A REMARK ON LOCAL WELL-POSEDNESS FOR NONLINEAR SCHR¨ODINGER EQUATIONS WITH POWER NONLINEARITY –AN ALTERNATIVE APPROACH

. We study the nonlinear Schr¨odinger equation (NLS) ∂ t u + i ∆ u = iλ | u | p − 1 u in R 1+ n , where n ≥ 3, p > 1, and λ ∈ C . We prove that (NLS) is locally well-posed in H s if 1 < s < min { 4; n/ 2 } and max { 1; s/ 2 } < p < 1+4 / ( n − 2 s ). To obtain a good lower bound for p , we use fractional order Besov spaces for the time variable. The use of such spaces together with time cut-oﬀ makes it diﬃcult to derive positive powers of time length from nonlinear estimates, so that it is diﬃcult to apply the contraction mapping principle. For the proof we improve Pecher’s inequality (1997), which is a modiﬁcation of the Strichartz estimate, and apply this inequality to the nonlinear problem together with paraproduct formula.

1. Introduction. In this paper we study the Cauchy Problem for the following nonlinear Schrödinger equation where u : R 1+n → C , and f (u) = iλ|u| p−1 u with p > 1, λ ∈ C . The solvability of (1.1)-(1.2) in the Sobolev space H s = H s (R n ) has been studied in a large amount of literature. Let 0 ≤ s < n/2. It is well-known that the Cauchy problem (1.1)-(1.2) is time locally well-posed in H s if s < p ≤ p * (s), where p * (s) = 1 + 4/(n − 2s), see e.g. [4, 6-8, 11-13, 15, 21]. On the one hand, the condition p ≤ p * (s) comes from scaling; the upper bound p * (s) is the critical exponent in H s from the scaling point of view. On the other hand, the condition s < p comes from the regularity of the nonlinear term. When we solve (1.1)- (1.2) in H s , we usually take spatial derivatives of order s of the equation. Namely this lower bound is the condition for the nonlinear term to be differentiable at least s times. However, this lower bound for p is not necessarily optimal. For example, (1.1)-(1.2) is time locally wellposed in H 2 if 1 < p ≤ p * (2) = 1 + 4/(n − 4). (For simplicity we only consider the case n ≥ 5.) This result was first proved by Tsutsumi [20] in the case where 1 < p < 1+4/(n−2) with λ ∈ R, generalized by Kato [11,12] in the subcritical case 1 < p < 1 + 4/(n − 4) with λ ∈ C , and recently settled by Cazenave-Fang-Han [3] in the critical case p = 1 + 4/(n − 4). The point is that we can first evaluate ∂ t u instead of ∆u, since the Schrödinger equation is second order in x and first order in t. Once we obtain the estimate of ∂ t u, then using the equation itself we can recover spatial regularity. For 1 < s < 2, Pecher [17] treated similar problem and showed that (1.1)-(1.2) is time locally well-posed in H s if 1 < s < p * (s) (see also Fang-Han [5]). One of main ingredients in his result is a modification of Strichartz estimates by which we can replace fractional order spatial derivatives with half the numbers of time derivatives in terms of Besov spaces. The result in [17] was extended to the case where 2 < s < 4 and s/2 < p < p * (s) by Uchizono-Wada [23].
We also need the definition of admissible pairs.
Now we can state modified Strichartz estimates by Pecher [17] as follows. The statement includes a slight improvement by Uchizono-Wada [22,23].
There are several equivalent definitions of the Besov space. For simplicity, let 1 ≤ p < ∞, 1 ≤ α ≤ ∞ and 0 < θ < 1. Firstly, we can define the Besov space B θ q,α (R; V ) by the Littlewood-Paley decomposition as above; we denote this space by B 1 (R; V ) in the introduction. Secondly, we can define the Besov space by real interpolation; namely we define B 2 (R; V ) = (L q (R; V ), W 1 q (R; V )) θ,α . Thirdly, we can define the Besov space by finite difference; namely we define ) and the norms of these spaces are mutually equivalent (see [18]). To consider the time local theory, we need Besov spaces on intervals. Let I ⊂ R be an interval. We can define the Besov space B θ q,α (I; V ) in several ways. Firstly, we can define this space by restriction, namely we define Secondly, we can define B 2 (I; V ) = (L q (I; V ), W 1 q (I; V )) θ,α . Thirdly, we can define For fixed I, we again have B 1 (I; V ) = B 2 (I; V ) = B 3 (I; V ) with equivalence of the norms (see [19]). However, it is not clear whether the norms on these spaces are uniformly equivalent with respect to |I|, namely the length of I. To prove the time local well-posedness of (1.1)-(1.2) for large data, we should take |I| small enough so that the contraction mapping principle works. Therefore it is important to observe how various constants in both linear and nonlinear estimates depend on |I|. In the preceding works [5,17,22,23], the proof of Theorem A is based on the the restriction method and real interpolation, on the other hand the nonlinear estimates are based on the finite difference. Hence it is important to ensure the uniform equivalence of the norms in B i (I; V ), i = 1, 2, 3, with respect to |I|.
Alternatively, we can only use restriction method, but if we take this approach, we should multiply time cut-off by the nonlinear term, so that negative powers of |I| appear from time derivatives of the cut-off function, which makes it difficult for the contraction mapping principle to work.
However, in the preceding works do not seem to take this point into account. Therefore, in the present paper, we will give an alternative proof of Theorem 1.1 below, which has already appeared in [5,17,23], in order to ensure the time local well-posedness really holds: Theorem 1.1. Let n ≥ 3, 1 < s < min{4; n/2} and max{1; s/2} < p < 1 + 4/(n − 2s). Then for any u 0 ∈ H s , there exists This paper is organized as follows. In §2, we first introduce the definition of the Besov space and summarize basic properties thereof. Next we introduce Lemma 2.4, which is the key estimate in the proof of Theorem 1.1. This lemma is essentially obtained in the preceding work [16], but we modify it so that we can directly apply this estimate to our problem. In §3, we prove Theorem 1.1 when 1 < s < 2. The proof for 2 < s < 4 is given in §4.

Preliminaries.
We first review the definition of Besov spaces. For the detail, we refer the reader to [2,19]. We need Littlewood-Paley functions {φ j } ∞ j=−∞ on R; namely, let φ be a function whose Fourier transformφ is a non-negative even function which satisfies suppφ ⊂ {τ ∈ R; 1/2 ≤ |τ | ≤ 2} and ∞ j=−∞φ (τ /2 j ) = 1 for τ = 0. For j ∈ Z , we setφ j (·) =φ(·/2 j ) andψ j = j k=−∞φ k . If j = 0, we simply write ψ = ψ 0 . We also need Littlewood-Paley functions on R n . For x ∈ R n , we define ψ j (x) and φ j (x) by respectively. If n = 1, then these functions coincide with previous ones. For s ∈ R and 1 ≤ r, α ≤ ∞, the Besov space B s r,α (R n ) is defined by Here * x denotes the convolution with respect to the variables in R n . We next prepare the Besov space of vector-valued functions. Let θ ∈ R, 1 ≤ q, α ≤ ∞ and V a Banach space. We put with trivial modification as above if α = ∞. Here * t denotes the convolution in R. In most cases, V is a function spaces on R n like L r (R n ), so that B θ q,α (R; V ) = B θ q,α (R; L r (R n )), whose elements are regarded as functions defined on the spacetime with variables (t, x) ∈ R × R n . This is why we use the symbols * t and * x . For vector-valued Besov spaces, see [1,18]. [16].
In what follows, we write L q (I; L r ) = L q (I; L r (R n )) etc. for short. Especially, if I = R, then we simply write L q (L r ) = L q (R; L r ).
The proof is essentially the same as that of Claim 4.3 in [16].
We can estimate the difference Φ(u) − Φ(v) more easily. Let [u], [v] ∈ B R . By the Strichartz estimate together with the inequality we obtain the following: Therefore, if T is sufficiently small, then Φ is a contraction mapping on (B R , d).
By the contraction mapping principle, there exists a unique fixed point of Φ in B R . Therefore we have proved the existence of the solution to (1.1)-(1.2) in X s (I). The uniqueness of the solution in C(I; H s ) was proved in [13].