SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n 2

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1 Acta Mathematica Scientia 1,3B(6): SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n Li Dong ( ) Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, 54, USA mpdongli@gmail.com Zhang Xiaoyi (ΞΛΠ) Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, 54, USA Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 119, China yzhang@amss.ac.cn Abstract The usual Kato smoothing estimate for the Schrödinger propagator in 1D takes the form 1 e it u L L u L t. In dimensions n the smoothing estimate involves certain localization to cubes in space. In this paper we focus on radial functions and obtain Kato-type sharp smoothing estimates which can be viewed as natural generalizations of the 1D Kato smoothing. These estimates are global in the sense that they do not need localization in space. We also present an interesting countereample which shows that even though the time-global inhomogeneous Kato smoothing holds true, the corresponding time-local inhomogeneous smoothing estimate cannot hold in general. Key words Schrödinger equation; smoothing estimate 1 MR Subject Classification 35Q55 1 Introduction In this paper we consider Kato type smoothing estimates for the Schrödinger propagator e itδ in dimensions n. Kato smoothing estimates play an important role in local theory for semi-linear nonlinear Schrödinger equations. Typically one employs smoothing estimates to overcome the loss of derivatives caused by the semi-linear terms from the nonlinearity. The first such estimate dates back to [3], where Kato studied the Cauchy problem for the (generalized) Korteweg-de Vries equation t u u + u u, (, t) R R, u(, ) u (). Received October 11, 1. D. Li is supported in part by NSF Grant-983 and a start-up funding from University of Iowa. X. Zhang is supported by an Alfred P. Sloan fellowship and also a start-up funding from University of Iowa.

2 14 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B He proved a smoothing estimate for the linear propagator e t which takes the form T T l l ( u)(, t) ddt C(T,l, u L ). (1.1) Here and below we shall use the notation C(A, B,,) to denote constants which depend on quantities A, B,. For the Schrödinger propagator e it, the corresponding version of (1.1) was obtained in [1], [8], [9], and has the form T T l l (1 ) 1 4 e it u ddt C(T,l, u L ). (1.) The multi-dimensional analogue of (1.) for the Schrödinger propagator e itδ was also obtained in [1], [8], [9]. To state the results, denote {Q α } α Z n as the family of non-overlapping cubes of size l such that R n Q α.thenforn, α Z n ( sup α Z n Q α ) 1 1 e itδ u ddt C(n, l, u L ). (1.3) The sharp versions of (1.), (1.3) were proven by Kenig, Ponce and Vega [4] and take the following form: for n 1, ( ) 1 sup 1 e it u dt C1 u L, (1.4) and for n, ( ) 1 sup 1 e itδ u dtd C l u L. (1.5) α Z n Q α Here C 1 is an absolute constant and C depends only on the dimension n. To study the initial value problem for semilinear nonlinear Schrödinger equations, one needs time-inhomogeneous version of (1.4) (1.5). The following estimates were first proven by Kenig, Ponce and Vega [5]: for n 1, ( ) sup e i(t s) F (,s) 1 dt C 3 F L t, (R R), (1.6) and for n, ( sup α Z n Q α C 4 l α Z n ( e i(t s)δ F (,s)ds dtd Q α ) 1 ) 1 F (, t) dtd. (1.7) Here C 3 is an absolute constant and C 4 depends only on the dimension n. Note that due the averaging effect in time, the gain of derivative in (1.6) (1.7) is twice than that of (1.4) (1.5). Let χ be a smooth cut-off function so that its Fourier transform ˆχ Cc (Rn ). A slightly variant of (1.5), often referred to as the Kato 1/-smoothing estimate in R n, n takes the form χ()(1 1 Δ) 4 e itδ L u C 5 u L t, (R Rn ) (R n ). (1.8)

3 No.6 D. Li & X.Y. Zhang: SMOOTHING ESTIMATES OF SCHRÖDINGER PROPAGATOR 15 Here the constant C 5 depends on χ and n. Roughly speaking, the meaning of (1.8) is that if u L,then 1 e itδ u L loc (Rn R). The Kato smoothing estimate is sharp. For eample, one can consider a Gaussian wave packet with frequency localized to ξ ξ. The average time that the wave packet spent in a O(1) compact space region is roughly O(1/ ξ ). Therefore the L t norm is O( ξ 1 )which precisely cancels the ξ 1 factor induced by the 1/-derivative. The estimates (1.1) (1.8) are by now very well-known and considered to be standard smoothing estimates. However one should notice some qualitative differences between the n 1 case and n case. In particular (1.4) and (1.6) are L L t type estimates while (1.5), (1.7) and (1.8) are L L t type estimates. This naturally motivates us to consider whether there are genuine L L t type estimates in dimensions n which are in a sense more natural analogues of (1.4) and (1.6). The purpose of this paper is to answer this question at the epense of radial assumptions. Here are our main results. Theorem 1.1 (Kato smoothing for radial functions in dimension n ) Let the dimension n. Then for any radial function f on R n we have n 1 1 e itδ f f. (1.9) L L t (Rn R) Remark 1. One should view Theorem 1.1 as the natural generalization of the usual one-dimensional Kato smoothing (1.4) to higher dimensions under the radial assumption. Our intention here is to obtain global weighted estimates (for positive weights). The usual Kato smoothing in higher dimensions involves localization in space. To have an impression, compare (1.9) with (1.5). From duality and a T T argument, (1.9) implies Theorem 1.3 (Kato smoothing, time-inhomogeneous global version) Let the dimension n. Then for any radial function f : R R n C we have n 1 e i(t s)δ y n 1 L f(s, y)ds f L 1 L t R). (1.1) R (Rn L t (Rn R) Theorem 1.1 and Theorem 1.3 have applications in our recent work [6, 7]. On the other hand, one may wonder the time retarded estimate, where in (1.1) the integrating in time is from to t, instead of the whole R still hold. Note that the retarded estimate is used in the nonlinear problem. Surprisingly, we shall show that this is not the case. In other words, we shall prove in Section 3 that the following estimate n 1 e i(t s)δ y n 1 L f(s, y)ds f L 1 L t (Rn R) (1.11) L t (Rn R) fails in general. On the other hand, an ɛ-version of (1.11) holds, namely n 1 ɛ 1 ɛ e i(t s)δ y n 1 ɛ f(s, y)ds f L 1 L L L t R). (1.1) (Rn t (Rn R) The proof of (1.1) uses a similar analysis used in the disproof of (1.11). We therefore leave it as an eercise to interested readers. This paper is organized as follows. In Section we give the proofs of Theorem 1.1 and Theorem 1.3. Section 3 is devoted to the disproof of (1.11).

4 16 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Kato Smoothing Estimate for Radial Functions In this section we give detailed proofs of Theorem 1.1 and Theorem 1.3. As we shall show, the proofs are not surprisingly low-tech since the original Kato smoothing proof is also very simple. We begin with ProofofTheorem1.1 By passing to radial coordinates, we can write ( ) n 1 1 e itδ f n 1 k 1 e itk ˆf(k)k n 1 e ik ω1 dσ(ω) ω 1 dk, (.1) where ˆf is the Fourier transform of the function f and dσ(ω) is the the surface measure on S n 1. Since the Fourier transform of a radial function is still radial, we can slightly abuse the notation ˆf(k) to denote the Fourier transform of f. Now consider the function h(ρ) : e iρω1 dσ(ω). It is clear that ρ n ω 1 h(ρ) J n (ρ), where J n is the usual Bessel function of order n. Then by using the asymptotics for Bessel functions, it is not difficult to see that sup ρ n 1 h(ρ) C1 <, (.) ρ> where C 1 is a constant depending only on the dimension n. By Plancherel, one can show that for any one-dimensional function F,wehave e itk F (k)dk 1 F (k) dk k. (.3) Now by (.1), (.), (.3), we obtain n 1 1 e itδ f L t 1 1 C 1 f. L t ˆf(k) n 1 k (n 1) h(k ) dk ( ˆf(k) k n 1 dk ) sup ρ n 1 h(ρ) ρ> The theorem is proved. We shall establish the inhomogeneous smoothing estimate similar to Theorem 1.1. ProofofTheorem1.3 By Theorem 1.1 and duality, we have 1 e isδ y n 1 L f(s, y)ds f L 1 L. (.4) t R Now define the linear operator T as (Tf)(t, ) n 1 R e i(t s)δ y n 1 f(s, y)ds. (.5)

5 No.6 D. Li & X.Y. Zhang: SMOOTHING ESTIMATES OF SCHRÖDINGER PROPAGATOR 17 Then (.4) implies for any f, f,tf,t f L. (.6) 1 L t With (.6) in hand, we now establish (1.11) by a polarization argument. Namely for any h, g with h L 1 L g t L 1, substituting f h + g and f h + ig, weobtain 1 L t h, T g,t 1. Then for any h, g, weobtain h, T g,t h L 1 L g t L. 1 L t This immediately implies (1.11). 3 Time-local Inhomogeneous Flow Smoothing is not True: A Disproof From Theorem 1.3, it is tempting to invoke the Christ-Kiselev Lemma [] to obtain the retarded estimate n 1 e i(t s)δ y n 1 L f(s, y)ds f L 1 L t R). (3.1) (Rn L t (Rn R) However, the Christ-Kiselev lemma fails to be applicable since we are considering the case with the endpoint eponent. More surprisingly, we shall show that (3.1) is not true in general! To see this it is enough consider the D case (higher dimensional computations are similar). Since f is radial, we write 1 e i(t s)δ y 1 f(s, y)ds 1 ξ e iξ e i(t s) ξ y 1 f(s, y )e iξ y dydsdξ 1 k J (k ) e i(t s)k ρ 1 f(s, ρ)j (kρ) ρkdsdρdk. Now pass everything to radial coordinates and write as one-dimensional, we can reformulate (3.1) as the following equivalent one-dimensional problem: 1 k J (k) e i(t s)k ρ 1 f(s, ρ)j (kρ) kdsdρdk L L t (R R) f L 1 L t (R R). (3.) Note here that we regard f(ρ)ρ as f(ρ) to absorb the volume factor since originally f is a twodimensional function. In (3.) everything is one-dimensional. By conveity, to check whether (3.1) is true, it suffices to consider f of the form f δ(ρ ρ )η(t), where η L t and ρ is arbitrary. So equivalently we need to check 1 k J (k) e i(t s)k ρ 1 η(s)j (kρ) kdsdk η L t (R). (3.3) L ρ L L t (R R)

6 18 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Net write the Fourier transform of η(t) as. We compute e i(t s)k η(s)ds e i(t s)k e isμdsdμ e itμ e itk k + μ e itk dμ k + μ dμ }{{} (a) + e itμ k + μ dμ. (3.4) } {{ } (b) We estimate separately the contributions of (a) and(b) to the LHS of (3.3). For (a), we have e itk k + μ dμ 1 k J (k)j (kρ)ρ 1 dk e itk( ( ) k + μ) dμ) (ρ) 1 k J (k)j (kρ) dk k + μ dμ (ρ) k 3 J (k) J (kρ) dk k k + μ dμ dk k + μ dμ dk 1 μ ˆη L H(ˆη) k L k η L. t Therefore the contribution of (a) to the LHS of (3.3) is acceptable. compute (ρ) 1 k e itμ J (k)j (kρ) k + μ dkdμ L t (ρ) 1 k 1 J (k)j (kρ) k + μ dk dμ. }{{} G(, μ, ρ) L t L t Net consider (b), we To finish the proof of (3.3), we would have to show that G(, μ, ρ) is uniformly bounded. However one can check using the asymptotics of Bessel functions that this is not true! Remark 3.1 Although (3.1) is not true, we can show that an ɛ-perturbation of (3.1) is true. Namely for ɛ>, we have n 1 ɛ 1 ɛ e i(t s)δ y n 1 ɛ f(s, y)ds f L 1 L. (3.5) L L t t (Rd R) We leave it as an eercise to interested readers.

7 No.6 D. Li & X.Y. Zhang: SMOOTHING ESTIMATES OF SCHRÖDINGER PROPAGATOR 19 References [1] Constantin P, Saut J -C. Local smoothing properties of dispersive equations. J Amer Math Soc, 1988, 1(): [] Christ M, Kiselev A. Maimal functions associated to filtrations. J Funct Anal, 1, 179: [3] Kato T. On the Cauchy problem for the (generalized) Korteweg-de Vries equation//studies in applied mathematics, 93 18, Adv Math Suppl Stud 8. New York: Academic Press, 1983 [4] Kenig C, Ponce G, Vega L. Oscillatory integrals and regularity of dispersive equations. Indiana Univ Math J, 1991, 4(1): [5] Kenig C, Ponce G, Vega L. Small solutions to nonlinear Schröinger equations. Ann Inst H Poincaré Anal Non Linéaire, 1993, 1(3): [6] Li D, Zhang X. On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data. Electron J Differential Equations, 9, (78): 1 19 [7] Li D, Zhang X. On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d. Submitted [8] Sjölin P. Regularity of solutions to the Schröinger equation. Duke Math J, 1987, 55(3): [9] Vega L. Schröinger equations: pointwise convergence to the initial data. Proc Amer Math Soc, 1988, 1(4):