On $L^{2}$-boundedness of Fourier integral operators

Abstract

Let $T_{a,\varphi }$ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions $a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$ and $\varphi \in L^{\infty }\varPhi ^{2}$, the authors show that $T_{a,\varphi }$ is bounded from $L^{2}(\mathbb{R}^{n})$ to $L^{2}(\mathbb{R}^{n})$.

1 Introduction and main results

Fourier integral operator on $\mathbb{R}^{n}$ has been studied extensively and is related to many areas in analysis and PDEs. In [1], Sogge considered the Cauchy problem of the hyperbolic equations via the $L^{p}$-estimates theory of the Fourier integral operators (also see, for the local smoothing estimates of wave equations, e.g., [2, 3] and the references therein for some recent developments). For the Fourier integral operators with smooth amplitude, the $L^{2}$-regularity theory is comparably more progress. In [4] and [5], Eskin and Hörmander found the local and global $L^{2}$-regularity theory for Fourier integral operators, respectively. There are also some results for the $L^{p}$ boundedness of Fourier integral operators with classical symbol and phase (see Littman [6], Miyachi [7], Peral [8], and Beals [9]).

Let f̂ be the Fourier transform of f. A Fourier integral operator T is a linear operator of the form

$$ T_{a,\varphi }f=\frac{1}{(2\pi )^{n}} \int _{\mathbb{R}^{n}} e^{ i \varphi (x,\xi )}a(x,\xi ) \hat{f}(\xi )\,d\xi $$

(1.1)

with symbol $a(x,\xi )$ and phase $\varphi (x,\xi )$, respectively. In particular, for $\varphi (x,\xi )=\langle x,\xi \rangle$, the operator $T_{a}$ is a so-called pseudo-differential operator. In [10], Hörmander showed that $T_{a}$ is bounded in $L^{2}(\mathbb{R}^{n})$, when $a\in S^{m}_{\rho ,\delta }$, $\delta <1$ and $m\leq n(\rho -\delta )/2$. For $a\in S^{0}_{1,1}$, Ching [11] proved that $T_{a}$ is not bounded in $L^{2}(\mathbb{R}^{n})$. Moveover, for $a\in S^{m}_{\rho ,1}$, Rodino [12] showed that $T_{a}$ is bounded in $L^{2}(\mathbb{R}^{n})$ if and only if $m< n(\rho -1)/2$. However, the operator $T_{a}$ is not always $L^{2}$-bounded for $a\in S^{n(\rho -1)/2}_{\rho ,1}$; see, for example, [10–12]. The necessary and sufficient conditions of $L^{2}$-boundedness of $T_{a}$ were obtained by Higuchi [13] as $m= n(\rho -1)/2$. It is natural to ask if the corresponding results hold for the Fourier integral operators. Recently, Kenig, David, Salvador, and Wolfgang [14–16] have studied the Fourier integral operators with rough symbol and rough phases, both of which behave in the spatial variable x like an $L^{\infty }$-function. More precisely, the symbol belongs to the class $L^{\infty }S^{m}_{\varrho }$ whose constituent element a obeys

$$ \bigl\Vert \partial ^{\alpha }_{\xi }a(\cdot ,\xi ) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n})} \leq C_{\alpha }\langle \xi \rangle ^{m-\varrho \vert \alpha \vert }. $$

Under this condition, for $m=\min \{0,\frac{n}{2}(\rho -\delta )\}$, $0\leq \rho \leq 1$, $0\leq \delta <1$, and $a\in S^{m}_{\rho ,\delta }$, Wolfgang [14] proved the global continuity on $L^{p}$-space with $p\in [1,\infty ]$ of Fourier integral operators. A natural question is $L^{2}$-boundedness of Fourier integral operators for $\delta =1$ and $m= n(\rho -1)/2$. In this paper, we answer the question and prove the results for the Fourier integral operators.

Our main result could be stated as follows.

Theorem 1.1

Let$T_{a,\varphi }$be a Fourier integral operator given by (1.1) with symbol$a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$and phase function$\varphi \in L^{\infty }\varPhi ^{2}$satisfying the Lipschitz rough non-degeneracy condition. Then, for$0\leq \rho \leq 1$, there exists a positive constantCsuch that

$$ \Vert T_{a,\varphi }u \Vert _{L^{2}}\leq C \Vert u \Vert _{L^{2}} . $$

Here, the symbol class $L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$ is defined by Definition 2.2, the phase class $L^{\infty }\varPhi ^{2}$ is given by Definition 2.5, and the Lipschitz rough non-degeneracy condition is defined by Definition 2.6.

Remark 1.1

Here we remark that, for $a\in S^{n(\rho -1)/2}_{\rho ,1}$, Higuchi and Nagase [13] pointed out that the boundedness of the pseudo-differential operator $T_{a}$ from $L^{2}(\mathbb{R}^{n})$ to $L^{2}(\mathbb{R}^{n})$ is not always true. As the main result in this paper, we give an answer for this problem for the Fourier integral operator $T_{a,\varphi }$. The main idea of our approach is treating the symbol class $L^{\infty }S^{m}_{\rho }(\omega )$, where $m=n(\rho -1)/2$. In particular, our results of $L^{2}(\mathbb{R}^{n})$-boundedness for $T_{a,\varphi }$ are also the best as far as we know. We also remark that our methods are different from the previous methods; see, for example, [13].

Finally, we make some conventions on notation. Throughout this article, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. We sometimes write $A\lesssim B$ as shorthand for $A\leq CB$. Let $\mathbb{R}^{n}$ be an n-dimensional Euclidean space, $x=(x_{1},\ldots ,x_{n})$ be a point in $\mathbb{R}^{n}$, $\mathbb{R}^{n}_{*}=\mathbb{R}^{n}\setminus \{0\}$, $\mathbb{N}=\{1, 2,\ldots \}$, $\mathbb{Z}_{+}=\mathbb{N}\cup \{0\}$, and $\mathbb{Z}_{+}^{n}=(\mathbb{Z}_{+})^{n}$. For any multi-index $\alpha = (\alpha _{1},\ldots ,\alpha _{n})$ and $\beta = (\beta _{1},\ldots ,\beta _{n})\in \mathbb{Z}_{+}^{n}$, we let

$$ \vert \alpha \vert =\sum_{j=1}^{n} \alpha _{j}, \qquad \alpha +\beta =(\alpha _{1}+ \beta _{1},\ldots ,\alpha _{n}+\beta _{n}) ,\qquad \partial _{x}^{\alpha }= \frac{\partial ^{\alpha }}{\partial _{x_{1}}^{\alpha _{1}} \cdots \partial _{x_{n}}^{\alpha _{n}}}, $$

and $\nabla _{\xi }=(\partial _{\xi _{1}},\ldots ,\partial _{ \xi _{n}})$. Also, in the sequel we use the notation

$$ \vert \xi \vert = \Biggl(\sum_{j=1}^{n} \xi _{j}^{2} \Biggr)^{1/2} \quad \text{and}\quad \langle \xi \rangle =\bigl(1+ \vert \xi \vert ^{2} \bigr)^{1/2}. $$

2 Definitions, notations, and preliminaries

The following definition is just [17].

Definition 2.1

Let $m\in \mathbb{R}$ and $0\leq \delta$, $\rho \leq 1$. For any two multi-indices α and β, we assume that the function $a(x,\xi )$ satisfies the following condition:

$$ \bigl\vert \partial ^{\beta }_{x}\partial ^{\alpha }_{\xi }a(x,\xi ) \bigr\vert \leq C_{ \alpha ,\beta } \langle \xi \rangle ^{m-\rho \vert \alpha \vert +\delta \vert \beta \vert }, $$

(2.1)

where $C_{\alpha \beta }$ is a positive constant only dependent on α and β. Let the smooth amplitude $S^{m}_{\rho ,\delta }$ be the set of all smooth functions $a(x,\xi )$ satisfying condition as in (2.1). Then the pseudo-differential operator $T_{a}$ with the symbol $a(x,\xi )\in S^{m}_{\rho ,\delta }$ is given formally by

$$ (T_{a} f) (x)= \int _{\mathbb{R}^{n}} e^{i x\cdot \xi }a(x,\xi )\hat{f}( \xi )\,d\xi . $$

The following definition for the class $L^{\infty }S^{m}_{\rho }(\omega )$ plays an important role in our setting.

Definition 2.2

Let m be a real number. A function $a(x,\xi )$, which is smooth in the frequency variable ξ and bounded measurable in the spatial variable x, belongs to the symbol class $L^{\infty }S^{m}_{\rho }(\omega )$ if, for all multi-indices α, it satisfies

$$ \bigl\Vert \partial ^{\alpha }_{\xi }a(x,\xi ) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n})} \leq C_{\alpha }\langle \xi \rangle ^{m-\rho \vert \alpha \vert } \omega \bigl( \langle \xi \rangle \bigr), $$

where $\omega (t)$ satisfies

$$ \int _{1}^{\infty }\frac{\omega (t)^{2}}{t}\,dt< \infty , $$

(2.2)

and $\omega (t)$ is a nonnegative and decreasing function on $[1,\infty )$.

Remark 2.1

If $\omega (t)$ satisfies (2.1), then $\sum_{j=0}^{\infty }\omega ^{2}(2^{j})<\infty$.

David and Wolfgang [14] gave the class $\varPhi ^{k}$ as follows.

Definition 2.3

([14], $\varPhi ^{k}$)

A real-valued function $\varphi (x,\xi )$ belongs to the class $\varPhi ^{k}$ if $\varphi (x,\xi )\in C^{\infty }(\mathbb{R}^{n}\times \mathbb{R}^{n}_{*})$ is positively homogeneous of degree 1 in the frequency variable ξ and satisfies the following condition: for any pair of multi-indices α and β, satisfying $|\alpha |+|\beta |\geq k$, there exists a positive constant $C_{\alpha ,\beta }$ such that

$$ \sup_{(x,\xi )\in \mathbb{R}^{n}\times \mathbb{R}^{n}_{*}} \vert \xi \vert ^{-1+ \alpha }\big|\partial _{\xi }^{\alpha }\partial _{x}^{\beta } \varphi (x, \xi )\big|\leq C_{\alpha ,\beta }. $$

In connection to the problem of local boundedness of Fourier integral operators, one considers phase functions $\varphi (x,\xi )$ that are positively homogeneous of degree 1 in the frequency variable ξ for which

$$ \biggl\vert \det \frac{\partial ^{2}\varphi (x,\xi )}{\partial {x_{j}} \partial {\xi _{k}}} \biggr\vert \neq 0. $$

The latter is referred to as the non-degeneracy condition. However, for the purpose of proving global regularity results, we require a stronger condition than the non-degeneracy condition above.

Definition 2.4

([14], The strong non-degeneracy condition)

A real-valued function $\varphi (x,\xi )\in C^{2}(\mathbb{R}^{n}\times \mathbb{R}^{n}_{*})$ satisfies strong non-degeneracy condition if there exists a positive constant c such that

$$ \biggl\vert \det \frac{\partial ^{2}\varphi (x,\xi )}{\partial {x_{j}} \partial {\xi _{k}}} \biggr\vert \geq c $$

for all $(x,\xi )\in \mathbb{R}^{n}\times \mathbb{R}^{n}_{*}$.

Remark 2.2

The phases in class $\varPhi ^{2}$ satisfying the strong non-degeneracy condition arise naturally in the study of the equations of hyperbolic type, namely

$$ \varphi (x,\xi )= \vert \xi \vert +\langle x,\xi \rangle $$

belongs to the class $\varPhi ^{2}$ and satisfies the strong non-degeneracy condition.

In [14], they introduced the nonsmooth version of the class $\varPhi ^{k}$ which will be used in our setting.

Definition 2.5

([14], $L^{\infty }\varPhi ^{k}$)

A real-valued function $\varphi (x,\xi )$ belongs to the phase class $L^{\infty }\varPhi ^{k}$ if it is positively homogeneous of degree 1 and smooth on $\mathbb{R}^{n}_{*}$ in the frequency variable ξ, bounded measurable in the spatial variable x, and if for all multi-indices $|\alpha |\geq k$ it satisfies

$$ \sup_{(x,\xi )\in \mathbb{R}^{n}\times \mathbb{R}^{n}_{*}} \vert \xi \vert ^{-1+ \alpha }\big|\partial _{\xi }^{\alpha }\varphi (x,\xi )\big|\leq C_{\alpha }. $$

Motivated by [14], we also need a Lipschitz rough non-degeneracy condition as follows.

Definition 2.6

(The Lipschitz rough non-degeneracy condition)

A real-valued function satisfies Lipschitz rough non-degeneracy condition if it is $C^{\infty }$ on $\mathbb{R}^{n}_{*}$ in the frequency variable ξ, bounded measurable in the spatial variable x, and there exist positive constants $C_{1}$ and $C_{2}$ such that, for all $x,y\in \mathbb{R}^{n}$ and $\xi \in \mathbb{R}^{n}_{*}$,

$$\begin{aligned}& \bigl\vert \partial _{\xi }\varphi (x,\xi )-\partial _{\xi }\varphi (y,\xi ) \bigr\vert \geq C_{1} \vert x-y \vert , \\& \bigl\vert \partial ^{\alpha }_{\xi }\varphi (x,\xi )- \partial ^{\alpha }_{\xi } \varphi (y,\xi ) \bigr\vert \leq C_{2} \vert x-y \vert \quad \text{for } \vert \alpha \vert \geq 2. \end{aligned}$$

3 Proof of the main result

In this section, we shall prove the main result, i.e., Theorem 1.1.

First we need a dyadic partition of unity. Let A be the annulus $A=\{\xi \in \mathbb{R}^{n};\frac{1}{2}\leq |\xi |\leq 2\}$ and

$$ \chi _{0}(\xi )+\sum_{j=1}^{\infty } \chi _{j}(\xi )=1 \quad \text{for all } \xi \in \mathbb{R}^{n}, $$

where $\chi _{0}(\xi )\in C^{\infty }_{0}(B(0,2))$ and $\chi _{j}(\xi )=\chi (2^{-j}\xi )$ when $j\geq 1$ with $\chi (\xi )\in C^{\infty }_{0}(A)$. Now we decompose the operator $T_{a,\varphi }$ as follows:

$$ T_{a,\varphi }=T_{ \chi _{0}}(D)+\sum _{j=1}^{\infty }T_{\chi _{j}}(D)=T_{0}(D)+ \sum_{j=1}^{\infty }T_{j}(D). $$

(3.1)

The first term in (3.1) is bounded on $L^{2}(\mathbb{R}^{n})$ from Theorem 1.1.8 in [14]. After a change of variables, we have

$$\begin{aligned} T_{j}(D) =&\frac{1}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{ i \varphi (x, \xi )}\chi _{j}(\xi )a(x, \xi ) \hat{u}(\xi )\,d\xi \\ =&\frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{ i \cdot 2^{j\varrho } \varphi (x,\xi )}\chi _{j} \bigl(2^{j\varrho }\xi \bigr)a\bigl(x,2^{j \varrho }\xi \bigr) \hat{u} \bigl(2^{j\varrho }\xi \bigr)\,d\xi \\ =& \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{ i \cdot 2^{j\varrho } \varphi (x,\xi )}\chi _{j} \bigl(2^{j\varrho }\xi \bigr)a\bigl(x,2^{j \varrho }\xi \bigr) \int _{\mathbb{R}^{n}}e^{ -i2^{j\varrho } \xi \cdot y}u(y)\,dy\,d\xi \\ =& \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}\times \mathbb{R}^{n}}e^{ i\cdot 2^{j\varrho }( \varphi (x,\xi )-y\cdot \xi )} \chi _{j} \bigl(2^{j\varrho }\xi \bigr)a\bigl(x,2^{j\varrho }\xi \bigr) u(y)\,d\xi \,dy. \end{aligned}$$

The kernel of the operator $T_{j}(D)$ is given by

$$ T_{j}(x,y)= \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{ i\cdot 2^{j\varrho }( \varphi (x,\xi )-y\cdot \xi )}\chi _{j} \bigl(2^{j \varrho }\xi \bigr)a\bigl(x,2^{j\varrho }\xi \bigr)\,d\xi . $$

Let

$$ a_{j}(x,\xi )=\chi \bigl(2^{j(\varrho - 1) }\xi \bigr)a \bigl(x,2^{j\varrho }\xi \bigr). $$

Then

$$ A_{j}=\mathop{\operatorname{Supp}} _{\xi }a_{j} \subset \bigl\{ \xi ;2^{-1}2^{j(1- \varrho )}< \vert \xi \vert < 2 \cdot 2^{j(1-\varrho )} \bigr\} $$

and it satisfies

$$ \bigl\vert \partial ^{\alpha }_{\xi }a_{j}(x, \xi ) \bigr\vert \leq C_{\alpha }\cdot 2^{jn(\rho -1)/2}. $$

(3.2)

We can confine ourselves to dealing with the high frequency component $T_{j}$ of $T_{a,\varphi }$. Here we shall use a $S_{j}=T_{j}T_{j}^{*}$ argument, and therefore,

$$\begin{aligned} S_{j}u(x) &= \frac{1}{(2\pi )^{n}} \int _{\mathbb{R}^{n}\times \mathbb{R}^{n}} e^{ i (\varphi (x,\xi )-\varphi (y,\xi ))}\chi _{j}^{2}( \xi )a(x,\xi )\overline{a(y,\xi )} u(y)\,dy\,d\xi \\ &= \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}\times \mathbb{R}^{n}}e^{ i2^{j\varrho } (\varphi (x,\xi )-\varphi (y,\xi ))} \\ &\quad {} + \chi _{j}^{2}\bigl(2^{j\varrho }\xi \bigr)a \bigl(x,2^{j\varrho }\xi \bigr) \overline{a\bigl(y,2^{j\varrho }\xi \bigr)} u(y)\,d\xi \,dy. \end{aligned}$$

The kernel of the operator $S_{j}=T_{j}T_{j}^{*}$ reads

$$ S_{j}(x,y)=\frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}} e^{ i2^{j\varrho } (\varphi (x,\xi )-\varphi (y,\xi ))}\chi _{j}^{2} \bigl(2^{j \varrho }\xi \bigr)a\bigl(x,2^{j\varrho }\xi \bigr)\overline{a \bigl(y,2^{j\varrho }\xi \bigr)} \,d\xi . $$

Let $b_{j}(x,y,\xi )= \chi _{j}^{2}(2^{j\varrho }\xi )a(x,2^{j\varrho } \xi )\overline{a(y,2^{j\varrho }\xi )}$. Then

$$ \operatorname{Supp} b_{j}\subset \biggl\{ \xi : \frac{2^{j(1-\varrho )}}{2}< \vert \xi \vert < 2\cdot 2^{j(1-\varrho )} \biggr\} . $$

We claim that

$$ \bigl\vert \partial ^{\alpha }_{\xi }b_{j}(x,y, \xi ) \bigr\vert \leq C_{\alpha }2^{jn(\rho -1)} \omega ^{2}\bigl(2^{j}\bigr). $$

In fact,

$$\begin{aligned} \bigl\vert \partial ^{\alpha }_{\xi }b_{j}(x,y, \xi ) \bigr\vert =& \bigl\vert \partial ^{\alpha }_{\xi } \bigl[\chi _{j}^{2}\bigl(2^{j\varrho }\xi \bigr)a \bigl(x,2^{j\varrho }\xi \bigr) \overline{a\bigl(y,2^{j\varrho }\xi \bigr)}\bigr] \bigr\vert \\ =&\sum_{\alpha _{1}+\alpha _{2}=\alpha } \bigl\vert \partial ^{\alpha _{1}}_{\xi }\bigl[a\bigl(x,2^{j\varrho }\xi \bigr) \overline{a\bigl(y,2^{j\varrho }\xi \bigr)}\bigr] \bigr\vert \bigl\vert \partial ^{\alpha _{2}}_{\xi }\chi ^{2} \bigl(2^{-j(1-\varrho ) }\xi \bigr) \bigr\vert \\ \lesssim & \sum_{\alpha _{1}+\alpha _{2}=\alpha }\bigl(2^{j\varrho } \bigr)^{ \vert \alpha _{1} \vert } \bigl\vert \bigl(\partial ^{\alpha _{1}}_{\xi }(a \cdot \bar{a}) \bigr) \bigl(x,2^{j \varrho }\xi \bigr) \bigr\vert \omega ^{2}\bigl(2^{j}\bigr) \\ &{} \times 2^{-j(1-\varrho ) \vert \alpha _{2} \vert } \bigl\vert \bigl( \partial ^{\alpha _{2}}_{\xi } \chi \bigr) \bigl(2^{-j(1-\varrho ) }\xi \bigr) \bigr\vert \\ \lesssim & \sum_{\alpha _{1}+\alpha _{2}=\alpha }2^{j\varrho \vert \alpha _{1} \vert }\bigl\langle 2^{j\varrho }\xi \bigr\rangle ^{n(\rho -1)-\varrho \vert \alpha _{1} \vert }2^{-j(1-\varrho ) \vert \alpha _{2} \vert } \omega ^{2}\bigl(2^{j}\bigr) \\ \lesssim & \sum_{\alpha _{1}+\alpha _{2}=\alpha }2^{j\varrho \vert \alpha _{1} \vert }2^{j(n(\rho -1)-\varrho \vert \alpha _{1} \vert )}2^{-j(1-\varrho ) \vert \alpha _{2} \vert } \omega ^{2}\bigl(2^{j}\bigr) \\ =& \sum_{\alpha _{1}+\alpha _{2}=\alpha }2^{jn(\rho -1)-j(1- \varrho ) \vert \alpha _{2} \vert }\omega ^{2}\bigl(2^{j}\bigr) \\ =&2^{jn(\rho -1)}\sum_{\alpha _{2}}2^{-j(1-\varrho ) \vert \alpha _{2} \vert } \omega ^{2}\bigl(2^{j}\bigr) \\ \lesssim & 2^{jn(\rho -1)}\omega ^{2}\bigl(2^{j} \bigr). \end{aligned}$$

(3.3)

Next we consider the following differential operators for $j\in \mathbb{N}$:

$$ L_{j}(x,y,D)= \frac{\nabla _{\xi }\varPhi \nabla _{\xi }}{i2^{j\varrho } \vert \nabla _{\xi }\varPhi \vert ^{2}}, $$

(3.4)

where $\varPhi (x,y,\xi )=\varphi (x,\xi )-\varphi (y,\xi )$. So $L_{j}^{N}(x,y,D)e^{i2^{j\varrho }\varPhi }=e^{i2^{j\varrho }\varPhi }$ and

$$ L_{j}^{*}(x,y,D)=-\nabla _{\xi }\frac{\nabla _{\xi }\varPhi }{i2^{j\varrho } \vert \nabla _{\xi }\varPhi \vert ^{2}}. $$

(3.5)

From this and (3.4), it follows that

$$\begin{aligned} S_{j}(x,y) =& \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}} e^{ i2^{j\varrho } (\varphi (x,\xi )-\varphi (y,\xi ))}\chi _{j}^{2} \bigl(2^{j \varrho }\xi \bigr)a\bigl(x,2^{j\varrho }\xi \bigr)\overline{a \bigl(y,2^{j\varrho }\xi \bigr)} \,d\xi \\ =& \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}} \bigl(L_{j}^{N}e^{i2^{j \varrho }\varPhi } \bigr)b_{j}(x,y \xi )\,d\xi \\ =& \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{i2^{j \varrho }\varPhi }\bigl(L_{j}^{*} \bigr)^{N}b_{j}\,d\xi . \end{aligned}$$

Moreover, by (3.5), we see that

$$\begin{aligned} \partial _{\xi _{\mu _{1}}} \biggl[ \frac{\nabla _{\xi }\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}}b_{j} \biggr] =& \biggl( \partial _{\xi _{\mu _{1}}}\biggl[ \frac{\nabla _{\xi }\varPhi b_{j}}{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr] \biggr) \\ =& \biggl[\frac{\partial _{\xi _{1}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr]\partial _{\xi _{\mu _{1}}}b_{j}+ \partial _{\xi _{\mu _{1}}} \biggl[\frac{\partial _{\xi _{1}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr]b_{j}, \ldots , \biggl[ \frac{\partial _{\xi _{n}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr] \partial _{\xi _{\mu _{1}}}b_{j} \\ &{} +\partial _{\xi _{\mu _{1}}} \biggl[ \frac{\partial _{\xi _{n}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr]b_{j}, \end{aligned}$$

which implies that

$$\begin{aligned} L^{*}_{j}b_{j} =& \nabla _{\xi } \biggl[ \frac{\nabla _{\xi }\varPhi }{-i2^{j\varrho } \vert \nabla _{\xi }\varPhi \vert ^{2}}b_{j} \biggr] \\ =&\frac{1}{-i2^{j\varrho }} \sum_{l=1}^{n} \biggl\{ \biggl[ \frac{\partial _{\xi _{l}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr] \partial _{\xi _{l}}b_{j} +\partial _{\xi _{l}} \biggl[ \frac{\partial _{\xi _{l}}\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}} \biggr]b_{j} \biggr\} . \end{aligned}$$

Thus

$$\begin{aligned} \bigl(L_{j}^{*}\bigr)^{N}b_{j} &= \frac{1}{(-i2^{j\varrho })^{N}}\nabla _{\xi }\underbrace{ \biggl\{ \frac{\nabla _{\xi }\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}}\cdots \biggl[\nabla _{\xi }\biggl( \frac{\nabla _{\xi }\varPhi }{ \vert \nabla _{\xi }\varPhi \vert ^{2}}b_{j}\biggr) \biggr] \biggr\} }_{N} \\ & =\frac{1}{(-i2^{j\varrho })^{N}}\sum_{\alpha _{1}+\cdots +\alpha _{N}+ \beta =N}\partial ^{\alpha _{1}} \biggl( \frac{\varPhi _{\mu _{1}}}{ \vert \nabla \varPhi \vert ^{2}} \biggr)\cdots \partial ^{ \alpha _{N}} \biggl(\frac{\varPhi _{\mu _{N}}}{ \vert \nabla \varPhi \vert ^{2}} \biggr)\partial ^{\beta }b_{j}, \end{aligned}$$

(3.6)

where $\varPhi _{\mu _{k}}=\partial _{\xi _{\mu _{k}}}\varPhi$. Because of the following equation

$$ \partial ^{k}_{\xi } \biggl(\frac{\varPhi _{\mu }}{ \vert \nabla \varPhi \vert ^{2}} \biggr)= \sum_{k_{0}+k_{1}+\cdots +k_{j}=k} \frac{C_{k_{0},\ldots ,k_{j}}\partial ^{k_{0}}_{\xi }\varPhi _{\mu } \partial ^{k_{1}}_{\xi } \vert \nabla \varPhi \vert ^{2}\cdot \partial ^{k_{j}}_{\xi } \vert \nabla \varPhi \vert ^{2}}{ \vert \nabla \varPhi \vert ^{2+2j}}, $$

by Definition 2.6, we see that

$$ \bigl\vert \partial ^{k_{0}}_{\xi }\varPhi _{\mu } \bigr\vert \lesssim \vert x-y \vert \quad \text{and} \quad \bigl\vert \partial ^{k}_{\xi }|\nabla \varPhi \vert ^{2}\bigr| \lesssim \vert x-y \vert ^{2}. $$

From this, together with (3.3) and (3.6), we further obtain

$$\begin{aligned} \bigl\vert \bigl(L_{j}^{*}\bigr)^{N}b_{j} \bigr\vert \lesssim & \frac{1}{2^{j\varrho N}}\cdot \frac{1}{ \vert x-y \vert ^{N}}2^{jn(\rho -1)} \omega ^{2}\bigl(2^{j}\bigr). \\ =& \frac{1}{ [2^{j\varrho } \vert x-y \vert ]^{N}}2^{jn(\rho -1)} \omega ^{2} \bigl(2^{j}\bigr). \end{aligned}$$

Integration by parts yields

$$\begin{aligned} \begin{aligned} S_{j}(x,y) &= \frac{2^{j\varrho n}}{(2\pi )^{n}} \int _{\mathbb{R}^{n}}e^{i2^{j \varrho }\varPhi }\bigl(L_{j}^{*} \bigr)^{N}b_{j}\,d\xi \\ &\lesssim \frac{2^{j\varrho n}}{ [2^{j\varrho } \vert x-y \vert ]^{N}}2^{jn( \rho -1)}\omega ^{2} \bigl(2^{j}\bigr)2^{j(1-\varrho )n} \\ &= \frac{2^{j\varrho n}}{ [2^{j\varrho } \vert x-y \vert ]^{N}} \omega ^{2}\bigl(2^{j}\bigr). \end{aligned} \end{aligned}$$

Thus,

$$ \bigl\vert S_{j}(x,y) \bigr\vert \sum _{l=0}^{N}\bigl(2^{j\varrho } \vert x-y \vert \bigr)^{l}\lesssim 2^{j \varrho n}\omega ^{2} \bigl(2^{j}\bigr), $$

which implies that

$$ \bigl\vert S_{j}(x,y) \bigr\vert \lesssim \frac{2^{j\varrho n}}{(1+2^{j\varrho } \vert x-y \vert )^{N}} \omega ^{2}\bigl(2^{j}\bigr). $$

This further gives

$$\begin{aligned} \sup_{x} \int _{\mathbb{R}^{n}} \bigl\vert S_{j}(x,y) \bigr\vert \,dy & \lesssim 2^{j \varrho n}\omega ^{2}\bigl(2^{j} \bigr) \int _{\mathbb{R}^{n}} \frac{1}{(1+2^{j\varrho } \vert x-y \vert )^{N}}\,dy \\ & \lesssim \omega ^{2}\bigl(2^{j}\bigr) \int _{\mathbb{R}^{n}} \frac{1}{(1+ \vert z \vert )^{N}}\,dz\lesssim \omega ^{2}\bigl(2^{j}\bigr). \end{aligned}$$

By Young’s inequality, we obtain

$$ \bigl\Vert S_{j}u(x) \bigr\Vert _{L^{2}}\lesssim \omega ^{2}\bigl(2^{j}\bigr) \bigl\Vert u(x) \bigr\Vert _{L^{2}}. $$

Therefore, we have

$$\begin{aligned} \bigl\Vert T^{*}_{j}u \bigr\Vert ^{2}_{L^{2}} =& \bigl\langle T^{*}_{j}u, T^{*}_{j}u\bigr\rangle = \bigl\langle u,T_{j}T^{*}_{j}u\bigr\rangle \\ \leq & \Vert u \Vert _{L^{2}} \Vert S_{j}u \Vert _{L^{2}} \\ \leq & C \omega ^{2}\bigl(2^{j}\bigr) \Vert u \Vert _{L^{2}}^{2}. \end{aligned}$$

Namely,

$$ \Vert T_{j}u \Vert _{L^{2}}\leq C \omega \bigl(2^{j}\bigr) \Vert u \Vert _{L^{2}}. $$

(3.7)

Next we need a Littlewood–Paley decomposition. Let $\psi _{0}:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a smooth radial function which is equal to one on the unit ball centric at the origin and supported on its concentric double. Set $\psi (\xi )=\psi _{0}(\xi )-\psi _{0}(2\xi )$ and $\psi _{k}(\xi )=\psi (2^{-k}\xi )$. Then

$$ \psi _{0}(\xi )+\sum_{k=1}^{\infty } \psi _{k}(\xi )=1\quad \text{for all } \xi \in \mathbb{R}^{n}, $$

and $\operatorname{supp}\psi _{k}(\xi )\subset \{\xi :2^{k-1}\leq |\xi |\leq 2^{k+1} \}$ for $k\geq 1$. And we further have

$$ \hat{u}(\xi )=\sum_{k=0}^{\infty }\hat{u}( \xi )\psi _{k}(\xi )=\sum_{k=0}^{ \infty } \hat{u}_{k}(\xi ). $$

Then

$$ T_{a,\varphi }u =\sum_{j=1}^{\infty }T_{j}u= \sum_{k=0}^{\infty }\sum _{j=1}^{\infty } \int _{\mathbb{R}^{n}}e^{i\varphi (x, \xi )}a_{j}(x,\xi ) \hat{u}_{k}(\xi )\,d\xi . $$

For simplicity of notation, we write

$$ \sum_{k=0}^{\infty }\sum _{j=1}^{\infty } \int _{\mathbb{R}^{n}}e^{i \varphi (x,\xi )}a_{j}(x,\xi ) \hat{u}_{k}(\xi )\,d\xi =\sum_{j=1}^{ \infty }T_{j}u_{j}, $$

where

$$ u_{j}=\sum_{k=0}^{\infty }a_{j}(x, \xi )\hat{u}_{k}(\xi ). $$

From this, (3.7), Cauchy–Schwartz’s inequality, and Remark 2.1, it follows that

$$\begin{aligned} \Vert T_{a,\varphi }u \Vert _{L^{2}}&= \Biggl\Vert \sum _{j=0}^{\infty }T_{j}u_{j} \Biggr\Vert _{L^{2}} \\ &\lesssim \sum_{j=0}^{\infty }\omega \bigl(2^{j}\bigr) \Vert u_{j} \Vert _{L^{2}} \\ &\lesssim \Biggl(\sum_{j=0}^{\infty }\omega ^{2}\bigl(2^{j}\bigr) \Biggr)^{1/2} \Biggl( \sum_{j=0}^{\infty } \Vert u_{j} \Vert _{L^{2}}^{2} \Biggr)^{1/2} \\ &\lesssim \Vert u \Vert _{L^{2}}. \end{aligned}$$

This finishes the proof of Theorem 1.1.