Extremals for $\alpha$-Strichartz Inequalities

Abstract

A necessary and sufficient condition on the precompactness of extremal sequences for one-dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.

1 Introduction

For $\alpha >1$, we investigate the following symmetric $\alpha$-Strichartz inequality

$$\begin{aligned} \left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]u\right\Vert_{L_{t,x}^6({\mathbb {R}}^{2})}\le {\textbf{M}}_{\alpha } \Vert u\Vert _{L_x^2({\mathbb {R}})}, \end{aligned}$$

(1)

where

$$\begin{aligned}{\textbf{M}}_{\alpha }:=\sup \left\rbrace \left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]u\right\Vert_{L_{t,x}^{6}({\mathbb {R}}^{2})}: \Vert u\Vert _{L_x^2({\mathbb {R}})}=1\right\lbrace ,\end{aligned}$$

is the sharp constant and

$$\begin{aligned}{} & {} \big [e^{it|\nabla |^{\alpha }}\big ]u(x):={\mathscr {F}}^{-1}e^{-it|\xi |^{\alpha }}{\mathscr {F}}[u](x),\quad [D^s]u(x):={\mathscr {F}}^{-1}|\xi |^s{\mathscr {F}}[u](x),\\{} & {} \quad {\mathscr {F}}[u](\xi ):=\int _{{\mathbb {R}}} e^{-ix\xi }u(x)\text {d}x,\end{aligned}$$

with ${\mathscr {F}}$ denoting the spatial Fourier transform. This estimate (1) comes from Kenig et al. [24, Theorem 2.3] which is also named Fourier extension estimate. Moreover it says that, for every $\alpha >1$, there holds the (mixed norm) asymmetric $\alpha$-Strichartz inequality

$$\begin{aligned} \left\Vert\bigg [D^{\frac{\alpha -2}{q}}\bigg ]\bigg [e^{it|\nabla |^{\alpha }}\bigg ]u\right\Vert_{L_t^q L_x^r({\mathbb {R}}^2)} \le \tilde{{\textbf{M}}}_{\alpha , q,r} \Vert u\Vert _{L_x^2({\mathbb {R}})}, \end{aligned}$$

(2)

where $2/q+1/r=1/2$ with the sharp constant $\tilde{{\textbf{M}}}_{\alpha , q,r}$ defined by

$$\begin{aligned}\tilde{{\textbf{M}}}_{\alpha ,q,r}:=\sup \left\rbrace \left\Vert\bigg [D^{\frac{\alpha -2}{q}}\bigg ]\bigg [e^{it|\nabla |^{\alpha }}\bigg ]u\right\Vert_{L_t^q L_x^r({\mathbb {R}}^2)}: \Vert u\Vert _{L_x^2({\mathbb {R}})}=1\right\lbrace .\end{aligned}$$

We call the pairs $(q,r)=(\infty ,2)$ and $(q,r)=(4,\infty )$ endpoint pairs. Otherwise, the pairs (q, r) are called non-endpoint pairs.

The symmetries for these $\alpha$-Strichartz inequalities, on the $L_x^2$ side, are time–space translations and scaling as follows:

$$\begin{aligned}\big [g_n^{\textrm{sym}}\big ]u:=\bigg [e^{it_n|\nabla |^{\alpha }}\bigg ]\left[(h_n)^{-1/2}u\left(\frac{\cdot -x_n}{h_n}\right)\right], \quad (h_n,x_n,t_n)\in {\mathbb {R}}_{+}\times {\mathbb {R}}\times {\mathbb {R}},\end{aligned}$$

and the associated group $G^{\textrm{sym}}$ is defined by

$$\begin{aligned}G^{\textrm{sym}}:=\Big \{\big [g_n^{\textrm{sym}}\big ]: (h_n,x_n,t_n)\in {\mathbb {R}}_{+}\times {\mathbb {R}}\times {\mathbb {R}}\Big \}.\end{aligned}$$

To state the results more precisely, we say a sequence of functions $(f_n)$ in $L^2({\mathbb {R}})$ is precompact up to symmetries if there exists a sequence of symmetries $([g_n^{\textrm{sym}}])$ in $G^{\textrm{sym}}$ such that $\left([g_n^{\textrm{sym}}]f_n\right)$ has convergent subsequence in $L^2({\mathbb {R}})$. On the other hand, a sequence of functions $(f_n)$ in $L^2({\mathbb {R}})$ concentrates at a point $x_0\in {\mathbb {R}}$ if for arbitrary $\varepsilon ,\rho >0$, there exists $N\in {\mathbb {N}}_{+}$ such that for every $n>N$, there holds

$$\begin{aligned}\int _{|x-x_0|\ge \rho }|f_n(x)|^2\text {d}x\le \varepsilon \Vert f_n\Vert _{L^2({\mathbb {R}})}^2.\end{aligned}$$

Meanwhile a sequence of functions $(f_n)$ in $L^2({\mathbb {R}})$ is an extremal sequence for $\tilde{{\textbf{M}}}_{\alpha , q,r}$ if it satisfies

$$\begin{aligned}\Vert f_n\Vert _{L^2({\mathbb {R}})}=1,\quad \lim _{n\rightarrow \infty } \left\Vert\bigg [D^{\frac{\alpha -2}{q}}\bigg ]\bigg [e^{it|\nabla |^{\alpha }}\bigg ]f_n\right\Vert_{L_t^q L_x^r({\mathbb {R}}^2)}=\tilde{{\textbf{M}}}_{\alpha , q,r}.\end{aligned}$$

And a function $f_0\ne 0$ in $L^2({\mathbb {R}})$ is called an extremal function for $\tilde{{\textbf{M}}}_{\alpha ,q,r}$ if $f_0$ can make the inequality (2) an equality.

The sharp Fourier restriction theory, more generally the sharp constant theory, has been an important part in harmonic analysis. Readers are referred to the survey [15] and the references therein for some recent progress on sharp Fourier restriction theory. One of the recent results is [7, Theorem 1.3]. In our setting, we rephrase this theorem as follows.

Theorem A

([7]) All the extremal sequences for ${\textbf{M}}_{\alpha }$ are precompact up to symmetries if and only if

$$\begin{aligned} {\textbf{M}}_{\alpha }>\left[\sqrt{3}\alpha (\alpha -1)\right]^{-\frac{1}{6}}. \end{aligned}$$

(3)

In particular, if the strict inequality (3) holds, then there exists an extremal for ${\textbf{M}}_{\alpha }$. If on the contrary, the equality holds in (3), then given any $x_0\in {\mathbb {R}}$, there exists an extremal sequence for ${\textbf{M}}_{\alpha }$ which concentrates at $x_0$.

This result is previously obtained by Brocchi et al. [7, Theorem 1.3]. The proof there uses a variant of Lions’ concentration-compactness lemma from [30, 31] together with a variant of Brézis–Lieb lemma from [6, 29]. As pointed out in [7], various results with a similar condition to (3) have been studied in recent literature. In our paper, this condition comes from the asymptotic Schrödinger  behavior Lemma 6.1¹, see also Remarks 6.2 and 7.3. Roughly speaking, to get the existence of extremals, there may be some strict inequality conditions like (3) to rule out some concentrate-type situations which deduce the loss of compactness. We refer to [10, 16, 17] for more discussions on these type of conditions in the low-dimensional sphere and cubic curve cases.

The main purpose of this article is investigating the extremal problems for $\alpha$-Strichartz inequalities by means of profile decomposition arguments. One of our results, Theorem 1.1 below, generalizes the aforementioned Theorem A to asymmetric cases. As an application of our profile decomposition consequences, for $\alpha \ge 2$, we also give the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities (14) which will be presented later as Theorem 1.8². One key ingredient to establish this generalized profile decomposition Proposition 1.5 is a conditional dislocation property consequence Proposition 1.3 on the weak operator topology convergence for some $L^2$-unitary operators. Now, we state our first main result as follows.

Theorem 1.1

For the non-endpoint pairs (q, r), all the extremal sequences for $\tilde{{\textbf{M}}}_{\alpha , q,r}$ are precompact up to symmetries if and only if

$$\begin{aligned} \tilde{{\textbf{M}}}_{\alpha , q,r}>\left(\frac{\alpha ^2-\alpha }{2}\right)^{-\frac{1}{q}} \tilde{{\textbf{M}}}_{2,q,r}. \end{aligned}$$

(4)

In particular, if the strict inequality (4) holds, then there exists an extremal for $\tilde{{\textbf{M}}}_{\alpha , q,r}$. If on the contrary, the equality holds in (4), then given any $x_0\in {\mathbb {R}}$, there exists an extremal sequence for $\tilde{{\textbf{M}}}_{\alpha , q,r}$ which concentrates at $x_0$.

As we have mentioned above, Theorem 1.1 extends the previous result [7, Theorem 1.3]. Meanwhile by taking some symmetries,³ on the Fourier side, Theorem 1.1 claims that if the equality holds in (4), then there exists an extremal sequence which concentrates at one fixed frequency. Thereby, this Theorem 1.1, in some sense, also coincides with the result in [16] where the extremal sequence concentrates at two opposite frequencies due to some symmetries of the odd curves.

Here, we make some historical remarks first. For the case $\alpha =2$ in (1), the classical Strichartz inequality (Stein–Tomas inequality for the paraboloid), abundant conclusions have been made: the existence of extremals is proved by Kunze [28] for one-dimensional case and by Shao [36] for general dimensions; in low dimensions, up to symmetries, the only extremals are shown to be Gaussians by Foschi [13] and Hundertmark–Zharnitsky [21] independently. Extremals are conjectured to be Gaussians in all dimensions [21]. Meanwhile, on the Stein–Tomas inequality for the sphere, we briefly mention that Christ-Shao [10, 38] give the existence of extremals in low dimensions, and Foschi [14] shows that the extremals are constants for two-dimension sphere ${\mathbb {S}}^2$. We refer to [15, 34] and the references therein for more recent results on the sharp Fourier restriction theory in the sphere situation.

As for the case $\alpha =4$ in (1), Jiang et al. [22, 23] give some dichotomy results on the existence of extremals by using the profile decomposition from [1, 2, 8, 26, 32]. For more general case $\alpha >1$ in (1), Brocchi et al. [7] resolves the dichotomy in [22] by using a geometric comparison principle developed in [33] which resolves the dichotomy in [23]. As far as we know, there is no extremal result on the asymmetric $\alpha$-Strichartz inequality (2) with general $\alpha >1$, except for the classical $\alpha =2$ case in (2) which has been studied in some papers such as [3, 9, 18, 36]. Meanwhile, it should be mentioned that Frank–Sabin [16] has studied the existence of extremals for Airy–Strichartz inequality (odd cubic curve), of which result is also valid for non-endpoint asymmetric cases, by using the missing mass method.

Note that $\alpha >1$ may not be a natural number in our setting and this fact leads to some barriers. In order to establish the desired linear profile decomposition, one of the main results we should establish is the conditional dislocation property Proposition 1.3 for some unitary operators on $L^2({\mathbb {R}})$. We begin with the definitions for the dislocation group and the $L^2$-unitary operators that, maybe non-compact, we are concerned about. For parameters $(h_0, x_0, \xi _0, \theta _0)\in {\mathbb {R}}_{+} \times {\mathbb {R}}^d \times {\mathbb {R}}^d\times {\mathbb {R}}$, the unitary operators $[g_0]$ on $L_x^2({\mathbb {R}}^d)$ is defined by

$$\begin{aligned}\phi (x):=g_{\theta _0, \xi _0, x_0, h_0}[\phi ](x):=e^{i\theta _0}h_0^{-\frac{d}{2}}e^{i x\cdot \xi _0}\phi \left(\frac{x-x_0}{h_0}\right).\end{aligned}$$

We should point out that the parameter $\theta _0$ is inessential and we use it just because, on the Strichartz space $L_t^q L_x^r({\mathbb {R}}^2)$, it may be deduced from other parameters.

Definition 1.2

(Dislocation group [35]) Let H be a separable Hilbert space and let G be a group of unitary operators on H. We said G is a group of dislocations if it satisfies the following condition: for every sequence, $([g_n])\subset G$ does not converge weakly (in weak operator topology) to zero, there exists a renamed strongly convergent subsequence of $([g_n])$ such that the strong limit (in strong operator topology) is not zero.

In the classical case $\alpha =2$, due to the Galilean invariance of classical Schrödinger  equations, the dislocation property for the group generated by non-compact $L^2$-unitary operators is obvious. This potentially crucial fact, when establishing the classical profile decomposition, deduces the orthogonality of these decomposed profiles in Strichartz spaces. Hence, it is a natural idea to generalize this dislocation property to the $\alpha$-Strichartz setting. On the other hand, we may do some adaption along the way we generalize it. The following conditional dislocation property proposition comes from an application of stationary phase method, which is contained in [39, Chapter 8] and [46, Chapter 6], or more precisely the classical van der Corput Lemma [39, p. 332, Proposition 2].

Proposition 1.3

(Conditional dislocation property) When $d=1$, if we assume that for fixed $j\ne k$ either

$$\begin{aligned}\lim _{n\rightarrow \infty }\left(\frac{h_n^j}{h_n^k}+\frac{h_n^k}{h_n^j}+(h_n^j+h_n^k)\left|\xi _n^j-\xi _n^k\right|\right)=\infty ,\end{aligned}$$

or $(h_n^j,\xi _n^j)\equiv (h_n^k,\xi _n^k)$. Then the group G, generated by the $L^2$-symmetries

$$\begin{aligned} \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ], \end{aligned}$$

(5)

is a group of dislocations for all real numbers $\alpha >1$.

Remark 1.4

As we will see in Lemma 3.3, the assumptions in Proposition 1.3 arise naturally during the construction of linear profile decomposition. Analogous assumptions can also be seen in [22, Theorem 1.3] and [23, p. 10] as well as some earlier papers such as [1, 8, 26]. To deal with the case that $\alpha >1$ is a real number rather than integers, we make use of the stationary phase method and Taylor’s theorem. Further details are shown in Sect. 2.

The profile decomposition results are intensively studied and widely used in many topics. Besides some of the aforementioned references such as [1, 2, 8, 26, 32] which establish these profile decompositions in different analysis situations, the profile decomposition may also be called bubble decomposition in the literature due to some geometric background. We refer to [27, p. 359] for a historical discussion, see also [27, p. 373]. For fractional Schrödinger  equations, a profile decomposition without frequency parameters has appeared in [19] which considered the Cauchy problem for the energy-critical fractional nonlinear Schrödinger  equation in the radial case. Here, with the conditional dislocation property Proposition 1.3 in place, we are able to show the following $\alpha$-Strichartz version linear profile decomposition.

Proposition 1.5

(Linear profile decomposition for the $\alpha$-Strichartz version) Let $(u_n)$ be a bounded sequence in $L^2({\mathbb {R}})$. Then, up to subsequences, there exists a sequence of operators $([T_n^j])$ defined by

$$\begin{aligned}\phi (x):=\big [e^{-it_n^j|\nabla |^{\alpha }}\big ]\left[(h_n^j)^{-\frac{1}{2}}e^{i(x-x_n^j)\xi _n^j}\phi \left(\frac{x-x_n^j}{h_n^j}\right)\right],\end{aligned}$$

with $(h_n^j, x_n^j, \xi _n^j, t_n^j) \in {\mathbb {R}}_{+}\times {\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}$ and a sequence of functions $(\phi ^j)\subset L^2({\mathbb {R}})$ such that for every $J\ge 1$, we have the profile decomposition

$$\begin{aligned} u_n=\sum _{j=1}^{J} \big [T_n^j\big ]\phi ^j+\omega _n^{J}, \end{aligned}$$

(6)

where the decomposition possesses the following properties: first, the remainder term $\omega _n^{J}$ has vanishing Strichartz norm

$$\begin{aligned} \lim _{J\rightarrow \infty }\limsup _{n\rightarrow \infty }\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^{J}\right\Vert_{L_{t,x}^6({\mathbb {R}}^{2})}=0, \end{aligned}$$

(7)

second, the sequence of operators $[T_n^j]$ satisfies that if $j\ne k$, then there holds the limit-orthogonality property

$$\begin{aligned} \big [T_n^k\big ]^{-1}\big [T_n^j\big ]\rightharpoonup 0, \end{aligned}$$

(8)

as n goes to infinity in the weak operator topology of ${\mathcal {B}}(L^2)$; moreover, for each $J\ge 1$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left[\Vert u_n\Vert _{L^2({\mathbb {R}})}^2-\left(\sum _{j=1}^{J}\Vert \phi ^j\Vert _{L^2({\mathbb {R}})}^2\right)-\Vert \omega _n^{J}\Vert _{L^2({\mathbb {R}})}^2\right]=0. \end{aligned}$$

(9)

Remark 1.6

We should point out that the limit orthogonality (8) of the operators $[T_n^j]$ is crucial and powerful, especially when combined with the conditional dislocation property Proposition 1.3. By the $L^2$-almost orthogonal identity (9), we can deduce that for every $j\ne k$ in Proposition 1.5, there holds

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\langle [T_n^j]\phi ^j, [T_n^k]\phi ^k\right\rangle _{L^2({\mathbb {R}})}=0, \end{aligned}$$

(10)

and for each $j\le J$, there holds

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\langle [T_n^j]\phi ^j, \omega _n^{J}\right\rangle _{L^2({\mathbb {R}})}=0. \end{aligned}$$

(11)

Meanwhile, the $\alpha$-Strichartz version profile decomposition Proposition 1.5 is equipped with the following Strichartz orthogonality for the decomposed linear profiles.

Proposition 1.7

(Strichartz orthogonality of profiles) Furthermore, in the linear profile decomposition Proposition 1.5, for $j\ne k$, there holds

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j \cdot \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^k\big ]\phi ^k\right\Vert_{L_{t,x}^{3}({\mathbb {R}}^{2})}=0. \end{aligned}$$

(12)

Thus, for each $J\ge 1$, by Hölder’s  inequality, there holds

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left(\left\Vert\sum _{j=1}^J\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_{t,x}^6}^{6} -\sum _{j=1}^J\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_{t,x}^6}^{6}\right)=0.\nonumber \\ \end{aligned}$$

(13)

Finally, as an application of our profile decomposition results, we investigate the following estimates. For $\alpha \ge 2$, the result of Kenig et al. [24, Theorem 2.3] and Sobolev inequalities imply the following non-endpoint $\alpha$-Strichartz estimates

$$\begin{aligned} \left\Vert[e^{it|\nabla |^{\alpha }}]u\right\Vert_{L_{t,x}^{2\alpha +2}({\mathbb {R}}^2)}\le \dot{{\textbf{M}}}_{\alpha } \Vert u\Vert _{L_x^2({\mathbb {R}})}, \end{aligned}$$

(14)

where $\dot{{\textbf{M}}}_{\alpha }$ is the sharp constant

$$\begin{aligned}\dot{{\textbf{M}}}_{\alpha }:=\sup \left\rbrace \left\Vert[e^{it|\nabla |^{\alpha }}]u\right\Vert_{L_{t,x}^{2\alpha +2}({\mathbb {R}}^{2})}: \Vert u\Vert _{L_x^2({\mathbb {R}})}=1\right\lbrace .\end{aligned}$$

See, for instance, [24, Theorem 2.4] for analogous arguments. In [20], Hundertmark and Shao give the existence of extremals for some similar non-endpoint Airy–Strichartz inequalities based on the Airy–Strichartz version profile decomposition. Moreover, by using a bootstrap argument, they also establish some analyticity of these extremals on the Fourier space. In the spirit of their work and based on the generalized profile decomposition consequences obtained above, we show the existence of extremals for ${\dot{M}}_{\alpha }$ as a short incidental result.

Theorem 1.8

For every $\alpha \ge 2$, there exists an extremal for $\dot{{\textbf{M}}}_{\alpha }$.

The outline of this paper is as follows. In Sect. 2, we begin with proving the conditional dislocation property Proposition 1.3 which is one of the key ingredients in our paper. Then we extract the frequency and scaling parameters for the $\alpha$-Strichartz version linear profile decomposition in Sect. 3. After that, by using Proposition 1.3, we are able to obtain the time and space translation parameters in Sect. 4 and further present the desired linear profile decomposition in Sect. 5. Then Sects. 6 and 7 contain the extremal results for symmetric $\alpha$-Strichartz estimates Theorem A and asymmetric $\alpha$-Strichartz estimates Theorem 1.1, respectively. Finally, the proof of Theorem 1.8 is provided in Sect. 8.

We end this section with some notations. First, we use the familiar notation $x\lesssim y$ to denote that there exists a finite constant C such that $|x|\le C|y|$, similarly for $x > rsim y$ and $x\sim y$. Sometimes we may show the dependence such as $x\lesssim _{\alpha } y$ for the constant $C=C(\alpha )$ if necessary. Occasionally we may write ${\hat{u}}:={\mathscr {F}}[u]$ or $u^{\wedge }:={\mathscr {F}}[u]$, similarly for the inverse Fourier transform ${\check{u}}=u^{\vee }:={\mathscr {F}}^{-1}[u]$. In addition, since there may be different topologies throughout this paper, we use the notation $\rightarrow$ to denote strong convergence and the notation $\rightharpoonup$ to denote weak convergence. More precisely, for a sequence of functions $(f_n)\subset L^p$, we write $f_n\rightarrow f_0$ for the fact that $f_n$ converge to $f_0$ as n goes to infinity in the norm (strong) topology of $L^p$, and write $f_n \rightharpoonup f_0$ for the fact that $f_n$ converge to $f_0$ as n goes to infinity in the weak topology of $L^p$. As for a sequence of operators $([T_n])$ on the space H which means $([T_n])\subset {\mathcal {B}}(H)$, similarly $[T_n]\rightarrow [T_0]$ and $[T_n]\rightharpoonup [T_0]$ denote the convergence in the strong operator topology and weak operator topology of ${\mathcal {B}}(H)$, respectively.

2 Dislocation Property from van der Corput Lemma

Before to give the linear profile decomposition, we show the conditional dislocation property Proposition 1.3 first since it will be used in the forthcoming work of extracting time–space translation parameters in Sect. 4. As what we have said before this property is, in some sense but not directly, a generalization of the classical Schrödinger  dislocation property which comes from the Galilean invariance. Note that the conditional dislocation property Proposition 1.3 has been adapted to the desired profile decomposition Proposition 1.5 when we establish it.

Proof of Proposition 1.3

By a standard approximation argument together with the symmetry of j and k, it suffices to prove that if

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle \ne 0, \end{aligned}$$

(15)

for some Schwartz functions $\phi$ and $\psi$ whose Fourier supports are compact, then there exist one unitary operator $[G^{jk}]\in {\mathcal {B}}(L_x^2)$ and a subsequence for n (also denoted by n) such that

$$\begin{aligned} \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]f\rightarrow [G^{jk}]f, \end{aligned}$$

(16)

as $n\rightarrow \infty$ in the $L_x^2$ norm topology for all Schwartz functions f. Note that a direct computation shows

$$\begin{aligned}&2\pi \left|\big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi (x)\right| \nonumber \\&\quad =\big [g_n^j\big ]^{-1} (h_n^k)^{-\frac{1}{2}}\left|e^{ix\xi _n^k} \int _{{\mathbb {R}}} e^{i\xi \frac{x-x_n^k}{h_n^k}-i|\xi +h_n^k\xi _n^k|^{\alpha }\frac{t_n^j-t_n^k}{(h_n^k)^{\alpha }}}{\hat{\phi }}(\xi )\text {d}\xi \right| \nonumber \\&\quad =\left(\frac{h_n^j}{h_n^k}\right)^{\frac{1}{2}}\left|e^{ix h_n^j(\xi _n^k-\xi _n^j)} \int _{{\mathbb {R}}} e^{i\xi \frac{h_n^j x+x_n^j-x_n^k}{h_n^k}-i|\xi +h_n^k\xi _n^k|^\alpha \frac{t_n^j-t_n^k}{(h_n^k)^{\alpha }}} {\hat{\phi }}(\xi ) \text {d}\xi \right|. \end{aligned}$$

(17)

Step 1 We eliminate the case $\lim _{n\rightarrow \infty }\left(h_n^j/h_n^k+h_n^k/h_n^j\right)=\infty$. Due to the fact that the operators in condition (15) are unitary operators on $L_x^2$, it is not hard to conclude

$$\begin{aligned}&2\pi \left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle , \end{aligned}$$

(18)

$$\begin{aligned}&\quad =(h_n^jh_n^k)^{-\frac{1}{2}}\left\langle \int _{{\mathbb {R}}} e^{i\frac{x-x_n^j}{h_n^j}\xi +i\frac{t_n^j}{(h_n^j)^{\alpha }}|\xi |^{\alpha }} \big [e^{i(\cdot )h_n^j\xi _n^j}\phi \big ]^{\wedge }(\xi )\text {d}\xi ,\right. \nonumber \\&\qquad \left. \int _{{\mathbb {R}}} e^{i\frac{x-x_n^k}{h_n^k}\xi +i\frac{t_n^k}{(h_n^k)^{\alpha }}|\xi |^{\alpha }} \big [e^{i(\cdot )h_n^k\xi _n^k}\psi \big ]^{\wedge }(\xi )\text {d}\xi \right\rangle , \end{aligned}$$

(19)

$$\begin{aligned}&\quad =:(h_n^j h_n^k)^{-\frac{1}{2}}\left\langle \Phi _n^j\left(\frac{x-x_n^j}{h_n^j}\right), \Phi _n^k\left(\frac{x-x_n^k}{h_n^k}\right)\right\rangle . \end{aligned}$$

(20)

Notice that $\Phi _n^j\in L^2$ which implies

$$\begin{aligned} \lim _{R\rightarrow \infty } \int _{|y|>R}|\Phi _n^j(y)|^2\text {d}y=0. \end{aligned}$$

(21)

Hence, if we setting

$$\begin{aligned}B_n^j(R):=\left\rbrace x: \left|\frac{x-x_n^j}{h_n^j}\right|\le R\right\lbrace , \quad B_n^k(R):=\left\rbrace x: \left|\frac{x-x_n^k}{h_n^k}\right|\le R\right\lbrace ,\end{aligned}$$

and considering (18) with the integral on ${\mathbb {R}}\setminus B_n^j(R)$, Hölder’s  inequality will give a bound as follows:

$$\begin{aligned}{} & {} 2\pi \left\langle \big [g_n^j]^{-1}[e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle _{{\mathbb {R}}\setminus B_n^j(R)}\\{} & {} \quad \le \left(\int _{|y|>R}|\Phi _n^j(y)|^2\text {d}y\right)^{\frac{1}{2}}\left(\int _{{\mathbb {R}}}|\Phi _n^k(y)|^2 \text {d}y\right)^{\frac{1}{2}}.\end{aligned}$$

Similar approach also works for the integral on ${\mathbb {R}}\setminus B_n^k(R)$ in (18). Thus, by the fact that $\Phi _n^j$ and $\Phi _n^k$ are $L_x^{\infty }$ functions, we aim to show the following estimate:

$$\begin{aligned} \lim _{n\rightarrow \infty } (h_n^j h_n^k)^{-\frac{1}{2}} \Big |B_n^j(R) \cap B_n^k(R)\Big |=0, \end{aligned}$$

(22)

which will lead to a contradiction to the assumption (15). One observation we need is

$$\begin{aligned}\Big |B_n^j(R) \cap B_n^k(R)\Big | \le C_R \min \left\rbrace h_n^j, h_n^k\right\lbrace .\end{aligned}$$

Then we obtain the desired estimate (22) immediately since $h_n^j/h_n^k$ goes to either zero or infinity. Consequently, we can assume $h_n^j\sim h_n^k$ from now on.

Step 2 We eliminate the case $\lim _{n\rightarrow \infty } \big (h_n^j+h_n^k\big )\big |\xi _n^j-\xi _n^k\big |=\infty$. By the Plancherel theorem and the fact that these operators are unitary operators in on $L^2({\mathbb {R}})$, we conclude

$$\begin{aligned} \left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle _{x}&=\left\langle \big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \big [e^{-it_n^j|\nabla |^{\alpha }}\big ]\big [g_n^j\big ]\psi \right\rangle _{x} \\&\sim \left\langle e^{-it_n^k|\cdot |^{\alpha }} \widehat{\big [g_n^k\big ]\phi }, e^{-it_n^j|\cdot |^{\alpha }} \widehat{\big [g_n^j\big ]\psi }\right\rangle _{\xi } \\&=\big (h_n^jh_n^k\big )^{1/2} \left\langle {\hat{\phi }}\left(h_n^k\xi -h_n^k\xi _n^k\right), {\hat{\psi }}\left(h_n^j\xi -h_n^j\xi _n^j\right)\right\rangle _{\xi }. \end{aligned}$$

Thus, the assumption $h_n^j\sim h_n^k$ gives

$$\begin{aligned} \left\langle [g_n^j]^{-1}[e^{it_n^j|\nabla |^{\alpha }}][e^{-it_n^k|\nabla |^{\alpha }}][g_n^k]\phi , \psi \right\rangle _{x}&\sim \left\langle {\hat{\phi }}\left(\frac{h_n^k}{h_n^j}(\xi -h_n^j\xi _n^k)\right), {\hat{\psi }}\left(\xi -h_n^j\xi _n^j\right)\right\rangle _{\xi } \\&\sim \left\langle {\hat{\phi }}\left(\xi -h_n^k\xi _n^k\right), {\hat{\psi }}\left(\frac{h_n^j}{h_n^k}(\xi -h_n^k\xi _n^j)\right)\right\rangle _{\xi }. \end{aligned}$$

Then, up to subsequences, the condition (15) and the compact supports assumption imply

$$\begin{aligned}\lim _{n\rightarrow \infty } (h_n^j+h_n^k)\left|\xi _n^k-\xi _n^j\right|=c_2, \quad c_2<\infty .\end{aligned}$$

Hence, we can assume that $(h_n^j,\xi _n^j) \equiv (h_n^k,\xi _n^k) \equiv (h_n,\xi _n)$ from now on.

Step 3 We prove the conclusion for integer $\alpha \in {\mathbb {Z}}_{+}$ where $\alpha >1$. With the assumptions for $\xi _n^{\cdot }$ and $h_n^{\cdot }$ at hand, we can turn the expression (20) into

$$\begin{aligned}\left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle = (2\pi h_n)^{-1}\left\langle \Phi _n^j\left(\frac{x-x_n^j}{h_n}\right), \Phi _n^k\left(\frac{x-x_n^k}{h_n}\right)\right\rangle .\end{aligned}$$

Then just as what we have done above, recalling the condition (21), a similar changing of variables argument and the assumption (15) imply, up to subsequences, which

$$\begin{aligned}\lim _{n\rightarrow \infty } \frac{x_n^j-x_n^k}{h_n}=c_3, \quad |c_3|<\infty .\end{aligned}$$

On the other hand, we can turn the expression (19) into

$$\begin{aligned}{} & {} \left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi , \psi \right\rangle \nonumber \\{} & {} \quad =(2\pi h_n)^{-1}\left\langle {\tilde{\Phi }}_n^j\left(\frac{x}{h_n}+\frac{t_n^j}{(h_n)^{\alpha }}\right), {\tilde{\Phi }}_n^k\left(\frac{x}{h_n}+\frac{t_n^k}{(h_n)^{\alpha }}\right)\right\rangle , \end{aligned}$$

(23)

where the function ${\tilde{\Phi }}_n^j$ is defined by

$$\begin{aligned}{\tilde{\Phi }}_n^j\left(\frac{x}{h_n}+\frac{t_n^j}{(h_n)^{\alpha }}\right):= e^{i\left(\frac{x}{h_n}+\frac{t_n^j}{(h_n)^{\alpha }}\right)\xi +i\frac{t_n^j}{(h_n^j)^{\alpha }}|\xi |^{\alpha }} e^{-i\left(\frac{x_n^j}{h_n}+\frac{t_n^j}{(h_n)^{\alpha }}\right)\xi } \big [e^{i(\cdot )h_n^j\xi _n^j}\phi \big ]^{\wedge }(\xi )\text {d}\xi ,\end{aligned}$$

and similarly for the definition of ${\tilde{\Phi }}_n^k$. Then we still have the fact ${\tilde{\Phi }}_n^j\in L^2$ and further

$$\begin{aligned}\lim _{R\rightarrow \infty } \int _{|y|>R}|{\tilde{\Phi }}_n^j|^2\text {d}y=0.\end{aligned}$$

Analogously, the expression (23) and a changing of variables argument imply, up to subsequences, which

$$\begin{aligned}\lim _{n\rightarrow \infty }\frac{t_n^j-t_n^k}{(h_n)^{\alpha }}=c_4, \quad |c_4|<\infty ,\end{aligned}$$

based on the assumption (15). Moreover, we can turn the expression (17) into

$$\begin{aligned} \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi (x)=\frac{e^{i\theta _n^{j,k}}}{2\pi }\int _{{\mathbb {R}}} e^{i\Phi _n^{jk}(x,\xi )} {\hat{\phi }}(\xi ) \text {d}\xi , \end{aligned}$$

(24)

where $\theta _n^{j,k}$ depends on the parameters $\big (h_n^k, t_n^k, x_n^k, \xi _n^k, h_n^j, t_n^j, x_n^j, \xi _n^j\big )$ and

$$\begin{aligned} \Phi _n^{jk}(x,\xi ):=\xi \left(x+\frac{x_n^j-x_n^k}{h_n}\right)-\frac{t_n^j-t_n^k}{(h_n)^{\alpha }}|\xi +h_n\xi _n|^\alpha . \end{aligned}$$

It is obvious that $\int _{{\mathbb {R}}} e^{i\Phi _n^{jk}(x,\xi )} {\hat{\phi }}(\xi ) \text {d}\xi \in L_x^{\infty }$. Next we are going to use the method of stationary phase to obtain the decay estimates of this oscillatory integral. To begin with, analyzing piece by piece if necessary, we can assume $\xi +h_n\xi _n>0$ without loss of generality and rewrite $\Phi _n^{jk}(x,\xi )$ as follows:

$$\begin{aligned} \Phi _n^{jk}(x,\xi )&=\xi \left(x+\frac{x_n^j-x_n^k}{h_n}\right)+\sum _{m=0}^{\alpha }\frac{\left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (t_n^k-t_n^j)(\xi _n)^{\alpha -m} (\xi )^m}{(h_n)^{m}} \\&=:\xi x+\sum _{m=0}^\alpha a_n^{m,j,k} (\xi )^m, \nonumber \end{aligned}$$

(25)

where $a_n^{m,j,k}$ are the coefficients of the m-order term $(\xi )^m$ in the expression of $\Phi _n^{jk}$, except for the case $m=1$ where $x+a_n^{1,j,k}$ is the coefficient of $\xi$. We are going to prove that for every fixed $m\ge 1$, after passing to a subsequence, each of the coefficients $a_n^{m,j,k}$ goes to some constant $c^{m,j,k}\ne \infty$ as n goes to infinity. Then this result gives the desired function:

$$\begin{aligned}\Phi ^{jk}(x,\xi ):=\xi x+\sum _{m=1}^\alpha c^{m,j,k} (\xi )^m,\end{aligned}$$

which satisfies $\lim _{n\rightarrow \infty }\Phi _n^{jk}(x,\xi )- a_n^{0,j,k}=\Phi ^{jk}(x,\xi )-a_n^{0,j,k}$. Since ${\mathbb {S}}^1$ is compact and $\left|e^{i\theta }\right|=1$ for real numbers $\theta$, we take the parameter $\theta ^{jk}$ as follows:

$$\begin{aligned}e^{i\theta ^{jk}}:=\lim _{n\rightarrow \infty } e^{i\big (\theta _n^{jk}+a_n^{0,j,k}\big )}.\end{aligned}$$

Then we get the desired operator $[G^{jk}]$ defined as follows:

$$\begin{aligned}{}[G^{jk}]f(x):=\frac{e^{i\theta ^{jk}}}{2\pi }\int _{{\mathbb {R}}} e^{i\Phi ^{jk}(x,\xi )} {\hat{f}}(\xi ) \text {d}\xi . \end{aligned}$$

(26)

Indeed, on Fourier space, the identity (24) and dominated convergence theorem imply that

$$\begin{aligned}\widehat{\big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]f}\rightarrow \widehat{[G^{jk}]f},\end{aligned}$$

as $n\rightarrow \infty$ in the $L_{\xi }^2$ norm topology for all Schwartz functions f. Then Plancherel theorem implies that the operator $[G^{jk}]$ satisfies the estimate (16).

Now, after passing to a subsequence, we are going to show that for each (m, j, k) with $m\ge 1$, there exists a constant $|c^{m,j,k}|<\infty$ which satisfies the following relation

$$\begin{aligned}\lim _{n\rightarrow \infty }a_n^{m,j,k}=c^{m,j,k}.\end{aligned}$$

As we argued above, this result can give the desired operator $[G^{j,k}]$ defined in (26). Here, we use proof by contradiction. If on the contrary for some fixed m, there holds $\lim _{n\rightarrow \infty }a_n^{m,j,k}=\infty$. Take

$$\begin{aligned} m_0:=\max \{m: \lim _{n\rightarrow \infty }a_n^{m,j,k}=\infty \}. \end{aligned}$$

(27)

We break the proof into two cases $m_0=1$ and $m_0>1$. For the case $m_0=1$, we have the following limit relation:

$$\begin{aligned} \Phi _n^{jk}(x,\xi )=\big (x+a_n^{1,j,k}\big )\xi +\Phi _n^{1,j,k}(\xi )\;\rightarrow (x+\infty )\xi +\Phi ^{1,j,k}(\xi ), \end{aligned}$$

(28)

as n goes to infinity. Here, the function $\Phi ^{1,j,k}(\xi )$ of which coefficients are bounded is the limit function of $\Phi _n^{1,j,k}(\xi )$. Since the parameters $a_n^{1,j,k}$ just deduce spatial translations for x, the assumption that $\phi$ and $\psi$ are Schwartz functions can imply

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\langle \big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi (x), \psi (x)\right\rangle= & {} \lim _{n\rightarrow \infty } \left\langle \phi _{j,k}(x+a_n^{1,j,k}), \psi (x)\right\rangle \nonumber \\= & {} 0, \end{aligned}$$

(29)

where $\phi _{j,k} (x):= (2\pi )^{-1} \int _{{\mathbb {R}}} e^{ix\xi + i\Phi ^{1,j,k}(\xi )} {\hat{\phi }}(\xi ) \text {d}\xi$. This is a contradiction to the condition (15). For the case $m_0>1$, by the compactness of $\text {supp}({\hat{\phi }})$, we deduce the following estimates:

$$\begin{aligned}\left|\left(\frac{\text {d}}{\text {d}\xi }\right)^{m_0}\Phi _n^{jk}(x,\xi )\right|\sim \left|a_n^{m_0,j,k}\right|,\end{aligned}$$

for all $\xi \in \text {supp}({\hat{\phi }})$ and n large enough. Therefore, the classical van der Corput Lemma [39, p. 334, Corollary] implies

$$\begin{aligned}\left\Vert\big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi \right\Vert_{L_x^{\infty }}\lesssim _{\phi } \big |a_n^{m_0,j,k}\big |^{-\frac{1}{m_0}}\rightarrow 0,\end{aligned}$$

as $n\rightarrow \infty$, and thus, $\big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi \rightharpoonup 0$ in $L_x^2$ as n goes to infinity. This is a contradiction to (15). Therefore, we have obtained the desired operator $[G^{j,k}]$ defined in (26) which satisfies (16) and established the desired conclusion for all integers $\alpha \in {\mathbb {Z}}_{+}$ with $\alpha >1$.

Step 4 We generalize the arguments in Step 3 to all real numbers $\alpha >1$. Indeed, we still can argue as Step 3 to get the expression (24) even if $\alpha >1$ is a real number. Then we divide the proof into two parts: up to subsequences, either

$$\begin{aligned} \lim _{n\rightarrow \infty }|h_n\xi _n|\rightarrow \infty , \end{aligned}$$

(30)

or $\lim _{n\rightarrow \infty }h_n\xi _n=c_5$ with $|c_5|<\infty$. For the latter case, we can directly define the desired operator $[G^{jk}]$ as follows

$$\begin{aligned}f(x):=\frac{e^{i\theta ^{jk}}}{2\pi }\int _{{\mathbb {R}}} e^{i(x+c_3)\xi -ic_4|\xi +c_5|^{\alpha }}{\hat{f}}(\xi )\text {d}\xi .\end{aligned}$$

Hence, without loss of generality, our last step is to deal with the case $h_n\xi _n\rightarrow +\infty$ as n goes to infinity. This time, due to the compact Fourier supports assumption, there exists some $N_0$ large enough such that $|\xi /(h_n\xi _n)|<1/2$ holds for all $n>N_0$. Thus, by selecting the terms with $n>N_0$, we can take this subsequence for n and use Taylor’s theorem to conclude the following series⁴

$$\begin{aligned} \Phi _n^{jk}(x,\xi )&=\xi \left(x+\frac{x_n^j-x_n^k}{h_n}\right)+\frac{(h_n\xi _n)^{\alpha }\big (t_n^k-t_n^j\big )}{(h_n)^{\alpha }}\left|1+\frac{\xi }{h_n\xi _n}\right|^\alpha \nonumber \\&=\xi \left(x+\frac{x_n^j-x_n^k}{h_n}\right)+\sum _{m=0}^{\infty }\frac{\left( {\begin{array}{c}\alpha \\ m\end{array}}\right) \big (t_n^k-t_n^j\big )(h_n\xi _n)^{\alpha -m} (\xi )^m}{(h_n)^{\alpha }} \nonumber \\&=:\xi x+\sum _{m=0}^{\infty } a_n^{m,j,k} (\xi )^m. \end{aligned}$$

(31)

Notice that for $m\ge 2$, there holds

$$\begin{aligned} \frac{a_n^{m,j,k}}{a_n^{m+1,j,k}}=\frac{\left( {\begin{array}{c}\alpha \\ m\end{array}}\right) h_n\xi _n}{\left( {\begin{array}{c}\alpha \\ m+1\end{array}}\right) }=\frac{(m+1)h_n\xi _n}{\alpha -m}. \end{aligned}$$

(32)

Therefore, by the choosing of $N_0$, we obtain that the power series (31) converges uniformly for all $\xi$ in the compact Fourier support. Furthermore, for the second-order derivative of $\Phi _n^{jk}$, there holds

$$\begin{aligned} \left(\frac{\text {d}}{\text {d}\xi }\right)^2 \Phi _n^{jk}(\xi )&= \frac{\alpha (\alpha -1)(h_n\xi _n)^{\alpha -2}\big (t_n^k-t_n^j\big )}{(h_n)^{\alpha }} \left|1+\frac{\xi }{h_n\xi _n}\right|^{\alpha -2} \nonumber \\&=2a_n^{2,j,k} \left|1+\frac{\xi }{h_n\xi _n}\right|^{\alpha -2} \nonumber \\&>2^{3-\alpha } a_n^{2,j,k}. \end{aligned}$$

(33)

Similarly, we use proof by contradiction and define $m_0$ as in (27). For the case $m_0>1$, the relation (32) and condition (30) imply that

$$\begin{aligned}\lim _{n\rightarrow \infty }a_n^{2,j,k}=\infty .\end{aligned}$$

Hence, as shown earlier in Step 3, the estimate (33) and classical van der Corput Lemma give the following decay estimate

$$\begin{aligned}\left\Vert\big [g_n^j\big ]^{-1}\big [e^{it_n^j|\nabla |^{\alpha }}\big ]\big [e^{-it_n^k|\nabla |^{\alpha }}\big ]\big [g_n^k\big ]\phi \right\Vert_{L_x^{\infty }}\lesssim _{\phi } \big |a_n^{2,j,k}\big |^{-\frac{1}{2}} \;\;\xrightarrow {n\rightarrow \infty } 0,\end{aligned}$$

which deduces a contradiction to (15). For the case $m_0=1$, there exists $c^{2,j,k}$ with $|c^{2,j,k}|<\infty$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }a_n^{2,j,k}=c^{2,j,k}, \end{aligned}$$

(34)

and the relation (32) further implies that

$$\begin{aligned}\lim _{n\rightarrow \infty }a_n^{m,j,k}=0, \quad \text {for} \quad m\ge 3.\end{aligned}$$

By (34), for every fixed x and $\xi$, we know from Taylor’s theorem that

$$\begin{aligned} \Phi _n^{jk}(x,\xi )-\big (x+a_n^{1,j,k}\big )\xi - a_n^{2,j,k}\xi ^2 =O\left(\frac{|\xi ^3|}{h_n\xi _n}\right). \end{aligned}$$

(35)

This estimate and the condition (30) further imply the following limit relation

$$\begin{aligned}\Phi _n^{jk}(x,\xi ) \;\;\xrightarrow {n\rightarrow \infty } (x+\infty )\xi +c^{2,j,k}(\xi )^2.\end{aligned}$$

Therefore, as shown earlier in Step 3, the Schwartz functions assumption further leads to (29) which is a contradiction to (15). In summary, we conclude that both $a_n^{1,j,k}$ and $a_n^{2,j,k}$ are bounded. Then, up to subsequences, we can assume that

$$\begin{aligned}\lim _{n\rightarrow \infty }a_n^{1,j,k}=c^{1,j,k}, \quad \lim _{n\rightarrow \infty }a_n^{2,j,k}=c^{2,j,k}.\end{aligned}$$

Thus the desired operator $[{\tilde{G}}^{jk}]$, similar to the expression (26), is given by

$$\begin{aligned}\big [{\tilde{G}}^{jk}\big ]f(x):=\frac{e^{i{\tilde{\theta }}^{jk}}}{2\pi }\int _{{\mathbb {R}}} e^{i{\tilde{\Phi }}^{jk}(x,\xi )} {\hat{f}}(\xi ) \text {d}\xi , \quad {\tilde{\Phi }}^{jk}(x,\xi ):=\big (x+c^{1,j,k}\big )\xi +c^{2,j,k}(\xi )^2.\end{aligned}$$

Here, due to the relation (35), our function ${\tilde{\Phi }}^{jk}(x,\xi )$ satisfies the following pointwise estimate:

$$\begin{aligned}\lim _{n\rightarrow \infty } \Phi _n^{jk}(x,\xi )={\tilde{\Phi }}^{jk}(x,\xi ).\end{aligned}$$

Hence, as the arguments after (26), the Plancherel theorem and dominated convergence theorem imply that our operator $[{\tilde{G}}^{jk}]$ satisfies (16). This completes the proof.

3 First-Step Decomposition: Frequency and Scaling

Usually the profile decomposition results are obtained by following two steps: first finding the frequency-scaling parameters based on some refinement of Strichartz estimates, which can be deduced by the bilinear restriction estimates from [41, 44, 45], and second, finding the time–space translations by using some weak convergence arguments, which will be further discussed in Sect. 4 later. There may be some papers providing slightly different procedures by using similar ingredients such as [43, Appendix A] and [27, Theorem 4.26]. We refer to [40] for a brief discussion on the $L^2$-based linear profile decomposition and a generalization in the $L^p$ setting, see also [4] for some recent results on the $L^p$-generalization.

In this section, we present the first-step decomposition by following the proofs in [7, 22], similar method can also be seen in some earlier papers [8, 25]. To do some dyadic analysis, it is convenient to give the following dyadic intervals in ${\mathbb {R}}$.

Definition 3.1

(Dyadic intervals) Given $j\in {\mathbb {Z}}$, the dyadic intervals of length $2^j$ in ${\mathbb {R}}$ is defined by

$$\begin{aligned}{\mathcal {D}}_j:=\left\rbrace 2^j[k,k+1): k\in {\mathbb {Z}}\right\lbrace ,\end{aligned}$$

and we use ${\mathcal {D}}:=\cup _{j\in {\mathbb {Z}}}{\mathcal {D}}_j$ to denote the set of all the dyadic intervals in ${\mathbb {R}}$.

Proposition 3.2

($\alpha$-refined Strichartz) For any $p>1$, we have

(36)

where $|\tau |$ denotes the length of the interval $\tau$.

Proof of Proposition 3.2

We adapt the proofs in [22, Lemma 1.2] and [7, Sect. 2] by using the Whitney decomposition and Hausdorff–Young inequality instead of the aforementioned bilinear restriction estimates, since we are dealing with the one-dimensional case now. See also [8, 25] for different methods using Fefferman–Phong’s weighted inequality from [12].

Notice that we can normalize $\sup _{\tau \in {\mathcal {D}}}|\tau |^{1/2-1/p}\Vert {\hat{f}}\Vert _{L^p(\tau )}=1$ for given $p>1$. This implies that the following inequality

$$\begin{aligned} \int _{I} |{\hat{f}}|^p \text {d}\xi \le |I|^{1-p/2}, \end{aligned}$$

(37)

holds for all dyadic intervals $I\in \{2^j[k,k+1): j\in {\mathbb {Z}}, k\in {\mathbb {Z}}\}$. In our proof, we aim to show that

$$\begin{aligned} \left\Vert\big [D^{\frac{(\alpha -2)}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]f \big [D^{\frac{(\alpha -2)}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]g\right\Vert_{L^3({\mathbb {R}}^2)}^{\frac{3}{2}} \lesssim \int _{{\mathbb {R}}^2} \frac{|{\hat{f}}(\xi ){\hat{g}}(\eta )|^{\frac{3}{2}}}{|\xi -\eta |^{\frac{1}{2}}}\text {d}\xi \text {d}\eta . \end{aligned}$$

(38)

Then, based on the estimates (37) and (38), one can follow the same arguments in the proof of [7, Proposition 2.7] to get the desired result (36). Indeed, it is a direct application of Whitney decomposition. We omit the details on this application of Whitney decomposition for avoiding too much repetition.

Now, let us turn to prove (38). Define the extension operator $[E_{\alpha }]$ by

$$\begin{aligned}f(t,x):=2\pi \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]f(x) =\int _{{\mathbb {R}}}e^{ix\xi -it|\xi |^{\alpha }}|\xi |^{\frac{\alpha -2}{6}} {\hat{f}}(\xi )\text {d}\xi .\end{aligned}$$

Then we investigate the following bilinear forms:

$$\begin{aligned}f[E_{\alpha }]g(t,x)=\int _{{\mathbb {R}}^2}e^{ix(\xi +\eta )-it\big (|\xi |^{\alpha }+|\eta |^{\alpha }\big )}|\xi |^{\frac{\alpha -2}{6}}|\eta |^{\frac{\alpha -2}{6}} {\hat{f}}(\xi ){\hat{g}}(\eta ) \text {d}\xi \text {d}\eta .\end{aligned}$$

Consider the changing of variables

$$\begin{aligned} (\xi , \eta )\mapsto (u,v):=(\xi +\eta , -|\xi |^{\alpha }-|\eta |^{\alpha }). \end{aligned}$$

(39)

Recall that for fixed $(u_0,v_0)$, the graph of the function $u_0=\xi +\eta$ is a line and the graph of $v_0=-|\xi |^{\alpha }-|\eta |^{\alpha }$ is a “circle” in some sense. This implies that the map defined in (39) is an at most 2-to-1 map from ${\mathbb {R}}^2$ to the region $Q:=\{(u,v): -v\ge 2^{1-\alpha }|u|^{\alpha }\}$ which comes from the convexity. Further the Jacobian is given by

$$\begin{aligned}J(u,v)=J^{-1}(\xi , \eta )=\frac{\partial (u,v)}{\partial (\xi , \eta )}=\alpha \big (\xi |\xi |^{\alpha -2}-\eta |\eta |^{\alpha -2}\big ).\end{aligned}$$

Thus, we conclude

$$\begin{aligned}\Big |[E_{\alpha }]f[E_{\alpha }]g(t,x)\Big |\le 2\left|\int _{Q} e^{ixu+itv} |\xi \eta |^{\frac{\alpha -2}{6}} {\hat{f}}(\xi ){\hat{g}}(\eta ) J^{-1}(u,v)\text {d}u\text {d}v\right|,\end{aligned}$$

where $(\xi ,\eta )$ is a function of (u, v) via the change of variables (39) above. By the symmetry, we can assume $|\eta |\le |\xi |$ without loss of generality. Using the Hausdorff–Young inequality and then changing variables back to $(\xi ,\eta )$ we deduce the following

$$\begin{aligned} \left\Vert[E_{\alpha }]f\cdot [E_{\alpha }]g\right\Vert^{3/2}_{L_{t,x}^3({\mathbb {R}}^2)}&\lesssim \left\Vert|\xi \eta |^{\frac{\alpha -2}{6}}{\hat{f}}(\xi ){\hat{g}}(\eta ) J^{-1}(u,v)\right\Vert^{3/2}_{L_{u,v}^{3/2}({\mathbb {R}}^2)} \nonumber \\&=\left\Vert|\xi \eta |^{\frac{\alpha -2}{6}} \left|J(\xi ,\eta )\right|^{\frac{1}{3}} {\hat{f}}(\xi ){\hat{g}}(\eta )\right\Vert^{3/2}_{L_{u,v}^{3/2}({\mathbb {R}}^2)}. \end{aligned}$$

(40)

To estimate the norm above, our next target is the Jacobian factor

$$\begin{aligned}{\tilde{J}}(\xi ,\eta ):=|\xi \eta |^{\frac{\alpha -2}{4}} |J(\xi ,\eta )|^{\frac{1}{2}}=\frac{|\xi \eta |^{\frac{\alpha -2}{4}}}{\left[\alpha (\xi |\xi |^{\alpha -2}-\eta |\eta |^{\alpha -2})\right]^{1/2}}.\end{aligned}$$

If $\xi \eta \le 0$, it is easy to see that

$$\begin{aligned}{\tilde{J}}(\xi ,\eta )=\frac{|\xi \eta |^{\frac{\alpha -2}{4}}}{\left[\alpha (|\xi |^{\alpha -1}+|\eta |^{\alpha -1})\right]^{1/2}}\lesssim _{\alpha } (|\xi |+|\eta |)^{-\frac{1}{2}}=|\xi -\eta |^{-\frac{1}{2}}.\end{aligned}$$

If $\xi \eta >0$ and $|\xi |\ge |\eta |$, then we have

$$\begin{aligned}|\xi |^{\alpha -1}-|\eta |^{\alpha -1}\sim _{\alpha } (|\xi |-|\eta |)|\xi |^{\alpha -2}.\end{aligned}$$

This estimate leads to

$$\begin{aligned}{\tilde{J}}(\xi ,\eta )=\frac{|\xi \eta |^{\frac{\alpha -2}{4}}}{\left[\alpha \big (|\xi |^{\alpha -1}-|\eta |^{\alpha -1}\big )\right]^{1/2}}\lesssim _{\alpha } \frac{|\xi \eta |^{\frac{\alpha -2}{4}}}{|\xi |^{\frac{\alpha -2}{2}}|\xi -\eta |^{\frac{1}{2}}}\le |\xi -\eta |^{-\frac{1}{2}}.\end{aligned}$$

If $\xi \eta >0$ and $|\xi |<|\eta |$, by the symmetry, analogously as above we can obtain ${\tilde{J}}(\xi ,\eta )\lesssim _{\alpha } |\xi -\eta |^{-\frac{1}{2}}$. In summary, we know that

$$\begin{aligned}{\tilde{J}}(\xi ,\eta )\lesssim _{\alpha } |\xi -\eta |^{-\frac{1}{2}},\end{aligned}$$

holds uniformly in $\xi$ and $\eta$. Taking this into the expression (40), we get the desired estimate (38).

Based on the refined Strichartz estimate Proposition 3.2, we can extract the frequency and scaling parameters by following a standard approach in [22], and similar argument can also be seen in [8]. We omit the detailed proof of the following Lemma 3.3 here, since it is too long but essentially the same as [22, Lemma 5.1] and [8, Lemma 3.3].

Lemma 3.3

Let $\{u_n\}_{n\ge 1}$ be a sequence of functions with $\Vert u_n\Vert _{L_x^2({\mathbb {R}})}\le 1$. Then up to subsequences, for any $\delta >0$, there exist

$$\begin{aligned}N=N(\delta ),\quad \left\rbrace \big (\rho _n^{\beta },\xi _n^{\beta }\big )_{1\le \beta \le N}\right\lbrace \subset (0,\infty )\times {\mathbb {R}}, \quad \left\rbrace (f_n^{\beta })_{1\le \beta \le N} \right\lbrace \subset L_x^2({\mathbb {R}}),\end{aligned}$$

such that

$$\begin{aligned} u_n=\sum _{\beta =1}^{N} f_n^{\beta }+q_n^N, \end{aligned}$$

(41)

and there exists a compact set $K=K(N)$ in ${\mathbb {R}}$ such that for every $1\le \beta \le N$, there holds

$$\begin{aligned} (\rho _n^{\beta })^{\frac{1}{2}}\left|{\hat{f}}_n^\beta (\rho _n^{\beta }\xi +\xi _n^{\beta })\right|\le C_{\delta } \mathbb {1}_K(\xi ). \end{aligned}$$

(42)

Here, the sequence $(\rho _n^{\beta },\xi _n^{\beta })$ satisfies that if $\beta \ne \gamma$ then

$$\begin{aligned} \lim _{n\rightarrow \infty }\left(\frac{\rho _n^{\beta }}{\rho _n^\gamma } +\frac{\rho _n^\gamma }{\rho _n^{\beta }} +\frac{|\xi _n^{\beta }-\xi _n^\gamma |}{\rho _n^{\beta }}+\frac{|\xi _n^{\beta }-\xi _n^\gamma |}{\rho _n^{\gamma }}\right)=\infty . \end{aligned}$$

(43)

The remainder term $q_n^N$ has a negligible Strichartz norm:

$$\begin{aligned} \left\Vert[D^{\frac{\alpha -2}{6}}][e^{it|\nabla |^{\alpha }}]q_n^N\right\Vert_{L_{t,x}^6}\le \delta , \end{aligned}$$

(44)

and furthermore, if for each $1\le N'\le N$, we generally define

$$\begin{aligned}q_n^{N'}:=q_n^N+f_n^N+f_n^{N-1}+\cdots +f_n^{N'+1},\end{aligned}$$

then we have the $L^2$-almost orthogonal identity

$$\begin{aligned} \lim _{n\rightarrow \infty }\left(\Vert u_n\Vert _{L^2}^2-\left(\sum _{\beta =1}^{N'}\Vert f_n^{\beta }\Vert _{L^2}^2+\Vert q_n^{N'}\Vert _{L^2}^2\right)\right)=0. \end{aligned}$$

(45)

Remark 3.4

We should remark that in the proof of Lemma 3.3, by the construction, we know that the Fourier supports of $f_n^{\beta }$ and $q_n^N$ are mutually disjoint. This crucial fact also implies the conclusion (45). On the other hand, define operators $[{\tilde{G}}_n^{\beta }]$ on the Fourier side by

$$\begin{aligned}\big [{\tilde{G}}_n^\beta \big ]\big [{\hat{f}}\big ](\xi ):=\big (\rho _n^{\beta }\big )^{\frac{1}{2}}{\hat{f}}\big (\rho _n^{\beta }\xi +\xi _n^{\beta }\big ).\end{aligned}$$

Then the conclusion (43) means that, in view of the conditional dislocation property Proposition 1.3, the sequence of operators satisfy

$$\begin{aligned}\big [{\tilde{G}}_n^\beta \big ]\big [{\tilde{G}}_n^\gamma \big ]^{-1}\rightharpoonup 0\;\; \text {and}\;\; \big [{\tilde{G}}_n^\beta \big ]^{-1}\big [{\tilde{G}}_n^\gamma \big ]\rightharpoonup 0,\end{aligned}$$

as $n\rightarrow \infty$ for every $\beta \ne \gamma$. Or equivalently on the spatial side, define

$$\begin{aligned}\big [G_n^\beta \big ]f(x):={\mathscr {F}}^{-1}\big [{\tilde{G}}_n^{\beta }\big ]{\mathscr {F}}f(x) =\big (\rho _n^{\beta }\big )^{-\frac{1}{2}}e^{-ix\big (\rho _n^{\beta }\big )^{-1}\xi _n^{\beta }}f\left(\frac{x}{\rho _n^{\beta }}\right).\end{aligned}$$

Then the conclusion (43) implies that $\big [G_n^\beta \big ]\big [G_n^\gamma \big ]^{-1}$ and $\big [G_n^\beta \big ]^{-1}\big [G_n^\gamma \big ]$ goes to zero as n go to infinity in the weak operator topology of ${\mathcal {B}}(L^2)$ for $\beta \ne \gamma$. This comes from the dual approach on $L^2({\mathbb {R}})$ and Plancherel theorem as follows:

$$\begin{aligned} \left\langle \big [{\tilde{G}}_n^{\beta }\big ]\big [{\tilde{G}}_n^{\gamma }\big ]^{-1}[{\hat{f}}], {\hat{g}}\right\rangle _{\xi }&=\left\langle \big [{\tilde{G}}_n^{\gamma }\big ]^{-1}{\hat{f}}, \big [{\tilde{G}}_n^{\beta }\big ]^{-1}{\hat{g}}\right\rangle _{\xi } =\left\langle {\mathscr {F}}\big [G_n^\gamma \big ]^{-1}f, {\mathscr {F}}[G_n^\beta ]^{-1}g\right\rangle _{\xi } \\&\sim \left\langle [G_n^\gamma ]^{-1}f, [G_n^\beta ]^{-1}g\right\rangle _x = \left\langle [G_n^\beta ][G_n^\gamma ]^{-1}f, g\right\rangle _x. \end{aligned}$$

4 Second-Step Decomposition: Time and Space Translations

After the first-step decomposition Lemma 3.3, we have obtained the desired frequency and scaling parameters. Hence, in this section, we are devoted to getting the time and space translation parameters. Recall that the dislocation property (or equivalently the Galilean invariance) always play an important role in the classical case [2, 5, 8, 32]. However, this Galilean invariance is not valid in our $\alpha$-Strichartz setting and also note that $\alpha$ may not be a natural number. Thus, our strategy is using the conditional dislocation property Proposition 1.3 obtained in Sect. 2.

To begin this section, one ingredient we need is the following local restriction Lemma 4.1. Then we are ready to further decompose the functions $f_n$ obtained in the first-step decomposition and to get the time–space translation parameters in Lemma 4.3.

Lemma 4.1

(Localized restriction) For $4<q<6$ and ${\hat{F}}\in L^{\infty }\big (B(\xi _0,R)\big )$ with some $R>0$, we have

$$\begin{aligned}\left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ] F\right\Vert_{L_{t,x}^q}\le C_{q,R} \Vert {\hat{F}}\Vert _{L^{\infty }(B(\xi _0, R))}.\end{aligned}$$

Proof of Lemma 4.1

Similarly as what we have done in the proof of Proposition 3.2, the desired estimate is equivalent to the following bilinear form:

$$\begin{aligned}{} & {} \left\Vert\int _{B(\xi _0,R)}\int _{B(\xi _0,R)} e^{ix(\xi +\eta )-it(|\xi |^{\alpha }+|\eta |^{\alpha })}|\xi |^{\frac{\alpha -2}{q}}|\eta |^{\frac{\alpha -2}{q}} {\hat{F}}(\xi ){\hat{F}}(\eta ) \text {d}\xi \text {d}\eta \right\Vert_{L_{t,x}^{\frac{q}{2}}}\nonumber \\{} & {} \quad \lesssim _{q,R} \Vert {\hat{F}}\Vert ^2_{L^{\infty }(B(\xi _0,R))}. \end{aligned}$$

(46)

By changing of variables

$$\begin{aligned}(u,v):=(\xi +\eta , -|\xi |^{\alpha }-|\eta |^{\alpha }),\end{aligned}$$

using the Hausdorff–Young inequality and then changing the variables back, we conclude that the left-hand side of (46) is bounded by

$$\begin{aligned} C \left(\int _{B(\xi _0, R)\times B(\xi _0, R)} |{\hat{F}}(\xi ){\hat{F}}(\eta )|^{r'} |\xi |^{\frac{(\alpha -2)r'}{2r}}|\eta |^{\frac{(\alpha -2)r'}{2r}} |J(\xi ,\eta )|^{r'-1} \text {d}\xi \text {d}\eta \right)^{\frac{1}{r'}}, \end{aligned}$$

(47)

where

$$\begin{aligned}r:=\frac{q}{2}\in (2,3),\quad J(\xi ,\eta )^{-1}:=\alpha (\xi |\xi |^{\alpha -2}-\eta |\eta |^{\alpha -2}).\end{aligned}$$

We then consider the Jacobian factor

$$\begin{aligned}{\tilde{J}}(\xi ,\eta ):=|\xi \eta |^{\frac{(\alpha -2)r'}{2r}}|J(\xi ,\eta )|^{r'-1}= \frac{|\xi \eta |^{\frac{(\alpha -2)(r'-1)}{2}}}{\big |\xi |\xi |^{\alpha -2}-\eta |\eta |^{\alpha -2}\big |^{r'-1}}.\end{aligned}$$

It is easy to see that ${\tilde{J}}$ can only have singularity at the following singular line:

$$\begin{aligned}\xi =\eta .\end{aligned}$$

By investigating the order of the singularity of ${\tilde{J}}$ at this singular line, we know that

$$\begin{aligned}\left\Vert{\tilde{J}}(\xi ,\eta )\right\Vert_{L^1_{\xi ,\eta }(B(\xi _0,R)\times B(\xi _0,R))}\lesssim _{R,r'} 1.\end{aligned}$$

Therefore, we can control (47) by

$$\begin{aligned}C_{q,R} \Vert {\hat{F}}\Vert ^2_{L^{\infty }(B(\xi _0,R))},\end{aligned}$$

which leads to the desired result (46), and thereby, the proof is completed.

Definition 4.2

(Limit-orthogonality for sequences of operators) For fixed $j\ne k$, we say that two sequences of operators $\bigg (\big [g_n^j\big ]\bigg )$ and $\bigg (\big [g_n^k\big ]\bigg )$ in ${\mathcal {B}}(L^2)$ are limit-orthogonal if

$$\begin{aligned}\big [g_n^j\big ]^{-1}\big [g_n^k\big ]\rightharpoonup 0, \quad n\rightarrow \infty .\end{aligned}$$

Lemma 4.3

(Time-space translations) Let ${\mathbb {F}}:=(f_n)_{n\ge 1}$ be a sequence of $L^2({\mathbb {R}})$ functions. Define the unitary operators $[{\tilde{G}}_n]$ and $[G_n]$ on $L^2({\mathbb {R}})$ by

$$\begin{aligned}f(x):= & {} (\rho _n)^{\frac{1}{2}}f(\rho _nx+\xi _n),\quad [G_n]f(x)\\:= & {} {\mathscr {F}}^{-1}[{\tilde{G}}_n]{\mathscr {F}}f(x)=(\rho _n)^{-\frac{1}{2}} e^{-ix\frac{\xi _n}{\rho _n}} f\left(\frac{x}{\rho _n}\right).\end{aligned}$$

If we assume that the following condition:

$$\begin{aligned}\left|\big [{\tilde{G}}_n\big ]\big [\hat{f_n}\big ](\xi )\right|\le {\hat{F}}(\xi ),\quad {\hat{F}}\in L^{\infty }(K),\end{aligned}$$

holds for some compact set $K\subset {\mathbb {R}}$ independent of n. Then up to subsequences, there exist

$$\begin{aligned}\bigg \{(s_n^j, y_n^j)_{j\ge 1}\bigg \}\subset {\mathbb {R}}\times {\mathbb {R}},\quad \bigg \{(\phi ^j)_{j\ge 1}\bigg \}\subset L^2({\mathbb {R}}),\quad \big [g_n^j\big ]\phi (x):=\big [e^{-is_n^j|\nabla |^{\alpha }}\big ]\phi (x-y_n^j),\end{aligned}$$

such that the operators $[g_n^j][G_n]^{-1}$ satisfy the following limit-orthogonality property:

$$\begin{aligned}{}[G_n]\big [g_n^j\big ]^{-1}\big [g_n^k\big ][G_n]^{-1}\rightharpoonup 0, \quad n\rightarrow \infty , \end{aligned}$$

(48)

for every $j\ne k$. Meanwhile, for every $M\ge 1$, there exist $e_n^M\in L^2({\mathbb {R}})$ and the decomposition

$$\begin{aligned} f_n(x)=\sum _{j=1}^{M} \big [g_n^j\big ][G_n]^{-1}\phi ^j(x)+e_n^M(x), \end{aligned}$$

(49)

with the vanishing Strichartz norm estimate for the remainder

$$\begin{aligned} \lim _{M\rightarrow \infty }\lim _{n\rightarrow \infty } \left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]e_n^M\right\Vert_{L_{t,x}^6}=0. \end{aligned}$$

(50)

Furthermore, for every $M\ge 1$, we have the $L^2$-almost orthogonal identity

$$\begin{aligned} \lim _{n\rightarrow \infty }\left(\Vert f_n\Vert _{L^2}^2-\left(\sum _{j=1}^{M}\Vert \phi ^j\Vert _{L^2}^2+\Vert e_n^M\Vert _{L^2}^2\right)\right)=0. \end{aligned}$$

(51)

Proof of Lemma 4.3

We adopt some ideas from [22, Lemma 5.2] and [8, Sect. 3], while similar approaches also arise in earlier papers [1, 26] and some of the references aforementioned. However, as we have stated before, to generalize these classical arguments into our $\alpha$-Strichartz setting, we should use the conditional dislocation property Proposition 1.3. Take ${\mathbb {P}}:=(P_n)_{n\ge 1}$ with

$$\begin{aligned}{\hat{P}}_n(\xi )=[{\tilde{G}}_n][{\hat{f}}_n](\xi ).\end{aligned}$$

Let ${\mathscr {W}}({\mathbb {P}})$ be the set of weak limits in $L^2({\mathbb {R}})$ for subsequences of $[G_n][g_n]^{-1}[G_n]^{-1}{\mathbb {P}}$. In other words,

$$\begin{aligned}{\mathscr {W}}({\mathbb {P}}):= & {} \left\rbrace w\!\!-\!\!\!\lim _{n\rightarrow \infty } [G_n][g_n]^{-1}[G_n]^{-1}P_n(x): (s_n,y_n)\in {\mathbb {R}}^2\right\lbrace , \quad [g_n]\phi (x)\\:= & {} [e^{-is_n|\nabla |^{\alpha }}]\phi (x-y_n),\end{aligned}$$

and then define

$$\begin{aligned}\mu ({\mathbb {P}}):=\sup \left\rbrace \Vert \phi \Vert _{L^2}: \phi \in {\mathscr {W}}({\mathbb {P}})\right\lbrace .\end{aligned}$$

To get the desired decomposition (49), our strategy is to get the decomposition for $P_n$ as follows:

$$\begin{aligned} P_n(x)=\sum _{j=1}^{M} [G_n]\big [g_n^j\big ][G_n]^{-1}\phi ^j(x)+p_n^M(x), \end{aligned}$$

(52)

and then set $e_n^M(x):=[G_n]^{-1} p_n^M(x)=\sqrt{\rho _n} e^{ix\xi _n}p_n^M(\rho _n x)$. Similarly define the following notations:

$$\begin{aligned}{\mathbb {P}}^M:=\big (p_n^M\big )_{n\ge 1},\quad {\mathbb {E}}^M:=\big (e_n^M\big )_{n\ge 1}.\end{aligned}$$

Firstly, we claim that if the conclusion (50) in Lemma 4.3 is replaced by

$$\begin{aligned} \lim _{M\rightarrow \infty }\mu ({\mathbb {E}}^M)=\lim _{n\rightarrow \infty }\mu ({\mathbb {P}}^M)=0, \end{aligned}$$

(53)

then this lemma is true even if we do not have the assumption that K is a compact set independent of n. We show this claim as follows.

Indeed, if $\mu ({{\mathbb {P}}})=0$, then we can take $\phi ^j=0$ for all j and the claim is proved. Otherwise if $\mu ({\mathbb {P}})>0$, we take $\phi ^1\in {\mathscr {W}}({\mathbb {P}})$ such that

$$\begin{aligned}\Vert \phi ^1\Vert _{L^2}\ge \frac{\mu ({\mathbb {P}})}{2}>0.\end{aligned}$$

By the definition of ${\mathscr {W}}({\mathbb {P}})$, there exists a sequence $(s_n^1,y_n^1)\in {\mathbb {R}}^2$ such that, up to extracting a subsequence, we have

$$\begin{aligned}{}[G_n][g_n^1]^{-1}[G_n]^{-1}P_n\rightharpoonup \phi ^1, \end{aligned}$$

(54)

in $L^2({\mathbb {R}})$ as n goes to infinity. Setting $p_n^1:=P_n-[G_n]\big [g_n^1\big ][G_n]^{-1}\phi ^1$, then we obtain

$$\begin{aligned}\lim _{n\rightarrow \infty }\left(\Vert P_n\Vert _{L^2}^2-\Vert \phi ^1\Vert _{L^2}^2-\Vert p_n^1\Vert _{L^2}^2\right)=0,\end{aligned}$$

due to the weak convergency (54) and the fact that $L^2({\mathbb {R}})$ is a Hilbert space. Notice that all these operators involved are unitary operators on $L^2$. Therefore, the almost orthogonal identity (51) holds for $M=1$. Next, we replace $P_n$ by $p_n^1$ and then do this process again. If $\mu ({\mathbb {P}}^1)>0$, we can get the function $\phi ^2$, the sequence of parameters $(s_n^2, y_n^2)$, and the sequence of functions ${\mathbb {P}}^2$. Moreover, we have one more conclusion as follows: the sequence of operators

$$\begin{aligned}\big [g_n^2\big ]^{-1}\big [g_n^1\big ][G_n]^{-1}\rightharpoonup 0,\end{aligned}$$

in ${\mathcal {B}}(L^2)$ as n goes to infinity. Indeed if this conclusion is not true, then the dislocation property Proposition 1.3 asserts that, up to subsequences, there exists an isometric $[g^{1,2}]$ on $L^2({\mathbb {R}})$ satisfying

$$\begin{aligned}^{-1}\big [g_n^2\big ]^{-1}\big [g_n^1\big ][G_n]\rightarrow \big [g^{1,2}\big ],\end{aligned}$$

in ${\mathcal {B}}(L^2)$ as n goes to infinity. Therefore, the following relation

$$\begin{aligned}\big [g_n^2\big ]^{-1}[G_n]^{-1}p_n^1=\left([G_n]\big [g_n^2\big ]^{-1}\big [g_n^1\big ][G_n]^{-1}\right) [G_n]\big [g_n^1\big ]^{-1}[G_n]^{-1}p_n^1,\end{aligned}$$

and the weak convergency fact (54) imply that $\phi ^2=0$, which means $\mu ({\mathbb {P}}^2)=0$. This is a contradiction. Iterating this process leads to

$$\begin{aligned}\begin{array}{ccc} p_n^1:= P_n -[G_n][g_n^1][G_n]^{-1}\phi ^1, &{} [G_n][g_n^1]^{-1}[G_n]^{-1}P_n\rightharpoonup \phi ^1, &{} \Vert \phi ^1\Vert _{L^2}\ge \frac{\mu ({\mathbb {P}})}{2}>0; \\ p_n^2:= p_n^1 -[G_n][g_n^2][G_n]^{-1}\phi ^2, &{} [G_n][g_n^2]^{-1}[G_n]^{-1}p_n^1\rightharpoonup \phi ^2, &{} \Vert \phi ^2\Vert _{L^2}\ge \frac{\mu ({\mathbb {P}}^1)}{2}>0; \\ \vdots &{}\vdots &{} \vdots \\ p_n^j:=p_n^{j-1}-[G_n][g_n^{j}][G_n]^{-1}\phi ^{j}, &{} [G_n][g_n^j]^{-1}[G_n]^{-1}p_n^{j-1}\rightharpoonup \phi ^j, &{} \Vert \phi ^j\Vert _{L^2}\ge \frac{\mu ({\mathbb {P}}^{j-1})}{2}>0; \\ \vdots &{}\vdots &{} \vdots \\ \end{array} \end{aligned}$$

Then a diagonal process yields a sequence of functions $(\phi ^j)_{j\ge 1}$ and a family of operators $[g_n^j]$ satisfying the orthogonal conclusion (48) for the case $j=k+1$. By the construction, we get the decomposition identity (52) and the almost orthogonal identities (51). To prove the desired claim, it remains to show the conclusion (48) for all $j\ne k$ and the estimate (53). We show the estimate (53) first. Recall that $\Vert f_n\Vert _{L^2}$ is uniformly bounded. Then (51) implies

$$\begin{aligned}\sum _{j=1}^{M}\Vert \phi ^j\Vert _{L^2}^2\le \limsup _{n\rightarrow \infty }\Vert f_n\Vert _{L^2}^2\le C.\end{aligned}$$

Hence, we know that the positive series $\sum _j \Vert \phi ^j\Vert _{L^2}^2$ is convergent and further $\lim _{n\rightarrow \infty }\Vert \phi ^j\Vert _{L^2}=0$. On the other hand, by the construction, we have

$$\begin{aligned}\mu ({\mathbb {P}}^M)\le 2\big \Vert \phi ^{M+1}\big \Vert _{L^2},\end{aligned}$$

which gives the desired estimate (53). Now we turn to prove the conclusion (48). Indeed, the more general case $j=k+m (m\in {\mathbb {Z}}_{+})$ comes from the basic case $j=k+1$, the following identity

$$\begin{aligned}p_n^{k+m-1}=p_n^k-[G_n][g_n^{k+1}][G_n]^{-1}\phi ^{k+1}- \cdots - [G_n][g_n^{k+m-1}][G_n]^{-1}\phi ^{k+m-1},\end{aligned}$$

and an inductive argument. For the case $j=k+m (m\in {\mathbb {Z}}_{-})$, if there does not hold the following

$$\begin{aligned}{}[g_n^j]^{-1}\big [g_n^k\big ][G_n]^{-1}\rightharpoonup 0,\end{aligned}$$

in ${\mathcal {B}}(L^2)$ as n goes to infinity, then by the dislocation property Proposition 1.3, we can assume

$$\begin{aligned}\big [g_n^j\big ]^{-1}\big [g_n^k\big ][G_n]^{-1}\rightarrow \big [g^{j,k}\big ], \quad \big [g^{j,k}\big ]\in {\mathcal {B}}(L^2), \quad \big [g^{j,k}\big ]\ne 0,\end{aligned}$$

in ${\mathcal {B}}(L^2)$ as n goes to infinity. In this case, we obviously have $\big [g^{j,k}\big ]^{-1}=[g^{k,j}]$. Hence, we can investigate the sequence $[G_n]\big [g_n^k\big ]^{-1}[g_n^j][G_n]^{-1}$, and turn the case $j=k+m (m\in {\mathbb {Z}}_{-})$ into the case $j=k+m (m\in {\mathbb {Z}}_{+})$ which we have already proved. Therefore, we complete the proof of the claim.

Second, to totally finish the proof of this Lemma 4.3, our next target is to get the desired conclusion (50) from the estimate (53) by using the localized restriction estimate Lemma 4.1.

Notice that we have the compact set K and the operators $\big [g_n^j\big ]$ do not change the support on the Fourier side. It means that when we get the above decomposition with conclusion (53), on the Fourier side, all the processes are taken place on this compact set K. Therefore, we conclude ${\hat{\phi }}^j\in L^{\infty }(K)$ and further ${\hat{p}}_n^M\in L^{\infty }(K)$. Since the Fourier support for $e_n^M$ is not ideal, we use some scaling skills as follows:

$$\begin{aligned} \left\Vert[D^{\frac{\alpha -2}{6}}][e^{it|\nabla |^{\alpha }}]e_n^M\right\Vert_{L_{t,x}^6}&=\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ] \big [\sqrt{\rho _n} e^{i(\cdot )\xi _n}p_n^M(\rho _n \cdot )\big ]\right\Vert_{L_{t,x}^6} \\&=\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ] [e^{i(\cdot )\frac{\xi _n}{\rho _n}}p_n^M]\right\Vert_{L_{t,x}^6}. \end{aligned}$$

Then we investigate the function

$$\begin{aligned}\omega _n^M(x):=e^{ix\frac{\xi _n}{\rho _n}}p_n^M(x),\end{aligned}$$

with the Fourier support information $\text {supp}({\hat{\omega }}_n^M)\subset K+(\rho _n)^{-1}\xi _n$. The Hölder inequality and the Bernstein inequality imply that

$$\begin{aligned}\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^6}\lesssim _K \left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^q}^{q/6} \left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^{\infty }}^{1-q/6}\end{aligned}$$

for $4<q<6$. Meanwhile, Lemma 4.1 gives the following estimate

$$\begin{aligned}\left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^q} \lesssim _K 1,\end{aligned}$$

which is independent of n and M. Hence, to get the desired result (50), it suffices to prove

$$\begin{aligned}\lim _{M\rightarrow \infty }\limsup _{n\rightarrow \infty }\left\Vert[e^{it|\nabla |^{\alpha }}]\omega _n^M\right\Vert_{L_{t,x}^{\infty }}=0.\end{aligned}$$

Moreover, by (53), it suffices to prove the following claim:

$$\begin{aligned} \limsup _{n\rightarrow \infty } \left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^{\infty }}\lesssim _{K} \mu ({\mathbb {E}}^M). \end{aligned}$$

(55)

Indeed, choose an even function $\mathbb {1}_K\in C_c^{\infty }({\mathbb {R}})$ satisfying $\mathbb {1}_K=1$ on K and choose $(a_m, b_m)$ such that

$$\begin{aligned}\left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M\right\Vert_{L_{t,x}^{\infty }}= \lim _{m\rightarrow \infty }\left|\big [e^{ia_m|\nabla |^{\alpha }}\big ]\omega _n^M(b_m)\right|.\end{aligned}$$

Define

$$\begin{aligned}\mathbb {1}_{K_n}(x):=\mathbb {1}_{K}\left(x-(\rho _n)^{-1}\xi _n\right), \quad \Omega _n^M(t,x):=\big [e^{it|\nabla |^{\alpha }}\big ]\omega _n^M(x).\end{aligned}$$

It follows that

$$\begin{aligned}\Omega _n^M(t,x)={\mathscr {F}}^{-1}e^{-it|\xi |^{\alpha }}\mathbb {1}_{K_n}(\xi ){\mathscr {F}}\omega _n^M(x), \quad \Omega _n^M(a_m, x+b_m)\in {\mathscr {W}}({\mathbb {P}}^M).\end{aligned}$$

Then using some basic properties for the spatial Fourier transform ${\mathscr {F}}$ and $\mathbb {1}_K$, by Hölder’s  inequality, we can control the $\left\Vert\Omega _n^M\right\Vert_{L_{t,x}^{\infty }}$ term as follows:

$$\begin{aligned} \left\Vert\Omega _n^M\right\Vert_{L_{t,x}^{\infty }}&= \lim _{m\rightarrow \infty }\left|\Omega _n^M(a_m,b_m)\right| = \lim _{m\rightarrow \infty }\left|{\mathscr {F}}^{-1}[\mathbb {1}_{K_n}{\widehat{\Omega }}_n^M](a_m,b_m)\right| \\&\sim \lim _{m\rightarrow \infty }\left|\check{\mathbb {1}}_{K_n}*\Omega _n^M(a_m, b_m)\right| \\&=\lim _{m\rightarrow \infty }\left|\int _{{\mathbb {R}}} \check{\mathbb {1}}_{K_n}(x) \Omega _n^M(a_m, x+b_m) \text {d}x\right| \\&\le \left\Vert\check{\mathbb {1}}_{K_n}\right\Vert_{L_x^2} \mu ({\mathbb {P}}^M)\lesssim _K \mu ({\mathbb {P}}^M). \end{aligned}$$

Therefore, we can obtain the desired result (55) and finish the proof.

Remark 4.4

As has been pointed out in [22, Remark 5.3], we can make a reduction in Lemma 4.3 when

$$\begin{aligned}\lim _{n\rightarrow \infty }(\rho _n)^{-1}\xi _n=a, \quad |a|<\infty .\end{aligned}$$

In this case, we can assume $\xi _n\equiv 0$ since we can replace $e^{ix(\rho _n)^{-1}\xi _n}\phi ^{j}$ by $e^{ixa}\phi ^j$, put the difference into the reminder term and then regard $e^{ixa}\phi ^j$ as the new $\phi ^j$.

5 Profile Decomposition of $\alpha$-Strichartz Version

In this section, with the two steps of decomposition Lemmas 3.3 and 4.3 at hand, we are able to show the desired $\alpha$-Strichartz version profile decomposition results Proposition 1.5 and the Strichartz orthogonality of profiles Proposition 1.7. It should be pointed out that, in the proof of Proposition 1.7, we use the conditional dislocation property Proposition 1.3 once more to coordinate the limit-orthogonal property conclusion (8) in Proposition 1.5.

Proof of Proposition 1.5

Using the Lemma 3.3 with $\frac{\delta }{2}$ and then using Lemma 4.3 properly, we can obtain the decomposition

$$\begin{aligned} u_n(x)=\sum _{\beta =1}^N\left(\sum _{j=1}^{M_{\beta }} \big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}\phi ^{\beta , j}(x)\right)+ e_{n}^{N, M_1,\ldots , M_{N}}(x), \end{aligned}$$

(56)

where the remainder term is

$$\begin{aligned}e_{n}^{N, M_1,\ldots , M_{N}}(x):=\sum _{\beta =1}^{N}e_n^{M_{\beta }}+q_n^N,\end{aligned}$$

and the operators in (56) are defined by

$$\begin{aligned}\quad \big [G_n^\beta \big ]f(x):= & {} \big (\rho _n^{\beta }\big )^{-\frac{1}{2}} e^{-ix\big (\rho _n^{\beta }\big )^{-1}\xi _n^{\beta }}f\left(\frac{x}{\rho _n^{\beta }}\right), \quad \big [g_n^{\beta ,j}\big ]f(x)\\:= & {} \big [e^{-is_n^{\beta ,j}|\nabla |^{\alpha }}\big ]f\big (x-y_n^{\beta ,j}\big ).\end{aligned}$$

Here, for each $1\le \beta \le N$, we choose $M_{\beta }$ to guarantee that for all $M\ge M_{\beta }$ there holds

$$\begin{aligned}\limsup _{n\rightarrow \infty } \left\Vert\big [D^{\frac{\beta -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]e_n^{M_{\beta }}\right\Vert_{L_{t,x}^6}\le \frac{\delta }{2N}.\end{aligned}$$

This is realizable since we have the vanishing Strichartz norm estimate (50) for the remainder in Lemma 4.3. Therefore, by combining the negligible Strichartz norm estimate (44) for the remainder in Lemma 3.3 with $\frac{\delta }{2}$, we have the following norm estimate for the remainder term:

$$\begin{aligned} \limsup _{n\rightarrow \infty } \left\Vert\big [D^{\frac{\beta -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]e_{n}^{N, M_1,\ldots , M_{N}}\right\Vert_{L_{t,x}^6}\le \delta . \end{aligned}$$

(57)

Firstly, for the sequence of operators $\big [G_n^{\gamma }\big ]\big [g_n^{\gamma ,k}\big ]^{-1}\big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}$, we will investigate the limit of this sequence in the weak operator topology of ${\mathcal {B}}(L^2)$ as n goes to infinity. If $\beta =\gamma$, then the limit-orthogonal conclusion (48) in Lemma 4.3 implies

$$\begin{aligned} \big [G_n^{\gamma }\big ]\big [g_n^{\gamma ,k}\big ]^{-1}\big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}\rightharpoonup 0, \end{aligned}$$

(58)

in ${\mathcal {B}}(L^2)$ as n goes to infinity. If $\beta \ne \gamma$, then the dual approach and Plancherel theorem give

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\langle \big [G_n^{\gamma }\big ]\big [g_n^{\gamma ,k}\big ]^{-1}\big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}f,g\right\rangle _x&=\lim _{n\rightarrow \infty } \left\langle \big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}f, \big [g_n^{\gamma , k}\big ]\big [G_n^{\gamma }\big ]^{-1}g\right\rangle _x \nonumber \\&\sim \lim _{n\rightarrow \infty } \left\langle {\mathscr {F}}\big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}f, {\mathscr {F}}\big [g_n^{\gamma , k}\big ]\big [G_n^{\gamma }\big ]^{-1}g \right\rangle _{\xi }, \end{aligned}$$

(59)

where f and g can be assumed to be Schwartz functions with compact Fourier supports. Note that the operators $[g_n^{\beta ,j}]$ and $[g_n^{\gamma ,k}]$ do not change the Fourier supports. Hence the conclusion (43) for the frequency and scaling parameters in Lemma 3.3, recalling the Remark 3.4, imply that the limit value in (59) is zero and further

$$\begin{aligned} \big [G_n^{\gamma }\big ]\big [g_n^{\gamma ,k}\big ]^{-1}\big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }]^{-1} \rightharpoonup 0, \end{aligned}$$

(60)

in ${\mathcal {B}}(L^2)$ as n goes to infinity.

Second, for the $L^2$-orthogonality, we can combine the identities (45) and (51) to conclude

$$\begin{aligned}\lim _{n\rightarrow \infty }\left(\Vert u_n\Vert _{L^2}^2-\sum _{\beta =1}^N\left(\sum _{j=1}^{M_{\beta }} |\phi ^{\beta , j}\Vert _{L^2}^2+\Vert e_n^{M_{\beta }}\Vert _{L^2}^2\right)-\Vert q_n^N\Vert _{L^2}^2\right)=0.\end{aligned}$$

Recall that the Fourier supports of $q_n^N$ and $e_n^{M_{\beta }}$ are mutually disjoint which comes from the fact that the operators $[g_n^{\beta , j}]$ do not change the Fourier support and the Remark 3.4. Therefore, we conclude

(61)

Finally, notice that the parameters N and $M_{\beta }$ depend only on $\delta$. Therefore, by enumerating the pairs $(\beta ,j)$ as

$$\begin{aligned} \big \{(\beta ,j)<(\gamma ,k)\big \}:=\big \{\beta +j<\gamma +k \;\;\text {or,}\;\; \beta +j=\gamma +k \;\;\text {and}\;\; \beta <\gamma \big \}, \end{aligned}$$

(62)

and then relabeling the pairs $(\beta ,j)$, we can define

$$\begin{aligned}\big (h_n^{{\tilde{j}}}, x_n^{{\tilde{j}}}, \xi _n^{{\tilde{j}}}, t_n^{{\tilde{j}}}\big ):= & {} \left(1/\rho _n^{\beta }, -y_n^{\beta ,j}, \xi _n^{\beta }, s_n^{\beta ,j}\right), \quad \bigg [T(h_n^{{\tilde{j}}}, x_n^{{\tilde{j}}}, \xi _n^{{\tilde{j}}}, t_n^{{\tilde{j}}})\bigg ]\\:= & {} \big [g_n^{\beta ,j}\big ]\big [G_n^{\beta }\big ]^{-1}, \quad \omega _n^{J_{\delta }}:=e_{n}^{N, M_1,\ldots , M_{N}}.\end{aligned}$$

Thus, after a classical diagonal process, we obtain the desired decomposition (6) by (56); the limit-orthogonality conclusion (8) comes from the weak operator topology convergency (60) and (58); meanwhile, the $L^2$-almost orthogonal identity (9) comes from (61). Therefore, it remains to prove the Strichartz norm estimate (7) in view of the enumeration (62). Actually, by using the Strichartz orthogonality of profiles Proposition 1.7, a standard $3\varepsilon$ trick will give this desired result (7). Here, we omit the detailed proof of (7) for simplicity, and the readers can find similar proofs in [37, p. 107] and [26, Page 371].

In order to investigate the extremal problem for the $\alpha$-Strichartz estimates and complete the proof for Proposition 1.5, we need to show the Strichartz orthogonality of profiles. Indeed, the limit-orthogonality property (8) for $[T_n^j]$ and the conditional dislocation property Proposition 1.3 imply this desired conclusion.

Proof of Proposition 1.7

Without loss of generality, we focus on the Schwartz functions $\phi ^j$ and $\phi ^k$ whose Fourier supports are compact. Based on the conclusion (43) for the frequency and scaling parameters in Lemma 3.3, we first deal with the case

$$\begin{aligned}\lim _{n\rightarrow \infty } \frac{h_n^j}{h_n^k}+\frac{h_n^k}{h_n^j}=\infty .\end{aligned}$$

Direct computation gives the following

$$\begin{aligned} \Phi _n^{j,k}(t,x):&=\big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j(x) \cdot \big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^k\big ]\phi ^k(x)\nonumber \\&= (h_n^j)^{-\frac{1}{2}-\frac{\alpha -2}{6}} \frac{e^{-ix_n^j\xi _n^j}}{2\pi } \int _{{\mathbb {R}}} |\xi |^{\frac{\alpha -2}{6}} e^{i\frac{x-x_n^j}{h_n^j}\xi -i\frac{t-t_n^j}{(h_n^j)^{\alpha }}|\xi |^{\alpha }} \big [e^{i(\cdot )h_n^j\xi _n^j}\phi ^j\big ]^{\wedge }(\xi )\text {d}\xi \nonumber \\&\quad \times (h_n^k)^{-\frac{1}{2}-\frac{\alpha -2}{6}} \frac{e^{-ix_n^k\xi _n^k}}{2\pi } \int _{{\mathbb {R}}} |\xi |^{\frac{\alpha -2}{6}} e^{i\frac{x-x_n^k}{h_n^k}\xi -i\frac{t-t_n^k}{(h_n^k)^{\alpha }}|\xi |^{\alpha }} \big [e^{i(\cdot )h_n^k\xi _n^k}\phi ^k\big ]^{\wedge }(\xi )\text {d}\xi \nonumber \\&=:\big (h_n^j\big )^{-\frac{\alpha +1}{6}}\Phi _n^j\left(\frac{t-t_n^j}{(h_n^j)^{\alpha }}, \frac{x-x_n^j}{h_n^j}\right) \cdot (h_n^k)^{-\frac{\alpha +1}{6}}\Phi _n^k\left(\frac{t-t_n^k}{(h_n^k)^{\alpha }}, \frac{x-x_n^k}{h_n^k}\right).\nonumber \\ \end{aligned}$$

(63)

By the $\alpha$-Strichartz estimate (1), we have

$$\begin{aligned}\lim _{R\rightarrow \infty }\int _{\{|s|+|y|> R\}} |\Phi _n^j(s,y)|^{6} \text {d}y\text {d}s=0.\end{aligned}$$

Therefore, setting

$$\begin{aligned}B_n^j(R):=\left\rbrace (t,x): \left|\frac{t-t_n^j}{(h_n^j)^{\alpha }}\right|+\left|\frac{x-x_n^j}{h_n^j}\right|\le R\right\lbrace ,\end{aligned}$$

Hölder’s  inequality gives

$$\begin{aligned}\int _{{\mathbb {R}}\setminus B_n^j(R)} |\Phi _n^{j,k}|^{3} \text {d}x\text {d}t \le \left(\int _{\{|x|+|t|> R\}} |\Phi _n^j|^{6} \text {d}x\text {d}t\right)^{\frac{1}{2}} \left(\int _{{\mathbb {R}}} |\Phi _n^k|^{6} \text {d}x \text {d}t\right)^{\frac{1}{2}},\end{aligned}$$

and analogously for ${\mathbb {R}}\setminus B_n^k(R)$. Thus, we are reduced to proving

$$\begin{aligned} \lim _{n\rightarrow \infty } (h_n^j h_n^k)^{-\frac{1+\alpha }{2}} \Big |B_n^j(R) \cap B_n^k(R)\Big |=0, \end{aligned}$$

(64)

due to the fact that $\Phi _n^j$ and $\Phi _n^k$ are $L_{t,x}^{\infty }$ functions. By the observation

$$\begin{aligned}\Big |B_n^j(R) \cap B_n^k(R)\Big | \le C_R \min \left\rbrace (h_n^j)^{1+\alpha }, (h_n^k)^{1+\alpha }\right\lbrace ,\end{aligned}$$

the desired estimate (64) follows immediately since $h_n^j/h_n^k$ goes to either zero or infinity. Hence, we can assume $h_n^j\sim h_n^k$ from now on.

Second, we turn to deal with the case

$$\begin{aligned}\lim _{n\rightarrow \infty }\left(h_n^j+h_n^k\right)|\xi _n^j-\xi _n^k|=\infty .\end{aligned}$$

By symmetry, we may assume $\lim _{n\rightarrow \infty }h_n^j|\xi _n^j-\xi _n^k|=\infty$; thus, from the expression (63), we conclude

$$\begin{aligned} \Phi _n^{j,k}(t,x)&= \big (h_n^j\big )^{-\frac{\alpha +1}{6}} \frac{e^{-ix_n^j\xi _n^j}}{2\pi }\int _{{\mathbb {R}}} |\xi |^{\frac{\alpha -2}{6}} e^{i\frac{x-\big (h_n^j\big )^2\xi _n^j}{h_n^j}\xi -i\frac{t-t_n^j}{\big (h_n^j\big )^{\alpha }}|\xi |^{\alpha }} e^{i\frac{\big (h_n^j\big )^2\xi _n^j-x_n^j}{h_n^j}\xi }[e^{i(\cdot )h_n^j\xi _n^j}\phi ^j]^{\wedge }(\xi )\text {d}\xi \nonumber \\&\quad \times (h_n^k)^{-\frac{\alpha +1}{6}} \frac{e^{-ix_n^k\xi _n^k}}{2\pi }\int _{{\mathbb {R}}} |\xi |^{\frac{\alpha -2}{6}} e^{i\frac{x-\big (h_n^j\big )^2\xi _n^k}{h_n^k}\xi -i\frac{t-t_n^k}{\big (h_n^k\big )^{\alpha }}|\xi |^{\alpha }} e^{i\frac{\big (h_n^j\big )^2\xi _n^k-x_n^k}{h_n^k}\xi }[e^{i(\cdot )h_n^k\xi _n^k}\phi ^k]^{\wedge }(\xi )\text {d}\xi \nonumber \\&=:(h_n^j)^{-\frac{\alpha +1}{6}}{\tilde{\Phi }}_n^j\left(\frac{t-t_n^j}{\big (h_n^j\big )^{\alpha }}, \frac{x-\big (h_n^j\big )^2\xi _n^j}{h_n^j}\right) \cdot \big (h_n^k\big )^{-\frac{\alpha +1}{6}}{\tilde{\Phi }}_n^k\left(\frac{t-t_n^k}{\big (h_n^k\big )^{\alpha }}, \frac{x-\big (h_n^j\big )^2\xi _n^k}{h_n^k}\right). \end{aligned}$$

(65)

Based on the expression (65), the assumption $h_n^j\sim h_n^k$ gives

$$\begin{aligned} \Vert \Phi _n^{j,k}\Vert _{L_{t,x}^3}&\sim \left\Vert{\tilde{\Phi }}_n^j\left(t-\frac{t_n^j}{(h_n^j)^{\alpha }}, x-{h_n^j\xi _n^j}\right) {\tilde{\Phi }}_n^k\left(\frac{\big (h_n^j\big )^{\alpha }t-t_n^k}{\big (h_n^k\big )^{\alpha }}, \frac{h_n^j}{h_n^k} \big (x-h_n^j\xi _n^k\big )\right)\right\Vert_{L_{t,x}^{3}}. \end{aligned}$$

Similarly, we still have the following estimate

$$\begin{aligned}\lim _{R\rightarrow \infty }\int _{\{|s|+|y|> R\}} \big |{\tilde{\Phi }}_n^j(s,y)\big |^{6} \text {d}y\text {d}s=0.\end{aligned}$$

By imitating the argument above, we can get the desired result (12) too. Hence, we turn to the case

$$\begin{aligned}\lim _{n\rightarrow \infty }\left({h_n^j}+{h_n^k}\right)\big |\xi _n^j-\xi _n^k\big |=a, \quad a<\infty .\end{aligned}$$

Recall the construction of the linear profile decomposition and the label in (62). Therefore, due to the conclusion (43) in Lemma 3.3, it remains to deal with the case $\beta =\gamma$ in view of the label (62). Consequently, we can assume $\big (h_n^j,\xi _n^j\big ) \equiv \big (h_n^k, \xi _n^k\big ) \equiv (h_n,\xi _n)$ from now on.

Thirdly, recalling the Remark 4.4, we may further assume and deal with the case

$$\begin{aligned} \lim _{n\rightarrow \infty }h_n\xi _n=\infty , \end{aligned}$$

(66)

since the case $\xi _n\equiv 0$ is much easier. By changing the variables in $\Vert \Phi _n^{j,k}\Vert _{L_{t,x}^3}$, we turn to investigate

$$\begin{aligned} {\tilde{\Phi }}_n^{j,k}(t,x)&:= \int _{{\mathbb {R}}} |\xi +h_n\xi _n|^{\frac{\alpha -2}{6}} e^{i\left(x+\frac{x_n^j-x_n^k}{h_n}\right)\xi -i\left(t+\frac{t_n^j-t_n^k}{(h_n)^{\alpha }}\right) |\xi +h_n\xi _n|^{\alpha }}{\hat{\phi }}^k(\xi ) \text {d}\xi \\&\quad \times \int _{{\mathbb {R}}} |\xi +h_n\xi _n|^{\frac{\alpha -2}{6}} e^{ix\xi -it|\xi +h_n\xi _n|^{\alpha }}{\hat{\phi }}^j(\xi )\text {d}\xi \\&=: \int _{{\mathbb {R}}} e^{i\left(x+\frac{x_n^j-x_n^k}{h_n}\right)\xi - i\Psi _{n}^{j,k,t}(\xi )} |\xi +h_n\xi _n|^{\frac{\alpha -2}{6}} {\hat{\phi }}^k(\xi ) \text {d}\xi \\&\quad \times \int _{{\mathbb {R}}} e^{ix\xi -it\Psi _n^t(\xi )} |\xi +h_n\xi _n|^{\frac{\alpha -2}{6}} {\hat{\phi }}^j(\xi )\text {d}\xi \\&=: A_n^{j,k}(t,x)\cdot B_n^j(t,x). \end{aligned}$$

To get the desired conclusion (12), it suffices to show that the following estimate

(67)

holds for all $j\ne k$. Just as what we have done in the Proof of Proposition 1.3, based on the assumption (66), we can rewrite

$$\begin{aligned}\Psi _{n}^{j,k,t}(\xi )=\sum _{m=1}^{\infty } a_{n}^{m,j,k,t} (\xi )^{m}, \quad \Psi _n^{t}(\xi )=\sum _{m=1}^{\infty } a_n^{m,t} (\xi )^m.\end{aligned}$$

Note that the differences of the coefficients

$$\begin{aligned} b_n^{m,j,k}:=a_n^{m,j,k,t}-a_n^{m,t}=\left( {\begin{array}{c}\alpha \\ m\end{array}}\right) \frac{t_n^j-t_n^k}{(h_n)^{\alpha }}\left|h_n\xi _n\right|^{\alpha -m}, \end{aligned}$$

are independent of t since the difference of the functions

$$\begin{aligned}\Psi _{n}^{j,k,t}(\xi )-\Psi _n^{t}(\xi )=\frac{t_n^j-t_n^k}{(h_n)^{\alpha }} |\xi +h_n\xi _n\big |^{\alpha },\end{aligned}$$

is independent of t. Meanwhile, the assumption (66) implies $b_n^{m+1,j,k}\ll b_n^{m,j,k}$ for n large enough. Hence, after passing to a subsequence, we have the following condition

$$\begin{aligned} \lim _{n\rightarrow \infty }\left(\left|{\tilde{b}}_n^{1,j,k}\right|+\left|b_n^{2,j,k}\right|\right)=\infty , \quad \quad {\tilde{b}}_n^{1,j,k}:=\frac{x_n^k-x_n^j}{h_n}+b_n^{1,j,k}, \end{aligned}$$

(68)

due to the limit-orthogonality property (8) and the conditional dislocation property Proposition 1.3. Again, the method of stationary phase will provide the decay estimates of $A_n^{j,k}(t,x)$ and $B_n^j(t,x)$. Combining the trivial size estimates and the oscillation estimates deduced by the classical van der Corput Lemma, we always have the following estimates

$$\begin{aligned} |A_n^{j,k}(t,x)|\lesssim _{\phi ^k} \min \left\rbrace |h_n\xi _n|^{\frac{\alpha -2}{6}}, \left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right|^{-\frac{1}{2}}\!\!\!\!|h_n\xi _n|^{-\frac{\alpha -2}{3}}\right\lbrace , \end{aligned}$$

(69)

and

$$\begin{aligned} |B_n^j(t,x)|\lesssim _{\phi ^j} \min \left\rbrace \big |h_n\xi _n\big |^{\frac{\alpha -2}{6}}, |t|^{-\frac{1}{2}}\big |h_n\xi _n\big |^{-\frac{\alpha -2}{3}}\right\lbrace . \end{aligned}$$

(70)

To get the non-stationary bounds, we decompose the spatial space into

$$\begin{aligned} A_t&:=\left\rbrace x: \left|x+\frac{x_n^j-x_n^k}{h_n}-a_n^{1,j,k,t}\right|\lesssim _{\phi ^k} \left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right| |h_n\xi _n|^{\alpha -2}\right\lbrace , \\ B_t&:=\left\rbrace x: \left|x-a_n^{1,t}\right|\lesssim _{\phi ^j} |t| |h_n\xi _n|^{\alpha -2}\right\lbrace , \\ C_t&:={\mathbb {R}}\setminus (A_t\cup B_t), \end{aligned}$$

where the implicit constants in the definitions of $A_t$ and $B_t$ may depend on the Fourier supports of $\phi ^k$ and $\phi ^j$. Note that

$$\begin{aligned}A_t=\left\rbrace x:\left|x-a_n^{1,t}-{\tilde{b}}_n^{1,j,k}\right|\lesssim _{\phi ^k} \left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right| |h_n\xi _n|^{\alpha -2}\right\lbrace ,\end{aligned}$$

by the definition of ${\tilde{b}}_n^{1,j,k}$. On the other hand if $x\in C_t$, we always have

$$\begin{aligned}|h_n\xi _n|^{\alpha -2} \left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right| \Big / \left|x+\frac{x_n^j-x_n^k}{h_n}-a_n^{1,j,k,t}\right| \lesssim _{\phi ^k} 1, \quad \quad \frac{|t| \big |h_n\xi _n\big |^{\alpha -2}}{\left|x-a_n^{1,t}\right|} \lesssim _{\phi ^j} 1.\end{aligned}$$

Hence, on ${\mathbb {R}}\times C_t$, we can use the classical van der Corput Lemma to obtain the non-stationary bounds:

$$\begin{aligned} |A_n^{j,k}(t,x)|\lesssim _{\phi ^k} \frac{\big |h_n\xi _n\big |^{\frac{\alpha -2}{6}}}{\left|x-a_n^{1,t}-{\tilde{b}}_n^{1,j,k}\right|}, \quad \big |B_n^j(t,x)\big |\lesssim _{\phi ^j} \frac{\big |h_n\xi _n\big |^{\frac{\alpha -2}{6}}}{\left|x-a_n^{1,t}\right|}. \end{aligned}$$

(71)

Finally, based on the aforementioned estimates (68–71) and the estimates of [22, (44) and (47)], we can follow an analogous argument in [22, Lemma 6.1, Case 2] to get the desired result (67). This will complete the proof.

The details for the remaining proof are very long but essentially the same as [22, Lemma 6.1, Case 2]. For the convenience of the reader and avoiding too much redundancy, we provide part of the details and the rest of the proof will be sketchy. Split the time space into ${\mathbb {R}}=\tau _0^{-}\cup \tau _0\cup {\tilde{\tau }}_n \cup \tau _n\cup \tau _n^{+}$ where

$$\begin{aligned}\tau _0:=\left[-|h_n\xi _n|^{2-\alpha }, |h_n\xi _n|^{2-\alpha }\right], \quad \tau _n:=\left[\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}-\big |h_n\xi _n\big |^{2-\alpha }, \frac{t_n^k-t_n^j}{(h_n)^{\alpha }}+\big |h_n\xi _n\big |^{2-\alpha }\right],\end{aligned}$$

and

$$\begin{aligned}{} & {} \tau _0^{-}:=(-\infty ,-|h_n\xi _n|^{2-\alpha }],\; {\tilde{\tau }}_n:=\left[|h_n\xi _n|^{2-\alpha }, \frac{t_n^k-t_n^j}{(h_n)^{\alpha }}-|h_n\xi _n|^{2-\alpha }\right],\; \\{} & {} \quad \tau _n^{+}:=\left[\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}+|h_n\xi _n|^{2-\alpha },\infty \right).\end{aligned}$$

For simplicity, we use the notation $I(\tau _0,A_t)$ to denote the integral of $|A_n^{j,k}B_n^j|^3$ on the domain $\tau _0\times A_t$. In other words, we define

$$\begin{aligned}I(\tau _0,A_t):=\int _{t\in \tau _0}\int _{x\in A_t}\big |A_n^{j,k}B_n^j\big |^3 \text {d}x\text {d}t.\end{aligned}$$

Similarly for the notations $I(\tau _n, B_t)$, $I(\tau _n^{+}, C_t)$ and so on. Taking (68) into consideration, the rest of this proof is divided into two parts.

Case A If there holds

$$\begin{aligned} \lim _{n\rightarrow \infty }\big |b_n^{2,j,k}\big |=\lim _{n\rightarrow \infty }\left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|=\infty , \end{aligned}$$

(72)

which means that $\frac{|t_n^j-t_n^k|}{(h_n)^{\alpha }}\gg |h_n\xi _n|^{-(\alpha -2)}$ for n large enough. Then the desired result comes from a similar process in the proof of [22, Lemma 6.1, Case 2aI]. By taking a subsequence, together with the symmetry of positive and negative cases, we may assume that $\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}>0$ without loss of generality. From the estimates (69) and (70), it is not hard to show that

$$\begin{aligned}I(\tau _0, B_t)\lesssim \left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|^{-\frac{3}{2}}, \quad I\big (\tau _n, B_t\big )\lesssim \left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|^{-\frac{1}{2}}.\end{aligned}$$

And meanwhile some computation gives

$$\begin{aligned}I\big (\tau _0^{-}, B_t\big )\lesssim \left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|^{-\frac{1}{2}}, \quad I(\tau _n^{+},B_t)\lesssim \left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|^{-\frac{1}{2}}.\end{aligned}$$

For the term $I({\tilde{\tau }}_n, B_t)$, recall the indefinite integral

$$\begin{aligned}\int t^{-1/2}(a-t)^{-3/2} \text {d}t= \frac{2\sqrt{t}}{a\sqrt{a-t}}+C.\end{aligned}$$

Therefore, by the condition $\frac{\big |t_n^j-t_n^k\big |}{(h_n)^{\alpha }}\gg |h_n\xi _n|^{-(\alpha -2)}$, we can obtain

$$\begin{aligned}I({\tilde{\tau }}_n, B_t)\lesssim |h_n\xi _n|^{-(\alpha -2)}\frac{2\sqrt{(t_n^j-t_n^k)/(h_n)^{\alpha }-\big (h_n\xi _n\big )^{-\alpha -2}}}{\big (t_n^j-t_n^k\big )/(h_n)^{\alpha }\sqrt{\big (h_n\xi _n\big )^{-(\alpha -2)}}} \lesssim \left|\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\big (h_n\xi _n\big )^{\alpha -2}\right|^{-\frac{1}{2}}.\end{aligned}$$

Hence, we conclude that $I({\mathbb {R}}, B_t)\rightarrow 0$ as $n\rightarrow \infty$ due to the condition (72). Analogous arguments give the estimate $I({\mathbb {R}}, A_t)\rightarrow 0$ as $n\rightarrow \infty$. Finally, by combining the non-stationary bounds (71) and further splitting ${\tilde{\tau }}_n$ into

$$\begin{aligned}{\tilde{\tau }}_n:=\left[\big |h_n\xi _n\big |^{2-\alpha }, \frac{t_n^k-t_n^j}{2(h_n)^{\alpha }}\right] \bigcup \left[\frac{t_n^k-t_n^j}{2(h_n)^{\alpha }}, \frac{t_n^k-t_n^j}{(h_n)^{\alpha }}-|h_n\xi _n|^{2-\alpha }\right],\end{aligned}$$

we can obtain the estimate $I({\mathbb {R}}, C_t)\rightarrow 0$ as $n\rightarrow \infty$ and finish the proof of this case.

Case B If $|b_n^{2,j,k}|\le C_0$ for some fixed $C_0>0$ and

$$\begin{aligned}\lim _{n\rightarrow \infty }\left|{\tilde{b}}_n^{1,j,k}\right|=\lim _{n\rightarrow \infty } \left|\frac{x_n^j-x_n^k-\alpha \big (t_n^j-t_n^k\big )(\xi _n)^{\alpha -1}}{h_n}\right|=\infty .\end{aligned}$$

Analogously the desired result comes from a similar process in the proof of [22, Lemma 6.1, Case 2aII]. We may assume ${\tilde{b}}_n^{1,j,k}>0$ at first. In this case, the corresponding decomposition for the time space is ${\mathbb {R}}:={\dot{\tau }}_K^{+}\cup {\dot{\tau }}_K^{-}$ for a large constant $K\gg C_0$, where

$$\begin{aligned}{\dot{\tau }}_K^{+}:=\left\rbrace t: (h_n\xi _n)^{\alpha -2}|t|\ge K\right\lbrace .\end{aligned}$$

Note that on ${\dot{\tau }}_K^{+}$, it holds $\big |(t_n^k-t_n^j)/(h_n)^{\alpha }\big |\ll |t|$ and on ${\dot{\tau }}_K^{-}\times B_t$, it holds

$$\begin{aligned}|x-\alpha (h_n\xi _n)^{\alpha -1}|\ll \left|{\tilde{b}}_n^{1,j,k}\right|,\end{aligned}$$

for n large enough. These two facts together with the stationary bounds (69) and (70) imply

$$\begin{aligned}I\big ({\dot{\tau }}_K^{+}, B_t\big )\lesssim K^{-1}, \quad I\big ({\dot{\tau }}_K^{-}, B_t\big )\lesssim K^{1/2}\left|{\tilde{b}}_n^{1,j,k}\right|^{-3}.\end{aligned}$$

The first estimate for $I({\dot{\tau }}_K^{+}, B_t)$ is uniform in all large n and is going to zero as K goes to infinity, while the second estimate for $I({\dot{\tau }}_K^{-}, B_t)$ is going to zero as n goes to infinity. Thereby, after a similar argument for $I({\mathbb {R}}, A_t)$, we can obtain the following two estimates

$$\begin{aligned}\lim _{n\rightarrow \infty }I\big ({\mathbb {R}}, A_t\big )=0, \quad \lim _{n\rightarrow \infty } I\big ({\mathbb {R}}, B_t\big )=0.\end{aligned}$$

For the term $I({\mathbb {R}}, C_t)$, we should use the non-stationary bounds (71) too. The result $I({\dot{\tau }}_K^{+}, C_t)\lesssim K^{-1}$ is not hard to obtain. Hence, it remains for us to estimate $I\big ({\dot{\tau }}_K^{-}, C_t\big )$. Just as what we have done in Case A, for $t\in {\dot{\tau }}_K^{-}$ if we split $C_t$ further into $C_t=B_t^{-}\cup B_t^{+}\cup A_t^{-}\cup A_t^{+}$ for n large enough where

$$\begin{aligned}B_t^{-}:=\left(-\infty , a_n^{1,t}-|t||h_n\xi _n|^{\alpha -2}\right], \quad B_t^{+}:=\left[a_n^{1,t}+|t||h_n\xi _n|^{\alpha -2}, a_n^{1,t}+\frac{{\tilde{b}}_n^{1,j,k}}{2}\right],\end{aligned}$$

and

$$\begin{aligned}A_t^{-}:=\left[a_n^{1,t}+\frac{{\tilde{b}}_n^{1,j,k}}{2}, a_n^{1,t}+{\tilde{b}}_n^{1,j,k}-\left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right||h_n\xi _n|^{\alpha -2}\right],\end{aligned}$$

with

$$\begin{aligned}A_t^{+}:=\left[ a_n^{1,t}+{\tilde{b}}_n^{1,j,k}+\left|t-\frac{t_n^k-t_n^j}{(h_n)^{\alpha }}\right|\big |h_n\xi _n\big |^{\alpha -2}, +\infty \right),\end{aligned}$$

then we can get the estimate $I({\mathbb {R}},C_t)\rightarrow 0$ as $n\rightarrow \infty$ by estimating the integral piece by piece. Therefore, we conclude the desired result (67).

6 Extremals for Symmetric $\alpha$-Strichartz Estimate

With the linear profile decomposition Proposition 1.5, the Strichartz orthogonality of profiles Proposition 1.7 and the following asymptotic Schrödinger  behavior Lemma 6.1 in place, we are ready to give the proof of the desired extremal result Theorem A for symmetric $\alpha$-Strichartz estimate. This arguments can be directly used in asymmetric cases, which will be shown in Sect. 7 later.

Lemma 6.1

(Asymptotic Schrödinger  behavior) If $\Vert \phi \Vert _{L_x^2({\mathbb {R}})}=1$ and $\lim _{n\rightarrow \infty }|\xi _n|=\infty$, then we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ]\right\Vert_{L_{t,x}^{6}({\mathbb {R}}^{2})} =\left(\frac{\alpha ^2-\alpha }{2}\right)^{-\frac{1}{6}}\left\Vert\big [e^{-it\Delta }\big ]\phi \right\Vert_{L_{t,x}^{6}({\mathbb {R}}^{2})}.\nonumber \\ \end{aligned}$$

(73)

Proof of Lemma 6.1

This is a standard consequence by changing of variables and the dominated convergence theorem, as well as the stationary phase method. The readers can see [22, Proposition 7.1] or [37, Remark 1.7] for similar arguments in other contexts. This idea can also be seen in some earlier papers [11] and [42]. However, in our situation, we should be careful to deal with the case for general real numbers $\alpha >1$ rather than integers. Similar techniques haven been shown in the Proof of Proposition 1.3 by using Taylor’s theorem. For the convenience of the reader, we provide the detailed proof as follows.

Without loss of generality, we can assume $\xi _n\rightarrow +\infty$ and $\phi$ is a Schwartz function with compact Fourier support

$$\begin{aligned}\text {supp}{\hat{\phi }}\subset \left\rbrace \xi \in {\mathbb {R}}: |\xi |<M \right\lbrace .\end{aligned}$$

We introduce some notations first. Define a sequence of functions as follows:

$$\begin{aligned}{\tilde{\Phi }}_n(\xi ):=(\xi _n)^2\left|1+{\xi }/{\xi _n}\right|^{\alpha }.\end{aligned}$$

As shown in Step 4 for Proof of Proposition 1.3, there exists some $N_0$ large enough such that the following two estimates:

$$\begin{aligned}\xi _n>2M, \quad (1+M/\xi _n)^{\alpha -2}<3/2,\end{aligned}$$

hold simultaneously for all $n> N_0$. Then we can select the terms with $n> N_0$ and investigate this subsequence. Now, we can use Taylor’s theorem to deduce the following identity:

$$\begin{aligned} {\tilde{\Phi }}_n(\xi )=(\xi _n)^2\sum _{m=0}^{\infty } \left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (\xi /\xi _n)^m=\sum _{m=0}^{\infty } \left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (\xi _n)^{2-m} (\xi )^m. \end{aligned}$$

Firstly, for every fixed $\xi$, we know that

$$\begin{aligned}{\tilde{\Phi }}_n(\xi )-1-\alpha \xi _n\xi -\frac{\alpha (\alpha -1)}{2}\xi ^2 =O\left(\frac{|\xi |^3}{\xi _n}\right),\end{aligned}$$

which implies the following limit relation:

$$\begin{aligned}\lim _{n\rightarrow \infty } \left[{\tilde{\Phi }}_n(\xi )-1-\alpha \xi _n\xi -\frac{\alpha (\alpha -1)}{2}\xi ^2\right] =0.\end{aligned}$$

For convenience, denote

$$\begin{aligned}\Phi _n(\xi ):={\tilde{\Phi }}_n(\xi )-1-\alpha \xi _n\xi -\frac{\alpha (\alpha -1)}{2}\xi ^2.\end{aligned}$$

Notice that $\Phi _n(\xi ) =\sum _{m=3}^{\infty } \left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (\xi _n)^{2-m} (\xi )^m$ and

$$\begin{aligned}\left|\frac{\left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (\xi _n)^{2-m}}{\left( {\begin{array}{c}\alpha \\ m+1\end{array}}\right) (\xi _n)^{1-m}}\right| =\left|\frac{(m+1)\xi _n}{\alpha -m}\right| \;\;\xrightarrow {m\rightarrow \infty } |\xi _n|.\end{aligned}$$

Therefore, the choosing of $N_0$ implies that the power series $\sum _{m=3}^{\infty } \left( {\begin{array}{c}\alpha \\ m\end{array}}\right) (\xi _n)^{2-m} (\xi )^m$ converges uniformly for all $\xi \in \text {supp}{\hat{\phi }}$. Moreover, for the derivatives of $\Phi _n(\xi )$, there holds

$$\begin{aligned} \Phi '_n(\xi )&=\alpha \xi _n (1+\xi /\xi _n)^{\alpha -1}-\alpha \xi _n -\alpha (\alpha -1)\xi \nonumber \\&=\alpha \sum _{m=2}^{\infty } \left( {\begin{array}{c}\alpha -2\\ m\end{array}}\right) \frac{\xi ^m}{(\xi _n)^{m-1}} \nonumber \\&\le \alpha M \sum _{m=2}^{\infty } \left( {\begin{array}{c}\alpha -2\\ m\end{array}}\right) 2^{1-m} \nonumber \\&=2\alpha M\left[(3/2)^{\alpha -2}-1-(\alpha -2)/2\right], \end{aligned}$$

(74)

where the inequality comes from the choosing of $N_0$; and similarly, there holds

$$\begin{aligned} \Phi ''_n(\xi ) =\alpha (\alpha -1)\big (1+\xi /\xi _n\big )^{\alpha -2}-\alpha (\alpha -1) \le \frac{\alpha (\alpha -1)}{2}. \end{aligned}$$

(75)

With these notations in place, we begin with investigating the function: $\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ]$. A direct computation gives

$$\begin{aligned} \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ](x)&=\frac{1}{2\pi } \int _{{\mathbb {R}}} e^{ix\xi -it|\xi |^{\alpha }} |\xi |^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi -\xi _n) \text {d}\xi \\&=\frac{e^{ix\xi _n}}{2\pi } \int _{{\mathbb {R}}} e^{ix\xi -it|\xi +\xi _n|^{\alpha }} |\xi +\xi _n|^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi ) \text {d}\xi \\&=\frac{e^{ix\xi _n}}{2\pi } |\xi _n|^{\frac{\alpha -2}{6}} \int _{{\mathbb {R}}} e^{ix\xi -i(\xi _n)^{\alpha -2}t {\tilde{\Phi }}_n(\xi )} \left|1+\frac{\xi }{\xi _n}\right|^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi ) \text {d}\xi . \end{aligned}$$

If we further set $c_{\alpha }:=\frac{\alpha (\alpha -1)}{2}$ and

$$\begin{aligned}X:=x-\alpha (\xi _n)^{\alpha -1} t, \quad T:=c_{\alpha }(\xi _n)^{\alpha -2}t,\end{aligned}$$

then this changing of variables deduces the following identity:

$$\begin{aligned}{} & {} \left\Vert[D^{\frac{\alpha -2}{6}}][e^{it|\nabla |^{\alpha }}][e^{i(\cdot )\xi _n}\phi ]\right\Vert_{L_{t,x}^{6}}\nonumber \\{} & {} \quad =c_{\alpha }^{-\frac{1}{6}} \left\Vert\frac{1}{2\pi } \int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} \left|1+\frac{\xi }{\xi _n}\right|^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi ) \text {d}\xi \right\Vert_{L_{T,X}^{6}}. \end{aligned}$$

(76)

By dominated convergence theorem, for every fixed X and T, we conclude that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} \left|1+\frac{\xi }{\xi _n}\right|^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi ) \text {d}\xi =\int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2} {\hat{\phi }}(\xi ) \text {d}\xi . \end{aligned}$$

(77)

Therefore, it remains to find a function $F(T,X)\in L_{T,X}^6$ such that

$$\begin{aligned} \left|\int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} \left|1+{\xi }/{\xi _n}\right|^{\frac{\alpha -2}{6}} {\hat{\phi }}(\xi ) \text {d}\xi \right| \lesssim F(T,X). \end{aligned}$$

(78)

Based on the identities (76) and (77), once the estimate (78) is established, our final conclusion (73) directly follows from dominated convergence theorem.

Now, let us establish the desired dominating function F(T, X) by using stationary phase method. Since $|1+\xi /\xi _n|<2$, we only need to investigate the following function:

$$\begin{aligned}I_n(T,X):=\left|\int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} {\hat{\phi }}(\xi ) \text {d}\xi \right|.\end{aligned}$$

Define the phase function

$$\begin{aligned}\varphi _{n}(T,X,\xi ):=X\xi -T\xi ^2-Tc_{\alpha }^{-1}\Phi _n(\xi ).\end{aligned}$$

First, the estimate (75) implies that

$$\begin{aligned}\left|\left(\frac{\text {d}}{\text {d}\xi }\right)^2 \varphi _{n}(T,X,\xi )\right| \ge |T|, \quad \forall \;\xi \in \text {supp}{\hat{\phi }}.\end{aligned}$$

Thus, van der Corput Lemma [39, p. 334, Corollary] gives the following decay estimate:

$$\begin{aligned} I_n(T,X)\lesssim _{\phi } (1+|T|)^{-1/2}. \end{aligned}$$

(79)

This estimate holds uniformly for $X\in {\mathbb {R}}$. Second, by the estimate (74), we denote

$$\begin{aligned}{\tilde{C}}_{\alpha ,M}:=2\alpha M\left[(3/2)^{\alpha -2}-1-(\alpha -2)/2\right],\end{aligned}$$

and then compute as follows:

$$\begin{aligned} \left|\frac{\text {d}}{\text {d}\xi } \varphi _{n}(T,X,\xi )\right|&=\left|X-T\left[2\xi +c_{\alpha }^{-1} \Phi '_n(\xi )\right]\right| \\&\ge |X|-\left|T\left(2M+c_{\alpha }^{-1}{\tilde{C}}_{\alpha ,M}\right)\right|. \end{aligned}$$

Therefore, we can choose a large constant $C_{\alpha ,M}>0$ such that: if $|X|>C_{\alpha ,M} |T|$, then there holds

$$\begin{aligned}\left|\frac{\text {d}}{\text {d}\xi } \varphi _{n}(T,X,\xi )\right| > rsim _{\alpha ,M} |X|, \quad \forall \; \xi \in \text {supp}{\hat{\phi }}.\end{aligned}$$

Hence, on the set $\left\rbrace (T,X)\in {\mathbb {R}}^2: |X|>C_{\alpha ,M}|T|\right\lbrace$, we can use the localization principle of oscillatory integrals [39, p. 331, Proposition 1] to obtain the following decay estimate:

$$\begin{aligned}I_n(T,X)\lesssim _{\alpha ,\phi } (1+|X|)^{-1}\lesssim _{\alpha ,\phi } \left[(1+|T|)(1+|X|)\right]^{-1/2}.\end{aligned}$$

On the other hand, for the set $\left\rbrace (T,X)\in {\mathbb {R}}^2: |X|\le C_{\alpha ,M}|T|\right\lbrace$, we can use the estimate (79) to obtain

$$\begin{aligned}I_n(T,X)\lesssim _{\phi } (1+|T|)^{-1/2} \lesssim _{\alpha ,\phi } \left[(1+|T|)(1+|X|)\right]^{-1/4}.\end{aligned}$$

Combining these aforementioned two decay estimates, we define the desired function F(T, X) as follows:

$$\begin{aligned} F(T,X):=\left\rbrace \begin{array}{cc} C_{\alpha ,\phi }\left[(1+|T|)(1+|X|)\right]^{-1/4}, &{} |X|\le C_{\alpha ,M} |T|;\\ C_{\alpha ,\phi }\left[(1+|T|)(1+|X|)\right]^{-1/2}, &{} |X|> C_{\alpha ,M} |T|. \end{array}\right. \end{aligned}$$

It routine to verify that this function F(T, X) belongs to $L_{T,X}^6({\mathbb {R}}^2)$ and the proof is completed.

Remark 6.2

Based on the existence of extremals for ${\textbf{M}}_2$, Lemma 6.1 implies

$$\begin{aligned}{\textbf{M}}_{\alpha }\ge \left(\frac{\alpha ^2-\alpha }{2}\right)^{-\frac{1}{6}}{\textbf{M}}_2.\end{aligned}$$

Meanwhile we pointed out that the non-precompactness, in view of Theorem A, is different from the non-existence of extremals. Indeed when $\alpha =2$, Foschi [13] and Hundertmark–Zharnitsky [21] independently show that the sharp constant ${\textbf{M}}_2=12^{-12}$ and the only extremals are Gaussians up to symmetries. Based on this result, Lemma 6.1 implies

$$\begin{aligned}{\textbf{M}}_{\alpha }\ge [\sqrt{3}\alpha (\alpha -1)]^{-\frac{1}{6}}.\end{aligned}$$

Proof of Theorem A

Let $\{u_n\}_{n\ge 1}$ be an extremal sequence for ${\textbf{M}}_{\alpha }$. Then, up to subsequences, by the profile decomposition Proposition 1.5, we can decompose $u_n$ into linear profiles as follows:

$$\begin{aligned}u_n=\sum _{j=1}^{J} \big [T_n^j\big ]\phi ^j+\omega _n^{J}.\end{aligned}$$

Due to the vanishing Strichartz norm estimate (7) for the remainder term, we obtain that for arbitrary $\epsilon >0$, there exists $N_{\epsilon }$ such that for all $N\ge N_{\epsilon }$ and all $n\ge N_{\epsilon }$

$$\begin{aligned}{\textbf{M}}_{\alpha }-\epsilon \le \left\Vert \big [D^{\frac{\alpha -2}{6}}\big ]\sum _{j=1}^{N} \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_{t,x}^{6}}.\end{aligned}$$

Hence, the Strichartz orthogonality of profiles Proposition 1.7 gives the following

$$\begin{aligned}{\textbf{M}}_{\alpha }^6-C_{\alpha }\epsilon \le \sum _{j=1}^{N}\left\Vert \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert^6_{L_{t,x}^6}.\end{aligned}$$

Take $j_0$ such that

$$\begin{aligned}\left\Vert\big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^{j_0}\big ]\phi ^{j_0}\right\Vert_{L_{t,x}^6}=\max \left\rbrace \left\Vert \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_{t,x}^6}: 1\le j\le N\right\lbrace .\end{aligned}$$

The $\alpha$-Strichartz estimate (1) implies

$$\begin{aligned} {\textbf{M}}_{\alpha }^6-C_{\alpha }\epsilon&\le \sum _{j=1}^{N}\left\Vert \big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j \right\Vert^{6}_{L_{t,x}^{6}}\nonumber \\&\le {\textbf{M}}_{\alpha }^6\sum _{j=1}^{N} \Vert \phi ^j\Vert _{L_x^2}^6 \le {\textbf{M}}_{\alpha }^6\left(\sum _{j=1}^{N} \Vert \phi ^j\Vert _{L_x^2}^2\right)^3 \le {\textbf{M}}_{\alpha }^6, \end{aligned}$$

(80)

where the last inequality comes from the fact that the $L^2$-orthogonal identity (9) which leads to

$$\begin{aligned} \sum _{j=1}^{\infty }\Vert \phi ^j\Vert _{L_x^2}^2\le \lim _{n\rightarrow \infty }\Vert u_n\Vert ^2_{L_x^2}=1. \end{aligned}$$

(81)

This fact also deduces $\lim _{j\rightarrow \infty }\Vert \phi ^j\Vert _{L_x^2}=0$. Consequently, we can choose $j_0$ independent of $\epsilon$. Hence, we may let $\epsilon \rightarrow 0$ in (80) to obtain

$$\begin{aligned}\big \Vert \phi ^{j_0}\big \Vert _{L_x^2}=1,\end{aligned}$$

which means $\phi ^j=0$ for all $j\ne j_0$ due to the inequality (81). Therefore, by the linear profile decomposition, we conclude

$$\begin{aligned}\lim _{n\rightarrow \infty }\left\Vert \big [D^{\frac{\alpha -2}{6}}\big ] \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^{j_0}\big ]\phi ^{j_0} \right\Vert_{L_{t,x}^{6}}=\lim _{n\rightarrow \infty }\left\Vert \big [D^{\frac{\alpha -2}{6}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )h_n^{j_0}\xi _n^{j_0}}\big ]\phi ^{j_0} \right\Vert_{L_{t,x}^{6}}={\textbf{M}}_{\alpha }.\end{aligned}$$

This suggests that $e^{ixh_n^{j_0}\xi _n^{j_0}}\phi ^{j_0}(x)\in L_x^2$ is an extremal sequence where either $\lim _{n\rightarrow \infty }|h_n^{j_0}\xi _n^{j_0}| \rightarrow \infty$ or $h_n^{j_0}\xi _n^{j_0}\equiv 0$. For the case $h_n^{j_0}\xi _n^{j_0}\equiv 0$, we get the desired extremal function $\phi ^{j_0}$. However, for the case $|h_n^{j_0}\xi _n^{j_0}|\rightarrow \infty$, we should do more investigation as follows. Indeed as we will see, Lemma 6.1 gives the desired conclusion and finishes the proof.

If we have the strict inequality (3), then the case $|h_n^{j_0}\xi _n^{j_0}|\rightarrow \infty$ is ruled out by Lemma 6.1, and hence, all the extremal sequences for ${\textbf{M}}_{\alpha }$ are precompact up to symmetries. On the other hand, if

$$\begin{aligned}{\textbf{M}}_{\alpha }=\big [\sqrt{3}\alpha (\alpha -1)\big ]^{-\frac{1}{6}},\end{aligned}$$

then, after normalizing, ${\tilde{u}}_n(x):=\sqrt{n}e^{in^2x}e^{-|n(x-x_0)|^2}$ will give an extremal sequence that is not precompact up to symmetries, since ${\tilde{u}}_n$ goes to zero up to symmetries in the weak topology of $L^2({\mathbb {R}})$. Finally, it is easy to check that $({\tilde{u}}_n)$ concentrates at $x_0$.

7 Extremals for Asymmetric $\alpha$-Strichartz Estimate

As what we have stated before, our method can produce similar extremal results for the asymmetric $\alpha$-Strichartz estimates (2) as well. This mainly because that the profile decomposition Proposition 1.5 is simultaneously equipped with the Strichartz-orthogonality Proposition 1.7 for all the profiles. By imitating the arguments in the proof of Theorem A, it is not hard to see that the desired asymmetric $\alpha$-Strichartz result Theorem 1.1 is a direct consequence of the following two lemmas which are generalizations of the estimates (13) and (73), respectively.

Lemma 7.1

Let (q, r) be non-endpoint pairs and ${\tilde{N}}\ge 1$. For the profiles in Proposition 1.5, if $q\ge r$, then

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\Vert\sum _{j=1}^{{\tilde{N}}}\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_t^q L_x^r}^r \le \sum _{j=1}^{{\tilde{N}}}\lim _{n\rightarrow \infty } \left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_t^q L_x^r}^r; \end{aligned}$$

meanwhile if $q\le r$, then

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\Vert\sum _{j=1}^{{\tilde{N}}}\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_t^q L_x^r}^q \le \sum _{j=1}^{{\tilde{N}}}\lim _{n\rightarrow \infty } \left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j\right\Vert_{L_t^q L_x^r}^q. \end{aligned}$$

Lemma 7.2

If $\Vert \phi \Vert _{L_x^2({\mathbb {R}})}=1$ and $\lim _{n\rightarrow \infty }|\xi _n|=\infty$, then we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\Vert\big [D^{\frac{\alpha -2}{q}}\big ]\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ]\right\Vert_{L_t^q L_x^r} =\left(\frac{\alpha ^2-\alpha }{2}\right)^{-\frac{1}{q}}\left\Vert[e^{-it\Delta }]\phi \right\Vert_{L_t^q L_x^r}. \end{aligned}$$

Remark 7.3

As in Remark 6.2, the non-precompactness up to symmetries, equivalently the equality

$$\begin{aligned}\tilde{{\textbf{M}}}_{\alpha , q,r}=\left(\frac{\alpha ^2-\alpha }{2}\right)^{-\frac{1}{q}} \tilde{{\textbf{M}}}_{2,q,r},\end{aligned}$$

does not mean the non-existence of extremals. For the sharp constant $\tilde{{\textbf{M}}}_{2,q,r}$ with asymmetric pairs (q, r), recalling that the existence of extremals has been proved in [36], the known results are pretty few even if there are many excellent works such as [3, 9, 18]. Indeed, up to now, we are only aware of the case $\tilde{{\textbf{M}}}_{2,8,4}=2^{-1/4}$ as shown in the aforementioned three papers. As far as we know, there is no higher-dimensional result for the asymmetric sharp constant $\tilde{{\textbf{M}}}_{2,q,r}$.

We are not planing to show all the detailed proofs of these two lemmas since the arguments are standard. Instead, there will be some useful references for the readers who are interested in further details. For the Lemma 7.1, indeed it is a corollary of the Strichartz-orthogonality estimate (12). This fact may be not as obvious as getting the estimate (13) from (12), since $q\ne r$ and they may be not natural numbers at all. However, this difficult has been overcame by using the interpolation arguments and some floor function techniques, see [36, Lemma 1.6] for more details. Also notice that the conclusions in Lemma 7.1 are inequalities instead of equities compared with the estimate (13).

While Lemma 7.2 may seem easy to accept since it is obvious an asymmetric generalization of Lemma 6.1. Here, we give the $L_t^qL_x^r$-dominating function as follows:

$$\begin{aligned} F(t,x):=\left\rbrace \begin{array}{cc} C_{\alpha ,\phi }(1+|t|)^{-\frac{3}{2q}}(1+|x|)^{-\frac{q-3}{2q}}, &{} |x|\lesssim _{\alpha ,\phi } |t|;\\ C_{\alpha ,\phi }(1+|t|)^{-\frac{3}{q}}(1+|x|)^{-\frac{q-3}{q}}, &{} |x| > rsim _{\alpha ,\phi } |t|. \end{array}\right. \end{aligned}$$

One can argue as the proof of Lemma 6.1 to establish this dominating function, which also follows from an application of van der Corput Lemma. The details are omitted here for avoiding to much repetition.

8 Extremals for Non-endpoint $\alpha$-Strichartz Estimate

In this section, we provide the proof of Theorem 1.8 by following the arguments in [20]. As mentioned before, the case $\alpha =2$ is well known. Thus, we investigate the case $\alpha >2$. It is obvious that we only need to prove the following non-endpoint $\alpha$-Strichartz profile decomposition Proposition 8.1, which indeed is a direct consequence of the aforementioned linear profile decomposition results Proposition 1.5 and Proposition 1.7.

Proposition 8.1

Let $\alpha >2$ and $(u_n)$ be a bounded sequence in $L^2({\mathbb {R}})$. Then, up to subsequences, there exist a sequence of operators $\big [T_n^j\big ]$ defined by

$$\begin{aligned}\big [T_n^{j}\big ]\phi (x):=\big [e^{-it_n^j|\nabla |^{\alpha }}\big ]\left[(h_n^j)^{-\frac{1}{2}}\phi \left(\frac{x-x_n^j}{h_n^j}\right)\right],\end{aligned}$$

with $(h_n^j, x_n^j, t_n^j) \in {\mathbb {R}}_{+}\times {\mathbb {R}}\times {\mathbb {R}}$ and a sequence of functions $\phi ^j\in L^2({\mathbb {R}})$ such that for every $J\ge 1$, we have the profile decomposition

$$\begin{aligned} u_n=\sum _{j=1}^{J} [T_n^j]\phi ^j+\omega _n^{J}, \end{aligned}$$

where the decomposition possesses the following properties: first, the remainder term $\omega _n^{J}$ has vanishing Strichartz norm

$$\begin{aligned} \lim _{J\rightarrow \infty }\limsup _{n\rightarrow \infty }\left\Vert[e^{it|\nabla |^{\alpha }}]\omega _n^{J}\right\Vert_{L_{t,x}^{2\alpha +2}({\mathbb {R}}^{2})}=0; \end{aligned}$$

(82)

second, the sequence of operators $[T_n^j]$ satisfies that if $j\ne k$, there holds the limit-orthogonality property

$$\begin{aligned}{}[T_n^k]^{-1}[T_n^j]\rightharpoonup 0, \end{aligned}$$

(83)

as n goes to infinity in the weak operator topology of ${\mathcal {B}}(L^2)$; for each $J\ge 1$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\left[\Vert u_n\Vert _{L^2({\mathbb {R}})}^2-\left(\sum _{j=1}^{J}\Vert \phi ^j\Vert _{L^2({\mathbb {R}})}^2\right)-\Vert \omega _n^{J}\Vert _{L^2({\mathbb {R}})}^2\right]=0; \end{aligned}$$

moreover, for every $j\ne k$, there holds the Strichartz orthogonality of profiles

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^j\big ]\phi ^j \cdot \big [e^{it|\nabla |^{\alpha }}\big ]\big [T_n^k\big ]\phi ^k\right\Vert_{L_{t,x}^{\alpha +1}({\mathbb {R}}^{2})}=0. \end{aligned}$$

(84)

Remark 8.2

In this case, without the frequency parameters $\xi _n$, the limit-orthogonality property (83) holds up to subsequences if and only if

$$\begin{aligned}\limsup _{n\rightarrow \infty }\left(\frac{h_n^j}{h_n^k}+\frac{h_n^k}{h_n^j} +\frac{|t_n^j-t_n^k|}{\big (h_n^j\big )^{\alpha }} + \frac{\big |t_n^j-t_n^k\big |}{\big (h_n^k\big )^{\alpha }} +\frac{\big |x_n^j-x_n^k\big |}{h_n^j}+\frac{\big |x_n^j-x_n^k\big |}{h_n^k}\right)=\infty .\end{aligned}$$

This conclusion can be seen in the proof of the conditional dislocation property Proposition 1.3. Note that the condition above is symmetric in the indices j and k.

Proof of Proposition 8.1

It is not hard to see that the vanishing norm estimate (82) follows from the remainder term estimate (7) in Proposition 1.5 and Sobolev inequalities. To eliminate the frequency parameters, as shown in the proof of [20, Theorem 2.4], the key point is to deduce the following estimate:

$$\begin{aligned} \lim _{|\xi _n|\rightarrow \infty }\left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ]\right\Vert_{L_{t,x}^{2\alpha +2}}=0. \end{aligned}$$

(85)

Then the highly oscillatory terms in Proposition 1.5, which mean the terms $[T_n^j]\phi ^j(x)$ with

$$\begin{aligned}\lim _{n\rightarrow \infty }|h_n^j\xi _n^j|=\infty ,\end{aligned}$$

can be reorganized into the remainder term. After that, the desired Strichartz orthogonality (84) of these profiles is much easier to established due to the lack of frequency parameters, see also [20, Lemma 2.7] for further details. Other conclusions come from Proposition 1.5 and Proposition 1.7 accordingly.

To obtain the estimate (85), we can follow similar arguments in the proof of Lemma 6.1, see also [20, Theorem 2.4]. Here, we provide the details as follows.

First, using the notations $\Phi _n(\xi )$ and ${\tilde{\Phi }}_n(\xi )$ in the proof of Lemma 6.1, a direct computation gives

$$\begin{aligned} \big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ](x)&=\frac{1}{2\pi } \int _{{\mathbb {R}}} e^{ix\xi -it|\xi |^{\alpha }} {\hat{\phi }}(\xi -\xi _n) \text {d}\xi \\&=\frac{e^{ix\xi _n}}{2\pi } \int _{{\mathbb {R}}} e^{ix\xi -it|\xi +\xi _n|^{\alpha }} {\hat{\phi }}(\xi ) \text {d}\xi \\&=\frac{e^{ix\xi _n}}{2\pi } \int _{{\mathbb {R}}} e^{ix\xi -i(\xi _n)^{\alpha -2}t {\tilde{\Phi }}_n(\xi )} {\hat{\phi }}(\xi ) \text {d}\xi . \end{aligned}$$

If we further set $c_{\alpha }:=\frac{\alpha (\alpha -1)}{2}$ and

$$\begin{aligned}X:=x-\alpha (\xi _n)^{\alpha -1} t, \quad T:=c_{\alpha }(\xi _n)^{\alpha -2}t,\end{aligned}$$

then this changing of variables deduces the following identity:

$$\begin{aligned} \left\Vert\big [e^{it|\nabla |^{\alpha }}\big ]\big [e^{i(\cdot )\xi _n}\phi \big ]\right\Vert_{L_{t,x}^{2\alpha +2}}=c_{\alpha }^{-\frac{1}{2\alpha +2}} \big |\xi _n\big |^{-\frac{\alpha -2}{2\alpha +2}} \left\Vert\frac{1}{2\pi } \int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} {\hat{\phi }}(\xi ) \text {d}\xi \right\Vert_{L_{T,X}^{2\alpha +2}}. \end{aligned}$$

From the proof of Lemma 6.1, we know that there exists $F(T,X)\in L_{T,X}^6({\mathbb {R}}^2)$ such that

$$\begin{aligned}\left|\int _{{\mathbb {R}}} e^{iX\xi -iT\xi ^2-iTc_{\alpha }^{-1}\Phi _n(\xi )} {\hat{\phi }}(\xi ) \text {d}\xi \right| \le F(T,X).\end{aligned}$$

Therefore, the dominated convergence theorem implies our desired conclusion (85).