) = (4, ) when d = 1, the ineuality (1.5) is referred to as the endpoint estimate, otherwise the non-endpoint estimate for the rest pairs. It has a long history to establish (1.5) for all Schrödinger admissible pairs in (1.6) epect when (, r) = (, ), in which case it follows from (1.4). For the symmetric eponent = r = + 4 d, Strichartz established this ineuality in [17] which in turn had precursors in [1]. The non-endpoints were established by Ginibre and Velo [8], see also [, Theorem.3] for a proof; the delicate endpoints in higher dimensions were treated by Keel and Tao [1]. When (, r, d) = (,, ), it has been known to fail, see e.g., [15] and [19]. A close relative of the Strichartz ineuality for the Schrödinger euation is the Sobolev-Strichartz ineuality: for <, and r < and u Ḣs(,r) (R d ) with s(, r) := d d r >, there eists a constant C d,,r > such that e it u t r (R Rd ) C d,,r u Ḣs(,r) (R d ), (1.7) which can be proven by using the usual Sobolev embedding and the Strichartz ineuality (1.5). In this paper, we are interested in the eistence of maimizers for the Strichartz ineuality (1.5) and the Sobolev-Strichartz ineuality (1.7), i.e., functions which optimize (1.5) and (1.7) in the sense that they become eual. The answer to the former is confirmed for the non-endpoints by an application of a recent powerful result, the profile decomposition for Schrödinger euations, which was developed in [4, 14, 5, ] and had many applications in nonlinear dispersive euations, see [1] and the reference within. The problem of the eistence of maimizers and of determining them eplicitly for the symmetric Strichartz ineuality when = r = + 4 d has been intensively studied. Kunze [13] treated the d = 1 case and showed that maimizers eist by an elaborate concentration-compactness method; when d = 1,, Foschi [7] eplicitly determined the best constants and showed that the only maimizers are Gaussians by using the sharp Cauchy-Schwarz ineuality and the space-time Fourier transform; Hundertmark and Zharnitsky [9] independently obtained this result by an interesting representation formula of the Strichartz ineualities; recently, Carneiro [6] proved a sharp Strichartz-type ineuality by following the arguments in [9] and found its maimizers, which derives the results in [9] as a corollary when d = 1, ; very recently, Bennett, Bez, Carbery and Hundertmark [3] offered a new proof to determine the best constants by using the method of heat-flow. The answer to the latter is true as well. The proof follows almost along similar lines as in the case if we have an analogous profile decomposition for initial data in the homogeneous Sobolev spaces. We offer a new proof for this fact, which we have not seen in the literature In this subsection, we investigate the eistence of maimizers for the nonendpoint Strichartz ineualities. To begin, we recall the profile decomposition result in [] in the notation of the symmetry group which preserves the mass and the Strichartz ineualities.

3 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 3 Definition 1.1 (Mass-preserving symmetry group). For any phase θ R/πZ, scaling parameter h >, freuency ξ R d, space and time translation parameters, t R d, we define the unitary transformation g θ,h,ξ,,t : (R d ) (R d ) by the formula [g θ,h,ξ,,t φ]() = e iθ e i ξ e it [ 1 h d/ φ( h )](). (1.8) We let G be the collection of such transformations; G forms a group. Definition 1.. For k, two seuences Γ n = (h n, ξn, n, t n) n 1 and Γ k n = (h k n, ξn, k n k, tk n) n 1 in (, ) R d R d R are said to be orthogonal if one of the followings holds: ( h k n lim n + h h n + h n h n ξ k n ξn k n lim n ( t n tk n (h n) + n k n h n ) =, ) =. + tk n (ξk n ξ n ) h n We rephrase the linear profile decomposition theorem in [] by using the notation in Definition 1.1. Theorem 1.3. et {u n } n 1 be a bounded seuence in. Then up to passing to a subseuence of (u n ) n 1, there eists a seuence of functions φ and group elements (gn) n 1, 1 = g,h n,ξn, n,t G with orthogonal (h n n, ξn, n, t n) such that for any N 1, there eists e N n, u n = gn(φ ) + e N n, (1.9) with the error term having the asymptotically vanishing Strichartz norm lim lim N n eit e N n +4/d =, (1.1) t, and the following orthogonality properties: for any N 1, for k, for any 1 N, lim n ( u n ( N ) φ + en n ) =, (1.11) lim n eit gn(φ )e it gn(φ k k ) 1+/d =, (1.1) t, lim n g n(φ ), gn(φ k k ) =, (1.13) lim n g n(φ ), e N n =. (1.14) The first main result in this paper concerns on the eistence of maimizers for the symmetric Strichartz ineuality +4/d t,. Theorem 1.4. There eists a maimizing function φ such that, with S := sup{ e it u + 4 d t, e it φ + 4 d t, = S φ : u = 1} being the sharp constant.

4 4 S. SHAO EJDE-9/3 The proof of this theorem uses Theorem 1.3 and the following crucial ineuality in []: for any N 1, lim N e it gn(φ ) +4/d n +4/d t, lim n eit φ +4/d. (1.15) +4/d Remark 1.5. The ineuality (1.15) is a conseuence of (1.1) by an interpolation argument in [], which we will generalize in the proof of emma 1.6. When d = 1,, one can actually show that (1.15) is an euality by using the fact that + 4/d is an even integer. The ineuality (1.15) suggests a way to obtain similar claims as in Theorem 1.4 for other non-endpoint Strichartz ineualities if the following lemma were established. emma 1.6. et (, r, d) be non-endpoint Schrödinger admissible pairs and N 1. If r, lim N e it gn(φ ) r n lim t r n eit φ r ; (1.16) t r if r, lim N e it gn(φ ) lim n t r n eit φ. (1.17) t r Indeed, this is the case. Together with Theorem 1.3 again, this lemma yields the following corollary. Corollary 1.7. et (, r, d) be non-endpoint Schrödinger admissible pairs. There eists a maimizing function φ such that, e it φ = S,r φ t r with S,r := sup{ e it u t r : u = 1} being the sharp constant. The proof of this corollary is similar to that used in Theorem 1.4 and thus will be omitted. Instead, we will focus on proving emma 1.6. Remark 1.8. When (, r) = (, ), from the conservation of mass (1.4), we see that every -initial data is a maimizer for the Strichartz ineuality. 1.. In this subsection we concern on the eistence of maimizers for the Sobolev- Strichartz ineuality (1.7) for the Schrödinger euation. Theorem 1.9. et (, r, d) be defined as in (1.7). Then there eists a maimizing function φ Ḣs(,r) for (1.7) with C d,,r being the sharp constant S,r := sup{ e it u : u t r Ḣs(,r) = 1}. As we can see, it suffices to establish a profile decomposition result for initial data in Ḣs(,r). Theorem 1.1. et s(, r) be defined as in (1.7) and {u n } n 1 be a bounded seuence in Ḣs(,r). Then up to passing to a subseuence of (u n ) n 1, there eists a t,

5 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 5 seuence of functions φ Ḣs and a seuence of parameters (h n, n, t n) such that for any N 1, there eists e N n Ḣs, u n = e ( 1 it n ( ) (h n) d/ s φ n ) + e N n, (1.18) with the parameters (h n, n, t n) satisfying the following constraint: for k, ( h lim n n h k + hk n n h + t n t k n n (h + n k ) n n) h =, (1.19) n and the error term having the asymptotically vanishing Sobolev-Strichartz norm lim lim N n eit e N n =, (1.) t r and the following orthogonality property: for any N 1, ( ) lim u n ( φ + e N n Ḣ s Ḣ s n =. (1.1) Ḣ) s When s = 1 and d 3, Keraani [11] established Theorem 1.1 for the Schrödinger euation based on the following Besov-type improvement of the Sobolev embedding f d/(d ) where Ḃ is the Besov norm defined via, h n Df 1 /d Df /d, (1.) Ḃ, f Ḃ := sup f k, with f k denoting the k-th ittlewood-paley piece defined via the Fourier transform ˆf k := ˆf1 k ξ 1, and D s the fractional differentiation operator defined via the k+1 inverse Fourier transform, D s f() := e iξ ξ s ˆf(ξ)dξ. R d He followed the arguments in [1] where Bahouri and Gérard established the profiledecomposition result in the contet of the wave euation with initial data in Ḣ1 (R 3 ). Recently under the same constraints on s and d, Killip and Visan [1] obtained the same result by relying on their interesting improved Sobolev embedding involving the critical d/(d ) -norm on the right-hand side: f d/(d ) Df 1 /d sup f k /d d/(d ). (1.3) Note that (1.3) implies (1.) by the usual Sobolev embedding. By following their approaches, we will generalize both Keraani s and Killip-Visan s improved Ḣ-Sobolev 1 embeddings to those with Ḣs norms where 1 r + s d = 1 and d 1 in the appendi of this paper. Conseuently almost same approaches as in [11] or [1] would yield Theorem 1.1 without difficulties but we choose not to do it in this paper for simplicity. However, we will offer a new proof of Theorem 1.1 by taking advantage of the eisting linear profile decomposition, Theorem 1.3. The idea can be roughly eplained as follows. 1 For a rigorous definition of the ittlewood-paley decomposition (or dyadic decomposition) in terms of smooth cut-off functions, see [16, p.41].

6 6 S. SHAO EJDE-9/3 For (u n ) n 1 Ḣs, we regard (D s u n ) n 1 as an seuence and then apply Theorem 1.3 to this new seuence. Then the main task is to show how to eliminate the freuency parameter ξ n from the decomposition. To do it, we have two cases according to the limits of the seuence (h nξ n) n 1 for each : when the limit of h nξ n is finite, we will change the profiles φ so that we can reduce to ξ n = ; on the other hand, when it is infinite, we will group this term into the error term since one can show that its Sobolev Strichartz norm is asymptotically small. We organize this paper as follows: in Section 1.3 we establish some notations; in Section we prove Theorems 1.4, 1.6; in Section 3 we prove Theorem 1.9; finally in Appendi, we include the arguments for the general Keraani s and Killip-Visan s improved Sobolev embeddings Notation. We use X Y, Y X, or X = O(Y ) to denote the estimate X CY for some constant < C <, which might depend on d, p and but not on the functions. If X Y and Y X we will write X Y. If the constant C depends on a special parameter, we shall denote it eplicitly by subscripts. Throughout the paper, the symbol lim n should be understood as lim sup n. The homogeneous Sobolev space Ḣs (R d ) for s can be defined in terms of the fractional differentiation: Ḣ s (R d ) := {f : f Ḣs (R d ) := Ds f (R d ) = ξ s ˆf ξ (R d ) < }. We define the space-time norm t r of f on R R d by ( ( ) /rd ) 1/, f t r (R Rd ) := f(t, ) r d t R R d with the usual modifications when or r are eual to infinity, or when the domain R R d is replaced by a small region. When = r, we abbreviate it by t,. Unless specified, all the space-time integration are taken over R R d, and the spatial integration over R d. The inner product, in the Hilbert space (R d ) is defined via f, g := f()ḡ()d, R d where ḡ denotes the usual comple conugate of g in the comple plane C.. Maimizers for the symmetric Strichartz ineualities Proof of Theorem 1.4. We choose a maimizing seuence (u n ) n 1 with u n = 1, and then, up to a subseuence, decompose it into linear profiles as in Theorem 1.3. Then from the asymptotically vanishing Strichartz norm (1.1), we obtain that, for any ɛ >, there eists n so that for all N n and n n, S ɛ e it gn(φ ) +4/d. t, Thus from (1.15), there eists n 1 n such that when n, N n 1, S +4/d ɛ e it φ +4/d. +4/d t,

7 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 7 Choosing [1, N] such that e it φ has the largest +4/d t, norm among 1 N, we see that, by the usual Strichartz ineuality, S +4/d ɛ e it φ 4/d e it φ S +4/d φ 4/d +4/d +4/d S +4/d t, t, since (1.11) gives φ (.1) lim u n n = 1. (.) This latter fact also gives that lim φ =, which together with (.1) shows that must terminate before some fied constant which does not depend on ɛ. Hence in (.1) we can take ɛ to zero to obtain φ = 1. This further shows that φ = for all but = from (.). Therefore φ is a maimizer. Thus the proof of Theorem (1.4) is complete. We will closely follow the approach in [, emma 5.5] to prove emma 1.6. Proof of emma 1.6. We only handle (1.16) since the proof of (1.17) is similar. By interpolating with (1.1), we see that for k and non-endpoint Schrödinger admissible pair (, r, d), lim n eit gn(φ )e it gn(φ k k ) / =. (.3) t r/ Now we epand the left hand side of (1.16) out, which is eual to ( ( N e it gn(φ ) ) /rdt ) r r/ d ( ( N e it gn(φ ) N e it gn(φ l l ) r + e it gn(φ ) e it gn(φ k k ) k ( ( e it gn(φ ) + k ( ( e it g n(φ ) e it g k n(φ k ) =: A + B. ) /rdt ) r/ e it gn(φ l l ) r d ) /rdt ) r/ e it gn(φ l l ) r d ) /rdt ) r/ e it gn(φ l l ) r d For B, since r = + r, 1 = r + r r, the Hölder ineuality yields B e it gn(φ )e it gn(φ k k ) / e it gn(φ l l ) r t, r k t r/ which approaches zero by (.3) as n approaches infinity. Hence we are left with estimating A.

8 8 S. SHAO EJDE-9/3 For A, since (, r, d) is in the non-endpoint region and r, we have < r + 4/d, i.e., < r 4/d. We let s := [r ], the largest integer which is less than r. Then because r s < 1, e it gn(φ ) N e it gn(φ l l ) r e it gn(φ ) N e it gn(φ k k ) s e it gn(φ l l ) r s. k=1 (.4) We now eliminate some terms in (.4). The first case we consider is l : since r s <, we write e it g n(φ ) e it gn(φ k k ) s e it gn(φ l l ) r s k=1 = e it g n(φ ) 4+s r e it gn(φ k k ) s e it gn(φ )e it gn(φ l l ) r s. k=1 Then the Hölder ineuality and (.3) show that the summation above goes to zero as n goes to infinity. So we may assume that l = and take out the summation in l in (.4). The second case we consider is when the terms in the epansion of N k=1 eit g k n(φ k ) s contain two distinct terms: e it gn(φ k k ) s k=1 e it gn(φ k k ) s + k=1 k 1 k, k 1+ +k s=s e it gn k1 (φ k1 )e it gn k (φ k )... e it gn ks (φ ks ). Again the interpolation argument and (.3) show that the second term above goes to zero as n goes to infinity. Combining these two cases, we reduce (.4) to e it g n(φ ) r + k e it g n(φ ) r s e it g k n(φ k ) s. For the second term above, we consider r s s and r s s; it is not hard to see that it goes to zero as epected when n goes to infinity. Therefore the proof of emma 1.6 is complete. 3. Maimizers for the Sobolev-Strichartz ineualities Proof of Theorem 1.1. It is easy to see that {u n } Ḣs (R d ) implies that {D s u n } (R d ). We then apply Theorem 1.3: there eists a seuence of (ψ ) 1 and orthogonal Γ n = (h n, ξ n, n, t n) so that, for any N 1 D s u n = gn(ψ ) + wn N, (3.1)

9 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 9 with wn N and all the properties in Theorem 1.3 being satisfied. Without loss of generality, we assume all ψ to be Schwartz functions. We then rewrite it as u n = D s gn(ψ ) + D s wn N. (3.) Writing e N n := D s wn N, we see that for, r in (1.7), the Sobolev embedding and (1.1) together give (1.). Net we show how to eliminate ξn in the profiles. Case 1. Up to a subseuence, lim n h nξn = ξ R d for some 1 N. From the Galilean transform, we see that e it n ( 1 (h n) e it (e i( )ξ φ)() = e iξ it ξ e it φ( tξ ), ei d/ = e it n ξ n e it n ( 1 (h n) n ξ n +it n ξ n [e i( )h n ξ n ψ ]( n ei( )ξ d/ = e iξ n e it ( 1 n ( (h n) d/ ψ n e it n ( 1 (h n) ei d/ h n h n ) ) ( t nξn) n ψ ( ) n ) ( t nξn) h n ) ) () = gn(ψ )(). On the other hand, since the symmetries defined in Definition 1.1 keep the -norm invariant, n ξ n +it n ξ n [e i( )h n ξ n ψ ]( ) n ) ( t nξn) e it n ( 1 (h n) ei d/ = (e ih n ξ n e iξ )ψ h n n ξ n +it n ξ n [e i( )ξ ψ ]( n h n ) ) ( t nξn) as n approaches infinity. Hence we can replace gn(ψ ) with e ( 1 it n ei( )ξ (h n +it n ξ n [e i( )ξ ψ ]( ) n ) ( t d/ nξn); n) for the differences, we put them into the error term. e iξ ψ as a new ψ, we can re-define g n(ψ ) := e it n ( 1 (h n) ei d/ h n n ξ n +it n ξ n ψ ( n h n Thus if further regarding ) ) ( t nξn). Hence in the decomposition (3.1), we see that ξn no longer plays the role of the freuency parameter and hence we can assume that ξn for this term. With this assumption, D s gn(ψ ) = e ( 1 it n (h n) d/ s (D s ψ )( ) n ) (). Setting φ := D s ψ, we see that this case is done. Case. lim n h nξ n =. It is clear that e it g n(ψ )() = e iξ n it ξ n 1 (h n) d/ ei t t n h n (h n ) ψ ( + n t nξn h ). n

10 1 S. SHAO EJDE-9/3 Hence if changing t = t t n and (h n) = + n t n ξ n, we obtain D s gn(ψ ) = t r (h n) + d r s d D s (e ih n ξ n e is ψ ) t r h n = D s (e ih n ξ n e is ψ ) t r. By the Hörmander-Mikhlin multiplier theorem [18, Theorm 4.4], for r <, D s e it g n(ψ ) t r 1 ( h nξ n ) s eis ψ t r ψ ( h nξ n ) s as n goes to infinity since s > and ψ is assumed to be Schwartz. In view of this, we will organize D s gn(ψ ) into the error term e N n. Hence the decomposition (1.18) is obtained. Finally the Ḣs -orthogonality (1.1) follows from (1.11), and the constraint (1.19) from Definition 1. since ξn for all, n. Therefore the proof of Theorem 1.1 is complete. 4. Proof of the improved Sobolev embeddings Here we include the arguments for the generalized Keraani s and Killip-Visan s improved Sobolev embeddings, which can be used to derive Theorem 1.1 as well. Firstly the generalization of (1.) is as follows: for any 1 < r <, s and 1 r + s d = 1, f r Ds f 1 s/d D s f s/d. (4.1) Ḃ, To prove it, we will closely follow the approach in [1] by Bahouri and Gérard. Proof of (4.1). For every A >, we define f <A, f >A via ˆf <A (ξ) := 1 ξ A ˆf(ξ), ˆf>A (ξ) = 1 ξ >A ˆf(ξ). From the Riemann-ebesgue lemma, f <A ˆf <A 1 ξ, k K 1 k ξ k+1 ˆf 1 ξ, where K is the largest integer such that K A. Then by the Cauchy-Schwarz ineuality, Since d s > and K A, 1 k ξ k+1 ˆf 1 ξ k( d s) D s f Ḃ,. for some C >. We write and set Thus we obtain f <A CA d s D s f Ḃ, f r λ r 1 { R d : f > λ} dλ r ( λ ) 1 A(λ) := C D s f Ḃ, f <A λ. d/ s.

11 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 11 On the other hand, the Chebyshev ineuality gives Hence we have { f > λ} {f >A(λ) > λ } ˆf >A(λ) λ. f r λ r 3 r D s f r Ḃ, ξ >A(λ) ˆf(ξ) dξdλ C D s ˆf(ξ) f Ḃ ξ d/ s, D s f Ds f r. Ḃ, Therefore the proof of (4.1) is complete. ξ s ˆf(ξ) dξ λ r 3 dλdξ Net we prove the generalized version of (1.3): For any 1 < r <, s and 1 r + s d = 1, f r Ds f 1 s/d sup f k s/d, (4.) r where f k is defined as above. For the proof, we will closely follow the approach in [1]. We first recall the ittlewood-paley suare function estimate [16, p.67]. emma 4.1. et 1 < p <. Then for any Schwartz function f, f p ( f k ) 1/ p, where f k is defined as in the introduction. Proof of (4.). By emma 4.1, we see that ( f r f r M ) r/ d. (4.3) M Z When r 4, we have d 4s. Then the Hölder ineuality and the Bernstein ineuality yield, ( f r f r M ) r/4( f N ) r/4 d M N M Z M N f M ( ( sup sup N Z f M r/ f N r/ d d d 4s f k r ) r f k r ) r f M r/ 1 d s f N r/ 1 d s f N M N f M d d 4s f N N s M s D s f M D s f N. M N

12 1 S. SHAO EJDE-9/3 Then the Schur s test concludes the proof when r 4. On the other hand, when r > 4, we let r = [r/], the largest integer which is less than r/. Still by (4.3), the Hölder ineuality and the Bernstein ineuality, we have f r r ( )( ) r 1( f M1 f M f M ) r/ r d M 1 M M f M1 f M1 f M f M3 f Mr 1 f M r r d M 1 M M r 1 M M 1 M M r 1 M f M1 r f M1 f M r f M3... f r M r 1 r ( ) r f k r sup f M r r f r M r/ M 1 M M d/ s 1 M s d/ D s f M1 D s f M. Since d/ s >, the Schur s test again concludes the proof. Hence the proof of (4.) is complete. Acknowledgments. The author is grateful to his advisor Terence Tao for many helpful discussions, to Professor Changing Miao for his comments on the earlier version of this paper, and to the anonymous referees for their valuable comments. References [1] H. Bahouri and P. Gérard. High freuency approimation of solutions to critical nonlinear wave euations. Amer. J. Math., 11(1): , [] P. Bégout and A. Vargas. Mass concentration phenomena for the -critical nonlinear Schrödinger euation. Trans. Amer. Math. Soc., 359(11):557 58, 7. [3] J. Bennett, N. Bez, A. Carbery, and D. Hundertmark. Heat-flow monotonicity of strichartz norms. arxiv: [4] J. Bourgain. Refinements of Strichartz ineuality and applications to D-NS with critical nonlinearity. Internat. Math. Res. Notices, (5):53 83, [5] R. Carles and S. Keraani. On the role of uadratic oscillations in nonlinear Schrödinger euations. II. The -critical case. Trans. Amer. Math. Soc., 359(1):33 6 (electronic), 7. [6] E. Carneiro. A sharp ineuality for the strichartz norm. arxiv: [7] D. Foschi. Maimizers for the Strichartz ineuality. J. Eur. Math. Soc. (JEMS), 9(4): , 7. [8] J. Ginibre and G. Velo. Smoothing properties and retarded estimates for some dispersive evolution euations. Comm. Math. Phys., 144(1): , 199. [9] D. Hundertmark and V. Zharnitsky. On sharp Strichartz ineualities in low dimensions. Int. Math. Res. Not., pages Art. ID 348, 18, 6. [1] M. Keel and T. Tao. Endpoint Strichartz estimates. Amer. J. Math., 1(5):955 98, [11] S. Keraani. On the defect of compactness for the Strichartz estimates of the Schrödinger euations. J. Differential Euations, 175():353 39, 1. [1] R. Killip and M. Visan. Nonlinear schrödinger euations at critical regularity. ecture notes for the summer school of Clay Mathematics Institute, 8. [13] M. Kunze. On the eistence of a maimizer for the Strichartz ineuality. Comm. Math. Phys., 43(1):137 16, 3. [14] F. Merle and. Vega. Compactness at blow-up time for solutions of the critical nonlinear Schrödinger euation in D. Internat. Math. Res. Notices, (8):399 45, [15] S. J. Montgomery-Smith. Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave euations. Duke Math. J., 91():393 48, 1998.

13 EJDE-9/3 MAXIMIZERS FOR THE STRICHARTZ INEQUAITIES 13 [16] E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [17] R. Strichartz. Restrictions of Fourier transforms to uadratic surfaces and decay of solutions of wave euations. Duke Math. J., 44(3):75 714, [18] T. Tao. Math 47a, fourier analysis. tao/47a.1.6f/. [19] T. Tao. A countereample to an endpoint bilinear Strichartz ineuality. Electron. J. Differential Euations, pages No. 151, 6 pp. (electronic), 6. [] T. Tao. Nonlinear dispersive euations, volume 16 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 6. ocal and global analysis. [1] P. Tomas. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 81: , Shuanglin Shao Department of Mathematics, University of California, CA 995, USA Current address: Institute for Advanced Study, Princeton, NJ 854, USA address: slshao@math.ias.edu