36 M.C. VILELA Denition. The pai (q; ) is an admissible pai if q;, (q; ;n=) 6= (; ; ), and () q + n = n : Theoem. estimates: (3) (4) (5) If (q; ) and

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1 Illinois Jounal of Mathematics Volume 45, Numbe, Summe, Pages S 9-8 REGULARITY OF SOLUTIONS TO THE FREE SCHRÖDINGER EQUATION WITH RADIAL INITIAL DATA M.C. VILELA Abstact. We deive weighted smoothing inequalities fo solutions of the fee Schödinge equation. As an application, we give a new poof of the endpoint Stichatzestimates in the adial case. We also conside geneal dispesive equations and obtain simila estimates in this case.. Intoduction Conside the homogeneous initial value poblem fo the fee Schödinge equation ( i@t u u =; (; t) IR n IR; () u(; ) = u (); and denote its solution by e it u. The poblem of nding values q; fo which the L q t L -nom of the solution to () is contolled bythel -nom of the initial data has been etensively studied by seveal authos. Hee L q t L denotes the space of functions F (; t) such that kf k L q t L = ψ + IR n jf (; t)j d q= dt! =q < +: In the case q = =n=(n + ), such an estimate was given by R. Stichatz [3], using ideas developed by E. Stein on the estiction popeties of the Fouie tansfom on cuved sufaces; see [, p. 374], and the papes [5] and [6] by P. Tomas. The estimate was etended to the case q 6= by J. Ginibe and G. Velo [5], who used these esults in an essential manne to study the Initial Value Poblem of semilinea petubations of (). To state the known esults we need the following denition. Received Januay, 999; eceived in nal fom Mach 3,. Mathematics Subject Classication. Pimay 4B99. Suppoted by a gant fom the Basque County Govement. 36 c Univesity of Illinois

2 36 M.C. VILELA Denition. The pai (q; ) is an admissible pai if q;, (q; ;n=) 6= (; ; ), and () q + n = n : Theoem. estimates: (3) (4) (5) If (q; ) and (~q; ~) ae admissible, then we have the following s<t IR ke it u k q» L c ku k t L L ; e is F ( ;s) ds e i(t s) F ( ;s) ds L kf k L q L q t L t L kf k L ~q ~ t L ; : The estimates (3) and (4) ae equivalent, and a scaling agument shows that in this case () is necessay. Fo the last inequality the natual estiction is q + n + ~q + n ~ = n; which is weake than (), so it is possible that (5) holds fo a lage ange of pais. The poblem of detemining the eact set of pais fo which this estimate holds is still open. The case when (q; ) o(~q; ~) is equal to the citical value P = ; n n (n 3) was ecently settled by M. Keel and T. Tao [8]. The noncitical case had been solved ealie; see [3] and [5] fo the estimates (3) and (4), and [9] and[3] fo (5). When» q <, it is easy to constuct an eample which shows that (3) and (4) ae false. The same counteeample poves that (5) fails when =q+=~q >: In the case n = the citical point P =(; ) is not admissible. In fact, (3), (4) and (5) do not hold fo this citical value; see [9]. In this pape we pove some weighted smoothing inequalities fo abitay solutions of the fee Schödinge equation. In the st section we conside the homogeneous poblem, and in the second section the inhomogeneous poblem. Finally, in the last section we ecove the endpoint estimates of Stichatz in the adial case fom pevious estimates by using only a adial vesion of the Sobolev embedding theoem.. The homogeneous case Given a function f and s IR we dene the homogeneous deivative of ode s of f by d D s f(ο) = c s jοj s ^f(ο), and the factional integal of ode s

3 REGULARITY OF SOLUTIONS TO THE SCHRÖDINGER EQUATION 363 of f as I s f = D s f. Fo any IR; we denote the L p -space with measue jj d dt by L p (jj ) ; and the L p -space with measue jj d by L p (jj ) : The main esult of this section is the following theoem. Theoem. We have (6) (7) Ds kd s eit u k L (jj ff ) ku k L ; e i F L ( ;) d» IR c kf k L ); (jjff if and only if ff =( s); <ff<nand n : Remak. Estimates of this type have also been studied in [8]. Remak. We have stated Theoem in tems of the solution to the Schödinge equation, but the theoem holds in a moe geneal setting. In fact, take u to be the solution to the poblem ( i@t u +( ) a= u =; (; t) IR n IR; a>; u(; ) = u (); which we denote by e it a= u. The esult in this case is that Ds kd s eit a= u k L (jj ff ) kd u k L ; e i a= L F ( ;) d» IR c kd F k L ); (jjff hold if and only if ff = ( s), <ff<n, = a=, and n. Poof of Theoem. By duality, (6) and (7) ae equivalent, and because of the scale the estiction ff = ( s) is necessay. In ode to pove (6) fo <ff<n;we use pola coodinates and a change of vaiable to wite D s eit u () = = = + e i ο D s eit u b(ο) dο IR n e it s + e itu ψ e i ο cu (ο) dff (ο) d Sp n u s e i ο cu (ο) dff p u(ο) u! du:

4 364 M.C. VILELA Using this identity togethe with Plancheel's identity in the vaiable t, we have ke it u k L (jj ff ) = c = c = + jj ff IR n ψ + jj ff IR n ψirn jj ff ψ + Sp n u s u Hence it is enough to pove that IRn jj ff e i ο cu (ο) dff (ο) e i ο cu (ο) dff p u(ο) s e i ο cu (ο) dff (ο) e i ο cu (ο) dff (ο) d s! d du d dd A =! = s d! = : jcu (ο)j dff (ο): Since ff = ( s), this inequality is invaiant unde dilations, so we may assume without loss of geneality that =. Then, IRn jj ff = jj» + When ff<n,we have I» ei ο g(ο) dff(ο) jj S ff ei ο g(ο) dff(ο) n d jj S ff ei ο g(ο) dff(ο) n jj> d =I+II: ei ο g(ο) dff(ο) jj L jj» ff d jg(ο)j dff(ο) Sn jg(ο)j dff(ο): To estimate the integal II, we need the following lemma. Lemma (Tace lemma; see []). sup ;R R B( o;r) e i ο f(ο) dff (ο) We have d jf(ο)j dff (ο);

5 REGULARITY OF SOLUTIONS TO THE SCHRÖDINGER EQUATION 365 whee the constant c is independent of, is the Euclidean sphee of adius, dff is the suface measue, o IR n,andr; >. Dividing the ange of integation in II diadically, we can wite II = +X j jj ff <jj»j+ j= +X j= j(ff ) j+ ei ο g(ο) dff(ο) d jj» j+ ei ο g(ο) dff(ο) d jg(ο)j dff(ο): The last inequality is a consequence of the tace lemma and the fact that ff>: When n =; the estimate (6) fails. To see this, take u such that cu is an even function; then (6) does not hold because cos = L (jj ff ) fo any ff IR: When n ; (6) fails wheneve ff» o ff n. To see this, take u such that cu is a adial function; then (6) fails because c dff = L (jj ff ) ; whee dff denotes the suface measue of the unit Euclidean sphee. Λ Remak 3. When ff =and s ==, then (6) also fails. Howeve, in this case we have the following substitute of this estimate: + sup D = R> R eit u () dt d ku k L: B(;R) This Kato type smoothing estimate was poved in [4], [] and [7]. (8). The inhomogeneous case In this section we conside the inhomogeneous Initial Value Poblem ( i@t u u = F (; t); (; t) IR n IR; u(; )=: Using Duhamel's fomula we can wite the solution of (8) in the fom u(; t) = i t Ou main esult is the following theoem. Theoem 3. e i(t ) F (; ) d: The solution to the IVP (8) satises (9) kd s uk L (jj ff ) ki s F k L (jj ff ); wheneve ff i =( s i ), <ff i <n(i =; ), and n.

6 366 M.C. VILELA Remak 4. As in the homogeneous case, Theoem 3 can be fomulated in a moe geneal setting fo the solution of the IVP (8), whee is eplaced by ( ) a=. In this case we have the estimate kd s uk L (jj ff ) kd I s F k L (jjff ); wheneve ff i =( s i );= a=, <ff i <n(i =; ) and n : Poof of Theoem 3. We follow the agument used in Theoem.3 of [7] and fomally wite the solution of (8) in the fom whee v(; t) = u(; t) =v(; t) e it v( ; ) (); IR IR n jοj bf ;t (ο; )e it+i ο dο d: Hee F b;t denotes the Fouie tansfom of F in both vaiables. To estimate the second tem, we use Theoem. To contol the st tem, we ewite this tem as v(; t) = + T bf t () ()e it d; whee T is the Helmholtz opeato dened by () d T f(ο) = ^f(ο): jοj Using Plancheel's identity in the vaiable t we have () kd s vk L (jj ff ) = = = D s T bf () t b L t (t) L t (jj ff ) D s T bf () t L L (jj ff ) D s T bf () t L L (jj : ff ) The following poposition will allow us to complete the poof. Poposition. The Helmholtz opeato T dened by() satises () kd s T fk L (jj ff ) ki s fk L (jj ff ) ; wheneve ff i = ( s i ), < ff i < n (i = ; ), and n. Hee c is a constant independent of.

7 REGULARITY OF SOLUTIONS TO THE SCHRÖDINGER EQUATION 367 Using this poposition in () and the Plancheel identity inthetvaiable, we have kd s vk L (jj ff ) = c = c I sf b t () L (jj ff ) L (Is F ) t () L L (jj ff ) L ki s F k L t (jj ff ) = c ki s F k L (jj ff ): The above fomal pocess can be justied by applying it to the equation i@ t u u + i"u = F (; t); (; t) IR n IR; ">: In this case the estimate (9) holds unifomly in " and the esult follows on letting "!. Λ To pove the poposition we need the following two lemmas. Lemma ([]). Let <<n; <p< and p n<<n(p ). Then ki fk Lp (jj p ) kfk Lp (jj ): Lemma 3 ([6], [7]). Let ' C (IR) with supp ' [ ; ], ' =on the inteval [ =; =] and» '». Given s IR, dene the opeato S by csf(ο) = If f has compact suppot, then jοjs jοj '((jοj )) ^f(ο): R ksfk L (B(;R))) d(supp f)kfk L; whee d(supp f) is the diamete of the suppot of f and R>: Poof of Poposition. By a scaling agument itis enough to pove () when = ±: In the case = we have no singulaity. Theefoe jd s T f()j» ci s jfj(), and the esult follows fom Lemma. When =, we take ' C (IR) with supp ' [ ; ], ' = on [ =; =] and» '» : Let ' (ο) ='((jοj )) and ' (ο) = ' (ο); and dene the opeatos T ; and T ; by Then we have (T ;i f)b(ο) =m i (ο) ^f(ο) = jοjs jοj ' i(ο) ^f(ο); i =; : kd s T fk L (jj ff )»kt ; fk L (jj ff ) + kt ; fk L (jj ff ) :

8 368 M.C. VILELA The second tem can be contolled as in the case = because m has no singulaity. To contol the st tem, we eplace the homogeneous weights jj ff and jj ff with the inhomogeneous weights hi ff and hi ff, espectively, whee hi = +jj = : This is possible, by the Littlewood- Paley localization and the estimates k fk L (jj ff ) kfk L (hi ff ) ; k fk L (hi ff ) kfk L (jj ff ) ; fo» ff i < n (i = ; ), whee is the Littlewood-Paley pojection to fequencies jοj ο: We now divide IR n and decompose f into IR n = +[ j= X j ; f = whee X = f : jj» g, X j = f : j < jj» j g fo j and f k = fχ Xk. Using these decompositions, Lemma 3, and the Cauchy-Schwaz inequality, wehave kt ; fk L (hi ff )»» +X +X k= kt ; f k k L (hi ff ) k= +X + k= wheneve <ff i <n(i =; ). j= +X + k= +X k= ψ X + k= j= f k ; jff kt ; f k k L (B(;j )) k= kf k k L j(ff ) k kf k k L kff kf k k L kfk L (hi ff ) ; 3. Application! = Fom Theoems and 3 we can deive the Stichatz estimates (3), (4) and (5) in the citical case and fo adial initial data wheneve n 3; using only the following adial vesion of the Hady-Littlewood-Sobolev theoem. A A = = Λ

9 REGULARITY OF SOLUTIONS TO THE SCHRÖDINGER EQUATION 369 Lemma 4. Let f be a adial function. Then wheneve p q» s» n p q ki s fk Lq (IR n ) kfk Lp (jj ff ); ; ff = p» n p q s ; <p<q<: This lemma can be poved using the ideas in []. The fact that f is adial allows us to educe s to =p =q: Given a adial initial data u ; e it u is adial too, so we can apply Lemma 4 with p = and =n=(n ) (n 3) to obtain ke it u k L kd s eit u k L (jj ff ); wheneve =n» s» ; ff= ( s): Taking the L -nom in time and using Theoem, we get the estimates (3) ke it u k L t L kds eit u k L (jj ff ) ku k L ; fo =n» s<=: The dual vesion of (3) is IR e is F ( ;s) ds» L c ki sf k L (jj ff ) kf k L t L and the analogous esult fo the solution u of the inhomogeneous poblem is (4) kuk L t L» c kds uk L (jj ff ) ki s F k L (jj ff ) kf k L t L : Hee F is a adial function in the -vaiable, and I s F denotes the factional integal in the vaiable. When n =and =, this method fails because Lemma 4 is false fo p =andq = : Howeve, Tao [4] ecently showed that the estimates (3), (4) and (5) hold in this case fo adial data wheneve (~q; ~) is an admissible pai. Estimates simila to (4) have ecently been used by Bougain [] to pove the global eistence fo the defocusing quintic nonlinea Schödinge equation with adial data and abitay lage enegy nom. In paticula, Bougain used estimates such as (4) to pove that solutions which cease to eist in nite time must concentate. This popety has not been established fo dimensions n 3, and data in L,even in the adial case. We shall study these questions elsewhee. Acknowledgements. I would like to thank Luis Vega fo suggesting this poblem to me and fo his patient help. ;

10 37 M.C. VILELA Refeences [] S. Agmon and L. Hömande, Asymptotic popeties of solutions of diffeential equations with simple chaacteistics, J. Analyse Math. 3 (976), 8. [] J. Bougain, Global wellposedness of defocusing 3D citical nonlinea Schödinge equation in the adial case, J. Ame. Math. Soc. (999), [3] T. Cazenave and F.B. Weissle, The Cauchy poblem fo the nonlinea Schödinge equation in H, Manuscipta Math. 6 (988), [4] P. Constantin and J.C. Saut, Local smoothing popeties of dispesive equations, J. Ame. Math. Soc. (988), [5] J. Ginibe and G. Velo, The global Cauchy poblem fo the nonlinea Schödinge equation evisited, Ann. Inst. H. Poincaé Anal. Non Linéaie (985), [6] L. Hömande, The Analysis of linea patial diffeential opeatos II, Spinge Velag, Belin, 983. [7] C. Kenig, G. Ponce, and L. Vega, Small solutions to nonlinea Schödinge equations, Ann. Inst. H. Poincaé, (993), [8] M. Keel and T. Tao, Endpoint Stichatz estimates, Ame. J. Math. (998), [9] S. J. Montgomey-Smith, Time Decay fo the Bounded Mean Oscillation of Solutions of the Schödinge and Wave equation, Duke Math. J. 9 (998), [] P. Sjölin, Regulaity of solutions to the Schödinge equation, Duke Math. J. 55 (987), [] E.M. Stein, Hamonic Analysis: Real Vaiable Methods, Othogonality, and Oscillatoy Integals, Pinceton Univesity Pess, Pinceton, NJ, 993. [] E.M. Stein and G. Weiss, Factional integals on n dimensional Euclidean espace, J. Math. Mech. 7 (958), [3] R.S. Stichatz, Restictions of Fouie tansfoms to quadatic sufaces and decay of solutions of wave equations, Duke Math. J. 44 (977), [4] T. Tao, Spheically aveaged endpoint Stichatz estimates fo the two dimensional Schödinge equation, Comm. Patial Diffeential Equations 5 (), [5] P.A. Tomas, A estiction theoem fo the Fouie tansfom, Bull. Ame. Math. Soc. 8 (975), [6], Restiction theoems fo the Fouie tansfom, Hamonic analysis in Euclidean spaces, Poc. Sympos. Pue Math., XXXV, Pat, Ame. Math. Soc., Povidence, RI, 979, pp. 4. [7] L. Vega, Schödinge equations: pointwise convegence to the initial data, Poc. Ame. Math. Soc. (988), [8] B.G. Walthe, An L -Estimate fo the solution to the time-dependent Schödinge equation, TRITA-MAT-996-MA-5, Royal Institut of Technology, Stockholm, 996. [9] K. Yajima, Eistence of solutions fo Schödinge evolution equations, Comm. Math. Phys. (987), Depatamento de Matemáticas, Facultad de Ciencias, Univesidad del Pa s Vasco (UPV-EHU), Aptdo 644, 488 Bilbao, Spain Cuent addess: Univesidad Alfonso X, Facultad de Estudios Sociales, D-6, CP 869 Villanueva de la Canada, Madid, Spain addess: maicuz@ua.es

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