CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO

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1 ON INHOMOGENEOUS STRICHARTZ ESTIMATES FOR FRACTIONA SCHRÖDINGER EQUATIONS AND THEIR APPICATIONS arxiv:5.5399v [math.ap] 8 Jul 5 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Abstract. In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger euation in the radial case. Then we apply them to the well-posedness theory for the euation i tu + α u = V(x,t)u, < α <, with radial Ḣ γ initial data below radial potentials V r t w x under the scaling-critical range α/r +n/w = α.. Introduction To begin with, let us consider the following Cauchy problem { i t u+ α u = F(x,t), < α <, u(x,) = f(x), associated with the fractional Schrödinger euation (.) i t u+ α u = V(x,t)u (.) where V : R n+ C is a potential. This euation has recently attracted interest from mathematical physics. This is because fractional uantum mechanics introduced by askin [6] is governed by the euation where it is conjectured that physical realizations may be limited to the cases of < α <. Of course, the case α = corresponds to the ordinary uantum mechanics. By Duhamel s principle, the solution of (.) is given by u(x,t) = e it α f(x) i t e i(t s) α F(,s)ds, (.3) where the propagator e it α is given by means of the Fourier transform, as follows: e it α f(x) = α f(ξ)dξ. (π) n R n e ix ξ+it ξ Then the stard approach to the problem (.) is to obtain the corresponding Strichartz estimates which control space-time integrability of (.3) in view of that of the initial datum f the forcing term F. In the classical case α =, the Strichartz estimates originated by Strichartz [3] have been extensively studied by many authors ([9, 4,,, 8, 4, 5, 7, 8, 5, 6, ]). Over the past several years, considerable attention has been paid to the fractional order where < α < in the radial case (see [,, 3] references Mathematics Subject Classification. Primary: 35B45, 35A; Secondary: 35Q4. Key words phrases. Strichartz estimates, Well-posedness, Schrödinger euations. The first author was supported by POSTECH Math BK Plus Organization. The second author was supported by NRF grant

2 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO therein). From these works, when n/(n ) α <, the homogeneous Strichartz estimate holds for radial functions f Ḣγ (R n ) if α + n p = n e it α f t p x f Ḣ γ (.4) 4n γ, (,p) (, ). (.5) n 3 Here the condition (.5) is optimal if n/(n ) < α <. But when α = n/(n ), (.4) is unknown for the endpoint (,p) = (,(4n )/(n 3)). Also, it is known that the estimate does not hold in general if f does not have radial symmetry. Now, by duality the Christ-Kiselev lemma ([4]), one may use (.4) with γ = to get some inhomogeneous estimates t e i(t s) α F(,s)ds t p x F p t x (.6) for (,p) (, p) which satisfy (.5) with γ = >. This means that (,p) (, p) lie on the segment AD in Figure. However, these trivial estimates are not enough to imply the well-posedness for the euation (.) with the initial data f Ḣγ (R n ) beyond the case γ =. When γ we need to obtain (.6) on a wider range of (,p) (, p). See Section for details. et us first mention the following necessary conditions for (.6): α( + ) n( ) = (.7) p p + n p < n, + ñ p < n. (.8) The first is just the scaling condition the second will be shown in Section 4. Our main result in this paper is the following theorem where we obtain (.6) on the open segment BC in Figure. Theorem.. et n n/(n ) α <. Assume that F(x,t) is a radial function with respect to the spatial variable x. Then we have t e i(t s) α F(,s)ds F (.9) x,t if, < n (n+) x,t + = n n+α. (.) Remark.. It should be noted that the range (.) is sharp. Namely, the second condition in (.) is the scaling condition for (.9) (see (.7)), the first one is the necessary condition (.8) when = p = p. From interpolation between (.6) (.9), we can directly obtain further estimates when (,p) (, p) are contained in the open uadrangle with vertices A, B, D, C. Precisely, we have the following corollary.

3 INHOMOGENEOUS STRICHARTZ ESTIMATES 3 Figure. The range for Corollary.3. Here the open segment (B,C) is the range for (.9), [A,D] is the range for (.6). Corollary.3. et n n/(n ) α <. Assume that F(x,t) is a radial function with respect to the spatial variable x, that (,p) (, p) satisfy the necessary conditions (.7) (.8). Then we have t if the following conditions hold: For (,p), e i(t s) α F(,s)ds t p x F p t x (.) n(n+ α) (α )n+α ( p ) < < n (α )n+α ( p n α n )+, (.) p > (α )n α n α ((α )n+α)n + α(n+ α) ((α )n+α). (.3) Similarly for (, p). et usgivemoredetailsabout the conditionsinthe abovecorollary. The linebd in Figure is when the euality holds in the first ineuality of (.). Similarly, the lines AC AB correspond to the second ineuality in (.) the ineuality (.3), respectively. Finally, the line CD is sharp because it is determined from the necessary condition (.8). Now we apply the above Strichartz estimates to the well-posedness theory for the fractional Schrödinger euation in the radial case: { i t u+ α u = V(x,t)u, < α <, (.4) u(x,) = f(x), where we assume that u, f V are radial functions with respect to the spatial variable x. We obtain the following well-posedness for (.4) with Ḣγ initial data f below potentials V r t w x under the scaling-critical range α/r+n/w = α. The Cauchy problem (.4) was studied in [7, 9] particularly when α = γ =.

4 4 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Theorem.4. et n n α < (α )n (n+) < γ for n. Assume that f Ḣγ (R n ) V r t([,t]; w x(r n )) for some T >. Then there exists a uniue solution u C([,T];Ḣγ (R n )) t ([,T];p x (Rn )) of (.4) if α + n p = n γ, α r + n = α, (.5) w n(n+α)(α )+γ((α )n+α) n(n+α)(α ) γ α < < γ (α )n +, (.6) < r < + γ(n+ α) (n+α)(α ). (.7) Remark.5. The condition α/r+n/w = α on the potential is critical in the sense of scaling. Indeed, u ǫ (x,t) = u(ǫx,ǫ α t) takes (.4) into i t u ǫ + α u ǫ = V ǫ (x,t)u ǫ with V ǫ (x,t) = ǫ α V(ǫx,ǫ α t). Hence the norm V ǫ r t w x = V ǫ r t (R; w x (Rn )) = ǫ α α/r n/w V r t w x is independent of ǫ precisely when α/r+n/w = α. Remark.6. In Proposition 3.9 of [], the inhomogeneous estimates were shown in certain region that lies below the segment ED in Figure. (Note that (/,/p) ED if only if,p / + (n )/p = n /. Also, the point E is the same as A when α = n/(n ).) By interpolation between these our estimates, we can also obtain further estimates in the triangle with vertices A,C,E. We omit the details since it does not affect the range (α )n (n+) < γ of γ in Theorem.4 (see Section ). The rest of the paper is organized as follows. In Section, we prove Theorem.4 by making use of the Strichartz estimates (.4) (.). Section 3 is devoted to proving Theorem., in Section 4 we show the necessary condition (.8). In the final section, Section 5, we show emma 3.5 which gives some estimates for Bessel functions is used for the proof of Theorem.. Throughout the paper, we shall use the letter C to denote positive constants which may be different at each occurrence. We also use the symbol f to denote the Fourier transform of f, denote A B A B to mean A CB CB A CB, respectively, with unspecified constants C >.. Application In this section we prove Theorem.4. The proof is uite stard but we need to observe that if (, p) (, p) satisfy the inhomogeneous estimate (.), then the midpoint of them lies on the segment AD. Note that (/,/p) AD if only if,p α/ +n/p = n/. Hence, if α/ +n/p = n/ γ for γ R, then (, p) should satisfy α/ +n/ p = n/+γ to give (.). In what follows, it will be convenient to keep in mind this key observation. By Duhamel s principle, the solution of (.4) is given by Φ(u) := e it α f(x) i t e i(t s) α V(,s)u(,s)ds. (.)

5 INHOMOGENEOUS STRICHARTZ ESTIMATES 5 Then the stard fixed-point argument is to choose the solution space on which Φ is a contraction mapping. The Strichartz estimates play a central role in this step. Indeed, by the estimates (.4) (.), we see that if Φ(u) t ([,T];p x) C f Ḣγ +C Vu t ([,T]; p x ) γ(n+ α) (n+α)(α ) (.) (α )n < γ (n+) (.3) α + n p = n γ, α + ñ p = n +γ, (.4) γ α < < γ (α )n +, (.5) n(n+α)(α )+γ((α )n+α) < <. (.6) n(n+α)(α ) Here, the conditions (.3) (.5) are given from that the line α + n p = n γ lies in the closedtrianglewith verticesa,c,d except the closedsegments[a,c],[c,d]. Note that γ = (α )n (n+) when this line passes through the point C. Similarly, the condition (.6) is given from that the line α + ñ p = n +γ lies in the closed triangle with vertices A,B,D except the closed segments [A,B],[B,D]. By Hölder s ineuality, we now get Φ(u) t ([,T];p x) C f Ḣγ +C V r t ([,T]; w x ) u t ([,T];p x) if α/r +n/w = α the condition (.7) holds. Indeed, when applying Hölder s ineuality to the second term in the right-h side of (.), the conditions α/r+ n/w = α (.7) follow from (.4) (.6), respectively. From the above argument the linearity, it follows that Φ(u) Φ(v) t ([,T];p x) C V r t ([,T]; w x ) u v t ([,T];p x) u v t ([,T];p x), whichsaysthatφisacontractionmapping, ift issufficiently small. Buthere, since the above process works also on time-translated small intervals if u(,t) Ḣγ (R n ) forall t, the smallnessassumption on T can be removedby iteratingthe process a finite number of times. For this we will show that u t Ḣ γ x f Ḣ γ + V r t w x u t p x. (.7) From (.), we first see that t u t Ḣx γ eit α D γ f t + e i(t s) α D γ (V(,s)u(,s))ds x t x. Since e it α is an isometry in, the first term in the right-h side is clearly bounded by f Ḣγ. On the other h, by the inhomogeneous estimate (.6) the second term is bounded by D γ (Vu) ũ, where ũ,ṽ α/ũ+n/ṽ = n/. t ṽ x Here we use the Sobolev embedding g b D β g a,

6 6 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO where /a /b = β/n with β < n/a < a <, Hölder s ineuality to get D γ (Vu) ũ Vu t ṽ x ũ t a x V r t w u x. t p x The reuired conditions here are summarized as follows: α ũ,ṽ, ũ + ñ v = n, a ṽ = γ n, γ < n a, < a <, ũ = r +, a = w + p. But, the ineualities ũ, ṽ < a < are satisfied automatically from the conditions on,r,p,w in Theorem.4. On the other h, the ineuality γ < n a is redundant because ṽ. The remaining four eualities is reduced to the following one euality α + n p n +γ = (α r + n w α) which is clearly satisfied from the condition (.5). Conseuently, we get (.7). 3. Inhomogeneous Strichartz estimates In this section we prove Theorem.. et us first consider the multiplier operators P k for k Z defined by P k f = φ( / k ) f, where φ : R [,] is a smooth cut-off function which is supported in (/,) satisfies k Z φ( /k ) =. Then we will obtain the following freuency localized estimates (Proposition 3.) which imply Theorem.. Proposition 3.. et n n/(n ) α <. Assume that F(x,t) is a radial function with respect to the spatial variable x. Then we have e i(t s) α P k F(,s)ds F (3.) x,t uniformly in k Z if, < R n (n+) x,t + = n n+α. (3.) Indeed, since > from the first condition in (.), by the ittlewood-paley theorem Minkowski integral ineuality, one can see that e i(t s) α F(,s)ds ( C e i(t s) α P k F(,s)ds ) / x,t R k Z C k Z R R e i(t s) α P k ( j k Now, by (3.), the right-h side in the above is bounded by C P j F. x,t k Z j k x,t P j F(,s) ) ds x,t.

7 INHOMOGENEOUS STRICHARTZ ESTIMATES 7 Since <,usingtheminkowskiintegralineualityittlewood-paleytheorem, this is bounded by C F. By this boundedness < <, one may now x,t use the Christ-Kiselev lemma ([4]) to obtain t e i(t s) α F(,s)ds F x,t x,t as desired. Now iemains to prove the above proposition. 3.. Proof of Proposition 3.. Since we are assuming the scaling condition in (3.), by rescaling (x,t) (λx,λ α t), we may show (3.) only for k =. et us first consider x = rx, y = λy ξ = ρξ for x,y,ξ S n, where r = x, λ = y ρ = ξ. Then by using the fact (see [], p. 347) that S n e irρx ξ dx = c n (rρ) n Jn (rρ), where J m denotes the Bessel function with order m, it is easy to see that e i(t s) α P F(,s)ds (3.3) R = e iρ(rx λy ) ξ +i(t s)ρ α φ(ρ)f(λy,s)(λρ) n dy dξ dρdλds R R R S n S ( n = r n λ n R R e i(t s)ρα Jn Now we define the operators T j h, j, as for j T h(r,t) = χ (,) (r) r n T j h(r,t) = χ [ j, j )(r) r n [λ (rρ)j n (λρ)ρφ(ρ)dρ) n F(λy,s) ] dλds. e itρα Jn (rρ)ϕ(ρ)h(ρ)dρ (3.4) e itρα Jn (rρ)ϕ(ρ)h(ρ)dρ, (3.5) where χ A denotes the characteristic function of a set A ϕ (ρ) = ρφ(ρ). Then the adjoint operator Tk of T k is given by T H(ρ) = ϕ(ρ) e isρα χ (,) (λ) λ n Jn (λρ)h(λ, s)dλds for k TkH(ρ) = ϕ(ρ) R R e isρα χ [ k, k )(λ) λ n Jn (λρ)h(λ, s)dλds. (3.6) Hence, since F(λy,s) is independent of y S n, by setting H(λ,s) = F(λy,s), it follows from (3.3) that e i(t s) α P F(,s)ds = T j Tk (λn H). R Now we are reduced to showing that T j Tk (λn H) j,k j,k t r where we denote by r the space (r n dr). H, (3.7)

8 8 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Figure. The range of, for emma 3.. Here the open segment (A,B) is the range for Proposition 3. (see (3.)). From now on, we will show (3.7) by making use of the following lemma which will be obtained in Subsection 3.. emma 3.. et n n/(n ) α <. Then we have for j,k T j T k(λ n H) t r if, 6 (see Figure ). j(n+ n 4 ) k(n+ n 4 ) j k ( max(, )) H The case n/(n ) < α <. We first decompose the sum over j,k into two parts, j k j k: T j Tk(λ n H) t r j,k j= k=j T j T k(λ n H) t r + When j k, using emma 3., we then have T j Tk(λ n H) t r j= k=j = j= k=j j= k= j=k T j T k(λ n H) t r. n+ j( n 4 ) k(n+ n 4 ) j k ( max(, )) H j(n+ n 4 max(, )) k=j Note here that the first condition in (3.) implies n+ k(n+ n + max(, )) H n + max(, ) (n+)max(, ) n <. (3.8).

9 INHOMOGENEOUS STRICHARTZ ESTIMATES 9 From this, it follows that j= j(n+ n 4 max(, )) k(n+ n + max(, )) k=j j= On the other h, the second condition in (3.) implies j(n+ ( + ) n ). n+ ( n + ) < (3.9) since α > n/(n ). Conseuently, we get T j Tk (λn H) j(n+ t ( + ) n ) H r j= k=j j= H as desired. The other part where j k follows clearly from the same argument. The case α = n/(n ). The previous argument is no longer available in this case, since the left-h side in (3.9) becomes zero. But here we deduce (3.7) from bilinear interpolation between bilinear form estimates which follow from emma 3.. This enables us to gain some summability as before. et us first define the bilinear operators B j,k by B j,k (H, H) = Tk(λ n H),Tj (λ n H), where, denotes the usual inner product on the space r,t. Then it is enough to show the following bilinear form estimate B j,k (H, H) H H. (3.) j,k In fact, from (3.) we get T j Tk (λn H) j,k t r r,t t r = sup T j Tk (λn H) r n H(r,t)drdt H = j,k r,t = sup H r,t =j,k sup H H = r,t T k (λn H),T j (r n H) r,t r n H = H t r as desired. For (3.), we first decompose the sum over j,k into two parts, j k j k: B j,k (H, H) B j,k (H, H) + B j,k (H, H). j,k j= k=j k= j=k

10 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Then we will use the following estimate which follows from Hölder s ineuality emma 3.: B j,k (H, H) = H(r,s)r n r n T j Tk(λ n H)drds H Tj Tk(λ n H) t r t r n+ j( n 4 ) k(n+ n 4 ) j k ( max(, )) H t p H t r for, 6. By using this (3.8), the first part where j k is now bounded as follows: B j,k (H, H) j= = k=j j= k=j j= j= j(n+ n 4 ) k(n+ n 4 ) j k ( max(, )) H j(n+ n 4 max(, )) j(n+ ( + ) n ) H k=j k(n+ n + max(, )) H H t r H t r H t r (3.) for (n+)/n <, 6. If one applies this bound directly for, satisfying the conditions in Proposition 3. as in the previous case, then one can not sum over j because n+ ( n + ) = when α = n/(n ). But here we will make use of the following bilinear interpolation lemma (see [], Section 3.3, Exercise 5(b)) together with (3.) to give B j,k (H, H) H H. (3.) j= k=j t r emma 3.3. For i =,, let A i,b i,c i be Banach spaces let T be a bilinear operator such that T : A B C, T : A B C, T : A B C. Then one has, for θ = θ +θ / +/r, T : (A,A ) θ, (B,B ) θ,r (C,C ) θ,. Here, < θ i < θ <,r. Indeed, let us first consider the vector-valued bilinear operator T defined by T(H, H) = { T j (H, H) } j, where T j = k=j B j,k. Then, (3.) is euivalent to T : t r t r l (C), (3.3)

11 INHOMOGENEOUS STRICHARTZ ESTIMATES where l a p (C), a R, p, denotes the weighted seuence space euipped with the norm { ( {x j } j l a p = j jap x j p) p if p, sup j ja x j if p =. Now, by (3.) we see that where (n+)/n <, 6 T(H, H) β(,) l H (C) β(,) = n H n+ ( + ). t r, (3.4) Also, for (,) satisfying (3.), we can take a sufficiently small ǫ > such that the ball B((, ),3ǫ) with center (, ) radius 3ǫ is contained in the region of (, ) given by (n+)/n <, 6 (see Figure ). Now, we choose,,, such that = ǫ, = +ǫ, = ǫ, = +ǫ. Then it is easy to check that β(, ) = (n+)ǫ, β(, ) = n+ we get from (3.4) the following three bounds T : t r l (n+)ǫ (C), T : t r l n+ ǫ, β(, ) = n+ ǫ, ǫ T : t r l n+ ǫ (C), (C). Then, by applying emma 3.3 with θ = θ = /3 = r =, we get T : ( t r, t r ) /3, ( t r, t r ) /3, (l (n+)ǫ (C),l n+ ǫ (C)) /3,. Since p t p r = p t,r(r n drdt), by applying the real interpolation space identities in the following lemma, one can easily deduce (3.3) from the above boundedness. emma 3.4 (Theorems in []). et < θ <,,. Then one has ( (w ), (w )) θ, = (w) if = θ + θ w(x) = w (x) θ w (x) θ, if s = ( θ)s +θs (l s,l s ) θ, = l s. It is clear from the same argument that the second part where j k is bounded as follows: B j,k (H, H) H H. k= j=k t r Conseuently, we have the desired estimate (3.).

12 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO 3.. Proof of emma 3.. Iemains to prove emma 3.. We have to prove the following estimate: For j,k, T j T k(λ n H) t r j(n+ n 4 ) k(n+ n 4 ) j k ( max(, )) H (3.5) if, 6. The proof is divided into the case j,k the cases where j = or k =. The case j,k. First we claim that for, 6, T j Tk(λ n H) j(n+ t n 4 ) k(n+ n 4 ) H r Indeed, we note that for j 6,. (3.6) T j h j(n+ t n 4 ) h r (3.7) which follows immediately from applying Proposition 3. in [3] with (ρ) = ρ α, R j, p =. Then by the usual TT argument, it is not difficult to see that (3.7) implies T j T k H t r j(n+ n 4 ) k(n+ n 4 ) Hλ (n ) s λ for j,k, 6. Replacing H with λ n H, this gives (3.6). When j k, (3.5) follows now directly from (3.6). So we are reduced to showing (3.5) when j k >. For this we will obtain T j Tk(λ n n n j H) t 4 k 4 4 j k H r t r (3.8) when j k >. By interpolation between the estimates (3.6) (3.8), we then get (3.5) when j k >. Indeed, when max(, ) =, by interpolation between (3.6) with = (3.8), it is easy to check that (3.5) holds for /3. This range of, is wider than what is given by, 6. When max(, ) =, one can similarly get (3.5) by interpolation between (3.6) with = (3.8). Now iemains to show the estimate (3.8). From (3.5) (3.6), we first write T j TkH(r,t) = K(r,λ,t s)h(λ,s)dλds, where K(r,λ,t) is given as K(r,λ,t) = χ [ j, j )(r) χ [ k, k )(λ) r n λ n e itρα Jn Then, (3.8) would follow from the uniform bound (rρ)j n (λρ)ϕ (ρ)dρ. n n j K r,λ,t 4 k 4 4 j k. (3.9) To show this bound, we will divide K into four parts based on the following estimates for Bessel functions J ν (r). emma 3.5. For r > Reν > /, J ν (r) = cos(r νπ πr π 4 ) (ν )Γ(ν + 3 ) (π) (r) 3 Γ(ν + ) sin(r νπ π 4 )+E ν(r),

13 INHOMOGENEOUS STRICHARTZ ESTIMATES 3 where E ν (r) C ν r 5 (3.) d dr E ν(r) C ν (r 5 +r 7 ). (3.) Assuming this lemma which will be shown in Section 5, we see that Jn Jn (λρ) = (c n (λρ) +cn (λρ) 3 )e ±iλρ +En (λρ) (3.) (rρ) = (c n (rρ) +cn (rρ) 3 )e ±irρ +En (rρ), where the letter c n sts for constants different at each occurrence depending only on n. Now we write where Jn (λρ)j n (rρ) = 4 J l (r,λ,ρ), J (r,λ,ρ) = (c n (λρ) +cn (λρ) 3 )e ±iλρ (c n (rρ) +cn (rρ) 3 )e ±irρ, l= J (r,λ,ρ) = (c n (λρ) +cn (λρ) 3 )e ±iλρ En (rρ), J 3 (r,λ,ρ) = (c n (rρ) +cn (rρ) 3 )e ±irρ En (λρ), J 4 (r,λ,ρ) = En (λρ)e n (rρ). Then, K is divided as K = 4 l= K l, where K l (r,λ,t) = χ [ j, j )(r) r n χ [ k, )(λ) k e itρα J l (r,λ,ρ)ϕ (ρ)dρ. λ n First, it follows easily from (3.) that n+3 j K 4 r,λ,t k n+3 n n j 4 k 4 4 j k. Next, we shall consider K. Since the factors (λρ) 3 (rρ) 3 in J would give a better boundedness than (λρ) (rρ), respectively, we only need to show the bound (3.9) for K (r,λ,t) = χ [ j, j )(r) χ [ k, k )(λ) (λr) e itρα ±iλρ±irρ ρ ϕ (ρ)dρ. r n et us now decompose K as λ n K (r,λ,t) = χ { m(j,k) 8 < t <8 m(j,k) } (t) K (r,λ,t) +( χ { m(j,k) 8 < t <8 m(j,k) } (t)) K (r,λ,t), where m(j,k) = max(j,k). When m(j,k) 8 < t < 8 m(j,k), by the van der Corput lemma (see [], Chap. VIII) it follows that e itρα ±iλρ±irρ ρ ϕ (ρ)dρ m(j,k).

14 4 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Hence, we get χ (t) K { m(j,k) 8 < t <8 m(j,k) } k n j n m(j,k) r,λ,t n n j = 4 k 4 4 j k. For the second part where t > 8 m(j,k) or t < m(j,k) 8, we first note that ( α(α )tρ α ) i(±λ±r+αtρ α ) e ±iλρ±irρ+itρα ρ ϕ (ρ)dρ [ ] ρ= = i(±λ±r+αtρ α ) e±iλρ±irρ+itρα ρ ϕ (ρ) ρ=/ d ( ) i(±λ±r +αtρ α ρ ϕ (ρ) dρ ) e±iλρ±irρ+itρα dρ by integration by parts. Since we are hling the case where j k >, we see that ±λ±r+αtρ α m(j,k) when t > 8 m(j,k) or t < m(j,k) 8. Hence, using this, we get e ±iλρ±irρ+itρα ρ ϕ α(α )tρ α (ρ)dρ (±λ±r+αtρ α ) ρ ϕ (ρ) dρ [ + (±λ±r+αtρ α ) ρ ϕ (ρ) d + (±λ±r+αtρ α ) dρ This implies m(j,k). ] ρ= ρ=/ ( ρ ϕ (ρ)) dρ ( χ (t)) K { m(j,k) 8 < t <8 m(j,k) } r,λ,t k n j n m(j,k) n n j 4 k 4 4 j k. Iemains to bound K K 3. We shall show the bound (3.9) only for K because the same type of argument used for K works clearly on K 3. Since the factor (λρ) 3 in J would give a better boundedness than (λρ), we only need to show the bound (3.9) for K (r,λ,t) = χ [ j, j )(r) χ [ k, k )(λ) r n et us now decompose K as λ n e itρα ±iλρ En (rρ)(λρ) ϕ (ρ)dρ. K (r,λ,t) = χ { m(j,k) 8 < t <8 m(j,k) } (t) K (r,λ,t) +( χ { m(j,k) 8 < t <8 m(j,k) } (t)) K (r,λ,t),

15 INHOMOGENEOUS STRICHARTZ ESTIMATES 5 where m(j,k) = max(j,k). When m(j,k) 8 < t < 8 m(j,k), by the van der Corput lemma as before, it follows that χ { m(j,k) 8 < t <8 m(j,k) } (t) K (r,λ,t) χ [ k, )(λ)χ k [ j, j )(r) λ m(j,k) sup ρ [/,] λ n r n { En (rρ)ρ ϕ (ρ), d ( En (rρ)ρ ϕ (ρ) )}. dρ By (3.) (3.) in emma 3.5, we see En (rρ) r 5 d (rρ) r 3 dρ En (3.3) for / ρ. Thus, we get χ{ (t) K m(j,k) 8 < t <8 m(j,k) } k n j n m(j,k) j 3 r,λ,t n n j 4 k 4 4 j k. For the second part where t > 8 m(j,k) or t < m(j,k) 8, we will use the following trivial bound when m(j,k) = j r j : e itρα ±iλρ En (rρ)ρ ϕ (ρ)dρ En (rρ)ρ ϕ (ρ) dρ 5 j = m(j,k) 3 j which follows from (3.3). On the other h, when m(j,k) = k r j, we will also show e itρα ±iλρ En (rρ)ρ ϕ (ρ)dρ m(j,k) 3 j. Indeed, by integration by parts we see that ( ) α(α )tρα i(±λ+αtρ α ) [ = e ±iλρ+itρα En i(±λ+αtρ α ) e±iλρ+itρα En i(±λ+αtρ α ) e±iλρ+itρα d dρ (rρ)ρ ϕ (ρ)dρ ( ] ρ= (rρ)ρ ϕ (ρ) En ρ=/ ) (rρ)ρ ϕ (ρ) dρ.

16 6 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO Since m(j,k) = k, one can easily check that ±λ+αtρ α m(j,k) when t > 8 m(j,k) or t < m(j,k) 8. Hence, using this (3.3), we get e ±iλρ+itρα En (rρ)ρ ϕ (ρ)dρ α(α )tρ α (±λ+αtρ α (rρ)ρ ϕ (ρ) dρ ) En [ ] ρ= + (±λ+αtρ α ) En (rρ)ρ ϕ (ρ) ρ=/ d ( + (±λ+αtρ α En (rρ)ρ ϕ dρ (ρ)) ) dρ m(j,k) 3 j as desired. Conseuently, if r j (rρ)ρ ϕ (ρ)dρ m(j,k) 3 j e itρα ±iλρ En when t > 8 m(j,k) or t < m(j,k) 8. This implies ( χ (t)) K { m(j,k) 8 < t <8 m(j,k) } r,λ,t k n j n m(j,k) 3 j n n j 4 k 4 4 j k. The cases where j = or k =. Now we consider the following cases where j = or k = in (3.5): ( T Tk(λ n H) k(n+ t n 4 ) k max( ) H, ) r, (3.4) T j T n+ (λn H) t j( n 4 ) j r ( max(, ) ) H, (3.5) T T (λn H) H t r, (3.6) where j,k, 6. Since the second estimate (3.5) follows easily from the first one using the dual characterisation of p spaces a property of adjoint operators, we only show (3.4) (3.6) repeating the previous argument. But here we use the following estimates(see[], p. 46)forBesselfunctionsinstead ofemma3.5: For r < Reν > /, J ν (r) C ν r ν d dr J ν(r) Cν r ν. (3.7) First we shall show (3.6). Recall that T h(t,r) = χ (,) (r)r n Then, by changing variables ρ = ρ α, we see that e itρα Jn (rρ)ϕ(ρ)h(ρ)dρ = α e itρα Jn (rρ)ϕ(ρ)h(ρ)dρ. e itρ Jn (rρ /α )ϕ(ρ /α )h(ρ /α )ρ /α dρ.

17 INHOMOGENEOUS STRICHARTZ ESTIMATES 7 Thus, using Plancherel s theorem in t (3.7), we get T h t = T h r r t = Cχ (,] (r)r n Jn ( = C r (n ) / / ( h(ρ) ρ α ( / Also, by Hölder s ineuality, ) / h(ρ) dρ (rρ /α )ϕ(ρ /α )h(ρ /α )ρ /α r ρ ) / Jn (rρ) h(ρ) ρ α dρr n dr ) / Cr(rρ) n drdρ = h. (3.8) T h t r h. By interpolation between this (3.8), we obtain T h t r h (3.9) for. Then, by the usual TT argument as before, this implies T T(λ n H) H t r for,. Now we turn to (3.4). First, by using (3.9) the dual estimate of (3.7), we see that T T k(λ n H) t r T k(λ n H) k(n+ n 4 ) H for 6. Then, (3.4) wouldfollowfrom interpolationbetween this the following estimate as before (see the paragraph below (3.8)): T T k(λ n H) t r k n H t r. (3.3) Now we are reduced to showing (3.3). From (3.4) (3.6), we first write T Tk H(r,t) = K(r,λ,t s)h(λ,s)dλds, where K(r,λ,t) = χ (,)(r) r n Then, we only need to show that Recall from (3.) that Jn χ [ k, k )(λ) λ n e itρα Jn (rρ)j n (λρ)ϕ (ρ)dρ. K(r,λ,t) r,λ,t k n. (3.3) (λρ) = (c n (λρ) +cn (λρ) 3 )e ±iλρ +En (λρ). (3.3)

18 8 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO By (3.3) (3.7), the part of K coming from En (λρ) in (3.3) is bounded as follows: χ (,)(r) r n χ [ k, )(λ) k λ n e itρα Jn (rρ)e n (λρ)ϕ (ρ)dρ k n+3 k n. Now we may consider only the part of K coming from (λρ), because the factor (λρ) 3 in (3.3) would give a better boundedness than (λρ). Namely, we have to show the bound (3.3) for K(r,λ,t) = χ (,)(r) χ [ k, k )(λ) r n λ n e itρα ±iλρ Jn (rρ)(λρ) ϕ (ρ)dρ. et us now decompose K as K(r,λ,t) = χ (t) K(r,λ,t)+( χ { k 8 < t <8 k } { (t)) K(r,λ,t). k 8 < t <8 k } When k 8 < t < 8 k, by the van der Corput lemma as before, it follows that χ { (t) K(r,λ,t) k 8 < t <8 k } By (3.7), we see χ [ k, )(λ)χ k (,) (r) λ k sup ρ [/,] λ n r n { Jn (rρ)ρ ϕ (ρ), dρ( d Jn (rρ)ρ ϕ (ρ) )}. Jn (rρ) Cr n d dρ Jn (rρ) n Cr for / ρ. Thus, we get χ{ k 8 < t <8 k } (t) K r,λ,t k n. On the other h, by integration by parts we see that ( ) α(α )tρα i(±λ+αtρ α ) [ = e ±iλρ+itρα Jn i(±λ+αtρ α ) e±iλρ+itρα Jn i(±λ+αtρ α ) e±iλρ+itρα d dρ (rρ)ρ ϕ (ρ)dρ ( ] ρ= (rρ)ρ ϕ (ρ) Jn ρ=/ ) (rρ)ρ ϕ (ρ) dρ.

19 INHOMOGENEOUS STRICHARTZ ESTIMATES 9 One can also easily check that ±λ+αtρ α k when t > 8 k or t < k 8. Hence, using this (3.7), we get e ±iλρ+itρα Jn (rρ)ρ ϕ (ρ)dρ α(α )tρ α (±λ+αtρ α (rρ)ρ ϕ (ρ) dρ ) Jn [ ] ρ= + (±λ+αtρ α (rρ)ρ ) Jn ϕ (ρ) ρ=/ d ( + (±λ+αtρ α Jn (rρ)ρ ϕ dρ (ρ)) ) dρ k r n when t > 8 k or t < k 8. This implies ( χ (t)) K n k { k 8 < t <8 k } r,λ,t k = k n. 4. Sharpness of Theorem. In this section we discuss the sharpness of Theorem.. We will show that (.8) is a necessary condition for (.6) (see Remark.). If (.8) is valid with a pair (,p) on the left a pair (, p) on the right, then it must be also valid when one switches the roles of (,p) (, p). By this duality relation, we only need to show the first condition n/p+/ < n/ in (.8). et φ be a smooth cut-off function supported in the interval [,]. et us now define F(y,s) by F(,s)(ξ) = χ (,) (s)φ( ξ ). Then one can easily see that F, s p y t e i(t s) α F(,s)ds = t p x e ix ξ+i(t s) ξ α φ( ξ )dξds e ix ξ+it ξ α( e i ξ α ) i ξ α by taking integration with respect to s as t ((N, );p x( t < x <t)) φ( ξ )dξ e is ξ α ds = e i ξ α i ξ α. Now we recall from [] (see p. 344 there) that I(λ) := e iλψ(ξ) ω(ξ)dξ λ n t ((N, );p x( t < x <t)) a j λ j wherea,ψhasanondegeneratecriticalpointatsomepointξ (i.e., ψ(ξ ) = the matrix [ ψ ξ i ξ j ] (ξ ) is invertible), ω is supported in a sufficiently small j=

20 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO neighborhood of ξ. Applying this with ψ(ξ) = t x ξ + ξ α we get ω(ξ) = ( e i ξ α ) i ξ α φ( ξ ), e ix ξ+it ξ α( e i ξ α ) i ξ α φ( ξ )dξ for sufficiently large t. Thus, if N is sufficiently large, e ix ξ+it ξ α( e i ξ α ) i ξ α φ( ξ )dξ ( t n( t Conseuently, if (.6) holds, ( N N t n t ((N, ;p x( t < x <t)) ( t n( p ) dt N ) t n(/p /) dt. dx < x <t ) But, this is not possible as N unless n(/p /) < which is euivalent to the condition n/p+/ < n/. 5. Appendix Here we shall provide a proof of emma 3.5 for estimates of Bessel functions J ν (r). It is based on easy but uite tedious calculations. First, we recall from [] (see p. 43 there) that for r > Reν > /, (r/) ν ( J ν (r) = Γ(ν + )Γ( ) ie ir e rt (t +it) ν dt ie ir e rt (t it) ν dt ), where Γ is the gamma function given by Γ(z) = Then, using the following identities. ) p dt ) x z e x dx, Rez >. (5.) ie ir (t +it) ν = (t) ν e i(r νπ π 4 ) ( it )ν ie ir (t it) ν = (t) ν e i(r νπ π 4 ) (+ it )ν together with ( it )ν = (ν )it +R ν(t) (5.) (+ it )ν = +(ν )it + R ν (t), (5.3)

21 INHOMOGENEOUS STRICHARTZ ESTIMATES one can rewrite J ν (r) = (π) r ν Γ(ν + ) (e i(r νπ π4 ) e rt t ν +e i(r νπ π 4 ) e rt t ν ( (ν ) )it +R ν(t) dt ( +(ν )it + R ) ) ν (t) dt. Here, R ν (t) R ν (t) are the remainder terms in Taylor series (5.) (5.3), respectively, which are given by R ν (t) = (ν )(ν 3 )(it ) ( it )ν 5 R ν (t) = (ν )(ν 3 )(it ) (+ it )ν 5 for some t t with < t,t < t. Now we decompose J ν (r) into three parts as J ν (r) = I +II +III, where r ν I = (π) Γ(ν + ) (e i(r νπ π 4 ) +e i(r νπ π 4 ) ) II = (ν )(π) r ν Γ(ν + ) ( ie i(r νπ π 4 ) +ie i(r νπ π 4 ) ) III = (π) r ν Γ(ν + ) ( e rt t ν dt, e rt t ν+ dt e rt t ν R ν (t) e rt t ν Rν (t) ) dt+ e i(r νπ π 4 ) e dt. i(r νπ π 4 ) Then, from the definition (5.) of Γ, we easily see that I = cos(r νπ πr π 4 ) II = (ν )Γ(ν + 3 ) (π) r 3 Γ(ν + ) sin(r νπ π 4 ). Now, the lemma is proved by taking E ν (r) = III. Indeed, to show (3.) (3.), we first note that when Reν 5/, when Reν 5/ >, R ν (t), R ν (t) C ν (ν )(ν 3 ) t, (5.4) R ν (t), R ν (t) C ν (ν )(ν 3 ) t max(t ν 5,). (5.5) Hence, when Reν 5/, by using (5.4) (5.), it follows that III C ν ν ν 3 Γ(ν + 5 ) Γ(ν + )r 5.

22 CHU-HEE CHO, YOUNGWOO KOH AND IHYEOK SEO On the other h, when Reν 5/ >, by using (5.5) (5.), we see that Conseuently, we get III C ν ν ν 3 max(γ(ν + 5 )r 5,Γ(ν)r ν ) Γ(ν + ) C ν ν ν 3 max(γ(ν + 5 ),Γ(ν)) Γ(ν + ) r 5. E ν (r) C ν r 5, (3.) is similarly shown by differentiating E ν (r) using (5.4) (5.5). References [] J. Bergh J. öfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 976. [] T. Cazenave F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger euation, Comm. Math. Phys. 47 (99), 75-. [3] Y. Cho S. ee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J. 6 (3), 99-. [4] M. Christ A. Kiselev, Maximal operators associated to filtrations, J. Funct. Anal. 79 (), [5] E. Cordero F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger euation, Math. Nachr. 8 (8), 5-4. [6] E. Cordero F. Nicola, Some new Strichartz estimates for the Schrödinger euation, J. Differential euations. 45 (8), [7] P. D Ancona, V. Pierfelice N. Visciglia, Some remarks on the Schrödinger euation with a potential in r t s x, Math. Ann. 333 (5), 7-9. [8] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Eu. (5), -4. [9] J. Ginibre G. Velo, The global Cauchy problem for the nonlinear Schrödinger euation revisited, Ann. Inst. H. Poincaré Anal. Non inéare (985), []. Grafakos, Classical Fourier analysis, Second edition, Graduate Texts in Mathematics, 49. Springer, New York, 8. [] Z. Guo Y. Wang, Improved Strichartz estimates for a class of dispersive eations in the radial case their applications to nonlinear Schrödinger wave euations, J. Anal. Math. 4 (4), -38. [] T. Kato, An,r -theory for nonlinear Schrödinger euations, Spectral scattering theory applications, Adv. Stud. Pure Math., vol. 3, Math. Soc. Japan, Tokyo (994), [3] Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl. 387 (), [4] M. Keel T. Tao, Endpoint Strichartz estimates, Amer. J. Math. (998), [5] Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger euation, J. Math. Anal. Appl. 373 (), [6] N. askin, Fractional uantum mechanics évy path integrals, Phys. ett. A 68 (), [7] S. ee I. Seo, A note on uniue continuation for the Schrödinger euation, J. Math. Anal. Appl. 389 (), [8] S. ee I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger euation, Rev. Mat. Iberoam. 3 (4), [9] V. Naibo A. Stefanov On some Schrödinger wave euations with time dependent potentials, Math. Ann. 334 (6), [] I. Seo, Uniue continuation for the Schrödinger euation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J. 6 (), 3-7. [] S. Shao, Sharp linear bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam. 5 (9), [] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, oscillatory integrals, Princeton Mathematical Series,

23 INHOMOGENEOUS STRICHARTZ ESTIMATES 3 [3] R. S. Strichartz, Restriction of Fourier transforms to uadratic surfaces decay of solutions of wave euations, Duke Math. J. 44 (977), [4] M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger euation, Trans. Amer. Math. Soc. 359 (7), Department of Mathematics, Pohang University of Science Technology, Pohang , Republic of Korea address: chcho@postech.ac.kr School of Mathematics, Korea Institute for Advanced Study, Seoul 3-7, Republic of Korea address: ywkoh@kias.re.kr Department of Mathematics, Sungkyunkwan University, Suwon , Republic of Korea address: ihseo@skku.edu