A note on bilinear wave-Schrödinger interactions
aa r X i v : . [ m a t h . C A ] M a y A NOTE ON BILINEAR WAVE-SCHR ¨ODINGER INTERACTIONS
TIMOTHY CANDY
Abstract.
We consider bilinear restriction estimates for wave-Schr¨odinger interactions and provided a sharpcondition to ensure that the product belongs to L qt L rx in the full bilinear range q + d +1 r < d +1, 1 q, r U type spaces that roughly implies that if an estimate holds for homogeneous solutions, thenit also holds in U . This transference argument can be used to obtain bilinear and multilinear estimates in U from the corresponding bounds for homogeneous solutions. Let u = e it |∇| f be a free wave, and let v = e it ∆ g be a homogeneous solution to the Schr¨odinger equation.Our goal is to understand for which 1 q, r ∞ we have the bilinear estimate k uv k L qt L rx ( R d ) . k f k L ( R d ) k g k L ( R d ) . (1)As a first step in this direction, assuming for instance that we have the support condition supp b f , supp b g ⊂{| ξ | ≈ } , then for any q + d − r d − with ( q , r , d ) = (2 , ∞ , q + dr d with ( q , r , d ) =(2 , ∞ ,
2) we have the linear Strichartz estimates k u k L q t L r x ( R d ) . k f k L ( R d ) , k v k L q t L r x ( R d ) . k g k L ( R d ) . Consequently an application of H¨older’s inequality and a short computation shows that the bilinear estimate(1) holds provided that2 q + dr d, q + d − r d − d , and ( q, d ) = (cid:16) , (cid:17) , (1 , . (2)The first condition in (2) is stronger in the region q > u ∈ L ∞ t L x and usingthe Strichartz estimate for v . Note that this explains the Schr¨odinger scaling of the first condition in (2).The second condition in (2) dominates in the region 1 q
2, where we are forced to use the Strichartzestimates on both u and v .A natural question now arises, is it possible to improve on the conditions (2)? This question is particularlyrelevant in applications to nonlinear PDE, where bilinear estimates such as (2) with q, r as small as possible,are extremely useful in controlling nonlinear interactions. Note that the wave-Schr¨odinger interactionsoccur naturally in important models, see for instance the Zakharov system [17]. In the case of wave-waveinteractions, it is possible to improve significantly on the range given by simply applying H¨older’s inequalityand the Strichartz estimate for the wave equation provided an additional transversality assumption is made. Theorem 1 (Bilinear restriction for wave [16, 14]) . Let d > and q, r with q + d +1 r < d + 1 . If f, g ∈ L ( R d ) and ω, ω ′ ∈ S d − with ∡ ( ω, ω ′ ) ≈ and supp b f ⊂ (cid:8) ξ ∈ R d (cid:12)(cid:12) | ξ | ≈ , ∡ ( ξ, ω ) ≪ (cid:9) , supp b g ⊂ (cid:8) ξ ∈ R d (cid:12)(cid:12) | ξ | ≈ , ∡ ( ξ, ω ′ ) ≪ (cid:9) (3) then k e it |∇| f e it |∇| g k L qt L rx ( R d ) . k f k L ( R d ) k g k L ( R d ) . Financial support by the Marsden Fund Council grant 19-UOO-142, and the German Research Foundation (DFG) throughthe CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their appli-cations” is acknowledged. Here ∡ ( x, y ) = (1 − x · y | x || y | ) is the angle between x, y ∈ R d \ { } . he first result beyond the linear Strichartz theory was obtained in [5]. The endpoint and extensionto more general frequency interactions is also known [13, 14, 15]. The range for ( q, r ) is sharp, and wasoriginally conjectured by Klainerman-Machedon. Theorem 1 is closely related to the restriction conjecturefor the cone, as the free wave e it |∇| f is essentially the extension operator for the cone. In particular, bilinearestimates of the form (1) were originally used to obtain restriction estimates for the cone, see for instance[12].Theorem 1 is truly a bilinear estimate as it relies crucially on the support assumption (3). This assumptionimplies that the two subsets of the cone, supp F [ e it |∇| f ] ⊂ R d and supp F [ e it |∇| g ] ⊂ R d , are transverse,where F denotes the space-time Fourier transform. Since the waves e it |∇| f and e it |∇| g propagate in thenormal directions to these surfaces, the two waves can only interact strongly for short times. Thus we shouldexpect the product e it |∇| f e it |∇| g to decay faster than say ( e it |∇| f ) .If we apply the above discussion to the bilinear estimate (1), since the normal direction to the cone is(1 , − ξ | ξ | ), and the normal direction to the paraboloid is (1 , ξ ), we should expect to improve on the range(2) obtained via the linear Strichartz estimates, by imposing a transversality condition of the form (cid:12)(cid:12)(cid:12) ξ | ξ | + 2 η (cid:12)(cid:12)(cid:12) & ξ ∈ supp b f and η ∈ b g (here b f denotes the spatial Fourier transform). Unfortunately, the simpletransversality condition (4) does not suffice due to the lack of curvature of the cone along the surface ofintersectionΣ wave ( a, z ) = (cid:8) ( τ, ξ ) ∈ supp F [ e it |∇| f ] (cid:12)(cid:12) ( a, z ) − ( τ, ξ ) ∈ supp F [ e it ∆ g ] (cid:9) , ( a, z ) ∈ R d . In fact it is well known that for certain surfaces, transversality alone is not sufficient to obtain the full bilinearrange, see for instance [11] for the example of the hyperbolic paraboloid, and the related discussion in [3, 6].However, imposing a stronger support condition gives the following.
Theorem 2 (Wave-Schr¨odinger bilinear restriction [6]) . Let d > , q, r , and q + d +1 r < d + 1 . Let ξ , η ∈ R d such that (cid:12)(cid:12)(cid:12)(cid:16) ξ | ξ | + 2 η (cid:17) · ξ | ξ | (cid:12)(cid:12)(cid:12) & (cid:12)(cid:12)(cid:12) ξ | ξ | + 2 η (cid:12)(cid:12)(cid:12) (5) and define λ = | η | , and α = | ξ | ξ | + 2 η | . If supp b f ⊂ (cid:8) | ξ | ≈ λ, ∡ ( ξ, ξ ) ≪ min { , α } (cid:9) , supp b g ⊂ {| ξ − η | ≪ α } then we have (cid:13)(cid:13) e it ∆ f e it |∇| g (cid:13)(cid:13) L qt L rx ( R d ) . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − k f k L x k g k L x . Theorem 2 is a consequence of a bilinear restriction estimate for general phases obtained in [6]. Thespecial case q = r and α = λ = 1 could also be deduced from [3]. As the precise conditions in [6] arecomplicated, the derivation is slightly nontrival and we give the details below in Section 1. The dependenceon the parameters α and λ is sharp, and this is particularly useful in applications to nonlinear PDE where α and λ roughly correspond to a derivative loss/gain. Clearly, applying Sobolev embedding and interpolatingwith the trivial case q = ∞ , r = 1 can extend the range to q, r > q + d +1 r < d + 1. However thedependence on α and λ would no longer be sharp (i.e. losses may occur).The condition (5) is necessary to obtain the full bilinear range q + d +1 r d + 1. Theorem 3 (Transverse counter example) . Suppose that the estimate (1) holds for all f, g ∈ L ( R d ) with supp b f ⊂ {| ξ − e | ≪ } , supp b g ⊂ {| ξ + e + e | ≪ } . Then q + d − r + 12 r d. (6) Here e j ∈ R d , j = 1 , . . . , n denote the standard basis vectors. r q ( , ) Figure 1.
The range of 1 q, r d = 3. The line corresponds to the sharp bilinearline q + d +1 r = d + 1 given by Theorem 2. If (4) holds but (5) fails, then Theorem 3 statesthat the bilinear estimate (1) can only hold to the left of the dotted line.Note that if we let ξ = e and η = − e − e , then | ξ | ξ | + 2 η | = 1 but ( ξ | ξ | + 2 η ) · ξ | ξ | = 0. Inother words the transversality condition (4) holds, but the stronger condition (5) fails. The range (6) isstronger than the bilinear range in Theorem 2 when q is close to 1 and d
5, see figure 1. If we drop thetransversality condition completely, then a similar counter example can be used to prove the following.
Theorem 4 (Non-transverse counter example) . Suppose that the estimate (1) holds for all f, g ∈ L ( R d ) with supp b f , supp b g ⊂ {| ξ | ≈ } . Then q + d − r d − , q d + 14 . (7)We give the proof of Theorem 3 and Theorem 4 in Section 2 below. In the positive direction, if (4) holds,but (5) fails, a naive adaption of the proof of Theorem 2 should give the region q + dr < d . The loss ofdimension corresponds to the lack of curvature in the radial direction (i.e. the cone only has d − q + dr < d and thecounter example given by Theorem 3. Similarly, even in the “linear” case when the transversality condition(4) is dropped, there is a gap between the counter example in Theorem 4 and the linear range given viaStrichartz estimates (2). It is an interesting open question to determine the precise range of ( q, r ) once thegeneral condition (5) is dropped. In particular, it is not clear to the author what the optimal range for ( q, r )should be. Presumably the counter examples used in the proof of Theorem 3 and Theorem 4 can be improved.In applications to nonlinear PDE, typically the homogeneous estimate in Theorem 2 is not sufficient, andit is more useful to have a version in suitable function spaces. One option is to work with X s,b type spaces.However, recently bilinear restriction estimates in the U p type spaces have proven useful, see for instance [7]and the discussion within. In the following we wish to give a general argument which can allow multilinearestimates for homogeneous solutions, to be upgraded to estimates in the adapted function spaces U . Theunderlying idea is straight forward. The first step is use the classical theorem of Marcinkiewicz-Zygmundthat given a bound for a linear operator, a standard randomisation argument via Khintchine’s inequalityimplies that a vector valued operator bound also holds. The second step is use the observation that a vectorvalued estimate immediately implies a U bound, see for instance [6, Section 1.2] or [8, Remark 5.2]. As anexample, we extend Theorem 2, and the multilinear restriction theorem [4] to U .We start with the definition of U . A function φ ∈ L ∞ t L x is an atom if we can write φ ( t ) = P I I ( t ) g I ,with the intervals I ⊂ R forming a partition of R , and the g I : R d → C satisfying the bound (cid:16) X I k g I k L x (cid:17) . he atomic space U is then defined as U = n X j c j φ j | φ j an atom and ( c j ) ∈ ℓ o with the induced norm k u k U = inf u = P j c j φ j X j | c j | where the inf is over all representations of u in terms of atoms. These spaces were introduced in unpublishedwork of Tataru, and studied in detail in [10, 9]. To obtain the adapted function spaces U |∇| and U adaptedto the wave and Schr¨odinger flows respectively, we define U |∇| = { u : R d → C | e − it |∇| u ∈ U } , U = { v : R d → C | e − it ∆ v ∈ U } . Note that since R ( t ) f ∈ U , we clearly have e it |∇| f ∈ U |∇| and e it ∆ f ∈ U . Thus the adapted functionspaces contain all homogeneous solutions. Running the argument sketched above implies the following U version of Theorem 2. Theorem 5 (Wave-Schr¨odinger bilinear restriction in U ) . Let d > , q, r , and q + d +1 r < d + 1 .Let ξ , η ∈ R d such that (cid:12)(cid:12)(cid:12)(cid:16) ξ | ξ | + 2 η (cid:17) · ξ | ξ | (cid:12)(cid:12)(cid:12) & (cid:12)(cid:12)(cid:12) ξ | ξ | + 2 η (cid:12)(cid:12)(cid:12) (8) and define λ = | η | , and α = | ξ | ξ | + 2 η | . If supp b u ⊂ (cid:8) | ξ | ≈ λ, ∡ ( ξ, ξ ) ≪ min { , α } (cid:9) , supp b v ⊂ {| ξ − η | ≪ α } then we have k uv k L qt L rx ( R d ) . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − k u k U |∇| k v k U . Proof.
Let u = P I ∈I e it |∇| f I be a U |∇| atom, and let v = e it ∆ g be a homogeneous solution to theSchr¨odinger equation. Assume that the support conditions (8) hold. Let ( ǫ I ) I ∈I be a family of independent,identically distributed random variables with ǫ I = 1 with probability , and ǫ I = − .The since the intervals I are disjoint, we have via Khintchine’s inequality | u | (cid:16) X I | e it |∇| f I | (cid:17) ≈ E h(cid:12)(cid:12)(cid:12) X I ǫ I e it |∇| f I (cid:12)(cid:12)(cid:12)i . Therefore, since q, r >
1, applying Theorem 2 gives k uv k L qt L rx . (cid:13)(cid:13)(cid:13) E h(cid:12)(cid:12)(cid:12) X I ǫ I e it |∇| f I (cid:12)(cid:12)(cid:12)i v (cid:13)(cid:13)(cid:13) L qt L rx . E h(cid:13)(cid:13)(cid:13) e it |∇| (cid:16) X I ǫ I f I (cid:17) v (cid:13)(cid:13)(cid:13) L qt L rx i . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − E h(cid:13)(cid:13)(cid:13) X I ǫ I f I (cid:13)(cid:13)(cid:13) L x i k g k L x . We now observe that H¨older’s inequality, together with another application of Khintchine’s inequality, impliesthat E h(cid:13)(cid:13)(cid:13) X I ǫ I f I (cid:13)(cid:13)(cid:13) L x i (cid:16) E h(cid:13)(cid:13)(cid:13) X I ǫ I f I (cid:13)(cid:13)(cid:13) L x i(cid:17) = (cid:16) X I k f I k L (cid:17) and consequently, applying the definition of the U |∇| norm, we obtain k uv k L qt L rx . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − k u k U |∇| k g k L x . (9)To replace the homogeneous solution v with a general U function follows by essentially repeating the aboveargument. In slightly more detail, suppose that v = P J ∈J e it ∆ g J is a U atom, and let ( ǫ J ) J ∈J be a family f i.i.d. random variables with ǫ J = ± k uv k L qt L rx . (cid:13)(cid:13)(cid:13) u E h(cid:12)(cid:12)(cid:12) X J ǫ J e it ∆ g J (cid:12)(cid:12)(cid:12)i(cid:13)(cid:13)(cid:13) L qt L rx . E h(cid:13)(cid:13)(cid:13) u X J e it ∆ ǫ J g J (cid:13)(cid:13)(cid:13) L qt L rx i . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − k u k U |∇| E h(cid:13)(cid:13)(cid:13) X J ǫ J g J (cid:13)(cid:13)(cid:13) L i . (min { α, λ, αλ } ) d +1 − d +1 r − q α r − λ q − k u k U |∇| (cid:16) X J k g J k L (cid:17) . Applying the definition the U norm, the required bound follows. (cid:3) Strictly speaking the above theorem can also be obtain via the vector valued version of Theorem 2 from[6], see for instance [6, Section 1.2]. However the above alternative argument is more direct, and has thedistinct advantage that it can be applied in more general situations. As an example, consider the followingspecial case of the multilinear restriction theorem [4].
Theorem 6 (Multilinear restriction for Schr¨odinger [4] ) . Let d > and ǫ > . Then for any R > andany f j ∈ L ( R d ) , j = 1 , . . . , d with supp b f j ⊂ {| ξ − e j | ≪ } we have (cid:13)(cid:13)(cid:13) Π j e it ∆ f j (cid:13)(cid:13)(cid:13) L d − t,x ( {| t | + | x | Let B R = {| t | + | x | < R } . We proceed as in the proof of Theorem 5. Thus suppose that u = P I e it ∆ f I is U atom, and let u j = e it ∆ f j for j = 2 , . . . , d . Let ǫ I be a family of i.i.d. random variables with ǫ I = ± | u | (cid:16) X I | e it ∆ f I | (cid:17) ≈ (cid:16) E h(cid:12)(cid:12)(cid:12) X I ǫ I e it |∇| f I (cid:12)(cid:12)(cid:12) d − i(cid:17) d − and hence Theorem 6 together with H¨older’s inequality gives (cid:13)(cid:13)(cid:13) u Π dj =2 u j (cid:13)(cid:13)(cid:13) L d − t,x ( B R ) . (cid:13)(cid:13)(cid:13)(cid:16) E h(cid:12)(cid:12)(cid:12) X I ǫ I e it ∆ f I (cid:12)(cid:12)(cid:12) d − i(cid:17) d − Π dj =2 u j (cid:13)(cid:13)(cid:13) L d − t,x ( B R ) . (cid:16) E h(cid:13)(cid:13)(cid:13) X I ǫ I e it ∆ f I Π dj =2 u j (cid:13)(cid:13)(cid:13) d − L d − t,x ( B R ) i(cid:17) d − . R ǫ (cid:16) E h(cid:13)(cid:13)(cid:13) X I ǫ I f I (cid:13)(cid:13)(cid:13) d − L i(cid:17) d − Π dj =2 k f j k L . R ǫ (cid:16) E h(cid:13)(cid:13)(cid:13) X I ǫ I f I (cid:13)(cid:13)(cid:13) L i(cid:17) Π dj =2 k f j k L ≈ R ǫ (cid:16) X I k f I k L (cid:17) Π dj =2 k f j k L . Applying the definition of the U norm, we conclude that (cid:13)(cid:13)(cid:13) u Π dj =2 u j (cid:13)(cid:13)(cid:13) L d − t,x ( B R ) . R ǫ k u k U Π dj =2 k f j k L . (10) s in the proof of Theorem 5, repeating this argument with Theorem 6 replaced with (10) and u replacedwith u gives (cid:13)(cid:13)(cid:13) u u Π dj =3 u j (cid:13)(cid:13)(cid:13) L d − t,x ( B R ) . R ǫ k u k U k u k U Π dj =3 k f j k L . The required bound follows by continuing in this manner. (cid:3) We have not attempted to write down the most general transference type argument that can be deducedfrom the above arguments. However the underlying idea is simple; if a estimate holds for free solutions,then via randomisation it should hold in the vector valued case, and consequently it will also hold in U . Ofcourse proving U p bounds, with p = 2 is substantially more challenging. Acknowledgements. The author would like to thank Sebastian Herr and Kenji Nakanishi for a number ofhelpful discussions, as well as the University of Bielefeld and MATRIX for their kind hospitality while partof this work was conducted. 1. Proof of Theorem 2 It suffices to check the conditions in [6]. Suppose that ξ , η ∈ R d such that (5) holds, and define λ = | η | ,and α = | ξ | ξ | + 2 η | . LetΛ = (cid:8) | ξ | ≈ λ, ∡ ( ξ, ξ ) ≪ min { , α } (cid:9) , Λ = {| ξ − η | ≪ α } and Φ ( ξ ) = | ξ | , Φ ( ξ ) = −| ξ | , H = λ − , H = 1 . In view of [6, Lemma 2.1 and Theorem 1.2], for { j, k } = { , } and ξ ∈ Λ j , η ∈ Λ k , it suffices to check thefollowing conditions:(i) for all v ∈ R d we have v · ( ∇ Φ j ( ξ ) − ∇ Φ k ( η )) = 0 = ⇒ (cid:12)(cid:12) ∇ Φ j ( ξ ) v ∧ (cid:0) ∇ Φ j ( ξ ) − ∇ Φ k ( η ) (cid:1)(cid:12)(cid:12) & H j α | v | , (ii) for ξ ′ ∈ Λ j and η ′ ∈ Λ k we have |∇ Φ j ( ξ ) − ∇ Φ j ( ξ ′ ) | + |∇ Φ k ( η ) − ∇ Φ k ( η ′ ) | ≪ α, (iii) the Hessian’s satisfy |∇ Φ j ( ξ ) − ∇ Φ j ( ξ ′ ) − ∇ Φ j ( ξ )( ξ − ξ ′ ) | ≪ H j | ξ − ξ ′ | , (iv) for 2 < m d we have the derivative bounds k∇ m Φ j k L ∞ (Λ j ) (min { α, λ, αλ } ) m − . H j , H j min { α, λ, αλ } . α. (v) we have the surface measure conditionsup ( a,h ) ∈ R d σ d − (cid:0)(cid:8) ξ ∈ Λ ∩ ( h − Λ ) (cid:12)(cid:12) Φ ( ξ ) + Φ ( h − ξ ) = a } (cid:1) . (cid:0) min { α, λ, αλ } (cid:1) d − where σ d − is the induced Lebesgue surface measure.To check the first property (i), by unpacking the definition, our goal is to show that for any ξ ∈ Λ and η ∈ Λ we have z · ( ω + 2 η ) = 0 = ⇒ (cid:12)(cid:12) ( z − ( ω · z ) ω ) ∧ ( ω + 2 η ) (cid:12)(cid:12) & | z || ω + 2 η | , where ω = ξ | ξ | . In view of the definition of the sets Λ j we have | ( ω + 2 η ) · ω | & | ω + 2 η | and hence as ( z · ω )( ω + 2 η ) · ω = 2 z · ( η − ( η · ω ) ω ) we get | z · ω | | z · ( η − ( η · ω ) ω ) || ( ω + 2 η ) · ω | . | z − ( ω · z ) ω | . Therefore (cid:12)(cid:12)(cid:0) z − ( ω · z ) ω (cid:1) ∧ ( ω + 2 η ) (cid:12)(cid:12) > (cid:12)(cid:12) z − ( ω · z ) ω (cid:12)(cid:12) | ( ω + 2 η ) · ω | & | z || ω + 2 ω | as required. he properties (ii), ... , (iv) follow by direct computation. Finally, to check the surface measure condition(v), we note that the vector N = ξ | ξ | + 2 η is essentially normal to the surface. On the other hand, from(5), N is roughly pointing in the direction η | η | . Hence the surface measure can be bounded by projectingonto the plane orthogonal to η | η | . Since this projection is contained in a ball of radius . min { α, λ, αλ } , thebound follows. 2. Counter Examples We first observe that by a randomisation argument, if the estimate (1) holds for all f, g ∈ L withsupp b f ⊂ Λ ⊂ R n and supp b g ⊂ Λ , then in fact we also have the vector valued version (cid:13)(cid:13)(cid:13)(cid:16) X j | e it |∇| f j | (cid:17) (cid:16) X k | e it ∆ g k | (cid:17) (cid:13)(cid:13)(cid:13) L qt L rx . (cid:16) X j k f j k L (cid:17) (cid:16) X k k g k k L x (cid:17) (11)for all supp b f j ⊂ Λ , supp b g k ⊂ Λ . This follows by noting that if ǫ j is an i.i.d. family of random variableswith ǫ j = ± with equal probability, then as in the proof of Theorem 5, we have via Khintchine’s inequalityand (1) (cid:13)(cid:13)(cid:13)(cid:16) X j | e it |∇| f j | (cid:17) e it ∆ g (cid:13)(cid:13)(cid:13) L qt L rx ≈ (cid:13)(cid:13)(cid:13) E h(cid:12)(cid:12)(cid:12) X j ǫ j e it |∇| f j (cid:12)(cid:12)(cid:12)i e it ∆ g (cid:13)(cid:13)(cid:13) L qt L rx . E h(cid:13)(cid:13)(cid:13) X j ǫ j e it |∇| f j e it ∆ g (cid:13)(cid:13)(cid:13) L qt L rx i . E h(cid:13)(cid:13)(cid:13) X j ǫ j e it |∇| f j (cid:13)(cid:13)(cid:13) L x i k g k L x . E h(cid:13)(cid:13)(cid:13) X j ǫ j e it |∇| f j (cid:13)(cid:13)(cid:13) L x i k g k L x ≈ (cid:16) X j k f j k L (cid:17) k g k L . Repeating this argument for the Schr¨odinger component then gives (11). Consequently, we see that thescalar version (1) holds, if and only if the vector valued version (11) holds. Thus to prove Theorem 3 andTheorem 4, it suffices to obtain vector valued counter examples.2.1. Proof of Theorem 3. Let N > b f , b g ∈ C ∞ withsupp b f ⊂ {| ξ − | ≪ , | ξ ′ | ≪ N − } , supp b g ⊂ {| ξ − e − e | ≪ N − } and k f k L ≈ N d − , k g k L ≈ N d . A short computation using integration by parts shows that we can choose f, g such that | u ( t, x ) | = | e it |∇| f ( x ) | > | t | N , | x + t | , | x ′ | N and | v ( t, x ) | = | e it ∆ g ( x ) | > | t | N , | x + t | N , | x + t | N , | x ′′ | N where we write x = ( x , x ′ ) = ( x , x , x ′′ ). In other words the free wave u is > N × × N d − oriented in the (1 , − e ) direction, with short direction e , while the free Schr¨odinger wave v is > N × N d oriented in the (1 , − e − e ) direction. Define the setΩ = {| t | N − , | x + t | N , | x ′ | N } . The support properties of u , implies that for any ( t, x ) ∈ Ω we have U ( t, x ) = (cid:16) X j ∈ Z | j | N | u ( t, x + je ) | (cid:17) & . imilarly, translating the free Schr¨odinger wave in both space and time gives for any ( t, x ) ∈ Ω V ( t, x ) = (cid:16) X ( j ,...,j d ) ∈ Z d − | j | ,..., | j d | . N X k ∈ Z | k | N (cid:12)(cid:12) v (cid:0) t + N k, x + N ( j e + · · · + j d e d ) (cid:1)(cid:12)(cid:12) (cid:17) & . Since the wave and Schr¨odinger equations are translation invariant, the bound (11) implies that N q N d − r + r . k Ω k L qt L rx . k U V k L qt L rx . (cid:16) X | j | N k f k L (cid:17) (cid:16) X | j | ,..., | j d | . N X | k | N k g k L (cid:17) . N d − + × N d + . Letting N → ∞ , we see that this is only possible if2 q + d − r + 12 r d. Proof of Theorem 4. Let 1 M N and b f , b g ∈ C ∞ withsupp b f ⊂ {| ξ − | ≪ , | ξ ′ | ≪ N − } , supp b g ⊂ {| ξ − e | ≪ M − } and k f k L ≈ N d − , k g k L ≈ M d . A short computation using integration by parts shows that we can choose f, g such that | u ( t, x ) | = | e it |∇| f ( x ) | > | t | N , | x + t | , | x ′ | N and | v ( t, x ) | = | e it ∆ g ( x ) | > | t | M , | x + t | M, | x ′ | M. In other words the free wave u is > N × × N d − oriented in the (1 , − e ) direction,with short direction e , while the free Schr¨odinger wave v is > M × M d orientedin the (1 , − e ) direction. Similar to the proof of Theorem (3), we consider a number of temporal translatedSchr¨odinger waves covering the setΩ = {| t | N − , | x + t | , | x ′ | M } . More precisely, we have for all ( t, x ) ∈ Ω V ( t, x ) = (cid:16) X j ∈ Z | j | N M | v ( t + jM , x ) | (cid:17) & . Since we clearly have | u | > N q M d − r . k Ω k L qt L rx . k uV k L qt L rx . k f k L (cid:16) X | j | N M k g k L (cid:17) . N d − × N M d − . Rearranging, we see that we must have N q − d +12 M d − r − d − . . Letting M = 1 and N → ∞ gives the restriction1 q d + 14 . 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Candy) Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, NewZealand E-mail address : [email protected]@maths.otago.ac.nz