On inhomogeneous Strichartz estimates for fractional Schr\"odinger equations and their applications

In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schr\"odinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.


Introduction
To begin with, let us consider the following Cauchy problem i∂ t u + |∇| α u = F (x, t), 1 < α < 2, u(x, 0) = f (x), (1.1) associated with the fractional Schrödinger equation where V : R n+1 → C is a potential. This equation has recently attracted interest from mathematical physics. This is because fractional quantum mechanics introduced by Laskin [16] is governed by the equation where it is conjectured that physical realizations may be limited to the cases of 1 < α < 2. Of course, the case α = 2 corresponds to the ordinary quantum mechanics. By Duhamel's principle, the solution of (1.1) is given by where the propagator e it|∇| α is given by means of the Fourier transform, as follows: Then the standard approach to the problem (1.1) is to obtain the corresponding Strichartz estimates which control space-time integrability of (1.3) in view of that of the initial datum f and the forcing term F .
Theorem 1.1. Let n ≥ 2 and 2n/(2n − 1) ≤ α < 2. Assume that F (x, t) is a radial function with respect to the space variable x. Then we have (1.10) Remark 1.2. It should be noted that the range (1.10) is sharp. Namely, the second condition in (1.10) is the scaling condition for (1.9) (see (1.7)), and the first one is the necessary condition (1.8) when q = p and q = p.
From interpolation between (1.6) and (1.9), we can directly obtain further estimates when (q, p) and ( q, p) are contained in the open quadrangle with vertices A, B, D, C. Precisely, we have the following corollary.
is a radial function with respect to the space variable x, and that (q, p) and ( q, p) satisfy the necessary conditions (1.7) and (1.8). Then we have if the following conditions hold: • For (q, p), . (1.13) • Similarly for ( q, p).
Let us give more details about the conditions in the above corollary. The line BD in Figure 1 is when the equality holds in the first inequality of (1.12). Similarly, the lines AC and AB correspond to the second inequality in (1.12) and the inequality (1.13), respectively. Finally, the line CD is sharp because it is determined from the necessary condition (1.8).
Now we apply the above Strichartz estimates to the well-posedness theory for the fractional Schrödinger equation in the radial case: (1.14) where we assume that f and V are radial functions with respect to the variable x. We obtain the following well-posedness for (1.14) withḢ γ initial data f below L 2 and potentials V ∈ L r t L w x under the scaling-critical range α/r + n/w = α. The Cauchy problem (1.14) was studied in [7,19] particularly when α = 2 and γ = 0.
Remark 1.5. The condition α/r + n/w = α on the potential is critical in the sense of scaling.
x is independent of ǫ precisely when α/r + n/w = α.
Remark 1.6. In Proposition 3.9 of [11], the inhomogeneous estimates were shown in certain region that lies below the segment ED in Figure 1. (Note that (1/q, 1/p) ∈ ED if and only if q, p ≥ 2 and 2/q + (2n − 1)/p = n − 1/2. Also, the point E is the same as A when α = 2n/(2n − 1).) By interpolation between these and 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−1)n 2(n+1) < γ ≤ 0 of γ in Theorem 1.4 (see Section 2).
The rest of the paper is organized as follows. In Section 2, we prove Theorem 1.4 by making use of the Strichartz estimates (1.4) and (1.11). Section 3 is devoted to proving Theorem 1.1, and in Section 4 we show the necessary condition (1.8).
In the final section, Section 5, we show Lemma 3.5 which gives some estimates for Bessel functions and is used for the proof of Theorem 1.1.
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 , and denote A B and A ∼ B to mean A ≤ CB and CB ≤ A ≤ CB, respectively, with unspecified constants C > 0.

Application
In this section we prove Theorem 1.4. The proof is quite standard but we need to observe that if (q, p) and ( q, p) satisfy the inhomogeneous estimate (1.11), then the midpoint of them lies on the segment AD. Note that (1/q, 1/p) ∈ AD if and only if q, p ≥ 2 and α/q + n/p = n/2. Hence, if α/q + n/p = n/2 − γ for γ ∈ R, then ( q, p) should satisfy α/ q + n/ p = n/2 + γ to give (1.11). In what follows, it will be convenient to keep in mind this key observation.
By Duhamel's principle, the solution of (1.14) is given by Then the standard 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 (1.4) and (1.11), we see that and Here, the conditions ( if α/r + n/w = α and the condition (1.17) holds. Indeed, when applying Hölder's inequality to the second term in the right-hand side of (2.2), the conditions α/r + n/w = α and (1.17) follow from (2.4) and (2.6), respectively. From the above argument and the linearity, it follows that , which says that Φ is a contraction mapping, if T is sufficiently small. But here, since the above process works also on time-translated small intervals if u(·, t) ∈Ḣ γ (R n ) for all t ≥ 0, the smallness assumption on T can be removed by iterating the process a finite number of times. For this we will show that (2.7) From (2.1), we first see that Since e it|∇| α is an isometry in L 2 , the first term in the right-hand side is clearly bounded by f Ḣγ . On the other hand, by the inhomogeneous estimate (1.6) the second term is bounded by where u, v ≥ 2 and α/ u + n/ v = n/2. Here we use the Sobolev embedding where 1/a − 1/b = β/n with 0 ≤ β < n/a and 1 < a < ∞, and Hölder's inequality to get x . The required conditions here are summarized as follows: But, the inequalities u, v ≥ 2 and 1 < a < ∞ are satisfied automatically from the conditions on q, r, p, w in Theorem 1.4. On the other hand, the inequality 0 ≤ −γ < n a is redundant because v ≥ 2. The remaining four equalities is reduced to the following one equality which is clearly satisfied from the condition (1.15). Consequently, we get (2.7).

Inhomogeneous Strichartz estimates
In this section we prove Theorem 1.1. Let us first consider the multiplier operators P k for k ∈ Z defined by where φ : R → [0, 1] is a smooth cut-off function which is supported in (1/2, 2) and satisfies k∈Z φ(·/2 k ) = 1. Then we will obtain the following frequency localized estimates (Proposition 3.1) which imply Theorem 1.1.
Indeed, since q > 2 from the first condition in (1.10), by the Littlewood-Paley theorem and Minkowski integral inequality, one can see that Now, by (3.1), the right-hand side in the above is bounded by Since q ′ < 2, using the Minkowski integral inequality and Littlewood-Paley theorem, this is bounded by C F 2 . By this boundedness and q ′ < 2 < q, one may now x,t as desired. Now it remains to prove the above proposition.
3.1. Proof of Proposition 3.1. Since we are assuming the scaling condition in (3.2), by rescaling (x, t) → (λx, λ α t), we may show (3.1) only for k = 0. Let us first consider x = rx ′ , y = λy ′ and ξ = ρξ ′ for x ′ , y ′ , ξ ′ ∈ S n−1 , where r = |x|, λ = |y| and ρ = |ξ|. Then by using the fact (see [22], p. 347) that where J m denotes the Bessel function with order m, it is easy to see that Now we define the operators T j h, j ≥ 0, as and for j ≥ 1 where χ A denotes the characteristic function of a set A and ϕ 2 (ρ) = ρφ(ρ). Then the adjoint operator T * k of T k is given by Now we are reduced to showing that j,k≥0 where we denote by L q r the space L q (r n−1 dr). From now on, we will show (3.7) by making use of the following lemma which will be obtained in Subsection 3.2. Figure 2).
The case 2n/(2n − 1) < α < 2. We first decompose the sum over j, k into two parts, j ≤ k and j ≥ k: When j ≤ k, using Lemma 3.2, we then have Note here that the first condition in (3.2) implies On the other hand, the second condition in (3.2) implies as desired. The other part where j ≥ k follows clearly from the same argument.
The case α = 2n/(2n − 1). The previous argument is no longer available in this case, since the left-hand side in (3.9) becomes zero. But here we deduce (3.7) from bilinear interpolation between bilinear form estimates which follow from Lemma 3.2. This enables us to gain some summability as before. Let us first define the bilinear operators B j,k by where , denotes the usual inner product on the space L 2 r,t . Then it is enough to show the following bilinear form estimate j,k≥0 (3.10) In fact, from (3.10) we get j,k≥0 For (3.10), we first decompose the sum over j, k into two parts, j ≤ k and j ≥ k: k (H, H) .
Then we will use the following estimate which follows from Hölder's inequality and Lemma 3.2: for 2 ≤ q, q ≤ 6. By using this and (3.8), the first part where j ≤ k is now bounded as follows: for 2(n + 1)/n < q, q ≤ 6. If one applies this bound directly for q, q satisfying the conditions in Proposition 3.1 as in the previous case, then one can not sum over j because 2n+1 2 ( 1 q + 1 q ) − 2n−1 2 = 0 when α = 2n/(2n − 1). But here we will make use of the following bilinear interpolation lemma (see [1], Section 3.13, Exercise 5(b)) together with (3.11) (3.12) Lemma 3.3. For i = 0, 1, let A i , B i , C i be Banach spaces and let T be a bilinear operator such that Then one has, for θ = θ 0 + θ 1 and 1/q + 1/r ≥ 1, Here, 0 < θ i < θ < 1 and 1 ≤ q, r ≤ ∞.
Indeed, let us first consider the vector-valued bilinear operator T defined by where T j = ∞ k=j B j,k . Then, (3.12) is equivalent to where ℓ a p (C), a ∈ R, 1 ≤ p ≤ ∞, denotes the weighted sequence space equipped with the norm Now, by (3.11) we see that (3.14) where 2(n + 1)/n < q, q ≤ 6 and Also, for ( q, q) satisfying (3.2), we can take a sufficiently small ǫ > 0 such that the ball B(( 1 q , 1 q ), 3ǫ) with center ( 1 q , 1 q ) and radius 3ǫ is contained in the region of ( 1 q , 1 q ) given by 2(n + 1)/n < q, q ≤ 6 (see Figure 2). Now, we choose q 0 , q 1 , q 0 , q 1 such that Then it is easy to check that and we get from (3.14) the following three bounds Then, by applying Lemma 3.3 with θ 0 = θ 1 = 1/3 and q = r = 2, we get Since L p t L p r = L p t,r (r n−1 drdt), by applying the real interpolation space identities in the following lemma, one can easily deduce (3.13) from the above boundedness. ). Let 0 < θ < 1 and 1 ≤ q 0 , q 1 , q ≤ ∞. Then one has It is clear from the same argument that the second part where j ≥ k is bounded as follows: Consequently, we have the desired estimate (3.10).

Proof of Lemma 3.2.
It remains to prove Lemma 3.2. We have to prove the following estimate: For j, k ≥ 0, if 2 ≤ q, q ≤ 6. The proof is divided into the case j, k ≥ 1 and the cases where j = 0 or k = 0.
Now it remains to show the estimate (3.18). From (3.5) and (3.6), we first write where K(r, λ, t) is given as Then, (3.18) would follow from the uniform bound To show this bound, we will divide K into four parts based on the following estimates for Bessel functions J ν (r).
Assuming this lemma which will be shown in Section 5, we see that and where the letter c n stands for constants different at each occurrence and depending only on n. Now we write Then, K is divided as K = 4 l=1 K l , where e itρ α J l (r, λ, ρ)ϕ 2 (ρ)dρ.
Hence, we get For the second part where t > 8 · 2 m(j,k) or t < 2 m(j,k)

8
. Hence, using this, we get This implies It remains to bound K 2 and K 3 . We shall show the bound (3.19) only for K 2 because the same type of argument used for K 2 works clearly on K 3 . Since the factor (λρ) − 3 2 in J 2 would give a better boundedness than (λρ) − 1 2 , we only need to show the bound (3.19) for Let us now decompose K 2 as where m(j, k) = max(j, k). When 2 m(j,k) 8 < t < 8 · 2 m(j,k) , by the van der Corput lemma as before, it follows that By (3.20) and (3.21) in Lemma 3.5, we see For the second part where t > 8 · 2 m(j,k) or t < 2 m(j,k) 8 , we will use the following trivial bound when m(j, k) = j and r ∼ 2 j : which follows from (3.23). On the other hand, when m(j, k) = k and r ∼ 2 j , we will also show Indeed, by integration by parts we see that Since m(j, k) = k, one can easily check that | ± λ + αtρ α−1 | 2 m(j,k) when t > 8 · 2 m(j,k) or t < 2 m(j,k) 8 . Hence, using this and (3.23), we get The cases where j = 0 or k = 0. Now we consider the following cases where j = 0 or k = 0 in (3.15): and where j, k ≥ 1 and 2 ≤ q, q ≤ 6. Since the second estimate (3.25) follows easily from the first one using the dual characterisation of L p spaces and a property of adjoint operators, we only show (3.24) and (3.26) repeating the previous argument. But here we use the following estimates (see [10], p. 426) for Bessel functions instead of Lemma 3.5: For 0 ≤ r < 1 and ν > − 1 2 , |J ν (r)| ≤ C ν r ν and d dr J ν (r) ≤ C ν r ν−1 . First we shall show (3.26). Recall that Then, by changing variables ρ = ρ α , we see that (rρ 1/α )ϕ(ρ 1/α )h(ρ 1/α )ρ 1/α−1 dρ.
Thus, using Plancherel's theorem in t and (3.27), we get Also, by Hölder's inequality, By interpolation between this and (3.28), we obtain for 2 ≤ q ≤ ∞. Then, by the usual T T * argument as before, this implies Now we turn to (3.24). First, by using (3.29) and the dual estimate of (3.17), we see that for 2 ≤ q ≤ ∞ and 2 ≤ q ≤ 6. Then, (3.24) would follow from interpolation between this and the following estimate as before (see the paragraph below (3.18)): Now we are reduced to showing (3.30). From (3.4) and (3.6), we first write Then, we only need to show that (λρ) in (3.32) is bounded as follows:

Sharpness of Theorem 1.1
In this section we discuss the sharpness of Theorem 1.1. We will show that (1.8) is a necessary condition for (1.6) (see Remark 1.2). If (1.8) is valid with a pair (q, p) on the left and a pair ( q, p) on the right, then it must be also valid when one switches the roles of (q, p) and ( q, p). By this duality relation, we only need to show the first condition n/p + 1/q < n/2 in (1.8).

Appendix
Here we shall provide a proof of Lemma 3.5 for estimates of Bessel functions J ν (r). It is based on easy but quite tedious calculations.
First, we recall from [10] (see p. 430 there) that for r > 1 and Re ν > −1/2, where Γ is the gamma function given by Then, using the following identities Here, R ν (t) and R ν (t) are the remainder terms in Taylor series (5.2) and (5.3), respectively, which are given by for some t * and t * with 0 < t * , t * < t. Now we decompose J ν (r) into three parts as J ν (r) = I + II + III, where