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On \(L^{2}\)-boundedness of Fourier integral operators


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

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

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, [1012]. 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 [1416] 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:

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

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

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

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:

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

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

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


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


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

and it satisfies

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

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,

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

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

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

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

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

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

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


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

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

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


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

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

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


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

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

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


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

For simplicity of notation, we write

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


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

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.


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The authors would like to thank the referees for their important comments and remarks.

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The research of the first author is supported by the Natural Science Foundation of Xinjiang Urgur Autonomous Region (2019D01C049, 62008031, 042312023) and the National Natural Science Foundation of China (11561065). The research of the second author is supported by the National Natural Science Foundation of China (11131005). The research of the third author is supported by the National Natural Science Foundation of China (11826202).

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Yang, J., Chen, W. & Zhou, J. On \(L^{2}\)-boundedness of Fourier integral operators. J Inequal Appl 2020, 173 (2020).

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