Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation 1

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1 Endpoin Sricharz esimaes and global soluions for he nonlinear Dirac equaion 1 Shuji Machihara, Makoo Nakamura, Kenji Nakanishi, and Tohru Ozawa Absrac. We prove endpoin Sricharz esimaes for he Klein-Gordon and wave equaions in mixed norms on he polar coordinaes in hree spaial dimensions. As an applicaion, global wellposedness of he nonlinear Dirac equaion is shown for small daa in he energy class wih some regulariy assumpion for he angular variable. 1. Inroducion Le us consider he Klein Gordon equaion in hree spaial dimensions : 2 u u + m 2 u =, (1.1) where u : R 1+3 C and m is he mass consan. The endpoin Sricharz esimae u L 2 L x E(u)1/2 (1.2) is known o be false in general [4, 16], where E(u) is he conserved energy defined by E(u) = E(u; ) := u 2 + u 2 + m 2 u 2 dx (= E(u; )). (1.3) R 3 Moreover, Mongomery-Smih [8] has shown ha even if we replace he L norm in (1.2) by BMO, he esimae does no hold. On he oher hand, Klainerman and Machedon [4] proved ha he esimae (1.2) holds if u is radial and m =. Then a naural quesion arises: To wha exen does he endpoin esimae depend on he radial symmery? Our heorem below answers ha i is very lile. We denoe he polar coordinaes by x = r, r = x, S 2. Theorem 1.1. (I) For any m, any 1 p < and any finie energy soluion u of (1.1), we have u L 2 L r L p C(pE(u))1/2, (1.4) where C is a posiive absolue consan. (II) The power p 1/2 in (1.4) is opimal in he following sense: For any m and 1 2 Mahemaics Subjec Classificaion. 35L5, 42B25. The hird auhor (K. N.) was parly suppored by JSPS Posdocoral Fellowships for Research Abroad (21-23). 1

2 2 any ε >, here exiss a finie energy soluion u of (1.1) saisfying lim u L p 2 L r L p/p1/2 ε =. (1.5) (III) For any finie energy soluion u of he wave equaion (1.1) wih m =, we have u L 2 L r H 3/4 where C is a posiive absolue consan. CE(u) 1/2, (1.6) The firs saemen implies ha if he iniial daa has sligh addiional regulariy for roaion H ε, ε >, hen he endpoin Sricharz L2 L x is recovered by he Sobolev embedding H ε,p L, p > 2/ε. Noice ha he opimal power p1/2 in (1.4) is he same as in he criical Sobolev embedding H 1 p1/2 L p. We do no know if H 3/4 in (1.6) can be improved o higher Sobolev norm H s. However we can show an upper bound s 5/6 (Theorem 5.1), and so he L p esimae (I) for p > 12 can no be recovered from H s and he Sobolev embedding. We remark ha as for he Schrödinger equaion in wo spaial dimensions, Tao [15] proved he following endpoin esimae: u L 2 L r H s u() L 2 x, (1.7) for some small s >. In his case we have an upper bound s 1/3 (Theorem 5.1), and so L p esimae for p > 6 can no be obained by he Sobolev embedding. I seems open if we can replace H s by Lp for all p < in he Schrödinger case (1.7). Our primary moivaion for he above endpoin esimaes was applicaion o nonlinear wave equaions. Indeed, he lack of he endpoin esimae causes in some cases serious difficulies o prove wellposedness; he following Cauchy problem for he nonlinear Dirac equaion is a good example. 3 iγ α α u mu = λ(γ u, u)u, (1.8) α= u(, x) = ϕ(x), where ϕ(x) : R 3 C 4 is he given, u(, x) : R 1+3 C 4 is he unknown funcion, m and λ C are given consans, (, 1, 2, 3 ) = (, x ) is he space-ime derivaives, (, ) denoes he inner produc on C 4, and γ α GL(C, 4) (α =, 1, 2, 3) denoe he Dirac marices given by ( I γ = I σ 1 = ( 1 1 ) ( σ, γ k k = σ k ) ( i, σ 2 = i ) ( 1, σ 3 = 1 ), (1.9) ). (1.1) In [7] we proved exisence of global soluion wih small H s daa ϕ H s for s > 1 and m >. Local exisence was proved by Escobedo and Vega in H s, s > 1 [3].

3 Here he value s = 1 is he scaling criical exponen for m =, see he inroducion in [3]. There are similar siuaions for nonlinear wave equaions wih derivaive nonlineariy. Lindblad [5] consruced counerexamples o disprove local wellposedness in H 1 of he following equaion: 2 u u + ( u 1 u)u =, (1.11) while i is easy o prove is local wellposedness in H 1+ε by using non-endpoin Sricharz esimaes (cf. [11]). Lindblad s counerexample of he iniial daa is concenraed in one direcion. Our endpoin esimaes imply ha if he daa had regulariy for he angular variable H ε, ε >, hen he blowup could no occur. Noice ha radial symmery is no preserved for mos equaions including he above examples, and so he endpoin esimae (1.2) for radial soluions is no direcly applicable. Bu our esimaes can be applied wihou any consideraion on special symmeries of given sysems. For he nonlinear Dirac equaion, we have he following global exisence for small H 1 daa wih a sligh regulariy for angular variables. Theorem 1.2. Le m, λ C and s >. Then here exiss δ > such ha if ϕ H 1 (R 3 ) saisfies ϕ H 1 (H s ) := ϕ L 2 r (H s ) + ϕ L 2 r (H s ) < δ (1.12) hen we have a unique global soluion u of (1.8) saisfying u() = ϕ and u C (R; H 1 (H s )) L 2 (R; L ). (1.13) In he case of m =, we may replace he above norm of H 1 (H s ) wih is homogeneous version, namely ϕ L 2 r (H s). We prove Theorem 1.2 by he sandard fixed poin argumens using he above endpoin esimaes ha hold uniformly on any ime inerval. Hence we can easily obain global wellposedness and scaering for small daa, as well as local exisence for large daa by he sandard argumens (see, e.g., [3]). The res of his paper is organized as follows. In Secion 2, we inroduce he noaions and basic esimaes on he fracional Sobolev spaces on he sphere S 2. In Secion 3, we prove our endpoin Sricharz esimaes. In Secion 4, we prove he global wellposedness for he nonlinear Dirac. In Secion 5, we make a number of remarks. Throughou his paper, we ofen use he noaion A B and D E which mean A CB and D/C E CD, respecively, where C is some posiive consan. We denoe x := (1 + x 2 ) 1/2. We idenify any se wih is characerisic funcion. Thus for any se A, A(x) = 1 if x A and A(x) = oherwise. 3

4 4 2. Fracional Sobolev spaces on he sphere In his secion, we recall some basic facs ha we need on he fracional Sobolev spaces on he uni sphere S 2. See [14, 17] for more general informaion. We denoe he polar coordinaes x = r, r = x and S 2. Le denoe he Laplace- Belrami operaor on S 2. For any funcion f(r), we have f(x) = x 2 f(x). (2.1) The Lebesgue and Sobolev spaces on S 2 are defined by he norms ( ) 1/p, f L p = f() p d f H s,p = (1 S 2 ) s/2 f L p. (2.2) Throughou his paper, we will use hese norms in he mixed form: ( 1/p f(x) L p r (X ) = f(r) p X r dr) 2. (2.3) The fracional power of can be wrien explicily by inroducing he spherical harmonics. Le F k ν (x) be a homogeneous polynomial of degree ν saisfying F k ν (x) =, such ha {F k ν ()} ν,k makes a complee orhonormal basis of L 2 (S 2 ). Then any funcion f(r) can be decomposed as f(r) = where a k ν(r) are deermined by f, and (1 ) s/2 f = ν,k ν= N(ν) k=1 a k ν(r)f k ν (), (2.4) (1 + ν(ν + 1)) s/2 a k ν(r)f k ν (), (2.5) where we used F k ν () = ν(ν + 1)F k ν (). In he case p = 2, we may use he orhogonaliy o deduce ha f 2 L 2 r(h s,2 ) ν,k ν s a k ν 2 L 2 r. (2.6) For nonlinear esimaes, we use he equivalen norms defined hrough local coordinaes. Le {(O j, Ψ j )} N j=1 be a sysem of coordinae neighborhoods, and {λ j } be a smooh pariion of uniy subordinae o {O j }. Le {χ j } C (R 2 ) saisfy χ j = 1 on Ψ j (supp λ j ) and supp χ j Ψ j (O j ). Then, for any funcions f : S 2 C and h : (R 2 ) N C, we define Sf : (R 2 ) N C and Rh : S 2 C by (Sf) j (x) := (λ j f)(ψ 1 j (x)), Rh(y) := Then we can define he Sobolev norms by N (χ j h)(ψ j (y)). (2.7) j=1 f H s,p (S 2 ) = Sf (H s,p (R 2 )) N. (2.8)

5 This gives an equivalen norm of H s,p for 1 < p < (see [17]). We do no deal wih he cases p = 1 or in his paper. I is easily seen ha RSf = f and SR is bounded from (H s,p (R 2 )) N ino iself, and so, R is a reracion from (H s,p (R 2 )) N o H s,p (S 2 ) wih a coreracion S. Therefore we have he same embeddings and inerpolaions for H s,p (S 2 ) as on R 2. We may inroduce anoher equivalen norm (S f) j (x) := χ j (x)f(ψ 1 j (x)), S f (H s,p (R 2 )) N Sf (H s,p (R 2 )) N. (2.9) Then he Hölder inequaliy and he Leibniz rule easily ransfers from he Euclidean case as follows. Le s and 1/p = 1/q 1 + 1/r 1 = 1/q 2 + 1/r 2, 1 < p <, q 1, r 2. We have 5 fg H s,p (S 2 ) j (Sf) j (S g) j H s,p (R 2 ) ( (Sf)j H s,q 1(R 2 ) (S g) j L r 1(R 2 ) + (Sf) j L q 2(R 2 ) (S g) j H s,r 2(R 2 )) j f H s,q 1(S 2 ) g L r 1(S 2 ) + f L q 2(S 2 ) g H s,r 2(S 2 ), (2.1) where we used he sandard esimae on poinwise muliplicaion on R 2 on he second line. Finally we check he equivalence of he following norms, (1 ) s/2 f H 1 f H 1 (H s ), (2.11) where he righ hand side was inroduced in (1.12). Noe ha and are no commuaive. Since (2.11) is obvious if we replace H 1 by L 2, i suffices o prove he homogeneous version, i.e., for Ḣ1 x. Since = commues wih, he above equivalence (2.11) reduces o he following one: f L 2 r (H s ) f L 2 r (H s ), (2.12) which is equivalen o he boundedness of he Riesz operaors: / : L 2 r(h s ) L 2 r(h s ) bounded. (2.13) This is easily checked when s is an (even) ineger by compuing he commuaors of x and. Then he remaining case is covered by inerpolaion. 3. Endpoin Sricharz esimaes In his secion, we consider he endpoin Sricharz esimae. Alhough one migh expec ha he esimaes in Theorem 1.1 were easier for he Klein-Gordon (m > ) because of he faser decay ( 3/2 ), he esimae for he Klein-Gordon acually implies ha for he wave. In fac, suppose ha we have an esimae of he form: u L 2 L r X CE(u)1/2 (3.1)

6 6 for a fixed m = m >. Then we obain he same esimae for all m > jus by rescaling u u(m/m, xm/m ). Taking he limi m, we obain he same esimae for m = as well. On he oher hand, i is no rivial o exend such an esimae from m = o m >. The res of his secion is devoed o he proof Sharpness of p. Firs we prove he opimaliy of p in (1.5). Le m =. We consider he funcion given by g(x) = χ A (x)g (x), g (x) = x 2 (1 + log x ) α, (3.2) where α (1/2, 1), χ A is he characerisic funcion of A = K \B, K is a sufficienly large cube, and B is a ball angen o he boundary of K from is inside a he origin: K := [, 1] [ 5, 5] [ 5, 5], B := {x R 3 x e 1 < 1}, (3.3) where e 1 = (1,, ). This funcion is a sligh modificaion of ha given by T. Tao [16] as a counerexample for he endpoin Sricharz esimae u L 2 L x g L 2 for free soluions wih daa u() = and u() = g. In fac, by a simple calculaion we know ha he above funcion g saisfies g L 2 bu he free soluion, which is given by u(, x) = g(x + )d, (3.4) 4π S 2 saisfies u(, e 1 ) = for all 1 < < 2. This funcion also shows sharpness of our L p esimae as we see in he following. Le < 1 1 and x = wih l := e 1 < ε 1. We wan o esimae u(, x) given by (3.4) from below. Firs we consider he resricion on he inegral region of due o he cu-off χ A. Le y = x + and we denoe he region for y by S := {x + S 2 }. I is easily seen ha y S is conained in he cube K when y l. Since he radius of S is greaer han ha of B, i is clear ha R 3 \ B conains a leas one of he hemispheres of S divided by a plane conaining and x. Thus we can esimae, by aking he inegral wih respec o ρ = y, u(, x) 1 Cl g (ρ)ρdρ log l /2 s α ds log l 1 α, (3.5)

7 7 where we changed he inegral variable as s = log ρ. Then we esimae he L p norm for sufficienly large p wih x =, l = e 1 as u(, ) p L p ε log l (1 α)p ldl log ε s (1 α)p e 2s ds p (1 α)p e 2s ds = (p (1 α) e 2 ) p /2, p (3.6) where we changed he inegral variable as s = log l and assumed ha p > log ε. Therefore we have u(, ) L p p1 α (3.7) for any α > 1/2 and large p. This complees he proof of (1.5) for m =. Nex we consider he Klein-Gordon case m >. Fix m >, ε > and suppose ha we have he esimae of he form sup u L 2 L r L p/p1/2 ε CE(u) 1/2. (3.8) p>1 Then by he rescaling argumen a he beginning of his secion, we have he same esimae for m =, which we have jus disproved. Therefore (3.8) is false, which means ha here exiss a finie energy soluion for which he lef hand side is infinie T T argumen. Now we sar o prove he main Sricharz esimaes. Firs of all, we conver hem ino he T T versions. Our desired esimaes can be rewrien as ω 1 m e ±iω m ϕ L 2 L r X ϕ L 2 x, ω m := m 2, (3.9) where X denoes some Banach space (L p for (I) and Hs for (III)). We apply he T T argumen o he operaors T ± := ωm 1 (e iωm ± e iωm ). We have T ± T±u = 2 R ω 2 m {cos(ω m ( s)) ± cos(ω m ( + s))}u(s)ds. (3.1) Hence, by ime reversibiliy, i suffices o prove cos ω m ( s)u(s)ds u L 2 L 2 L r X L 1 rx, (3.11) R ω 2 m where X denoes he L2 dual of X. I is imporan for our laer argumen ha we do no have sin bu cos above. We denoe he operaor in (3.11) by L m () := ωm 2 cos(ω m ) and is kernel funcion by L m (, x) = F 1 ξ 2 m cos ξ m, ξ m := ξ 2 + m 2. (3.12)

8 8 We use he following T T version of he Hardy Lilewood maximal operaor as he key esimae on (, r). In he lemma below, we forge abou he polar coordinaes and so L p r denoes he sandard L p ((, ); dr) wihou weighs. Lemma 3.1. Le g(r) be a nonnegaive nonincreasing inegrable funcion on (, ). Then he following esimae holds 1 ( s ) r l g h(s, l)dsdl g L 1 r l r h L 2 L. (3.13) 1 r where r l = max(r, l). R L 2 L r Proof. The Hardy-Lilewood maximal funcion heorem shows he boundedness of he operaor Mϕ(, r) = 1 ϕ(s)ds : L 2 L 2 L r. (3.14) r So MM is bounded and i is wrien explicily by where MM h(, r) = s <r MM : L 2 L 1 r L 2 L r, (3.15) R 1 rl I( s, r, l)h(s, l)dsdl, (3.16) 2 min(r, l), ( < r l ), I(, r, l) = r + l, ( r l < < r + l),, (r + l < ). Denoe he operaor in (3.13) by M(g, h). Since (3.17) 1 1 I(, r, l) { < < r l}, (3.18) rl r l he boundedness of MM implies he desired esimae for M([, 1], h), and by rescaling, for any inerval M([, a], h). (Remember ha we idenify any se wih is characerisic funcion.) Then he general case follows by slicing g ino inervals: M(g, h) L 2 L = r g (a)m([, a], h)da L 2 L r (3.19) g (a)a h L 2 L da = g 1 r L 1 h r L 2. L1 r

9 3.3. L p esimae (3.9) for he wave. We fix and esimae L () poinwise. By symmery, we may assume ha >. Using he well known formula for he fundamenal soluion, we obain L () = L (, x) = ω 1 sin ω sds, 1 1 δ(s r)ds = { < r}. 4πs 4πr Here again we idenify he se wih is characerisic funcion. coordinaes we may wrie i as L ()ϕ = 9 (3.2) Using he polar Ω[ϕ(l)l 2 ]dl, (3.21) where Ω is an operaor on S 2 defined by Ωϕ() = F ( r lα )ϕ(α)dα, F (r) = (4πr) 1 { < r}. (3.22) S 2 We esimae he L p norm of Ω as follows. Firs we have he rivial L bound: Ωϕ L F ( r lα ) L α ϕ L 1 1 { < r + l} ϕ L 1. (3.23) For he L 2 esimae, we apply he Young inequaliy for he convoluion on SO(3). Using he ideniy f()d = C f(ae)da, e S 2, (3.24) S 2 SO(3) we esimae Ωϕ L 2 F ( re lbe )ϕ(abe)db SO(3) L 2 A ϕ(ae) L 2 A F ( re lbe )db (3.25) SO(3) ϕ L 2 S 2 F ( re l )d, where we changed he variables as Ae and α ABe. The las inegral of F is dominaed by { < r + l} re l 1 d { < r + l}(r l) 1. (3.26) S 2 Inerpolaing hese esimaes, we obain Ωϕ L p 2/p 1 (r l) 2/p { < r + l} ϕ L p, (3.27) for 2 p, where p = p/(p 1) is he dual exponen. Plugging his esimae ino L (), we obain ( ) 1 s r l g p f(s, l)l 2 r l L p dlds, L f(, r) L p R (3.28)

10 1 where g p () = 2/p 1 { < < 2}. (3.29) Then he desired L p esimae (1.4) for m = follows from Lemma 3.1 ogeher wih he esimae g p L 1 p. The case p < 2 is covered by he embedding L 2 Lp L p esimae (3.9) for he Klein-Gordon. Nex we exend he above resul o he Klein-Gordon m >. Since our esimae is global in ime and he large ime behavior is essenially differen beween he wave and he Klein-Gordon, i seems meaningless o approximae he laer by he former. Neverheless, we will show ha he T T operaor L m () for he Klein-Gordon can be dominaed by he wave correspondence and a dispersive par, which is smooh and decays fas in ime. By he rescaling argumen, i suffices o prove he esimae for m = 1. We may assume > by symmery. We calculae he kernel L m by wriing he Fourier ransform in he polar coordinaes as L m (, x) = C ρ 2 m cos( ρ )eir ρα m ρ 2 dαdρ S 2 = C = C 1 ρ 2 m cos( ρ m ) cos(rρλ)ρ2 dλdρ cos(rν) ν m cos l dldν, l (3.3) where we changed he variables as λ = cos( α), ν = ρλ and l = ρ m. Then we obain a uniform bound L 1 (, x) L (, x) ν ν dl l dν 1. (3.31) Inegraing by pars afer changing he variable l l/ ν m, we furher rewrie (3.3) as L m (, x) = C 1 K m (, x) + C K m (l, x)l 2 dl, (3.32) where K m () denoes he one-dimensional fundamenal soluion of he Klein-Gordon. When m = 1, we have K 1 (, r) = C ν 1 sin( ν ) cos(rν)dν = CJ ( 2 r 2 ){r < } 1/2{r 2 r 2 < }, (3.33)

11 where J is he Bessel funcion of order and we used he esimae J (s) s 1/2 [12, p. 98]. Hence we have for < r, L 1 (, x) When /2 < r <, we esimae K 1 (, r) 1 and L 1 (, x) 1 + r (l 2 r 2 ) 1/4 l 2 dl r 3/2. (3.34) When r < /2, we have 2 r 2 and so 11 l 2 dl 1 r 1. (3.35) L 1 (, x) 3/2 + 1/2 l 2 dl 3/2. (3.36) Gahering he esimaes (3.2), (3.31), (3.35) and (3.36), we conclude L 1 (, x) L (/2, x) + 3/2. (3.37) Thus we have reduced he desired esimae for m = 1 o ha for m = and he L 2 L x esimae for he dispersive par 3/2, which follows simply from he Young inequaliy H 3/4 esimae (1.6) for m =. Firs we derive an expression of L () resriced o each spherical harmonic (2.4), using he ideniies (3.2). Since we have (a(r)h ν ()) = ( ν a(r))h ν (), ν = ν(ν + 1)r 2 (3.38) for any spherical harmonic H ν () of order ν, we have he same relaion for any funcion of, and in paricular ω 1 sin ω (a(r)h ν ()) = (K ν ()a(r))h ν () (3.39) wih a cerain operaor K ν () on radial funcions. Choosing H ν () = P ν (e ) [9, Theorem 3], where P ν (s) = (2 ν ν!) 1 [(d/ds) ν (s 2 1) ν ] is he Legendre polynomial, and hen leing = e and using ha P ν (1) = 1, we obain K ν ()a(r) = 1 4π δ( r) (a(r)p ν(e ))(re) = 1 (3.4) P ν (cos β)a(l)ldl, 2r (l,,r) where (l,, r) denoes resricion o he region where a riangle holds wih side lenghs l, and r, i.e. 2 max(l,, r) l + + r, and he respecive opposie angles are denoed by α, β and γ. r α β l γ Figure. (l,, r)

12 12 Hence we have by (3.2) (l,s,r) L ()(a(r)h ν ()) = (L ν ()a(r))h ν (), L ν ()a(r) := 1 P ν (cos β)a(l)l 2 s 2 rl dlds s. (3.41) (l,s,r) The T T argumen (3.11) and he orhogonal decomposiion (2.4) reduce our desired esimae o R L ν( s )v(s)ds ν 3/2 v(, r) L 2 L 2 L L, (3.42) 1 r r where we used he ime symmery of L ν (). Since he esimae for ν = follows from he endpoin esimae (1.4) wih p = 2, we assume ha ν 1 as well as > in he following. In order o derive he decay in ν, we exploi he oscillaory propery of he Legendre polynomial in (3.41). We inegrae by pars for he variable s. Using he ideniy ν(ν + 1)P ν (x) = ((x 2 1)P ν(x)) and he relaion s 2 = r 2 + l 2 2rl cos β by he riangle, we obain P ν (cos β)h(s) sds sin 2 β = rl ν(ν + 1) P ν(cos β)h (s)ds, (3.43) (l,s,r) where we pu h(s) = 1/s{s > }, hence h (s) = δ(s )/ 1/s 2 {s > }. Applying he classical esimae (see [6]) and he sine heorem we dominae (3.43) by ν 3/2 In conclusion we have sin β 3/2 P ν(cos β) ν 1/2 (3.44) sin β = sin α = sin γ 1 s l r r l, (3.45) (l,s,r) s r l h 3/2 { < r + l} (s) ds ν. (3.46) (r l) L ν ()a(r) ν 3/2 { < r + l} (r l) a(l)l 2 dl (3.47) and he desired (3.42) follows from Lemma 3.1 wih g() = 1/2 { < < 2}. 4. Global soluions for he nonlinear Dirac equaion In his secion, we prove Theorem 1.2. We rewrie he equaion (1.8) as he following inegral equaion: u = U m ()ϕ + U m ( s)f (u(s))ds, (4.1)

13 where F (u) = iλγ (γ u, u)u and U m () denoes he propagaor of he free Dirac equaion given by U m () = cos(ω m ) γ ( 3 j=1 13 γ j j + im)ω 1 m sin(ω m ), (4.2) where ω m = m 2. We se Φu = R.H.S of (4.1) and apply he conracion mapping heorem. For he linear erm, we use he Sricharz esimaes (1.4). We see from (4.2) ha ωm 1 U m () is a linear combinaion of ωm 1 e ±iωm wih bounded Fourier mulipliers. So we have esimaes for m, 1 p < as U m ()ϕ L 2 L r L p ϕ H1. (4.3) Moreover, from he fac ha is commuaive wih, i follows ha U m ()ϕ L 2 L r H s,p Therefore puing X = L H 1 (H s) L2 L r H s,p we have Φu X ϕ H 1 (H s ) + (1 ) s/2 ϕ H 1 ϕ H 1 (H s ). (4.4) ϕ H 1 (H s ) + F (u) L 1 H 1 (H s ). By (2.1), we esimae he nonlinear erm F (u) as F (u) H s u 2 L u H s, F (u) H s u H s,p u L wih p sufficienly large as p > 2/s, U m ( s)f (u(s)) X ds u L q + u 2 L u H s (4.5) (4.6) wih 1/p + 1/q = 1/2. By he embeddings H s,p L for s > 2/p, H s Lq for s 2/p, and he Hölder inequaliy for variables and r, we have Analogously we have F (u) L 1 H 1 (H s) u 2 L 2 L r H s,p u L H 1 (H s). (4.7) Φu Φv X ( u 2 X + v 2 X) u v X. (4.8) Therefore Φ is a conracion map on a small closed ball in X. For he uniqueness of soluions in he class of (1.13), we consider he L L 2 x meric. By he L 2 invariance of U(), we have u v L L 2 x ( u 2 L 2 L x + u 2 L 2 L x ) u v L L2 x. (4.9) We can conclude u = v ime locally, so ha for he enire ime inerval by he repeiion.

14 14 5. Discussion Theorem 1.2 implies ha if he iniial daa is spherically symmeric and small in H 1, hen he soluion is global. For anoher examples, we can find some of sudies on he following form of soluions for Dirac equaions in [1], [2], [13] ec., f(, r) x 1 = r sin ω cos φ, u(, x) = g(, r) cos ω, x 2 = r sin ω sin φ, (5.1) x g(, r) sin ωe iφ 3 = r cos ω. We can apply Theorem 1.2 o his ype soluion, namely, if he iniial daa akes he form ϕ = (f (r),, g (r) cos ω, g (r) sin ωe iφ ) and f H 1 (R 3 ), g H 1 (R 3 ) are sufficienly small, hen here exiss a global soluion of he form (5.1). Indeed, since U m () and he nonlinear erm (γ u, u)u preserve he form of (5.1), he funcions given by he ieraion argumen which sars from he free soluion U m ()ϕ have he form and he limiing funcion which is he soluion of (1.8) also has he form. Moreover, our argumen also applies o nonlinear Klein-Gordon equaions of he form u u + m 2 u + F ( u, mu) =, (5.2) where denoes he space-ime derivaives. We can deduce local wellposedness in H 1 (H s) for F = u u, global wellposedness for small daa in H1 (H s) for F = u2 u, local wellposedness in H 2 (H s) for F = ( u)2, global wellposedness for small daa in H 2 (H s) for F = ( u)3, ec. Compare wih [5],[11]. Noice ha sysems of nonlinear wave equaions in mos cases do no possess radial symmeric soluions bu have a cerain class of soluions wih he Lorenz covariance, jus as in he above case of he nonlinear Dirac. The radial endpoin Sricharz esimae does no simply apply o such classes, since he reduced equaions for he radial par of soluions would have erms of he form u/r 2. Bu one can apply our argumen direcly o such classes o have wellposedness, say in H 1, wihou even knowing algebraic properies of he symmery. Finally we give upper bounds for s in he L 2 L r H s esimae for boh he Klein- Gordon (on R 3 ) and he Schrödinger (on R 2 ). This implies ha we can no recover he L p esimae for all p < from Hs esimae and he Sobolev embedding. Theorem 5.1. (i) Le m, s R and suppose ha we have he esimae of he form u L 2 L r H s E(u)1/2, (5.3) for any finie energy soluion u of he Klein-Gordon equaion (1.1). Then we have s 5/6.

15 (ii) Le s R and suppose ha we have he esimae for he Schrödinger equaion on R 2 of he form Then s 1/3. e i ϕ L 2 L r H s ϕ L 2 (R 2 ). (5.4) Proof. Firs we consider he Klein-Gordon case. By he scaling argumen, we may assume m = wihou loss of generaliy. Then by he Sricharz esimae and he dualiy we have (ψ(x)f(), ω 1 e iω ϕ(x)),x ψ L 1 r H s f L 2 ϕ L 2 x, (5.5) where (, ),x denoes he L 2 inner produc on R 1+3. We can rewrie he inner produc by using he Plancherel for (, x) ( ψ(x) f( x ), x 1 ϕ(x)) x, (5.6) where ψ denoes he Fourier ransform of ψ, and (, ) x denoes he inner produc on R 3. Thus we obain We can decompose any g L 1 rl 2 as hen we have Plugging his ino (5.7), we obain ( ψ(x), r 1 f(r)ϕ(x)) x ψ L 1 r H s f L 2 ϕ L 2 x. (5.7) g(r) = r 1 f(r)ϕ(x), f(r) = g(r) 1/2 r, (5.8) L 2 g L 1 r L 2 = f L 2 ϕ L 2 x. (5.9) ( ψ(x), g(x)) x ψ L 1 r H s g L 1 r L 2. (5.1) Hence he Plancherel and he dualiy imply ha g L r H s g L 1 rl 2. (5.11) Now le g(r) = a(r)h ν () where H ν () is a spherical harmonic of order ν. Then we have g(r) = CH ν ()i ν a(ρ)j ν (rρ)ρ 2 dρ, (5.12) where J ν (r) = r 1/2 J ν+1/2 (r), (5.13) and J ν (r) denoes he Bessel funcion (see [1, p. 164]). Then (5.11) implies ha a(ρ)j ν (rρ)ρ 2 dρ ν s a L 1 r, (5.14) L r 15

16 16 which is equivalen by dualiy o J ν (r) L r ν s. (5.15) By choosing r = ν + 1/2 and using he asympoic behavior of he Bessel funcion J ν (ν) ν 1/3 for ν [19, p. 231], we conclude ha s 5/6. The proof in he Schrödinger case is almos he same. By he same argumen we obain insead of (5.7) ( ψ(x)f(r 2 ), ϕ(x)) x ψ L 1 r H s f L 2 ϕ L 2 x, (5.16) on R 2. By using he following decomposiion for any g L 1 rl 2 : g(r) = f(r 2 )ϕ(r), f(r 2 ) = g(r) 1/2, (5.17) L 2 and he dualiy, we arrive a he same esimae as above g L r H s g L 1 rl 2. (5.18) Again we assume g is a spherical harmonic g(r) = a(r)e iν. In his case we have and (5.18) implies ha g(r) = Ce iν i ν a(ρ)j ν (rρ)ρdρ, (5.19) Then we obain s 1/3 by he asympoic of J ν (ν). J ν (r) L r ν s. (5.2) References 1. J.D. Bjorken and S.D. Drell, Relaivisic Quanum Mechanics, McGraw-Hill, New York, J.P. Dias and M. Figueira, On he exisence of weak soluions for a nonlinear ime dependen Dirac equaion, Proc. Royal Soc. of Edinburgh 113A (1989), M. Escobedo and L. Vega, A semilinear Dirac equaion in H s (R 3 ) for s > 1, SIAM J. Mah. Anal. 28 (1997), no. 2, S. Klainerman and M. Machedon, Space-ime esimaes for null forms and he local exisence heorem, Comm. Pure Appl. Mah. 46 (1993), no. 9, H. Lindblad, Counerexamples o local exisence for semi-linear wave equaions, Amer. J. Mah. 118 (1996), no. 1, G. Lohöfer, Inequaliies for he associaed Legendre funcions, J. Approx. Theory 95 (1998), no. 2, S. Machihara, K. Nakanishi and T. Ozawa, Small global soluions and he nonrelaivisic limi for he nonlinear Dirac equaion, Rev. Ma. Iberoamericana 19 (23), no. 1, S. Mongomery-Smih, Time decay for he bounded mean oscillaion of soluions of he Schrödinger and wave equaions, Duke Mah. J. 91 (1998), no. 2, C. Müller, Spherical Harmonics. Lecure Noes in Mahemaics, 17 Springer-Verlag, Berlin- New York, C. Müller, Analysis of Spherical Symmeries in Euclidean Spaces, Applied Mahemaical Sciences, 129. Springer-Verlag, New York, G. Ponce and T. Sideris, Local regulariy of nonlinear wave equaions in hree space dimensions, Comm. Parial Differenial Equaions 18 (1993), no. 1-2, M. Reed, Absrac Non-linear Wave Equaions, Lecure Noes in Mahemaics, 57. Springer- Verlag, Berlin-New York, 1976.

17 13. W. Srauss and L. Vazquez, Sabiliy under dilaions of nonlinear spinor fields. Phys. Rev. D (3) 34 (1986), no. 2, R.S. Sricharz, Analysis of he Laplacian on he complee Riemannian manifold. J. Func. Anal. 52 (1983), no. 1, T. Tao, Spherically averaged endpoin Sricharz esimaes for he wo-dimensional Schrödinger equaion, Comm. Parial Differenial Equaions 25 (2), no. 7-8, T. Tao, hp:// 17. H. Triebel, Spaces of Besov-Hardy-Sobolev ype on complee Riemannian manifolds, Ark. Ma. 24 (1986), no. 2, H. Triebel, Theory of Funcion Spaces. II, Monographs in Mahemaics, 84. Birkhäuser Verlag, Basel, G. Wason, A reaise on he heory of Bessel funcions, Reprin of he second (1944) ediion. Cambridge Mahemaical Library. Cambridge Universiy Press, Cambridge, Shuji Machihara Shimane Universiy, Shimane , Japan machihara@mah.shimane-u.ac.jp Makoo Nakamura Graduae School of Informaion Sciences (GSIS) Tohoku Universiy, Sendai , Japan m-nakamu@mah.is.ohoku.ac.jp Kenji Nakanishi Graduae School of Mahemaics Nagoya Universiy, Nagoya , Japan n-kenji@mah.nagoya-u.ac.jp Tohru Ozawa Deparmen of Mahemaics Hokkaido Universiy, Sapporo 6-81, Japan